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0 or Ap _> 0 if ~(ap) = 0.
In other words, a pillar exerts a reactive force on the plate only if they are in contact. rants: The problems found in (1) and (3) provide instances of constrained minimization problems, whose questions of existence and uniqueness of solutions are easily resolved. In both cases, the existence of the numbers Ap is more delicate to establish: W h e n equality constraints are imposed as in (1), it relies on the existence of Lagrange multipliers; when inequality constraints are imposed as in (3), it relies on the Kuhn-Tucker relations. For details about these questions, see, e.g., Ciarlet [1982, Thms. 7.2-3, 8.2-2, 9.2-4]. 1.12. Let the scaled displacements u~(c) 9ft ~ R be defined by -
for
all
-
e
i.e., the scaling factors are the same for the "horizontal" and "vertical" components (to fix ideas and facilitate the comparison with the text, c is chosen as the scaling factor; as often remarked, this is no loss of generality as the problem is linear); assume t h a t the Lam4 constants are independent of e and t h a t f~(x ~) - e3k(x) for all x ~ - 7r~x e f F , g~ (x ~) - e4g~(x) for all x ~ - 7r~x E F+~ U F~,
Linearly elastic plates
126
[Ch. 1
where the functions f~ E L2(ft) and 9~ E L~(F+u r _ ) are independent of e, i.e., the assumptions on the applied forces are the same for the
"horizontal" and "vertical" components. Show t h a t the "new" scaled unknown u*(c):= (u~(e))converges HI(f~) as c ~ 0. C o m p a r e the scaled limit problem obtained in this fashion, i.e. having lim u* (e) as its solution with t h a t found in in
6--*0
Thin. 1.4-1. Hint: The proof of convergence hinges on the following generalized Korn inequality (itself a special case of another generalized Korn's inequality, established in Ciarlet, Lods & Miara [1996, T h m . 4.1]): Given e > 0 and a vector field v = (v~) C H I ( ~ ) , let the s y m m e t r i c tensor e(c; v ) = (e~3(c; v)) be defined by
~ ( ~ ; ~) . - ~l(o~vz+Ozv~) 1
1
~(~; v ) . - ~ ( 0 ~ + - 0 ~ , ) , e33(c; v )
"- Osv3.
Let 70 C 7 be such t h a t length 7o > O. T h e n show t h a t there exists a constant C independent of ~ > 0 such t h a t
Ilvlll ~ < _c I~(~; v)io,~ for all v E H l ( f t ) t h a t vanish on F0 - 70 x [-1, 1]. 1.13 This problem, based on an idea of A. Raoult, provides another m a n n e r of establishing part (iii) of the proof of T h m . 1.11-2. T h e notations are as in this theorem. (1) Define the space
vk~(~) -
{ ( ( ~ . - x ~ O . ~ ) . ~ ) ; ~ , e H:(~). ~ e H~(~). r/~ -- oq~,r/3 -- 0 on 70}.
Show that, given any v e V ~ L (~t), there exist elements v(c) E V(~t),
127
Exercises c > O, such that v(c) -+ v in H I ( a ) , le~a(v(c)) --+ 0 in L2(f/), g
1
~e.o(v(c)) A+2#
+ 7eaa(v(c)) --+ 0 in L2(ft), as s --+ 0.
Hint: Given v - ( ( r / ~ - Xa0~Wa), r/a) E V~L(ft), let v~(c)"-- v~ and
X3
~(~)
V
-
,7~ -
~~(x~O~,7~
A+2#
-
-/x,7~).
2
(2) Show that the space V~L(ft ) is dense in V/eL(Ft). (3) Using (2), show that the conclusions of (1) hold in fact for any E VKL(~) hence that J ( v ) - lim J(c)(v). ~--+0
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CHAPTER 2 JUNCTIONS IN LINEARLY MULTI-STRUCTURES
ELASTIC
INTRODUCTION The modeling of elastic multi-structures, i.e., elastic bodies that comprise "clearly identified" substructures of possibly different "dimensions", such as three-dimensional substructures, plates, shells, rods, etc., usually made of different elastic materials, is a problem of outstanding practical importance, since such elastic multistructures are very common: They include folded plates, H-shaped beams, plates clamped in three-dimensional foundations, plates or shells with stiffeners, etc. (see Figs. 2.6-1 to 2.6-4). We describe and analyze in this chapter a systematic procedure recently devised for mathematically modeling such multi-structures. We consider in Sect. 2.1 a problem in three-dimensional linearized elasticity, posed over a domain consisting of a partially clamped plate with thickness 2~ inserted into a "three-dimensional" elastic body, (which the plate thus supports), these two bodies forming together a "canonical" multi-structure. If the Lam~ constants of the materim constituting the plate vary as c -3, those of the three-dimensional body are independent of ~, and the applied force densities vary as appropriate powers of c, we show (Sects. 2.3 to 2.5) that the solution of the three-dimensional problem, once appropriately scaled, converges as ~ approaches zero to the solution of a coupled, "multidimensional", problem of a new type, posed simultaneously over a three-dimensional open set with a slit and a two-dimensional open set (the middle surface of the plate). The asymptotic analysis employed here relies on the asymptotic analysis for a "single plate" already studied in Chap. 1 on the one hand, and on a particular technique for studying the asymptotic behavior as c ---, 0 of the scaled displacement field inside the portion of
Junctions in linearly elastic multi-structures
130
[Ch. 2
the plate that is inserted into the three-dimensional structure, on the other. More specifically, consider a linearly elastic plate with Lain6 constants A~, >~ occupying the set ~ - -w x I-e, e] and clamped on a portion F~ of its lateral face. The plate is inserted into a threedimensional linearly elastic body with Lam6 constants A~, /~ occupying a set {ft~}- (d denotes the depth of the insertion). These two-elastic bodies are "perfectly bonded" along their common bounddry, thus forming together an elastic multi-structure (Fig. 2.1-1). The unknown displacement u ~ - (u~) 9 S~ --+ R a, where S ~ = int { ( a ~ tO ft~)- }, satisfies U e C V ( S e) -- {V e -- (V~) C H I ( S e ) ;
v e --0
on P~)},
{)~eC;p(Ue)Cqq(Vg) + 2# e_.ij(ug)e_.ij(vg)} dx ~
+ [ {A~e;p(u~)eqq(V ~) + 2#~e~5(u~)eiS(v~)} dx ~ Jn c
- f
Jf~
f[v:dx~+ f
f [ v : d x ~ for all v ~ c V ( S ~ ) ,
where (f[) E L2(S ~) denotes the applied body force density. In Sect. 2.2, we transform this problem into an equivalent scaled problem, now posed over two sets ft and ft t h a t are both independent of e. We first let a = w x ] - 1, 1[ as for a "single plate" (Chap. 1). T h e n inside f~, the displacement is scaled as u(e) = (u~(e)), with
u~(x ~) - e2u~(c)(x)
and
u~(x ~) - ~Ua(e)(z),
for all x ~ - 7r~x E ~ , where "Ke(Xl, Z2, Z3) -- (Xl, Z2, CZ3). W e assume that A~ - ~ - 3 / k
f~(x ~) - e - l f ~ ( x )
and
and
#~-e-3#,
f~(x ~) - f3(x) for all x ~ - rr~x E ft ~,
where the constants A > 0, # > 0 and the functions fi E L2(ft) are independent of e. In other words, the scalings and assumptions inside the plate are as in Chap. 1 for a "single plate" (such assumptions on
