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The general theory of thin elastic shells
Sam& NO matter how thin they are, are three-dimensional bodies, and an “exact” theory of shells can not therefore be distinct of the three-dimensional continttum theory of the material of which the shell is made, In view of the known di&ulties connected with t~~~~e~s~on~ ~rob~ems~sliel~theory proposes to replace such ~ob~ms for thin she& by ~o~e~s~~~ ones, which thus shafl.re%ct with mare or tessaccuracythe ~hav~o~ of the ~~~e~~on~ body. The “derived” theories of thin she& consider the ~nema~~s and the co~s~tu~v~ equation of the tree-d~eu~ion~ medium, but eattrblishequations in which the number of i~d~~ndent space variables has heen reduced to two, the surface parameters of a reference surface (chosen very often as “middle surface”) being usuahy taken as these variables. The general laws of c~~t~n~~~ mechanics govern of course the problem of setting these aeon. It is however not rare that a more or kss tacit@estate abbe of virtual work in terms of two fables be used in such theories. The “direct””theories [ I-31, as opposed to derived theories, proceed in the opposite sense; i.e. they admit 4 p&n’ a represen~tion of the shell by a finite ~~~iun of surface, endows with some deformab~~typroperties, and possibly supplemented by a field of deformable vectors, called directors, defined over this portion of surface (Cosserat surface). Constitutive rehtions must then be postulated for this model, Assigned force vectors and assigned director couples are ~~~u~~ to represent the effect of external toads, while curve forces and curve director couples play a role similar to that of stress res~~nts and coupfe ~s~~n~ of the classic& (derived) theories. Of cotuse the f~~arnen~ jaws of me&&s must also be ~st~at~ in a manner applicable to the models of direct theories. An ~~~~t problem proper to any direct theory is the need an a “converse s~ment”[l], i.e. the liuk to be established back to the behaviour of the three-dimensional body which is thought to be represented by the model of the direct theory. Obtaining equations is of course just a part of the shell problem. There remains to show the existence of solutions, their ~ueness under certain ~~~~s~~ and &By to obtain efFe&veiy these sohstions with a desired accuracy. The maat progress made in the tast few decades in the wrirrtional methods and in their applications to mechanics and numericd analysis, as well as the progress in the theory of operators atlow now to attack efficiency these parts of the shell problem. The purpose of this paper is to present a unified picture of some advances achieved in shell theory during the past decade, by systematically applying variation9f and topological methods. The approach adopted here is that of derived theories, but the ~onn~~~ to direct theories is a&o es~~~~d. The ~~ is assume to be b~~~s~ isotropic and h~re~as~~ Both geometric and ~ons~utive ~~~e~~es are ~ons~d~red* 2. SOMB BASIC EQUATID~S OF T~REE-D~~ENS~~~AL IN CQNVWl%D COORDlNATES
ELASTICITY
We record here, for later use, and also with the purpose of iutroducing some notation, some basic reiations of the ~e~e~sion~ elasticity theory.
236
MLJRATDiKMEN
We define a system of convected coordinates (0’) (i = 1,2,3) in a body B, occupying in the three-dimensional euclidean space, at any instant t, a finite region V, with boundary 8V. For simplicity, V shall be assumed to be simply-connected. Position vectors r = r(8’, t)
(2.1)
emanating from some reference point fixed in space, describe the “actual configuration” (at time t) of the body. By choosing the origin of t to correspond to a “reference configuration”, we write R(8’) =
I-(#,0)
(2.2)
It(#) + u(ei, t).
(2.3)
and introduce the displacement vector u by
r(e’, t)
=
We shall assume that the vector-valued functions r, R and II are differentiable as many times as necessitated by the computations, with respect to each of their arguments. The base vectors and the metric tensors in the actual and reference configurations are
gij E
gi
’ gj
(2.4)
and Gk = R,k Gij=Gi
*Gj
(2.5)
respectively. Lower case latin indices have, here and henceforth, the range 1,2,3, and a comma followed by a subscript, say k, represents differentiation with respect to the variable t$, keeping all other variables fixed. The determinants of the metric tensors shall be denoted by g and G, respectively, and assumed not to vanish in B. The strain tensor is given by yij
E i
CT =
(gij -
U( rij).
(2.7)
It shall be assumed that the body is in unstressed state at the reference configuration and also that the material is isotropic. Thermal effects are not considered. The stresses are given by
au
?‘I=p a’yii
(2.8)
where p is the mass density and TVis the symmetric stress tensor (with u symmetrized in the components n) and referred to the 8’ coordinates in the actual confIguration and also measured
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237
per unit of area in the same configuration. Another symmetric stress tensor is defined by (2.9) and expresses stress again referred to the 8’ coordinates in the actual configuration, but measured per unit of the corresponding area in the reference configuration in which also the mass density p,,is measured. The relation between these stress tensors is sij= #
J5.
(2.10)
The fundamental dynamic law is expressed as a generalized Hamilton principle
s
Ldrdt+SW+SW*=O
(2.11)
(+2-c >
(2.12)
LV
where Lsp
is the lagrangean density function, and where the second and third terms are defined by
SW=
@b-u)dTdt
(2.13)
IV
and
SW*
(p.Su)dadt-
E
tJV
(+u)drl::
(2.14)
V
i = i is the velocity of a material point, b the body force per unit of mass, p the forces acting per unit of area of the boundary, I the momentum. dr is the element of volume and da is the element of area.
DESCRIPTION OF THE THIN SHELL Consider a body B, in its reference configuration, and let 8 be a surface which intersects B on a connected region Q = B f-12. It is convenient to choose the system of convected coordinates {e’) in such a way as to have Z represented by # = 0. Then, the points of fi are referred to a system of coordinates IOU}(a = 1,2), the Gaussian parameters on S. Henceforth, Greek indices will take the value 1 or 2. It will also be required that the e3-lines be chosen in such a way as to have the body B not extending outside of the “tube” 7 determined by 80 = 8B f-12 as dire&ix and the #-lines crossing &I as generators. It will furthermore be required that the #-lines intersect aB at most at two points. Of course, the configuration of B is assumed to permit such a construction of the system of convected coordinates {#}, or {fY, OS}. fl is then called reference surface. The intersection g = T 17aB is called the edge (and may reduce to an). The difference aB - 8 is disjoint and consists of two simply-connected parts, the faces Q+ and a- of B. In the next section, the system of coordinates shall conveniently be taken as a system of “normal coordinates”, i.e. in which the B3-linesare the normals to 2. The length of the segment 3.
238
MURAT D&MEN
intercepted, by the faces along the normal at a point {F) will be caiied thickness and denoted by h(P). In order to define the geometric quantities needed in the sequel, the surface Z shall be assumed sufficiently smooth; i.e. the position vector R(B”,O), or simply R(V), on the points {P, 0) of n shall be continuously differentiable up to the third order. Let RI and R2 be the radii of principal curvatures at (P,Oj and Iet Ir(&“) and &ffY) be the Iengths of arcs intercepted by 852 along the fines of curvature at the same point. The body B will be called a thin &if if h C rnk {R,, &}
(3.1)
u e max Ir, max iz
(3.2)
and
n
n
at every point of Q. Regularity properties concerning the faces, and hence h(V) shall also be assumed, as necessitated by the following computations. In particular, ribs are excluded.
4. SHELL GEOMETRYAND ~I~E~A~~~~ The base vectors, the metric tensor (first fundamental form) and its determinant on the reference surface 19~= 0 are
(4.11 in the actual configuration, and A, = G,(V, 0) A,=A,*A@ A = iAtlat
(4.2)
in the reference configuration. We record the relations
(4.31
where II is unit normal vector determined by a1 x 82, baB are the coefficients of the second fundamental form, and I’& are the Christoffel symbols of the second kind (metric connections with respect to a&. c,+ are the coefficients of the third f~~rnen~ form; they can be expressed a~ebrai~a~y in terms of baB and a,,. The semicobn indicates covariant diierentiation with respect to the metric tensor a,,. An analogous set of relations can be written for the reference configuration, simply replacing the latin symbols in (4.3) by capitals. To distinguish the covariant differentiation with
Recent advances in the generaltheory of thin elastic shells
239
respect to aaBfrom the covariant differentiations with respect to be, a colon shall be used for the latter. It is however convenient, as already noted in Section 2, to use a system of convected coordinates which is a system of normal c~rd~ates in the reference co~tion. Then G3 =
N, G, = &A,
G,@= /L:&&
Ge3 = 0,
G33 = 1
(4.4)
with f.&:=
s: - e313:
(4.5)
and I =
(;)“* .