Introduction
131
the Lam~ constants and the forces correspond to the choice t = - 3 in the class defined in Sect. 1.8). Let ~ denote the inserted portion of the plate. We define the set ~ - i n t ( { ~ U ~ } - ) , which is indeed a set independent of (Fig. 2.2-1); for technical reasons, ~t is rather a translation of the set i n t ( { ~ U ~t~}-), but this fact is ignored in this introduction. Then inside the three-dimensional body, and also inside the inserted portion of the plate, the displacement is scaled as ~t(c) - (~(~)) E H I ( ~ ) , where u~(x ~) - ~(~)(:~) for all x ~ - ~ C {~}- . In other words, the "inserted" portion ~t~ of the plate is mapped twice, once onto a subset of ~, once onto a subset of ~ (Fig. 2.2-1). Finally, we assume that the Lam~ constants and the applied forces inside the three-dimensional body are of the form ~--~
f.~(x ~)-cfi(~c)
and
y-/2,
for a l l x ~ - ~ E a
a,
where the constants i > 0, /~ > 0, and the functions ~ e L2(~) are
independent of c. The crucial idea for treating this multi-structure thus consists in
scaling its different parts independently of each other (in particular, the plate is scaled as is usually done in "single plate" theory), but counting the inserted portion twice. The scaled components of the displacement, which are defined in this fashion on two distinct domains, thus contain the information about the inserted portion twice. That they correspond to the same displacement of the whole structure then yields, after passing to the limit, the "junction conditions" that the solution of the limit problem must satisfy. In this fashion, we establish the main result of this chapter (Thm. 2.3-1), by showing that the family (~t(c), u(c))~>0 strongly converges in the space H~(~) • HI(Ft) and that (~t,u) - lim(~t(c) u ( ~ ) ) i s obtained as follows: (i) The vector field u = (u~) E H I ( ~ ) is a scaled Kirchhoff-Love displacement field: The function u3 is independent of the variable xa, and it can be identified with a function ~3 C H2(w) satisfying
Junctions in linearly elastic multi-structures
132
[Ch. 2
~a -- 0 ~ a - 0 on ~/0; the functions u~ are of the form u~ - ~ - x 3 0 ~ a with functions ~ E H 1(co) satisfying (~ - 0 on ~0. (ii) The vector field ~H -- ( ~ ) satisfies the same scaled m e m b r a n e equations as those found in Thin. 1.5-2 for a "single plate". (iii) Let Od -- ~ - - ~ ; hence Od is a three-dimensional open set with a t w o - d i m e n s i o n a l slit into which a portion ~a of the middle surface of the plate is inserted. Furthermore, let a~- and cod denote the upper and lower "faces" of the slit, a convenient way of distinguishing the traces "from above" and "from below" on the set Wd (Fig. 2.4-1). T h e n the vector field (s e n l ( g t ) x H2(co) satisfies, at least formally, the following b o u n d a r y value problem (which is independent of the problem solved by CH)" --Oj~rij(~_$ ) -- ~
5~j('g)5,j - 0
in Od, on OOd -- -gd,
--O~zm~z -- P3 + O~q~
+ Ext~ ~3-0.~3-0
on%,
maol]al] ~ -- 0
o n ~/1,
u31r
-~33(~)1w2 }
in co,
-- ?~3iw T -- ~3twd,
where Ext ~ 0 denotes the extension by 0 on (a~- coa) of any function O ' c d a ~ I~, ")'1 -- ~ ' - ~'0, and P3 -
/1 1
f3 dx3,
-
q~ -
?
1
x3f~dx3,
l(c~j~i + c~i~j), -
+
This b o u n d a r y value problem is "multi-dimensional", in t h a t the u n k n o w n ~i is defined over the three-dimensional set Od, while the
The three-dimensional equations
Sect. 2.1]
133
u n k n o w n r is defined over the two-dimensional set co; it is "coupled", in t h a t the traces of the functions ~i and r over the set cod satisfy specific junction conditions. It is also to be noted that, once this problem is appropriately descaled, it provides an instance of a "stiff problem", in t h a t different powers of c a p p e a r in its formulation (Sect. 2.6); the a s y m p t o t i c analysis of the associated eigenvalue and time-dependent problems likewise yield other examples of "stiff problems" (Sects. 2.8 and 2.9). We also show that, if the Lam~ constants of the "three-dimensional" s u b s t r u c t u r e approach + e c sufficiently rapidly as c ~ 0 (e.g., if they behave as ~-3), this s u b s t r u c t u r e becomes rigid in the limit; remarkably, this analysis provides a rigorous justification of the boundary conditions of a clamped plate (Sect. 2.7).
THE THREE-DIMENSIONAL EQUATIONS LINEARLY ELASTIC MULTI-STRUCTURE
2.1.
OF A
In this chapter, one exception is made to the rule governing Latin letters, whereby the index d denotes an arbitrary > 0 constant. Let a, b, c, d, e, f denote constants t h a t are all > 0, and assume t h a t d < a. For each e > 0, we let (Fig. 2.1-1): ( . d - {(Xl,X2) ~ I~2" 0 "~ Xl < a
Ix21 < b}
"7o - {(a, x2) e R2; [x2l < b}, (.dd __ {(Xl,X2) ~ ]~2; 0 < X 1 < d,
O-
Ix l
F;
ft ~ - w x ] -
c,c[
70 x [ - e , e ] ,
< b},
c,c[,
{(z~,x2, za) e Ra; - c < x~ < d, Ix2] < b, - e < xa < f}, O~-
O - --~ f~d,
S ~ - O U f~
we denote by x ~ - (x~) a generic point in the set S ~ and by 0~ the partial derivative O/Ox~. T h e set O~ is the reference configuration of a linearly elastic "three-dimensional" substructure with Lam6 constants ~ , /2~ and the set ~ is the reference configuration of a linearly elastic plate with Lam6 constants ~ , #~. T h e set S ~ is thus the reference configuration of a linearly elastic m u l t i - s t r u c t u r e comprising two s u b s t r u c t u r e s "perfectly bonded" together along their c o m m o n b o u n d a r y
134
[Ch. 2
Junctions in linearly elastic multi-structures
f i
~" ,
,,
I I I I
,,"I
''
,'"
I
LfS
s
,"
LI s s S, , "
i
s
s
,"
,'-
II II I~
sS
s S s s S 9 . . . .
s
sS
s
s"
s
s
,"
2~
,
I
I
i
I
_~
s
F~
:
,,.__'.,.. sS
ii:[
4. . . . . , z _ ~ . . . . . 1 I /3
I i i i i
| i
o
9
d
Fig. 2.1-1: A three-dimensional elastic multi-structure. The set ~e is the reference configuration of an elastic plate clamped on the portion F[~ of its lateral face, and inserted into a three-dimensional elastic body whose reference configuration is Od; the number d > 0 denotes the depth of the insertion. These two elastic bodies, "perfectly bonded" together along their common boundary t)g/~ C100~, form an "elastic multi-structure".
0f~ ~ ('1 00~, the plate being thus inserted into the three-dimensional substructure and d denoting the d e p t h o f t h e i n s e r t i o n . The u n k n o w n is the d i s p l a c e m e n t v e c t o r f i e l d u ~ - (u~) 9S ~ --~ Na; it is assumed to satisfy a b o u n d a r y c o n d i t i o n o f p l a c e u ~ - 0 on F~.
Sect. 2.1]
135
The t h r e e - d i m e n s i o n a l equations
In linearized elasticity, the displacement field u ~ - (uT) thus satisfies the following variational problem P(S~), which constitutes the t h r e e d i m e n s i o n a l e q u a t i o n s of t h e m u l t i - s t r u c t u r e "
Ue e
V ( S e) "-- {V e -- (v~) e H I ( S e ) ;
v e -
0
on F~},
+ s {A'e;p(u')eqq(V') + 2p'qh(u')qh(v')} dx" - J'o f.~v: dx ~ + s a
fly: dx ~
for all v~C V(S~),
where e~j(v ~) -~(0~ vj + O~v~) denote as in Chap. 1 the components of the linearized strain tensor e~(v~), and where the vector field (f[) E L2(S ~) represents the given applied body force density (for ease of exposition, we assume that there are no surface forces). By Korn's inequality with boundary conditions (Thm. 1.1-2) applied in the space V(S~), the bilinear form found in the variational equations of problem 7)(S ~) is V(S~)-elliptic, and thus (Thm. 1.2-1) problem P(S ~) has one and only one solution u ~. This solution can also be characterized as the unique solution of the minimization problem: Find u ~ such that: u ~ e V ( S ~) and J~(u ~) = 1
inf J~ (v~), where v~V(s ~)
~ e ~ ( v ~ ) " e~(v~)dx ~ + 2
~
and where
2flz~bijcia , A~B 9C "- A~bp~,Cqq+ 2#~b~jc~a ,
teB"
C
" - ~ebppcqq +
(v~) "
(v~)dx ~
136
Junctions in linearly elastic multi-structures
[Ch. 2
for all symmetric matrices B = (bij) and C = (cij), and
f.v:--kv~
if f - ( k ) ,
v--(v~).
T h e function u ~ also satisfies, at least formally, a classical "transmission problem" of three-dimensional linearized elasticity, which takes here the following form:
- d i v ~ { h ~ e ~ ( u ~ ) } - f~ in 0~, - d i v ~{A~e ~(u ~) } = f~ in f~, u~-
0 on F~,
~g
A e~(u~)h ~ -
0
on OO~ - Oft ~,
A~e ~(u~)n ~ - 0 on 0f~ ~ - 00~,
U~o: - u~a~ on OO~ r3 aft ~, h~e~(,.,~),i ~ + A ~ e ~ ( u ~ ) n ~ - 0 on 00~i n Oft ~,
where (div~E~)~ "- O~cr~ if E ~ -
(a~),
and ~ and n ~ denote the unit outer normal vectors along the boundary of the sets O~ and ~ , respectively. T h e relations along 0 0 ~ A 0fl ~, which formally express the continuity of the linearized displacement vectors and of the linearized stress vectors along the c o m m o n portion of the two boundaries, are called transmission conditions; details about such transmission problems are found in D a u t r a y & Lions [1984, p. 1245]. T h e first condition shows in particular t h a t we are modeling a situation where the inserted portion of the plate is "perfectly bonded" to the threedimensional structure. We are thus excluding here situations where the inserted portion could slide along the three-dimensional structure, or where an elastic adhesive would hold together the two substructures.
Sect. 2.2]
2.2.
137
Fundamental scalings and assumptions
TRANSFORMATION INTO A PROBLEM POSED OVER TWO DOMAINS INDEPENDENT OF c; THE FUNDAMENTAL S C A L I N G S OF T H E UNKNOWNS AND ASSUMPTIONS ON THE DATA
We describe in this section the basic p r e l i m i n a r i e s of the a s y m p totic a n a l y s i s o f an elastic m u l t i - s t r u c t u r e , as set forth in Ciarlet, Le D r e t & N z e n g w a [1989]. W i t h the sets ft ~ and O (defined in Sect. 2.1), which overlap over the inserted p o r t i o n ft~ of the "thin" set f~, we associate two d i s j o i n t sets f~ and ft, as follows. First, as in the case of a "single plate", we let ~ - a ~ x ] - 1, 1[; with each point x - (x~,x2, x3) C f~, we associate the point x ~ - (Xl,X2, Cx3) - 7r~x E (Fig. 2 2-1); and w i t h the r e s t r i c t i o n s (still denoted) u ~ - (u~)" ~ ---, R 3 and v --~ of the functions u ~, v E V ( S ~) to the set , we associate the functions u ( c ) (u{(c))" f~ --~ R 3 and v - (v{)" ft R a defined by the s c a l i n g s _
<
-
and :
and
u~(x~) = v[(x~) =
su3(s)(x), sv3(x),
for all x ~ = 7r~x E ~ .
Secondly, we define the translated set f~ - 0 + t, the vector t being such t h a t { f ~ } - A ft - 0. Then, with each point x ~ C 0 , we associate the t r a n s l a t e d p o i n t ~c - (x ~ + t) E { ~ } - (Fig. 2.2-1), and w i t h the restrictions (still denoted) u ~ - ( u ~ ) " O ~ R 3 and v ~ - (v~) 9 O ~ R 3 of the functions u ~ , v ~ E V ( S ~) to the set 0 , we associate the functions ~i(c) - (g~(e)) 9 { 9 } - ~ R 3 and v - (v~)" { ~ } - --~ R 3 defined by the s c a l i n g s u ~ ( x ~) - eg~(e)(2) and v~(z ~) - ev~(e)(~) for all x ~ - ( 2 -
t) r O.
138
[Ch. 2
]unctions in linearly elastic multi-structures
...... S X E X
----"
2~
(.L) " - - - -
~x ---co . . . .
I........
g~d i
Fig. 2.2-1" Transformation of the multi-structure problem into a scaled problem, posed over two sets ~ and ~ that are independent of E. The sets ~e and O, which overlap over the "inserted" part ~d of the "thin" part ~ , are mapped into two disjoint sets ~ and ( ~ } - . The "inserted" part ~ of the thin part is thus mapped twice, once onto ~d C ~ and once onto ( ~ } - C ( ~ } - .
Sect. 2.2]
139
Fundamental scalings and assumptions
It should be remembered that at this stage, the translation through the vector t is merely a "visual convenience" that affords to think of the two sets ft and 0 as being disjoint (Fig. 2.2-1). In fact, we shall later on identify the two sets ft and O (Sect. 2.4). The displacement u ~ C V ( S ~) is thus mapped through these scalings into a s c a l e d d i s p l a c e m e n t (~(c), u(s)), which belongs to the space H~(~) x H~(ft), which satisfies the boundary condition u(s) = 0 on F0 := " y 0 x ] - 1, 1[, and which satisfies the j u n c t i o n c o n d i t i o n s for t h e t h r e e - d i m e n s i o n a l p r o b l e m (Fig. 2.2-1):
g~(e)(~) - eu~(e)(x) and g3(c)(2) - u3(e)(x), at all points ~ E ~t~ "- ft} + t and x E ftd "-- a;d X ] - 1, 1[ corresponding to the same point x ~ C ft} An arbitrary function v ~ E V ( S ~) is likewise mapped into a s c a l e d f u n c t i o n (~, v) through the same scalings. Finally, we make the following a s s u m p t i o n s on t h e d a t a : There exist constants A > O,/5 > O, A > O, and # > 0 such that the Lamd constants of each substructure satisfy: A~-A
and
/2~-/2,
A~-e-3A and #~-c-3#, and there exist functions f~ e L2(Ft)and ~ E L 2 ( ~ ) i n d e p e n d e n t of s such that the applied body force densities in each substructure satisfy:
f:(x
-
-If~(x) and f ~ ( x ~) - 6f~(~)
f~(x ~) -
fa(x) for a11x~
- rcex C ~e,
for all x ~ - (5c- t) c 0 d.
Remark. For a given c > 0, the functions f~ need to be defined only over the set g t - {ft~}- in order that the last relations make
140
[Ch. 2
Junctions in linearly elastic multi-structures
sense; but e is arbitrarily small.