IPO’I =
(4.6)
Then R(P) t9”)= R(P, 0) + e3N(5”)
(4.7)
and (2.3)can be rewritten in the form r(P,
d3,t) = R(B”, 0)+ u(tP, 0, t) + fz
f fn
(4.8)
where f and f are functions of (8’) and t, and f” behaves as a contravariant surface vector. The base vectors in the actual con&nation, as obtained from (4.8),are then
The components of the metric tensor can be calculated by using (4.9),(2.4)*and (4.3)S,6, to find
a = a28-
2fbl
+2f&#'
+f*ca/3 +fmf9/J + 'tfLfw + (f'f,;&~ + Lbf
g3* =f%3+faf%v g33 =
ti3,*+
- 2fbT&l,,
‘f” +f#;*
(4.10) +(-~,3+f,~)~~~
fyY3fY,3
Round brackets are used to indicate the indices over which symmetrization must be performed, while the sign 1 1encloses indices which shall be singled out in the process. Now, according to a fundamental theorem of surface theory, if three functions u&V)cC2 and three functions ALEC’ (with symmetry in the indices a,$#)are given and if they obey to the con~tions err> 0, 022> 0 and 1~1 f 0, they determine in the ~~~ension~ euclidean space a surface uniquely to within its position in space and a mirror symmetry, provided the Mainardi-Codazziequations (4.3)8and the Gauss equations (4.11) be satisfied. The surface thus determined is represented by an equation r = r(F), rcC3.
240
MURAT DfKMEN
R so67in (4.11) is the covariant curvature tensor defined by
The regularity secured by this theorem corresponds exactly to the one which was imposed to the reference surface in the reference configuration. This suggests that, provided the displacement vector u be continuously differentiable at least up to third order, with respect to the variables 0”, the quantities
(4.13) can be used as strains defined over the reference surface. The velocity of a point of the shell (body B) is given by ti=i=v+f”a,+f”i,+fn+fi
(4.14)
where a superimposed dot indicates differentiation with respect to time, and where v(B”, t) = li(V, 0, t)
(4.15)
i.e. the velocity of a point (with coordinates 0”) of the reference surface. Thus, (4.14) can be written in the form
ir = k = v +f”v,= + l_P-f(n
- v,g)ascr}a, +fn.
(4.16)
The expressions (4.9) and (4.16) are exact, in the sense that besides the regularity condition imposed above to the displacement vector u, and therefore to the functions fcr(P, e3, t) and f(P, e3, t) no restriction has been assumed concerning these latter functions, except that they must of course vanish for B3= 0, in view of (4.8). We may, in particular, assume that these functions depend on 8“ and t through a finite number of functions (4.17) i.e.
(4.18) Motivated by a paper of L. Sedov[4], this is the position adopted by V. L. Berdichevskii[5]. The presence of the “parameters” $A means that some restrictions, i.e. internal constraints, are imposed to the deformation of the body B, by the structure of the functions f” and f. Here, we develop ab initio a theory deviating from that in [51inasmuch as (i) we avoid using an auxiliary Cartesian system of coordinates, which is not essential in the theory, (ii) we introduce a somewhat less general kinematic representation, namely (4.19) instead of (4.18), which allows a simpler form and interpretation of the equations of motion. In fact, Weierstrass’ polynomial approximation theorem suggests the structure p = P(e%;(e@,
,. f = P(e3h(e@,
t) t)
(4.19)
where the PA are appropriately chosen polynomials. It is understood that the summation
Recent advances in the general theory of thin elastic shells
241
convention applies to capital latin indices also. These indices run from 1 to N. Of course, the functions VAare subject to the same regularity conditions as the functions f” and f. The functions p and f defined this way may then be viewed either as a polynomial approx~ation to the functions f” resp. f, or they may be also viewed as elements of a subclass of functions (4.19), obtained by imposing restrictions, i.e. internal constraints to the kinematics of the medium. We shall use henceforth the structure (4.19) and omit the sign ( A). Back to (4.10) and (4.16), we see that the arguments on which the strains 7ij and velocity and the velocity ti depend, are the ones shown below:
5. EQUATIONS
OF MOTION
The energy integral in (2.11) can be written
(5.1) with dr = t/C de’ de* de3 = /L de3 dA dA = d/A de’ de*
(5.2)
and p defined by (4.6). Further,
e3 = b(e*) 83 = ate*)
(5.3)
b-a=h describe the upper resp. lower faces of the shell. They are also assumed (see Section 3) to be sufficiently smooth. The innermost integral afP’) A= Lp de3 (5.4) f bfe”f defines an energy density per unit area of 0, and is equal to the difference
w
A=K-S
of the kinetic energy density
(5.6) and strain energy density
s =I
b
a
&p&d.
de3.
(5.7)
Clearly, these quantities obtained by integration over [a, b] depend on the arguments listed in
K = Kb’,,,&fit
s =Sk@,xc+ in view of (4.20).
mA,+A) TA,
n;l,,)
(5.8)
MURAT DiKMEN
242
As concerns (2.13) and (2.14) we need to calculate &I E Sr. Using (4.8), we find
n,, stands for the displacement of points of the reference surface. This expression for Su can now be inserted into
If
SW=
I B
(pob * Su) dr dt = I, (I, [Jab @oh +WP de31 dA} dt
(5.10)
to calculate first the innermost integral in the right side b
f
(t pob *
p,,bPAp d03)S?rA
Suo+{a&"A+n&r~}*
{j-Obpobp
poPAbp
dd3)
-
c+a
* (lab p,,PAbp de3)]n)
- (Su,,,).
(5.11)
The terms in the right side of (5.11) suggest some definitions:
(5.12) With this notation, (5.11) can be summarized in the form (5.13) (2.14) can be rewritten as
sw* =
If I
an
= fff n+
(p.Su)dAdt-
(I,I+dr);; (p . Su) dA- dt +
(p . Su) dA+ dt +
(p.6u)pedB3dsdt-lleI.6ud7(;:
or
+
+
If If c
an
{Q * Sue + SA *(a&pS + n&A) + Ma - SUO~}dA dt
n {i.
Su,,+ TA * (a&r2 + n&A)
+ P
* c%Io,~}
-
?rAPA(a”
11
dA) ”
where the abbreviated notation ii = (p/i)+ + (Pa@AE (pP$)’
+ (pPAfi)-
p z {.lr;pPA - TAPA(a” ’
p)n}+
+ {r$PA
* p)$
(5.15)
243
Recent advances in the general theory of thin elastic shells b
Q=j-= pp=de3 b s*
s
I (1
i*
s
pPA$ de3
bIP*p
I (I
Ib{r:IP* 0
de3
- nAPA(aa’ I)n}p de3
(5.16)
and the geometric quantities
have been introduced. In (5.17)4,e and e. are the determinants of the metric tensor on rS,in the actual and reference configuration, respectively. #’ is the unit tangent vector to 0, in the reference configuration. There remains to evaluate the variations involved in the term with the energy integral. The intermediate steps shall not be shown here, and the final form taken by the variational law is given at once below:
-I,Ipl
@;@ - nu8b,&
+ y(naB;8 + q8b;)a., -(F+
I) + (P + ha);,, + (p&o+ &%A)
- (I@ + PA) . a. + (&ii + pp~~lr~) - a,
+
If,
dR{Q -
- q%)v, yWBaa
+ (P +
@pa
ST; dA dt
MURAT DiKMEN
244
where
(5.19)
and
(5.20)
v, is the normal to aa in 2.
J
y”
(5.21)
;.
The equations of motion and the boundary conditions, as well as the auxiliary conditions implied by the last integral can be obtained easily, by the usual argument of the calculus of variations. They shall therefore not be listed here separately. The expressions in (5.18) simplify significantly in the limiting case where p can be taken equal to one, by using a set of appropriate orthonormal polynomials PA. Then the quantities pp vanish. The vectors mAmay be viewed as a generalization of the directors of direct theories, thus establishing the link between these and the present derived theory. Using these equations, theories of various orders of accuracy for the description of the shell behaviour can be obtained systematically. Polynomials in O3for the description of the displacements have been used in [6] also, but in a different approach than in the present case. While (5.18) involves $I and r,., as unknown functions, in finite but unspecified number, a priori assumptions made with respect to these functions may considerably simplify the equations. The classical geometric hypothesis of the normal to the middle-surface, remaining normal during deformation is expressed by r;=O
(5.22)
(A=l,...,N)
and the reference surface is chosen as middle-surface, i.e.
If the normal is not permitted to stretch, then p’ = 03
WI=
PA=0
n*=o
(A = 2,. . ., N).