I
Using the scalings and the assumptions on the data, we can recast the variational problem of Sect. 2.1 in the following equivalent form:
T h e o r e m 2.2-1. The scaled displacement (~t(~),u(~)) satisfies the variational problem 7)(g; ~, ~)"
(u(c), u(c)) E V(c; ~,~)"-- {('v, v) e HI(~) • H I ( ~ ) ; = 0 on r0. ~ ( ~ ) = ~ ( ~ )
~nd ~ ( ~ ) :
at all corresponding points :~ ~ ~
~(~)
and x ~ ~d},
• ~(0~){~,,(~(~))6~(~)+ 2~j(~(~))~(~)} d~ I
+/o {~~(u(~))~.(~)+
1/2
+~
{~~(u(~))~(v)
2,~(u(~))~z(v)}
d~
+ ~(~(~))~..(v)
+ 4~tec~3(u(c))ea3(V) } dx
for M1 (~, v) e V(c; ~, ~), 1 ~ h ~ ~j(~) - ~(Sj~ + &~j), 4 - o / o ~ j , ~ j ( v ) - ~l(OjV i _~_ OiVj ) O~ = O/Ox~, x(A) denotes the characteristic function of a set A, and
O~ "- O~d + t.
I
Note that V(c; ~, Ft) is a subspace of H I (~) • H I (Ft) that depends on the parameter c. The scaled displacement (~t(c), u(c)) can also be characterized as the uniqu~ ~ol.tion of ~ , ~ , ~ z a t ~ o ~ p~obl~,% viz.. ~nd (~(~),~(~))
Sect. 2.3]
Convergence of the scaled displacements
141
such that (~(c), u(e)) C V(e; f~, f~) and J(e) (~(c), u(c)) -
inf J(c) (~, v), (,~,v)~v(~-fi,a)
where
J(~)(~, ~) .- ~1 j/~ x(O~i){.~6p(~)6q(~,) + 2gG(~)e,j(~)} d~ 1
{~,e~(~,)e~(~,) + 2~e~9(,,)e~,(~,)} dx
+ ~1 L{2~e~(~)~(~)
1L (~, + +57J4
2~)e~(~,)e~(~,)d~
-L x(Od)f~5~ d~
2.3.
CONVERGENCE DISPLACEMENTS
+ 4,e~(~)e~(~)} dx
-
f~v~ dz.
OF THE SCALED A S e ---, 0
We now etablish that the family ((~i(c), u(e)))~>0 strongly converges in the space H I ( ~ ) x Hl(f~) a8 e ~ O, and we also identify the "limit" variational problem that the limit of this family solves. We follow here Ciarlet, Le Dret & Nzengwa [1989]. We recall that I" 10,a and I1" II1,~ denote the norms in the spaces L2(f~) or L2(f~), and H l ( f t ) or Hl(ft), respectively, and that strong and weak convergences are denoted by --~ and ~ , respectively. In the next theorem, &d denotes the translated set (a;d + t); VIA denotes the trace of a function v on a set A in the sense of Sobolev spaces (for instance, the trace ~31~,~ is to be understood as a function in the space H1/2(&~), etc.); the equality v31~, - r/31~, is to be understood as holding up to a translation by the vector t; finally, 0, denotes the outer normal derivative operator along 0oz.
142
Junctions
[Ch. 2
in linearly elastic multi-structures
Theorem 2.3-1.
(a) As c ---, O, the family ((~t(~),u(c)))~>o converges strongly in the space HI(~) • HI(~) toward an element (s u) that satisfies the following relations: (b) The limit u = (u~) e H~(12) vanishes on Fo = 7 0 • 1, 1[ and is a scaled Kirchhoff-Love displacement field in ~, i.e., there exist functions ~ E H~(w) and ~3 C H2(~z), satisfying in addition ~i = 0~3 = 0 on 70, such that U a - - ~o~ - - X3C~c~3
and
U 3 -- ~3
in ~.
(c) The pair (s ~3) belongs to the space [H'(~) x V3(~)] d
"
{(V,?]3) E H'(~) x H2(w);
-
?73 -- G')u?~3 -- 0 o n 70,
"V315.,d -- T]3lWa, "Val&d - - 0 },
and it satisfies, and is the unique solution of, the variational equations:
f~
{~,(~)~(~)
+ 2 p ~ j ( ~ ) e ~ j ( ~ ) } d~
+ f~{ 3(A4AP+2#) A~3A~3+4~~O~z~30~z~13} dw
-s
§/
for all (~, r/3)e
/ [HI(~)
• V3(w)] d ,
where P3 "-
f
1
f3 dx3 , q~ "-
F 1
x3 f ~ dx3.
(d) The function r H "-- ( ~ ) belongs to the space VH(W) "-- {~/H --(r/a) E Hi(w); ~/H --0 on "Yo},
Sect. 2.3]
Convergence of the scaled displacements
143
and it satisfies, and is the unique solution of, the variational equations"
4A#
~eoo(~H)G.~-(rlH
) + 4pe.z((~H)e~z(Vl/~)} dw
= ~ p~r/~ da;
for all ?7/~ = (r/~) E V H(W),
where
1 (c9~ + 0 ~ ) )
~(r
- - -~
, p~'-
f
f~dx3. 1
(e) The variational problems found in (c) and (d) are independent.
The proof is long and technical and, for this reason, is broken into a series of ten parts, numbered (i) to (x) (a shorter proof, yet preserving the main features of this one, is proposed in Ex. 2.1 for a "model problem"). For conciseness, we henceforth let Proof.
v(~) .- v(~; a, a) denote the space defined in Thm. 2.2-1. We first show (part (i)) that the semi-norm
(~, v) ~ I(~, v)l - {1~(~)1 ~0,~ + le(v)10,~ }~/~ where ~(v) "- (~j(v)) and e ( v ) "- (e~3(v)), is a norm over the space V(~), and that this norm is in addition equivalent, uniformly with respect to e, to the product norm
(~, v) --, I1(~, v)ll - {ll~ll ~1,~ + Ilvlll,n } 1/2 This property will be in turn used for showing (part (ii)) that the family ((g(e), u(e)))~>0 is bounded in the space H I ( a ) x Hl(f~) and the family (~(e))~>0 is bounded in the space L~(f~) "- {(X~j) C
144
[Ch. 2
Junctions in linearly elastic multi-structures
L2(f/); Xij - Xj~}, where, for each e > O, the tensor e;(e) "- (~ij(c)) E L~(t2) is defined (as in Sect. 1.4) by 1 ~(~)-
~(u(~)),
~(~)-
1
-~(u(~)),
~(~)-
7~(u(~)).
(i) There exists a constant C independent of c such that the following generalized Korn inequality holds: I](~, v)l I _< CI(~ ,v)] for all ( ~ , v ) E V(r W i t h an arbitrary function (~, v) E V(e), we associate the "descaled" function v ~- E V ( S ~), defined by the relations:
v ; ( x ~) - e2v~(x) and v ; ( x ~) = eva(x) for all x ~ E fi~, v~(x ~) -- e ~ ( ~ ) for all x ~ E O. In this fashion, the components of the tensors e~(v~), a(v), and e(v) are related b y :
~;9(v~)(~ ~) - ~ ~ ( ~ ) ( ~ ) , ~;~(~)(~) = ~(v)(~),
e~aa(v~)(z ~) = eaa(v)(x)
for all x ~ E t2 ~,
and
e~(v~)(x ~) - eeij (v) (Y:) for all Y: E O. Hence I(~, v)l - 0 implies q~j(v ~) - 0 in S ~. Since v ~ ---+ le~(v~)lo,s~ is a norm on the space V ( S ~) (by Korn's inequality and the b o u n d a r y condition of place on F~; cf. Thin. 1.1-2), we conclude that the mapping (i~, v) ~ I(io, v)l is a n o r m on the space V ( c ) . If the stated inequality is false, there exist ck > 0 and (~k, v k) E V(ck), k > 1, such that the sequence (ek)~=l is bounded and:
I1(~~, ~ ) 1 1 -
1 for .11 k,
I(~ ~ ~ ) 1 ~ 0 ,~ k ~ o~
Since le(vk)10,a ~ 0 as k ---+ oc by the last relation, and since the functions v k vanish on F0, Korn's inequality with b o u n d a r y conditions on the set t2 shows that
Ilvklll,~ -+ 0 .s k ~ o~,
Sect. 2.3] and
Convergence of the scaled displacements
thus
1/2 (COd)a s k ~ e c ,
Vk
I~,,~OinH
on the one hand.
145
The relation I ~ ( ~ ) 1 0 , ~ 0 as k ~ o~ implies on
the other hand (Ex 9 2.2) that there exist vectors fik , ~/k C Ra and functions @k C H I ( ~ ) such that ~k (S:) _ gzk + ~k A o~ + @k (S:) for almost all S: E ~, I1~[11,~ -~ 0 as k ~
~,
By definition of the space V(e), -k k(x) v~(~)-ekv~
and 5~ (2) - vak (x)
at all corresponding points S: E t2} and x E ~d" We thus conclude that (the sequence (ok)is bounded) ~)k
1/2
I~~0inH
(~d) a s k ~ o e .
Since the functions S: E COd _ ~ (Sk + h k A o&)l~,, belong to a finitedimensional space, and since they converge to 0 in H1/2(&d) (~k[~, --, 0 and @kl~,, ---, 0 in H1/2(&d) as k --, co), we conclude that g:k ~ 0 and ~/k ---+ 0 in R 3. Hence (e k + (~k A o~) ---, 0 in H i (~) as well, and thus II'~klll,~ 4-4 0 as ]r ---+ oo,
which, together with the relation Ilvkl]l,~ ~ 0, contradicts the relations I](~ k, v k ) ] ] - 1. Therefore the desired inequality holds. (ii) The norms ]](~(c)]]l,a , ]]u(c)]]l,f~, ]~i;(c)]0,f2 are bounded independently of c. Thus there exists a subsequence, still indexed by c for notational convenience, and there exist elements r and ~ such that
u(c)--~u in HI(~) as c--~0, u(c) ~ u
in H I(f~) asc---+0, and l t - 0 o n F o , re(c) --~ m
in
L~(t2)
as c --, O.
146
[Ch. 2
Junctions in linearly elastic multi-structures
Let us introduce the notation AB "C
"-
)~bppcqq
-nL 2 f i t b i j c i j
,
A B " C " - )lbppcqq + 2#bijc{j,
for all symmetric matrices B - (b~j) and C - (c~j). The stratagem consists in splitting into two (equal for definiteness) parts the integral over the set ft~ that appears in the bilinear form of problem 7)(S ~)
(Sect. 2.1). Thanks to the scalings defined in Sect. 2.2, one part is mapped as an integral over the set f~, and the other is mapped as an integral over the set fte. In this fashion, we obtain the following equivalent expression of problem P(e; ft, Ft) (Thm. 2.2-1), where we let v)
.-
1 .-
c
1 v)
.-
for an arbitrary function v C H 1([1)" 1 jf~ X(f~})Age('g(e))" x(d,~)A~('g(e))" ~.('5)d~ + 2--75ea ~.('~)d2
+ fn{ ~X(f~d) + X(f~- f~e)} A~(e)" ~(e; v)dx 1 - / ( ~ X(O~d) f 9i~ dx + / a f . v dx for all (~, v) C V(e).
Let B " B "- bijbij and c "- 2min{ti, p}; then c B : B <_ A B " B and c B " B <_ A B " B for all symmetric matrices B - (bij) of order 3; hence letting (~, v) - ({t(e), u ( e ) ) i n these variational equations gives us
{X(O~) + 2-~eaX:(O~) 1 } e(,i(e))" e('g(e))d:~ {1~X(ad) + :)((a- ad) } ~(e)" ~(e)dx
_< ~ x(O~)if" {t(e)d~ + fa f" u(c)dx.
Sect. 2.3]
147
Convergence of the scaled displacements
Without loss of generality, we may restrict ourselves to values of that are < 1. in which case the last inequality implies
I~(a(:))l 0,~ ~ + I, ~ .(~ . : .())l~ .,~,~
<- I~(a(:))]0~ + I~(:)l 0,~ ~ ~<
2C-1 {Ifl0,~l~i(:)10,~ + Ifl0,~lu(:)10,~} 9
By (i), there exists a constant C independent of : such that
II~(:)ll ~1,ft
-t-11"tt(C)II21,Ft __ <
c ~ { le (a(:))l 0.5 ~ - t - l e (~(:))l" 0.fl }
;
hence we conclude that the family ((~t(e), u(e))):>0 is bounded independently of : in the space Hl(~t) x HI(f~), and that the family (tc(e))~>0 is bounded independently of : in the space L2(f~). As in the case of a "single plate" (Chap. 1), we next show that the weak limit u c HI(f~) found in part (ii) is a s c a l e d K i r c h h o f f - L o v e v e c t o r field over the set f~, in the sense that it belongs to the space VKL(ft) defined below (the same as in Thm. 1.4-1). (iii) The weak limit u belongs to the space V K L ( a ) "-- {v -- (v~) e H:(fl); ei3(v )
-
-
0 in ft, V -- 0 on C0},
which can also be defined as
VKL(ft) -- {V -- (Vi) e HI(Ft); v~ -- ~ - x30a?]3 , v 3 -- ?]3, with rl~ e H l(a~), ~3 e H2(a~), r/~- 0~r/3 - 0 Hence there exist functions ~ E ~ i - 0 . ~ 3 - 0 on ~/o such that
us-~-x3cg~3
Hi(w) and
on 70}.
and ~3 E H2(co) satisfying
u3-~3.
Since the sequence (~(:)),>0 is bounded in the space L2(ft) by (ii), the proof is the same as that for a "single plate" (cf. proof of Thin. 1.4-1, parts (ii) and (vii)). We next identify the junction conditions that the pair (~t, u) must satisfy. Note that the second equality in these conditions is to be understood as holding up to a translation by the vector t.