Il
(5.23)
The normal to the middle surface in the reference configuration may be required to remain straight but not necessarily normal during deformation. Such a kinematic constraint can also be easily introduced in the theory. M. Epstein and Y. Tene[7] have given a linear theory based on
Recent advances in the general theory of thin elastic shells
245
the more general hypothesis of the conservation of a non-normal straight-line. Such a hypothesis can also be included, but results then in more complicated expressions. The equations obtained from (5.18) contain the effect of the bending term. It has been shown by H. AMPI that in most of the two-dimensional shell equations derived by means of variational methods from approximate strain energy functions, these terms are not adequately accounted for. The energy density 5’ is a function of the ~guments Q, x4, ?ra and ?TA;a,as shown in Section 5. An approximation of this function may be obtained by application of the Taylor formula and retaining only the second order terms. The first order terms vanish, since there is no strain energy in the reference configuration, and the zero order terms are irrelevant. When this approximation is admitted to represent the strain energy with sufficient accuracy, it gives rise to the so-called physically linear theories. Finally, let us note a theory of slender bodies developed by C. WoiniaifS], starting from a t~eedimensional body subject to internal constraints, He applies, however, this theory only to membranes and chords. 6. EXISTENCE
OF SOLUTIONS
The proof of the existence of solutions of a shell theory is a difficult problem which has not been settled presently in some satisfactory generality. Considerable progress has been made however in recent years in the relatively simple case of the statics of the physically linear problems. The energy integral in the variational law can be used then to define an inner product for the vector functions w(&, ai, WA)with arguments satisfying the boundary conditions. Thus a Hilbert space H(Q) of vectors o is introduced. The work integrals constitute a functional in the vector function &&I,,, Sir:, &r,,)&(~). This functional is proved to be linear and continuous in H(a). Hence, the Riesz representation theorem can be invoked to write this functional in the form (Ko - iQH, where K is a nonlinear operator in the space H(Q). (S.18) becomes then in the case of eq~ib~um.
Clearly, w is a vector which shall turn (6.1) into an identity for all 6. It is called generalized solution. The problem of finding this generalized solution is equivalent to solving the operator equation w=Ko
(6.2)
in the space H(a). The solvability of the nonlinear operator equation (6.2) can be studied by investigating some invariant property. With this purpose, the equation (6.2) can be put in correspondence with some geometrical object which may be either the transformation K, or the vector field o - Ko in the function space in which the operator acts. Then every theorem on the existence of a solution of (6.2) is equivalent to a theorem on the existence of a fixed point of the transformation or to a theorem on the existence of a nullvector of the field. The topological invariant which provides with a base for such an inves~ation is the ~tut~~ of a vector field, an integer number y*. For a definition (which is not simple at all) of this invariant as it applies to completely continuous vector fields in a Banach space 3, we refer to [lo]. A vector field @a, - a - Ko is said to be completely continuous in 481if K is a completely continuous operator in 8. Two vector fields o -bar and 4) - Kto are said to be homotopic on the boundary of a bounded domain A, if there exists a completely continuous operator K(o, 1) (waA,O< T < 1) such that K&U,0) = t(ow I<(@,1) = K,u (MaA)
246
MURAT DiKMEN
and K(o, 7) # o (maA, 0 < r < 1). As an essential property, it can be shown (101 that fields homotopic on aA have the same rotation (on aA). Points weA at which &J = 0, are called fixed points of the vector field (of the operator K). Consider a field Cpwhich has only isolated fixed points We,not lying on aA. Consider the balls S(W~,c) about each of these fixed points, with radius l > 0 so small that the balls are contained in A and moreover do not intersect. The vector fields Qth thus induced on the boundaries of these balls are homotopi~ for different values of e. The rotation rk thus determined (independent of E) is called the index of the fixed point WeB A of the field @. Here are reproduced the main results concerning solvability: If a completely continuous field Q, has a finite number of nullvectors in A, then the sum of their indices is equal to the rotation of the field on aA. If the rotation of Q,on aA is zero, then there exists a completely continuous extension of the vector field which has no null vectors. In other words, every criterion for the rotation of a vector field on the boundary of some region to differ from zero is also a sufhcient criterion for the existence of fixed points of the field inside the region. This is the Leray-Schauder principle, expressed in terms of rotation. It is worth noting that the Leray-Schauder principle can be used not only to prove existence, but in some cases also uniqueness. Indeed, if the rotation of the boundary aA of some region A is equal to +l or -1, and if the indices of all fixed points of the field are known to be of the same sign, then evidently Qphas just one fixed point in A. In particular, if K is a completely continuous operator defined on a closed ball S of the Banach space 3, such that it maps the boundary t?S of S into S itself, i.e. K(aS) C S, and if furthermore the completely ~ont~uous field Cp= 1- K has no nulIvectors on S, then Q, is homotopic to the completely continuous field I, the rotation of which is 1. Hence the rotation of Q, an aS is also equal to 1, i.e. the field Qi,has at least one fixed point in the ball. Further theorems on the existence of fixed point can be found in [lo]. An efficient and quite general scheme for the proof of the existence of a solution follows from the last theorem stated above. This scheme requires primarily that K be shown to be a completely continuous operator in H. In ~ticu~, it suffices to show that the functional generating K is the gradient of a weakly continuous un~ormly d~erentiable functional. Then, a theorem due to E. S. TsitlanadzeI 10,111, guarantees complete continuity. There remains then to prove the existence of fixed points. In spite however of the simplicity in principle of this scheme, the actual proof may be quite tedious, requiring several intermediate steps. Here we briefly comment on some of the contributions on the solvability of shell equilibrium problems, which were published in the last decade and follow more or less the scheme outlined above.
L. S. S~bschc~ takes [12] a system of georne~~~y nonlinear equations (together with nonlinear boundary-conditions) given by E. Reissner for the axisymmetrical deformation of nonshallow shells of revolution and investigates the existence of a solution for a shell of constant thickness. A restriction imposed to the slope of the meridian (with respect to the axis of revolution) excludes however some types of shells, a.o, spherical domes. I. I. Vorovich and L. P. Lebedev[ll?] have investigated the existence of a generalized solution of a geometricalIy nonlinear theory of orthotropic shalIow shells of variable thickness. They consider a domain a (in our notation) whose homeomo~~c map into a plane of variables 8” is the finite sum of star-shaped domains, and impose a set of conditions to the geometric and elastic characteristics on 0 as well as to tlfl. Their investigation covers various types of boundary conditions. They prove the existence of a generalized solution of the shell equilibrium problem, and obtain also a Kom-type inequality. In 1141,I. I. Vorovich, L. P. Lebedev and Sh. M. Shlafman investigate the existence of a generalized solution of a geometricahy nonlinear theory of nonshallow shells of revolution with
Recent advances in the general theory of thin elastic shells
241
clamped edge. Here too, the planform of the shell is assumed to consist in a finite sum of star-shaped domains. The existence of the solution is shown by proving the existence of critical points of the functional J whose gradient generates I - K. In this context, the application of the Ritz and Bubnov-Galerkin methods [15] is discussed briefly and it is asserted that the set of approximate solutions o,, (obtained by the Ritz or BubnovGalerkin method) included in a sphere of sufficiently large radius in the space H, is infinite, strongly compact and contains a minimizing sequence, each limit point of this set being a generalized solution of the equilibrium problem. The particular case of the axisymmetric deformation is treated separately, and the existence of at least one generalized axisymmetric solution of the equilibrium of a nonshallow shell of revolution is proved by showing that the rotation of a vector equals +l. The authors refer also to some earlier work[l6,17] and note an erroneous proof of the basic estimate used therein. The solvability of a version of geometrically nonlinear equations for a symmetrically loaded nonshallow shell has been investigated by Vorovich and Shlafman[l8]. A relatively little explored area is that of the nonstiffness of shells, i.e. the property of having for given boundary conditions and no loads, equilibrium configurations different from the trivial one. L. S. Srubshchik[l9] has given a proof of the existence of a second equilibrium mode of a nonshallow spherical dome, using E. Reissner’s equations for axisymmetric deformation and introducing asymptotic expansions for small values of a parameter proportional to the ratio of the thickness to the radius of the middle surface. For the proof of the existence of solutions in the case of physically and geometrically linear theories, Kom-type inequalities can be used very efficiently, to establish in some class of admissible functions the existence of the minimizing function of the total energy functional, which is also the solution of the equilibrium equations. B. A. Shoikhet proved[20] existence theorems for two different sets of shell equations. V. G. Komeev[21] also derives some Korn-type inequalities. In his paper, he bases on a comparison between the strain energy of the shell viewed as a three-dimensional elastic body and the strain energy calculated for the shell subject to the conservation of the normal. 7. ERROR ESTIMATES
Two types of errors, different in their nature, arise in dealing with derived shell theories. The lirst of these has its roots in the additional hypotheses introduced in integrating the three-dimensional equations to reduce them down to two-dimensional equations, or in postulating some structure of the strain energy function (constitutive assumption). These errors can not be removed or compensated by increasing the accuracy somewhere else in the theory. The effect and order of magnitude of this kind of errors have been discussed by many authors. The fundamental paper of F. John[22] uses estimates for derivatives, to derive from the equations for a three-dimensional elastic solid, approximate two-dimensional equations, called “interior shell equations”, with concrete estimates for the errors of approximation. For further developments and discussions in this direction, we refer to [23-251. AW[8] follows a similar method Sensenig[6] reverses the procedure, to show that a solution to the shell theory approximately satisfies the equilibrium equations and stress boundary conditions of the theory of elasticity. The shell theory is derived in this paper by considering the strain energy to depend on displacements expressed by polynomials in the undeformed distance to the middle surface. Estimates for the difterence in the displacements obtained from these two theories are not discussed. Despite the increasing efforts however, the d&ult question of estimating the error involved by proposing a two-dimensional theory as a replacement of the three-dimensional theory of elasticity is far from having been settled. C. L. Ho and J. K. Knowles have obtained pointwise estimates, in the relatively simple case of the linear axisymmetric torsion problem of a shell of revolution of constant thickness, subject to end loads only[27]. A. L. Gol’denveizer [28] has discussed in general the adequacy of shell theories. In the present Section, we concentrate on the second kind of errors, namely those which occur in the approximnte solutions of a given set of shell equations and boundary conditions. We restrict ourselves to the physically and geometrically linear cases. It turns out that the Hilbert space H induced by the inner product introduced in Section 6, or an equivalent one, provides with the necessary mathematical structure[28].