Junctions in linearly elastic multi-structures
148
(iv) The weak limit (~t, u) satisfies the following ditions
[Ch. 2 junction
con-
9
~2~1~,, - 0
and
U31&~- -
U3lc,'a"
By definition of the space V(e), g~(e)(2) - eu~(c)(x) and g3(e)(2) - u3(e)(x) at all corresponding points 5: C ft~ and x C f~d. Hence, up to a translation by t, g~(e)l~. -- cu~(c)l~ . and ~23(c)1~,, - u3(c)l~,, for each e > 0. Since ~t(~)j~.--~ ul~,, in H1/2(c3a) and u(e)l~,, ~ ttl~ . in H1/2(Wa) (the trace operators from HI(Ft) onto H~/2(&d) and from HI(f~) onto H1/2(COd) are strongly continuous, and a linear mapping that is strongly continuous is also continuous with respect to the weak topologies; see, e.g., Brezis [1983, p. 39]), the relation g31co,, = u31~, follows from the relations ~3(c)1~ . = Ua(e)l~ ., c > 0. Since (u~(c)l~,)~>0 is a weakly convergent sequence, it is bounded; therefore each sequence (eu~(c)l~,)~>0 converges strongly to 0 in the space H1/2(&d). This fact, combined with the relations ~2~(e)la" = e'u~(e)l~,, e > 0, implies that (t~lCo,, = O. We next prove two independent technical results, which play a key r61e in the identification of the "limit" variational problem solved by the pair (~i, u). (v) The components of the weak limit ~ - (eqj) e L~2(ft) of the subsequence (~(c))~>o are given by -
-
0,
-
W i t h o u t loss of generality, we assume that e _< 1 and that 1 < min{e, f} (e and f are used in the definition of the set O at the
Convergence of the scaled displacements
Sect. 2.3]
149
beginning of Sect. 2.1). Since ec~9(c) - e~9(u(c)) and u(c) --~ u in
Hl(f~), it follows that
echo(e) ~ ~c~z - e~z(u) in L2(Ft). Given functions v~ C Hi(Q) that vanish on F0, there exist 5; E Hi(t)) such that 5;(2 1) - v~(x) at all corresponding points 2~1 ~ ~ and x C f/d, and such that supp 5; C {(:}i) ERa;
I2al < min{e,f}}.
To see this, first use an extension operator from H I ( F ~ ) i n t o Hl(f~) (see, e.g., Neeas [1967, Chap. 2, Thm. 3.10]); then multiply by a smooth function of 2a E R that is equal to 1 on [-1, 1] and vanishes for 12al >_ min{e,f}. Then for each e > 0, the function (~(e),v) defined by { ~(2~1, ~2, e-1:~3) if (:~1,2~2,E-l~3) ~ supp~;, 5~(e)(21,22,2a) "--
0 if (:~l,X2, g-lxa) E { ~ - supp~;},
5a(e) "--0 in f~, v := (vl, v2, 0) in f~, belongs to the space V(e). The variational equations found in problem 72(e; t2, t2) (Thm. 2.2-1) can be rewritten as s X~(0~)Age(,i (e))" ge(~)d9 + fa {Aecpp(c)5~ + 2#ec~ (e)}O~v~ dx +-
+
+Tg
{lec,~(e) + (t + 2#)ecaa(e)}0ava dx
1s C
= s )l(O~)fivi d2 + ~ fivi am for all (~, v) E V(e). Therefore, for fixed functions v~ C Hl(f~) that vanish on F0, we have
150
[Ch. 2
Junctions in linearly elastic multi-structures
a dx - - e fa{A~pp(e)5~z + 2tt~z(e)}O~vzdx
2/_t ~ Nc~3(r
9
where for each e > 0, the function (~(e), v) C V(e) is that constructed supra. As e ~ 0, the left-hand side converges to 2# fa ~30av~ dx, by definition of weak convergence, and the right-hand side converges to 0, since a weakly convergent sequence is bounded on the one hand, and since
[Va(r
~ -- O(r
[~Va (r [o,fi -- O(r
[~3Va(r
-- O(r
)
on the other. Hence by Thm. 1.4-3, we conclude that
/~3 "-- 0. Given a function v3 E HI(f~) that vanishes on F0, there likewise exists ~ E Hl(~t) such that ~?~(~1) _ Va(X) at all corresponding points 5:~ C ft~ and x E f~d, and such that supp~?~ C {(~{) C Ra;
1~3[ < min{e, f}}.
Then for each e > 0, the function (~(e), v) defined by @~(r " - 0 in ~,
"/'J3(r
X2, X3)
V~(Xl,X2,r o
if
(:Z:I,:Z:2,r
E supp~,
I
0 if (~1,~2,r
{f)-supp~},
v := (0, 0, v3) in ft, belongs to the space V(e). For a fixed v3 E Hl(f~) that vanishes on F0, we use this function (~(c), v) in the variational equations found in problem T~(s; a , a); we find in this fashion that, for each e > 0,
Sect. 2.3]
Convergence of the scaled displacements
151
{Anoo(s) + (A + 2#)na3(s)}O3va dx
=-ss
{2#n~a (e) }O~va dx
--C2 j/~ +s2 s X(O~) fa Oa(s) d2 + s 2 fa fa va dx. As s --+ 0, the left-hand side converges to
f{
a~
+ (A + 2#)naa}O3va dx - fa {Aeoo(u) + (A + 2#)na3}03va dx,
and the right-hand side converges to O, since Io~(C)lo,~ - o ( ~ / ~ ) , Hence by Thm.
IO,~(~)1o,<~
-
Io~(~)1o,~ - o(~-1/~) 9
0(s
1.4-3, we again conclude that
~~(~)
+ (~ + 2 , ) ~
= 0.
Parts (ii), (iii), (iv) thus imply that the weak limit (~t, u) belongs to the space (the space VKL(f~) has been defined in part (iii)):
z VKL(ft)]d 9= {(V,V) E HI(~) • VKL(a); Valid --0, V31~ --V31coa}" We next show that any function lying in two particular subspaces of the space H~(f~) x Vgc(f~ can be approximated as well as we d please by functions (O(s), v ( s ) ) i n the space V(s) whose component v(s) lies in addition in the space VKL(~ ). By taking limits as s ~ 0, this density property will later enable us (see part (vii)) to find the variational equations satisfied by (~i, u). In what follows, & designates the intersection of the set ~) by the plane that contains the set &d, and we assume that the origin 6 for the points ~ E ~ belongs to the "left" edge of the set &d.
152
Junctions in linearly elastic multi-structures
[Ch. 2
(vi) Let (go, v) be a function in the space [Hl(fi) • VKL(~)]d such that either supp ~ is contained in the set { 2 - (:~) E ~; :~ < O} and ,, - o, o ~ ~ e Hi(Z). T~ t~ ~x~t~ a ~ V ~ c ~ ((~(~),,,(~)))~>o such that" (~(~), v(~)) ~ V(r v(r
for all ~ > O,
~ VKL(~) for all ~ > O,
IIv(e)-
V]]l,~ --+ 0 as e --+ 0,
I1~(~) - ~111,~ --+ 0 as ~ -+ 0.
If supp ~ C {~ - (~) E ~; ~?1 ~< 0} and v - 0, it suffices to let ~ ( e ) - ~ and v ( c ) - o for all e > 0.
]
Assume next that a function (~ , v) E [H l ( f -~ ) x VKL(a) d is
such that vl~ E HI(a)); note that a function (~, v) in the space
[H1 (~) • VKL(~)]d a priori only satisfies ~1~ E H1/2(&) N HI(&d)
and ~31~ E H~/2(&)A H2(&d) (see parts (iii) and (iv)). Since v E VKc(Ft), part (iii) implies that there exist functions r/~ E H~(w) and r/3 E H2(co) such that Va -- Tic~ -- X3OQc~T]3a n d v 3 ~---7]3 in ~2.
Let ~(~) := r/~(x) at all corresponding points ~ E &d and x E cod. Since the set & has a Lipschitz-continuous boundary, the functions ~c~ E Hl(&d) and 113 E H 2 ( c ~ d ) c a n be e x t e n d e d to f u n c t i o n s (still denoted) ~ E H I ( ~ ) and ~a E H2(&) (see, e.g., Ne~as [1967, p. 80]). We then let v(e) = v in ~ for each e > 0, so that the requirements that v(e) E VKL(~) and ] i v ( e ) - vl[1,~ --+ 0 as g --+ 0 are certainly satisfied. Following an idea of Caillerie [1980], we next define a function ~(e) - (~(e)) in ~ by letting: ~ ( e ) "- e ~ - 2 a 0 ~ 3 + ~l~
and
~ 3 ( e ) " - ~31~ in ~ ,
i n h 2~ -- h ~,
Convergence of the scaled displacements
S e c t . 2.3]
e
val~ +
5a in
s
153
-
~(e) "- ~ in ( ~ - ( ~ , where ~'-&•
~2~._&•
2e[.
Since the function Vl~ belongs to the space H I (co) by assumption, the function ~(e) belongs to the space H*(~t); the assumption ~1~ E Hi(&) is thus crucially used here. Besides, a simple computation shows that 9 at all (~(e), It begin
cv~(x)
--
and v3(c)(x)
--
V3(X)
corresponding points $ E ft~ and x E ltd. Hence the function v(c)) constructed in this fashion belongs to the space V(g). thus remains to prove that ] ] ~ ( e ) - ~]ll,O --+ 0 as e --+ 0. To with, Lebesgue's dominated convergence .theorem shows that
[~(e) - v[0,fi ---' 0
and
[ c ~ j ( c ) - o5~5j]0,~ --~ 0 as c --~ 0,
- o ql0,a
0
0,
since no factor e-1 is introduced by partial differentiation with respect to ~ , nor by partial differentiation with respect to x3 in the set ~ (the assumption 61~ E Hi(&) is again crucially used here). It next follows from the definition of the functions 5~(e) in the set ~z~ _ ~t~ that
-~
(
)
+ 2 ~a - c c5~9a+ C
0a~ C
1 + -(~ C
-
- 5~1~) if 2a > 0, 0aS~
1 - - ( 5 ~ - v~l~) if 2a < 0, C
:~a - 2e cgaSa+ -(Sa - ~?al~) if :~a > O, g
g
( 23+E 2e) ~353 -- -ff I(~3 -- V3lub) if 23 < O,
154
Junctions in linearly elastic multi-structures
in the set ~e~ - ~ . 1
[Ch. 2
Hence it remains to prove that [ v i - vila[ ~d2 ~ 0 as r ~ 0,
since the other terms found in the differences ( 0 a ~ ( c ) - 0a~) can be again handled by Lebesgue's dominated convergence theorem. If ~3 is a smooth function, -
19
c~2, X3) -- 'V(Xl, X2, 0) 12 -- lf0 za ~V(Xl, X2, 8) dg[ 2 ~
/0
_< [ ~ a l
]&~(21, ~:2, ~)[2 da
2e and thus
, z2, za) -- v(:rl, :r2, 0)12 dtCl d:~2 _< [~3[ [[~[[~,(~, which in turn implies that, for any function ~ E HI(~t),
This last inequality then implies that e -2 f ( ~ - 5 ~ [vi - fhlco[2dye ~ 0 since [[~)[[~,5~ --~ 0 as E --~ 0. As a first step towards identifying the "limit" variational problem solved by the weak limit (s u), we obtain the variational equations that the weak limit should satisfy when the test-functions (~, v) are subjected to the same restrictions as in part (vi). (vii) Let (iJ, v) be a function in the space [ H I ( ~ ) z VKL(f~)j such that either supp~ is contained in the set { 2 . - (~c~) E f~; ~c1 <_ 0} and v O, or iJico C HI(&). Then the weak limit ( ~ i , u ) C
[HI(~) z VtcL(f~)]d" satisfies (the fonctions Pi, q~ are defined in the statement of the theorem):
Convergence of the scaled displacements
Sect. 2.3]
+ L { a 41# + 2it e~~
+ 4#e~n(~H)e~n(~TH) } dw =
.if "v 9 d2 +
with (see part (iii)) : Uo~ :
~c~ -- X 3 0 c ~ 3 ,
L
Pi'qi dm -
~c~ E H i ( w ) ,
U3 - - ~3~
L
~3 C
q<,0o<~3 do<.,
H2(~),
~i = Ou~3 = 0 o n Vc~ - - T]c~ - - X3~c~7"]3 ,
r162
155
70,
~ e H~(~), ~3 e H~(~),
V 3 = ?73 ~
1
.- 7 ( 0 ~ + %r
r
rh = 0~rl3 = 0 on 70, 1
.- 7(0.v~ + 0~v~).
We use the functions (~(e), v(c)) constructed in part (vi) for approximating the function (~, v) as test-functions in the variational equations of problem 7)(e; ft, ft) (Thm. 2.2-1). Since e~a(v(e)) = 0 in f~ by construction, these equations reduce to L x(O~),~('5(s))" ~('~ (e))d2 +
J~{2t,~(~,(~))~(~,(~)) + :~(~..(u(~)) + ,~(~))~.(v(~))} - Sfi
X(O<~),f""~(c)d.~+
dx
i, f" v(c)dx,
where naa(e) = e-2eaa(u(e)). Let then e approach 0 in these equations. Since (cf. parts (ii), (v), and (vi)) ~t(e) - - ~t in H I ( ~ )
and
u(e) -- u in H~(a),
naa(e) ~ ~ e ~ o ( u ( e ) ) i n L2(ft), ~+2# ~(e) ~ ~ in H I ( ~ ) and v(e) --, v in H I ( a ) , we can pass to the limit (whenever B is a strongly continuous bilinear form, un --~ u and vn ~ v implies B(u~, vn) --~ B(u, v); cf.,
156
[Ch. 2
Junctions in linearly elastic multi-structures
e.g., Vol. I, Thm. 7.1-5), and we obtain in this fashion the desired variational equations, after replacing the components of u and v by their expressions as functions of ~ and r/i. The weak convergence of the sequence (n3a(e)) is thus needed here; the weak convergence of the sequence (n~3(e)) will not be used until part (x). It turns out that the limit problem will eventually consist of two independent problems, one with (ft, ~a) as the unknown, the other one with (2H = ( ~ ) as the unknown. Accordingly, our identification of the limit problem comprises two stages (parts (viii) and (ix)). (viii) The pair (~t, r
belongs to the space
[HI(~)) x Va(w)] "-- {('v, r/3)C Hl(ff]) • H2(w); 773 - 0~r/3 - 0 on 70,
v31~ -
r/31~,
V~l~e-
0},
and it satisfies, and is the unique solution of, the variational equations: 9~.(~)d2 +
=s
4A~
3(a + 2~)
Ar
+ --~0~gG0~r/3
dee
for all (V, ~]3)E [HI(~) x V3(w)] d . The variational equations found in (vii) are satisfied in particular by any function of the form (~, (-Xa01r/a,-Xa02r/3, r/a)) such that (v, r/3) ~_ [Ha(a) x Va(a;)]a, and either s u p p 9 is contained in the set {X -- (:~i) E ~r-~;Xl ~ 0} and r/3 -- 0, or vl~ E H~(&), in which cases they reduce to the desired equations. Given an arbitrary function (~,r/a) E [Hl(ft) x V3(w)]e, let Oa E He(&) denote an extension of ~al~,, and let ~3(21,22, ~:a) - ~3(2"1, ~2) for 2 - (~:1, ~:2, 23) C ~. Since the function (w*, r / a ) " - ( ( 0 , 0, @3), r/3) belongs to the space [HI(~)) x Va(a;)]e and satisfies @1"~ E H2(&) C Hi(&), it also satisfies the desired variational equations. Since these equations are linear with respect to (~, r/a), it suffices to show that they are satisfied for all pairs of the form
Sect. 2.3]
157
Convergence of the scaled displacements
(v,0) E [Hl(f~) • V3(W)]d, with functions $ - (5~) E Ht(ft) satisfying 5~1~,, - 0. To this end, we show that, given any function 5 E Hl(f~) that satisfies 51~,~ -- O, there exist functions P and g~, n >_ 1, with the following properties: ~n C H l ( f i ) and ~1~-E H I ( ~ ) ,
(2~) E fi; 5Cl ~
g~ E H ~ ( ~ ) and suppg ~ C { 2 -
0},
( ~ + a n) ---, ~ in HI(f)) as n ---, oc.