MURAT DiKMEN
248
We consider the shell as a body represented by the surface portion a, i.e. all quantities occurring in the equations being referred to Q and to its boundary &l. Along this boundary, we distinguish a part &I, and a part a& over which dynamical resp. kinematical conditions alone are prescribed. One speaks of inhomogeneous statical conditions when not all of the “loads” in Q and along XI,, vanish. One speaks of inhomogeneous geometrical conditions when not all of the “displacements” along a& vanish. Homogeneous geometrical conditions characterize an elastic state which corresponds to an element w’ of the space H. Homogeneous statical conditions in turn characterize an elastic state corresponding to an element w” of H. These two elements are orthogonal in H: (w’,w”)H= 0.
(7.1)
Let now w- and w- be two estimates for an unknown w which is to be determined approximately, the former estimate satisfying all geometric conditions, while the latter satisfies all statical conditions. Obviously, (w--w,w--w)=0.
(7.2)
Then, we obtain at once the inequalities
~~w--w~~~~(w--w-~~B((w--wI(
(7.3)
and
Illwll’ - lb-II21 c lb- + w-11 *lb- - w-11 lllwl12 - llwEl121 s llw-+w-11 *llw--w-11.
(7.4)
The last set of inequalities shows that the approximation is the better, the smaller is IIw--w-/l. If w0 resp. w; satisfy the inhomogenous conditions, we may write the estimates WK=Wg+Wk
w;=w;+w;
(7.5)
with K
wk=
c c
ckgi
=I
K
wk=
d&.
(7.6)
=1
The norm Ilw- -w”/ can be minimized by variation of the factors ck and 4, thus obtaining the relations
c_, c K
ck(&,
gd = 6$,w;- wii)
K
d&k gl) = (d,
=I
wi - 6)
(1 = 1,. . ., K).
(7.7)
If w = w’ (i.e. w0 = 0), the optimized approximations ++kand +pZ obtained from (7.5) and (7.7) yield
i.e. upper and lower estimates for the norm of the solution to be approximated.
Recent advances in the general theory of thin elastic shells
249
Due to the definition cf the inner product in H, the estimates thus obtained are global estimates. Pointwise estimates, for some w at a given point can be obtained by introducing a “Green state” 8 * H. Antes has used these methods to obtain pointwise estimates for shallow shells [29] and global estimates for approx~ations of shell deformations [30]. 8. DISCUSSION
The variational equality (5.18) yields equations of motion, which formally agree with other shell theories [3]. Instead of the stress resultants and couple resultants of derived theories which establish first the equations of equilibrium or motion in terms of these resultants and then introduce a constitutive relation, the quantities n*@,q* and mQBwhich appear in (5.18) are obtained by d~erentiating the strain energy density S, known through (5.7). The recent achievements in using variational and function analytic methods in the actual numerical computation of mechanical systems, render most natural the use of a variational principle right from the beginning, instead of adopting first a set of differential equations and then reverting to some variational formulation. The kinematic hypotheses (4.19) are approximations, since they consist in finite sums. This excludes however any question of convergence of some series expansion across the thickness. On the other hand, the possib~ity remains to take any number of appropriate polynomials, in order to reach a satisfactory approx~ation, resulting in a finite number of equations. The variational formulation permits, in the case of physically linear problems, i.e. when an inner product can be defined to induce a Hilbert space, to prove existence theorems within a well established scheme. In the case of fully linear problems, lower and upper bounds for the exact solution (of the shell problem) can also be found. These results are of obvious value in approximate and numerical solutions.
[II M. DIKMEN, Direkte Schalentheorie,Teil I, Lecture Nofes, Technische Universitat Braunschweig,Braunschweig (1970). 121hf. DIKMEN, Stabilityof the CosseratSurface. Proc. ZCJTAMSymp. on instability of Continuous Systems. Springer, Berlin/Heidelberg/NewYork (1971). 131P. M. NAGHDI, The theory of shells and plates. Handbuch der Physik, Vol. VI-2. Springer,Berlin/Heidelberg/New York (1972). 141L. I. SEDOV, Models of continuousmedia with internaldegrees of freedom. Appl. Math. Mech. (PM&Z),32,803-819 (1%8). I51 V. L. BE~ICHEVSKII, V~ation~ methods of cons~cting models of shells. Appt. Math. Meek. (PM%), 36, 743-758(1972). 161C. B. SENSENIG, A shell theory comparedwith the exact three dimensionaltheory of elasticity. Inr.J. ErrgngSci. 6, 435-464(1968). 171hf. EPSTEIN and Y. TENE, A lineartheory of thin elastic shells, based on conservationof a non-normalstraightline, Int. J. Solids Structures, 9, 257-268(1973). 181H. ABE, On bendingterms in the linearthin shell equationsin terms of the displacementcomponents.Znt.J. Engng. Sci. 13, 1055-1065(1975). 191C. WOANIAI,On the analytical mechanics of slender bodies. Bull. Acad. PO/ON. Sci., Skrie Sci. Tech., 23, 193-X@ (1974). [lOI M. A. ~SNOSE~SKII, Topoiogica~ fetus in tke Theory of Noai~ear Integral ~uations. Pergamon Press, Oxford (1964). [Ill M. M. VAINBERG, Va~afionuf Methods for the Study of NonZinear Operators. Holden-Day, San Francisco/London/Amsterdam (1964). [12] L. S. SRUBSHCHIK,On solvability of nonlinearequations of Reissner type for nonshallow symmetricallyloaded shells of revolution.Aovl. Math. Mech. (PM&O.32.322-326 (1968. 1131I. I. VOROVICHand L: P. LEBEDEV, bn the’existenceof solutions of the nonlineartheory of shallow shells. Appl. Math. Mech. (PMM), 36,652-665 (1972). 1141I. I. VOROVICH,L. P. LEBEDEV and SH. M. SHLAFMAN, On some direct methodsand the existence of solution in the nonlineartheory of elastic nonshallow shells of revolution.Appl. Math. Me& (PM@, 38,310-319 (1974). [IS) M. A. K~SNOSE~SKII, G. hf. VAINIKKO, P. P. ZABREIKO,Ya. B. R~~ISKiI and V. Ya. STETSENKO, ~~~~rn~e Solr&urof Uperutor ~~~t~~. WoI~~N~r~ff, Groningen(1972). [161 I. I. VOROVICBand G. A. KOSUSHKIN, On tbe solvability of general problems for an elastic closed cylindrical shell in a nonlinearformulation.Appl. Math. Mech. (PM@), 33.6676 (1969). 1171G. A. KOSUSHKIN,On the solvabilityof the generalproblemfor an anisotropiclaminarshell within the framework of mediumde&&on theory. Appl. Math. Mech. (PMM),34,371-379 (1970). [181 I. I. VOROVICHand SH. M. SHLAFMAN, On tbe solvability of nonlinearequations for a symmetricallyloaded nonshaRowsphericaldome. Appl. Math. Me& (FM&f). 38,895-892 (1974). 1191L. S. SRUBSCHCHIK,NonstifInessof a non&allow sphericaldome. Appl. Math. Mech. (PMM), 32,435-445 (196tI). t2OlB. A. SHOIKHET,On existence theoremsin linear shell theory. Appt. Math. Mech. (PM&f),38,527-531(1974). Ut?SVd. 17. No. 3-R
250
MURAT DiKMEN
1211V. G. KORNEEV, Differential onerator for the equilibrium equations of the theorv of thin shells. Mechanics of _ Solids, 10,76-82 (1975). (221 F. JOHN, Estimates for the derivatives of the stresses in a thin shell and interior shell equations. Comm. Pure Appl. Math., IS, 235-267(1%5). [23] F. JOHN, Refined interior shell equations. Proc. Second ALTAR Symp. on the Theory of Thin Shells. Springer, Berli~He~el~rg~New York ( 1969). I241 W. T. KOITER, A comparison between John’s refined interior shell equations and classical shell theory. (With an appendix by F. John). ZAMP, 24642-653 (1969). [2S] W. T. KOITER, Foundations of shell theory. Rep. No. 473. Laboratory of Engineering Mechanics, Delft University of Technology, Delft (1972). [26] C. L. HO and J. K. KNOWLES, Energy inequalities and error estimates for torsion of elastic shells of revolution. ZAMP, 21,352377 (1970). 1271A. L. GOL’DENVEIZER, Methods for justifying and refining the theory of shells. Appl. Math. Mech. (PMM), 32. 784-718(1968). [Zsl G. RIEDER, E~~ungen in der Elastizitllts- und Potenti~th~~e. Z.UfM, 52, T342-T347(1972). 1291H. ANTES, Punktweise Ei~e~ung bei flachen Schalen. Acta ~ec~a~ica, W, 235-245 (1975). [30] H. ANTES, Uber pausehale Fehlersc~en fur Approximationen von Schalenverformungen. ZAMM, f&491-501 (1975).
ELASTICITY
We record here, for later use, and also with the purpose of iutroducing some notation, some basic reiations of the ~e~e~sion~ elasticity theory.
236
MLJRATDiKMEN
We define a system of convected coordinates (0’) (i = 1,2,3) in a body B, occupying in the three-dimensional euclidean space, at any instant t, a finite region V, with boundary 8V. For simplicity, V shall be assumed to be simply-connected. Position vectors r = r(8’, t)
(2.1)
emanating from some reference point fixed in space, describe the “actual configuration” (at time t) of the body. By choosing the origin of t to correspond to a “reference configuration”, we write R(8’) =
I-(#,0)
(2.2)
It(#) + u(ei, t).
(2.3)
and introduce the displacement vector u by
r(e’, t)
=
We shall assume that the vector-valued functions r, R and II are differentiable as many times as necessitated by the computations, with respect to each of their arguments. The base vectors and the metric tensors in the actual and reference configurations are
gij E
gi
’ gj
(2.4)
and Gk = R,k Gij=Gi
*Gj
(2.5)
respectively. Lower case latin indices have, here and henceforth, the range 1,2,3, and a comma followed by a subscript, say k, represents differentiation with respect to the variable t$, keeping all other variables fixed. The determinants of the metric tensors shall be denoted by g and G, respectively, and assumed not to vanish in B. The strain tensor is given by yij
E i
CT =
(gij -
U( rij).
(2.7)
It shall be assumed that the body is in unstressed state at the reference configuration and also that the material is isotropic. Thermal effects are not considered. The stresses are given by
au
?‘I=p a’yii
(2.8)
where p is the mass density and TVis the symmetric stress tensor (with u symmetrized in the components n) and referred to the 8’ coordinates in the actual confIguration and also measured
Recentadvances in the generaltheory of thin elastic shells
237
per unit of area in the same configuration. Another symmetric stress tensor is defined by (2.9) and expresses stress again referred to the 8’ coordinates in the actual configuration, but measured per unit of the corresponding area in the reference configuration in which also the mass density p,,is measured. The relation between these stress tensors is sij= #
J5.
(2.10)
The fundamental dynamic law is expressed as a generalized Hamilton principle
s
Ldrdt+SW+SW*=O
(2.11)
(+2-c >
(2.12)
LV
where Lsp
is the lagrangean density function, and where the second and third terms are defined by
SW=
@b-u)dTdt
(2.13)
IV
and
SW*
(p.Su)dadt-
E
tJV
(+u)drl::
(2.14)
V
i = i is the velocity of a material point, b the body force per unit of mass, p the forces acting per unit of area of the boundary, I the momentum. dr is the element of volume and da is the element of area.
DESCRIPTION OF THE THIN SHELL Consider a body B, in its reference configuration, and let 8 be a surface which intersects B on a connected region Q = B f-12. It is convenient to choose the system of convected coordinates {e’) in such a way as to have Z represented by # = 0. Then, the points of fi are referred to a system of coordinates IOU}(a = 1,2), the Gaussian parameters on S. Henceforth, Greek indices will take the value 1 or 2. It will also be required that the e3-lines be chosen in such a way as to have the body B not extending outside of the “tube” 7 determined by 80 = 8B f-12 as dire&ix and the #-lines crossing &I as generators. It will furthermore be required that the #-lines intersect aB at most at two points. Of course, the configuration of B is assumed to permit such a construction of the system of convected coordinates {#}, or {fY, OS}. fl is then called reference surface. The intersection g = T 17aB is called the edge (and may reduce to an). The difference aB - 8 is disjoint and consists of two simply-connected parts, the faces Q+ and a- of B. In the next section, the system of coordinates shall conveniently be taken as a system of “normal coordinates”, i.e. in which the B3-linesare the normals to 2. The length of the segment 3.
238
MURAT D&MEN
intercepted, by the faces along the normal at a point {F) will be caiied thickness and denoted by h(P). In order to define the geometric quantities needed in the sequel, the surface Z shall be assumed sufficiently smooth; i.e. the position vector R(B”,O), or simply R(V), on the points {P, 0) of n shall be continuously differentiable up to the third order. Let RI and R2 be the radii of principal curvatures at (P,Oj and Iet Ir(&“) and &ffY) be the Iengths of arcs intercepted by 852 along the fines of curvature at the same point. The body B will be called a thin &if if h C rnk {R,, &}
(3.1)
u e max Ir, max iz
(3.2)
and
n
n
at every point of Q. Regularity properties concerning the faces, and hence h(V) shall also be assumed, as necessitated by the following computations. In particular, ribs are excluded.
4. SHELL GEOMETRYAND ~I~E~A~~~~ The base vectors, the metric tensor (first fundamental form) and its determinant on the reference surface 19~= 0 are
(4.11 in the actual configuration, and A, = G,(V, 0) A,=A,*A@ A = iAtlat
(4.2)
in the reference configuration. We record the relations
(4.31
where II is unit normal vector determined by a1 x 82, baB are the coefficients of the second fundamental form, and I’& are the Christoffel symbols of the second kind (metric connections with respect to a&. c,+ are the coefficients of the third f~~rnen~ form; they can be expressed a~ebrai~a~y in terms of baB and a,,. The semicobn indicates covariant diierentiation with respect to the metric tensor a,,. An analogous set of relations can be written for the reference configuration, simply replacing the latin symbols in (4.3) by capitals. To distinguish the covariant differentiation with
Recent advances in the generaltheory of thin elastic shells
239
respect to aaBfrom the covariant differentiations with respect to be, a colon shall be used for the latter. It is however convenient, as already noted in Section 2, to use a system of convected coordinates which is a system of normal c~rd~ates in the reference co~tion. Then G3 =
N, G, = &A,
G,@= /L:&&
Ge3 = 0,
G33 = 1
(4.4)
with f.&:=
s: - e313:
(4.5)
and I =
(;)“* .
IPO’I =
(4.6)
Then R(P) t9”)= R(P, 0) + e3N(5”)
(4.7)
and (2.3)can be rewritten in the form r(P,
d3,t) = R(B”, 0)+ u(tP, 0, t) + fz
f fn
(4.8)
where f and f are functions of (8’) and t, and f” behaves as a contravariant surface vector. The base vectors in the actual con&nation, as obtained from (4.8),are then
The components of the metric tensor can be calculated by using (4.9),(2.4)*and (4.3)S,6, to find
a = a28-
2fbl
+2f&#'
+f*ca/3 +fmf9/J + 'tfLfw + (f'f,;&~ + Lbf
g3* =f%3+faf%v g33 =
ti3,*+
- 2fbT&l,,
‘f” +f#;*
(4.10) +(-~,3+f,~)~~~
fyY3fY,3
Round brackets are used to indicate the indices over which symmetrization must be performed, while the sign 1 1encloses indices which shall be singled out in the process. Now, according to a fundamental theorem of surface theory, if three functions u&V)cC2 and three functions ALEC’ (with symmetry in the indices a,$#)are given and if they obey to the con~tions err> 0, 022> 0 and 1~1 f 0, they determine in the ~~~ension~ euclidean space a surface uniquely to within its position in space and a mirror symmetry, provided the Mainardi-Codazziequations (4.3)8and the Gauss equations (4.11) be satisfied. The surface thus determined is represented by an equation r = r(F), rcC3.
240
MURAT DfKMEN
R so67in (4.11) is the covariant curvature tensor defined by
The regularity secured by this theorem corresponds exactly to the one which was imposed to the reference surface in the reference configuration. This suggests that, provided the displacement vector u be continuously differentiable at least up to third order, with respect to the variables 0”, the quantities
(4.13) can be used as strains defined over the reference surface. The velocity of a point of the shell (body B) is given by ti=i=v+f”a,+f”i,+fn+fi
(4.14)
where a superimposed dot indicates differentiation with respect to time, and where v(B”, t) = li(V, 0, t)
(4.15)
i.e. the velocity of a point (with coordinates 0”) of the reference surface. Thus, (4.14) can be written in the form
ir = k = v +f”v,= + l_P-f(n
- v,g)ascr}a, +fn.
(4.16)
The expressions (4.9) and (4.16) are exact, in the sense that besides the regularity condition imposed above to the displacement vector u, and therefore to the functions fcr(P, e3, t) and f(P, e3, t) no restriction has been assumed concerning these latter functions, except that they must of course vanish for B3= 0, in view of (4.8). We may, in particular, assume that these functions depend on 8“ and t through a finite number of functions (4.17) i.e.
(4.18) Motivated by a paper of L. Sedov[4], this is the position adopted by V. L. Berdichevskii[5]. The presence of the “parameters” $A means that some restrictions, i.e. internal constraints, are imposed to the deformation of the body B, by the structure of the functions f” and f. Here, we develop ab initio a theory deviating from that in [51inasmuch as (i) we avoid using an auxiliary Cartesian system of coordinates, which is not essential in the theory, (ii) we introduce a somewhat less general kinematic representation, namely (4.19) instead of (4.18), which allows a simpler form and interpretation of the equations of motion. In fact, Weierstrass’ polynomial approximation theorem suggests the structure p = P(e%;(e@,
,. f = P(e3h(e@,
t) t)
(4.19)
where the PA are appropriately chosen polynomials. It is understood that the summation
Recent advances in the general theory of thin elastic shells
241
convention applies to capital latin indices also. These indices run from 1 to N. Of course, the functions VAare subject to the same regularity conditions as the functions f” and f. The functions p and f defined this way may then be viewed either as a polynomial approx~ation to the functions f” resp. f, or they may be also viewed as elements of a subclass of functions (4.19), obtained by imposing restrictions, i.e. internal constraints to the kinematics of the medium. We shall use henceforth the structure (4.19) and omit the sign ( A). Back to (4.10) and (4.16), we see that the arguments on which the strains 7ij and velocity and the velocity ti depend, are the ones shown below:
5. EQUATIONS
OF MOTION
The energy integral in (2.11) can be written
(5.1) with dr = t/C de’ de* de3 = /L de3 dA dA = d/A de’ de*
(5.2)
and p defined by (4.6). Further,
e3 = b(e*) 83 = ate*)
(5.3)
b-a=h describe the upper resp. lower faces of the shell. They are also assumed (see Section 3) to be sufficiently smooth. The innermost integral afP’) A= Lp de3 (5.4) f bfe”f defines an energy density per unit area of 0, and is equal to the difference
w
A=K-S
of the kinetic energy density
(5.6) and strain energy density
s =I
b
a
&p&d.
de3.
(5.7)
Clearly, these quantities obtained by integration over [a, b] depend on the arguments listed in
K = Kb’,,,&fit
s =Sk@,xc+ in view of (4.20).
mA,+A) TA,
n;l,,)
(5.8)
MURAT DiKMEN
242
As concerns (2.13) and (2.14) we need to calculate &I E Sr. Using (4.8), we find
n,, stands for the displacement of points of the reference surface. This expression for Su can now be inserted into
If
SW=
I B
(pob * Su) dr dt = I, (I, [Jab @oh +WP de31 dA} dt
(5.10)
to calculate first the innermost integral in the right side b
f
(t pob *
p,,bPAp d03)S?rA
Suo+{a&"A+n&r~}*
{j-Obpobp
poPAbp
dd3)
-
c+a
* (lab p,,PAbp de3)]n)
- (Su,,,).
(5.11)
The terms in the right side of (5.11) suggest some definitions:
(5.12) With this notation, (5.11) can be summarized in the form (5.13) (2.14) can be rewritten as
sw* =
If I
an
= fff n+
(p.Su)dAdt-
(I,I+dr);; (p . Su) dA- dt +
(p . Su) dA+ dt +
(p.6u)pedB3dsdt-lleI.6ud7(;:
or
+
+
If If c
an
{Q * Sue + SA *(a&pS + n&A) + Ma - SUO~}dA dt
n {i.
Su,,+ TA * (a&r2 + n&A)
+ P
* c%Io,~}
-
?rAPA(a”
11
dA) ”
where the abbreviated notation ii = (p/i)+ + (Pa@AE (pP$)’
+ (pPAfi)-
p z {.lr;pPA - TAPA(a” ’
p)n}+
+ {r$PA
* p)$
(5.15)
243
Recent advances in the general theory of thin elastic shells b
Q=j-= pp=de3 b s*
s
I (1
i*
s
pPA$ de3
bIP*p
I (I
Ib{r:IP* 0
de3
- nAPA(aa’ I)n}p de3
(5.16)
and the geometric quantities
have been introduced. In (5.17)4,e and e. are the determinants of the metric tensor on rS,in the actual and reference configuration, respectively. #’ is the unit tangent vector to 0, in the reference configuration. There remains to evaluate the variations involved in the term with the energy integral. The intermediate steps shall not be shown here, and the final form taken by the variational law is given at once below:
-I,Ipl
@;@ - nu8b,&
+ y(naB;8 + q8b;)a., -(F+
I) + (P + ha);,, + (p&o+ &%A)
- (I@ + PA) . a. + (&ii + pp~~lr~) - a,
+
If,
dR{Q -
- q%)v, yWBaa
+ (P +
@pa
ST; dA dt
MURAT DiKMEN
244
where
(5.19)
and
(5.20)
v, is the normal to aa in 2.
J
y”
(5.21)
;.
The equations of motion and the boundary conditions, as well as the auxiliary conditions implied by the last integral can be obtained easily, by the usual argument of the calculus of variations. They shall therefore not be listed here separately. The expressions in (5.18) simplify significantly in the limiting case where p can be taken equal to one, by using a set of appropriate orthonormal polynomials PA. Then the quantities pp vanish. The vectors mAmay be viewed as a generalization of the directors of direct theories, thus establishing the link between these and the present derived theory. Using these equations, theories of various orders of accuracy for the description of the shell behaviour can be obtained systematically. Polynomials in O3for the description of the displacements have been used in [6] also, but in a different approach than in the present case. While (5.18) involves $I and r,., as unknown functions, in finite but unspecified number, a priori assumptions made with respect to these functions may considerably simplify the equations. The classical geometric hypothesis of the normal to the middle-surface, remaining normal during deformation is expressed by r;=O
(5.22)
(A=l,...,N)
and the reference surface is chosen as middle-surface, i.e.
If the normal is not permitted to stretch, then p’ = 03
WI=
PA=0
n*=o
(A = 2,. . ., N).
Il
(5.23)
The normal to the middle surface in the reference configuration may be required to remain straight but not necessarily normal during deformation. Such a kinematic constraint can also be easily introduced in the theory. M. Epstein and Y. Tene[7] have given a linear theory based on
Recent advances in the general theory of thin elastic shells
245
the more general hypothesis of the conservation of a non-normal straight-line. Such a hypothesis can also be included, but results then in more complicated expressions. The equations obtained from (5.18) contain the effect of the bending term. It has been shown by H. AMPI that in most of the two-dimensional shell equations derived by means of variational methods from approximate strain energy functions, these terms are not adequately accounted for. The energy density 5’ is a function of the ~guments Q, x4, ?ra and ?TA;a,as shown in Section 5. An approximation of this function may be obtained by application of the Taylor formula and retaining only the second order terms. The first order terms vanish, since there is no strain energy in the reference configuration, and the zero order terms are irrelevant. When this approximation is admitted to represent the strain energy with sufficient accuracy, it gives rise to the so-called physically linear theories. Finally, let us note a theory of slender bodies developed by C. WoiniaifS], starting from a t~eedimensional body subject to internal constraints, He applies, however, this theory only to membranes and chords. 6. EXISTENCE
OF SOLUTIONS
The proof of the existence of solutions of a shell theory is a difficult problem which has not been settled presently in some satisfactory generality. Considerable progress has been made however in recent years in the relatively simple case of the statics of the physically linear problems. The energy integral in the variational law can be used then to define an inner product for the vector functions w(&, ai, WA)with arguments satisfying the boundary conditions. Thus a Hilbert space H(Q) of vectors o is introduced. The work integrals constitute a functional in the vector function &&I,,, Sir:, &r,,)&(~). This functional is proved to be linear and continuous in H(a). Hence, the Riesz representation theorem can be invoked to write this functional in the form (Ko - iQH, where K is a nonlinear operator in the space H(Q). (S.18) becomes then in the case of eq~ib~um.
Clearly, w is a vector which shall turn (6.1) into an identity for all 6. It is called generalized solution. The problem of finding this generalized solution is equivalent to solving the operator equation w=Ko
(6.2)
in the space H(a). The solvability of the nonlinear operator equation (6.2) can be studied by investigating some invariant property. With this purpose, the equation (6.2) can be put in correspondence with some geometrical object which may be either the transformation K, or the vector field o - Ko in the function space in which the operator acts. Then every theorem on the existence of a solution of (6.2) is equivalent to a theorem on the existence of a fixed point of the transformation or to a theorem on the existence of a nullvector of the field. The topological invariant which provides with a base for such an inves~ation is the ~tut~~ of a vector field, an integer number y*. For a definition (which is not simple at all) of this invariant as it applies to completely continuous vector fields in a Banach space 3, we refer to [lo]. A vector field @a, - a - Ko is said to be completely continuous in 481if K is a completely continuous operator in 8. Two vector fields o -bar and 4) - Kto are said to be homotopic on the boundary of a bounded domain A, if there exists a completely continuous operator K(o, 1) (waA,O< T < 1) such that K&U,0) = t(ow I<(@,1) = K,u (MaA)
246
MURAT DiKMEN
and K(o, 7) # o (maA, 0 < r < 1). As an essential property, it can be shown (101 that fields homotopic on aA have the same rotation (on aA). Points weA at which &J = 0, are called fixed points of the vector field (of the operator K). Consider a field Cpwhich has only isolated fixed points We,not lying on aA. Consider the balls S(W~,c) about each of these fixed points, with radius l > 0 so small that the balls are contained in A and moreover do not intersect. The vector fields Qth thus induced on the boundaries of these balls are homotopi~ for different values of e. The rotation rk thus determined (independent of E) is called the index of the fixed point WeB A of the field @. Here are reproduced the main results concerning solvability: If a completely continuous field Q, has a finite number of nullvectors in A, then the sum of their indices is equal to the rotation of the field on aA. If the rotation of Q,on aA is zero, then there exists a completely continuous extension of the vector field which has no null vectors. In other words, every criterion for the rotation of a vector field on the boundary of some region to differ from zero is also a sufhcient criterion for the existence of fixed points of the field inside the region. This is the Leray-Schauder principle, expressed in terms of rotation. It is worth noting that the Leray-Schauder principle can be used not only to prove existence, but in some cases also uniqueness. Indeed, if the rotation of the boundary aA of some region A is equal to +l or -1, and if the indices of all fixed points of the field are known to be of the same sign, then evidently Qphas just one fixed point in A. In particular, if K is a completely continuous operator defined on a closed ball S of the Banach space 3, such that it maps the boundary t?S of S into S itself, i.e. K(aS) C S, and if furthermore the completely ~ont~uous field Cp= 1- K has no nulIvectors on S, then Q, is homotopic to the completely continuous field I, the rotation of which is 1. Hence the rotation of Q, an aS is also equal to 1, i.e. the field Qi,has at least one fixed point in the ball. Further theorems on the existence of fixed point can be found in [lo]. An efficient and quite general scheme for the proof of the existence of a solution follows from the last theorem stated above. This scheme requires primarily that K be shown to be a completely continuous operator in H. In ~ticu~, it suffices to show that the functional generating K is the gradient of a weakly continuous un~ormly d~erentiable functional. Then, a theorem due to E. S. TsitlanadzeI 10,111, guarantees complete continuity. There remains then to prove the existence of fixed points. In spite however of the simplicity in principle of this scheme, the actual proof may be quite tedious, requiring several intermediate steps. Here we briefly comment on some of the contributions on the solvability of shell equilibrium problems, which were published in the last decade and follow more or less the scheme outlined above.
L. S. S~bschc~ takes [12] a system of georne~~~y nonlinear equations (together with nonlinear boundary-conditions) given by E. Reissner for the axisymmetrical deformation of nonshallow shells of revolution and investigates the existence of a solution for a shell of constant thickness. A restriction imposed to the slope of the meridian (with respect to the axis of revolution) excludes however some types of shells, a.o, spherical domes. I. I. Vorovich and L. P. Lebedev[ll?] have investigated the existence of a generalized solution of a geometricalIy nonlinear theory of orthotropic shalIow shells of variable thickness. They consider a domain a (in our notation) whose homeomo~~c map into a plane of variables 8” is the finite sum of star-shaped domains, and impose a set of conditions to the geometric and elastic characteristics on 0 as well as to tlfl. Their investigation covers various types of boundary conditions. They prove the existence of a generalized solution of the shell equilibrium problem, and obtain also a Kom-type inequality. In 1141,I. I. Vorovich, L. P. Lebedev and Sh. M. Shlafman investigate the existence of a generalized solution of a geometricahy nonlinear theory of nonshallow shells of revolution with
Recent advances in the general theory of thin elastic shells
241
clamped edge. Here too, the planform of the shell is assumed to consist in a finite sum of star-shaped domains. The existence of the solution is shown by proving the existence of critical points of the functional J whose gradient generates I - K. In this context, the application of the Ritz and Bubnov-Galerkin methods [15] is discussed briefly and it is asserted that the set of approximate solutions o,, (obtained by the Ritz or BubnovGalerkin method) included in a sphere of sufficiently large radius in the space H, is infinite, strongly compact and contains a minimizing sequence, each limit point of this set being a generalized solution of the equilibrium problem. The particular case of the axisymmetric deformation is treated separately, and the existence of at least one generalized axisymmetric solution of the equilibrium of a nonshallow shell of revolution is proved by showing that the rotation of a vector equals +l. The authors refer also to some earlier work[l6,17] and note an erroneous proof of the basic estimate used therein. The solvability of a version of geometrically nonlinear equations for a symmetrically loaded nonshallow shell has been investigated by Vorovich and Shlafman[l8]. A relatively little explored area is that of the nonstiffness of shells, i.e. the property of having for given boundary conditions and no loads, equilibrium configurations different from the trivial one. L. S. Srubshchik[l9] has given a proof of the existence of a second equilibrium mode of a nonshallow spherical dome, using E. Reissner’s equations for axisymmetric deformation and introducing asymptotic expansions for small values of a parameter proportional to the ratio of the thickness to the radius of the middle surface. For the proof of the existence of solutions in the case of physically and geometrically linear theories, Kom-type inequalities can be used very efficiently, to establish in some class of admissible functions the existence of the minimizing function of the total energy functional, which is also the solution of the equilibrium equations. B. A. Shoikhet proved[20] existence theorems for two different sets of shell equations. V. G. Komeev[21] also derives some Korn-type inequalities. In his paper, he bases on a comparison between the strain energy of the shell viewed as a three-dimensional elastic body and the strain energy calculated for the shell subject to the conservation of the normal. 7. ERROR ESTIMATES
Two types of errors, different in their nature, arise in dealing with derived shell theories. The lirst of these has its roots in the additional hypotheses introduced in integrating the three-dimensional equations to reduce them down to two-dimensional equations, or in postulating some structure of the strain energy function (constitutive assumption). These errors can not be removed or compensated by increasing the accuracy somewhere else in the theory. The effect and order of magnitude of this kind of errors have been discussed by many authors. The fundamental paper of F. John[22] uses estimates for derivatives, to derive from the equations for a three-dimensional elastic solid, approximate two-dimensional equations, called “interior shell equations”, with concrete estimates for the errors of approximation. For further developments and discussions in this direction, we refer to [23-251. AW[8] follows a similar method Sensenig[6] reverses the procedure, to show that a solution to the shell theory approximately satisfies the equilibrium equations and stress boundary conditions of the theory of elasticity. The shell theory is derived in this paper by considering the strain energy to depend on displacements expressed by polynomials in the undeformed distance to the middle surface. Estimates for the difterence in the displacements obtained from these two theories are not discussed. Despite the increasing efforts however, the d&ult question of estimating the error involved by proposing a two-dimensional theory as a replacement of the three-dimensional theory of elasticity is far from having been settled. C. L. Ho and J. K. Knowles have obtained pointwise estimates, in the relatively simple case of the linear axisymmetric torsion problem of a shell of revolution of constant thickness, subject to end loads only[27]. A. L. Gol’denveizer [28] has discussed in general the adequacy of shell theories. In the present Section, we concentrate on the second kind of errors, namely those which occur in the approximnte solutions of a given set of shell equations and boundary conditions. We restrict ourselves to the physically and geometrically linear cases. It turns out that the Hilbert space H induced by the inner product introduced in Section 6, or an equivalent one, provides with the necessary mathematical structure[28].
MURAT DiKMEN
248
We consider the shell as a body represented by the surface portion a, i.e. all quantities occurring in the equations being referred to Q and to its boundary &l. Along this boundary, we distinguish a part &I, and a part a& over which dynamical resp. kinematical conditions alone are prescribed. One speaks of inhomogeneous statical conditions when not all of the “loads” in Q and along XI,, vanish. One speaks of inhomogeneous geometrical conditions when not all of the “displacements” along a& vanish. Homogeneous geometrical conditions characterize an elastic state which corresponds to an element w’ of the space H. Homogeneous statical conditions in turn characterize an elastic state corresponding to an element w” of H. These two elements are orthogonal in H: (w’,w”)H= 0.
(7.1)
Let now w- and w- be two estimates for an unknown w which is to be determined approximately, the former estimate satisfying all geometric conditions, while the latter satisfies all statical conditions. Obviously, (w--w,w--w)=0.
(7.2)
Then, we obtain at once the inequalities
~~w--w~~~~(w--w-~~B((w--wI(
(7.3)
and
Illwll’ - lb-II21 c lb- + w-11 *lb- - w-11 lllwl12 - llwEl121 s llw-+w-11 *llw--w-11.
(7.4)
The last set of inequalities shows that the approximation is the better, the smaller is IIw--w-/l. If w0 resp. w; satisfy the inhomogenous conditions, we may write the estimates WK=Wg+Wk
w;=w;+w;
(7.5)
with K
wk=
c c
ckgi
=I
K
wk=
d&.
(7.6)
=1
The norm Ilw- -w”/ can be minimized by variation of the factors ck and 4, thus obtaining the relations
c_, c K
ck(&,
gd = 6$,w;- wii)
K
d&k gl) = (d,
=I
wi - 6)
(1 = 1,. . ., K).
(7.7)
If w = w’ (i.e. w0 = 0), the optimized approximations ++kand +pZ obtained from (7.5) and (7.7) yield
i.e. upper and lower estimates for the norm of the solution to be approximated.
Recent advances in the general theory of thin elastic shells
249
Due to the definition cf the inner product in H, the estimates thus obtained are global estimates. Pointwise estimates, for some w at a given point can be obtained by introducing a “Green state” 8 * H. Antes has used these methods to obtain pointwise estimates for shallow shells [29] and global estimates for approx~ations of shell deformations [30]. 8. DISCUSSION
The variational equality (5.18) yields equations of motion, which formally agree with other shell theories [3]. Instead of the stress resultants and couple resultants of derived theories which establish first the equations of equilibrium or motion in terms of these resultants and then introduce a constitutive relation, the quantities n*@,q* and mQBwhich appear in (5.18) are obtained by d~erentiating the strain energy density S, known through (5.7). The recent achievements in using variational and function analytic methods in the actual numerical computation of mechanical systems, render most natural the use of a variational principle right from the beginning, instead of adopting first a set of differential equations and then reverting to some variational formulation. The kinematic hypotheses (4.19) are approximations, since they consist in finite sums. This excludes however any question of convergence of some series expansion across the thickness. On the other hand, the possib~ity remains to take any number of appropriate polynomials, in order to reach a satisfactory approx~ation, resulting in a finite number of equations. The variational formulation permits, in the case of physically linear problems, i.e. when an inner product can be defined to induce a Hilbert space, to prove existence theorems within a well established scheme. In the case of fully linear problems, lower and upper bounds for the exact solution (of the shell problem) can also be found. These results are of obvious value in approximate and numerical solutions.
[II M. DIKMEN, Direkte Schalentheorie,Teil I, Lecture Nofes, Technische Universitat Braunschweig,Braunschweig (1970). 121hf. DIKMEN, Stabilityof the CosseratSurface. Proc. ZCJTAMSymp. on instability of Continuous Systems. Springer, Berlin/Heidelberg/NewYork (1971). 131P. M. NAGHDI, The theory of shells and plates. Handbuch der Physik, Vol. VI-2. Springer,Berlin/Heidelberg/New York (1972). 141L. I. SEDOV, Models of continuousmedia with internaldegrees of freedom. Appl. Math. Mech. (PM&Z),32,803-819 (1%8). I51 V. L. BE~ICHEVSKII, V~ation~ methods of cons~cting models of shells. Appt. Math. Meek. (PM%), 36, 743-758(1972). 161C. B. SENSENIG, A shell theory comparedwith the exact three dimensionaltheory of elasticity. Inr.J. ErrgngSci. 6, 435-464(1968). 171hf. EPSTEIN and Y. TENE, A lineartheory of thin elastic shells, based on conservationof a non-normalstraightline, Int. J. Solids Structures, 9, 257-268(1973). 181H. ABE, On bendingterms in the linearthin shell equationsin terms of the displacementcomponents.Znt.J. Engng. Sci. 13, 1055-1065(1975). 191C. WOANIAI,On the analytical mechanics of slender bodies. Bull. Acad. PO/ON. Sci., Skrie Sci. Tech., 23, 193-X@ (1974). [lOI M. A. ~SNOSE~SKII, Topoiogica~ fetus in tke Theory of Noai~ear Integral ~uations. Pergamon Press, Oxford (1964). [Ill M. M. VAINBERG, Va~afionuf Methods for the Study of NonZinear Operators. Holden-Day, San Francisco/London/Amsterdam (1964). [12] L. S. SRUBSHCHIK,On solvability of nonlinearequations of Reissner type for nonshallow symmetricallyloaded shells of revolution.Aovl. Math. Mech. (PM&O.32.322-326 (1968. 1131I. I. VOROVICHand L: P. LEBEDEV, bn the’existenceof solutions of the nonlineartheory of shallow shells. Appl. Math. Mech. (PMM), 36,652-665 (1972). 1141I. I. VOROVICH,L. P. LEBEDEV and SH. M. SHLAFMAN, On some direct methodsand the existence of solution in the nonlineartheory of elastic nonshallow shells of revolution.Appl. Math. Me& (PM@, 38,310-319 (1974). [IS) M. A. K~SNOSE~SKII, G. hf. VAINIKKO, P. P. ZABREIKO,Ya. B. R~~ISKiI and V. Ya. STETSENKO, ~~~~rn~e Solr&urof Uperutor ~~~t~~. WoI~~N~r~ff, Groningen(1972). [161 I. I. VOROVICBand G. A. KOSUSHKIN, On tbe solvability of general problems for an elastic closed cylindrical shell in a nonlinearformulation.Appl. Math. Mech. (PM@), 33.6676 (1969). 1171G. A. KOSUSHKIN,On the solvabilityof the generalproblemfor an anisotropiclaminarshell within the framework of mediumde&&on theory. Appl. Math. Mech. (PMM),34,371-379 (1970). [181 I. I. VOROVICHand SH. M. SHLAFMAN, On tbe solvability of nonlinearequations for a symmetricallyloaded nonshaRowsphericaldome. Appl. Math. Me& (FM&f). 38,895-892 (1974). 1191L. S. SRUBSCHCHIK,NonstifInessof a non&allow sphericaldome. Appl. Math. Mech. (PMM), 32,435-445 (196tI). t2OlB. A. SHOIKHET,On existence theoremsin linear shell theory. Appt. Math. Mech. (PM&f),38,527-531(1974). Ut?SVd. 17. No. 3-R
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1211V. G. KORNEEV, Differential onerator for the equilibrium equations of the theorv of thin shells. Mechanics of _ Solids, 10,76-82 (1975). (221 F. JOHN, Estimates for the derivatives of the stresses in a thin shell and interior shell equations. Comm. Pure Appl. Math., IS, 235-267(1%5). [23] F. JOHN, Refined interior shell equations. Proc. Second ALTAR Symp. on the Theory of Thin Shells. Springer, Berli~He~el~rg~New York ( 1969). I241 W. T. KOITER, A comparison between John’s refined interior shell equations and classical shell theory. (With an appendix by F. John). ZAMP, 24642-653 (1969). [2S] W. T. KOITER, Foundations of shell theory. Rep. No. 473. Laboratory of Engineering Mechanics, Delft University of Technology, Delft (1972). [26] C. L. HO and J. K. KNOWLES, Energy inequalities and error estimates for torsion of elastic shells of revolution. ZAMP, 21,352377 (1970). 1271A. L. GOL’DENVEIZER, Methods for justifying and refining the theory of shells. Appl. Math. Mech. (PMM), 32. 784-718(1968). [Zsl G. RIEDER, E~~ungen in der Elastizitllts- und Potenti~th~~e. Z.UfM, 52, T342-T347(1972). 1291H. ANTES, Punktweise Ei~e~ung bei flachen Schalen. Acta ~ec~a~ica, W, 235-245 (1975). [30] H. ANTES, Uber pausehale Fehlersc~en fur Approximationen von Schalenverformungen. ZAMM, f&491-501 (1975).