Since the desired variational equations are separately satisfied by the functions ( ( ~ ) , 0) and ((g~), 0), and since they are linear and continuous with respect to ~ E Hl(ft), the assertion will follow. Given 5 E Hl(f~) that satisfies 51a,~ - 0, let the function 5 ~ E H I (~) be defined for each n _> 1 by '
5(Xl, x2, x3) for :~1 --~
2 n
1 5(2(21 + --), 22, 2a) for v n ( x l , X2, X3) --
2
n
1 v(21 + - , 2 2 , x3) for
l(g~ + d ) , 2 2 , 2 a ) "D('~ 7t
1
n
1 n
_< 21 _<
_< 21 <_ d - - ,
ford--2
n
n
2 n
<__21_
and let 0 ~ be a smooth enough function of the variable ~:1 that satisfies 0n(Xl)-
1 for Xl
a n d 0n(Xl) - - 0 for 0 _< 2C1.
Then the functions {n . _ ( 1 - On)~ n
E
HI(~)
&nd
clearly satisfy rl~- E H ' ( ~ ) (since rl~ - 0 )
~n . _ O n ~ n E
HI(~)
and supp gn C { : c - (2i) E
a; 21 _< 0}. It thus remains to show that ( P + g") ---, ~ in H i ( a ) as n ---, oc, i.e., that [[~" - ~lll,~ ~ 0 ~s ~ ~ o~. w e have
158
[Ch. 2
J u n c t i o n s in linearly elastic m u l t i - s t r u c t u r e s
1 +
1
]'v(2(a:l 4- --), x2, x3) -- '/~(Xl, X2, X3)] 2 d~
1
[?~(Xl + --), X2, X3) -- 'V(Xl, X2, X3)[ 2 d~: :~E~;_ • <_g:l<_d__2 } n
a(:~l + d), 5:2, :?a)
-
V(:~I, X2, X3)] 2 d~.
Since 1
[~(2(2~1 ~- - ) , 2~2,2~3)]2 d~ n 2
~e(~;--=<~
I~(~1, ~2, ~3)[ 2 d~
converges to zero as n --~ oo, the first and third integrals in the former inequality converge to zero; the second integral converges to zero since the translation operator is continuous in L2(f~) (see, e.g., He,as [1967, p. 57]). The norms [c3i~?~ - c3~]0,~ likewise converge to zero as n --~ oc, since the only effect of differentiating with respect to 2~ is the appearance of 1/2 or 2 as factors. As we shall see in part (i) of the proof of Thm. 2.4-1, the semin o ?wn
+
is a n o r m over the space HI((~) x Va(w) ]
(we anticipate here a result which, to our satisfaction, will be proved without recourse to the present proof!). Hence (~i, ~a) is the only solution of the variational equations. (ix) The f u n c t i o n ~H "-- ( ~ ) belongs to the space VH(.~) "-- {r/H -- (77~) E Hi(w); r/H - 0 it satisfies the variational equations:
on 70};
Sect. 2.3]
159
Convergence of the scaled displacements F
4A#
doJ
for all rlH E VH(CO), and it is their only solution.
The variational equations found in (vii) are satisfied in particular by any function of the form (0, (r/l, r/2, 0)) such that (rh, ~]2) E VH(cz) (since 01~ E HI(c~)), in which case they reduce to the above equations. We have seen in Thm. 1.5-2 that they have a unique solution. The next step is the final one.
(x) rh~ ~vho~ f ~ ~ y ((~i(~), ~(~)))~>0 ~o~v~g~ ~t~o~glr to (~,, u)
in the space Hi(h) X Hi(h).
By parts (viii)and (ix), the weak limit (ft, u ) i s unique; hence the whole family ((~(c), u(c)))~>0 converges weakly to ('5, u) in the space H I ( ~ ) x H I ( a ) , and strongly to (f,, u) in the space L2((~) x L2(f~), by the Rellich-Kondragov theorem. It thus suffices to show that the family (~.(~(c)), r strongly converges in the space L~2(~) x L~(ft), as the conclusion will then result from Korn's inequality without boundary conditions (Thm. 1.1-2) applied in the spaces HI(f)) and Hl(f~); in fact, we even prove the sharper result that the family ((5(s strongly converges to (e('g),m) in the space L~2(a) x L~2(ft), where the components of the tensor tr - (n~j) are defined by (cf. part (v)): N
~ -
~(~),
~ 3 - o,
~33-
A
+ 2 ~(~).
We recall that the weak convergence (fi('g(c)), re(c)) --" (fi(~), ~;) in L~2(~) x L~(~) has been established in parts (ii) and (v). Since there exists a constant c > 0 such that c B " B < A B " B and c B " B < A B " B for all symmetric matrices B of order 3, we infer that the following inequality holds for all c > 0"
[Ch. 2
Junctions in linearly elastic multi-structures
160
a(,i)l
_<
0,~
+
O,f~)
-
X(f~d)A(~(~i(e))
fi(~i))'(~(~i(e)) -- ~(~i))d2
+ ~ X(0~)A(~(~i(e)) - ~(~i)) 9(g~(~i(e)) - ~(~i))d2
+ ~ A(n(~) - n)
(n(~) 9 - ~,)dx.
Let us examine the behavior of the right-hand side of this inequality as e ~ 0. First, a simple computation based on the junction conditions for the three-dimensional problem shows that
X(t),~)A~('g(~))"
fi('g(~))db
E3 ~d A n ( e ) : n ( e ) d x -~ 0 as e ~ O,
-
since the weakly convergent family (~(~))~>0 is bounded in the space L~(f~). Next,
~ X(~)Aa(~)
: (fi(~i(e))
since
the family
-
fi(~i))d2
--+ 0 a s e ~
O,
(X(D~)e(~i) ) ~>0 converges strongly to 0 in L~(f~),
the family ( ~ ( ~ i ( r ~(~i))~>0 converges weakly to 0 in L~(~), and the inner product in the space L~(f~) is a continuous bilinear form (this argument will be used at several later places, but will not be repeated). Finally,
X(t)~)A@('g)" @('g)d~ --+ 0 as e ~ 0, since the d2-measure of the set ft~ approaches 0 as c + 0. Hence
X(~)A(fi('b,(e))
- fi('g))
9(fi('g(e)) - fi('g))ds ~ ~ 0 as ~ ---, O.
The remaining terms in the right-hand side of the inequality can be rewritten as
Sect. 2.3]
Convergence of the scaled displacements -
9
161
-
+ ./o A(~(e) - ~)"(~(c) - ~ ) d z
- .l~ x(0~)/~(~i) 9(~(~i) - 2~(~i(c)))d~ + .fo A~ "(~ - 2~(c))dz
First, we clearly have X(0~)/~,fi(ft)" (fi(gt)- 2fi(gt(e)))d:~--, {-f(~ ~,fi(gt)" g~(~i)d~:} &S~ ---+ 0~
/aA~" (~-2~(c))dx--~ {-/~A~'~dx}
as c-~ 0;
secondly, expressing that the variational equations found in problem P(e; t2, t2) (Thin. 2.2-1) are satisfied in particular by the pair (~i(e), u(c))E V(c), we find that L X(O})/~fi(~i(e))" fi(~(e))d2 +
fa An(e)"n(e)dx
-- ~ X(0})if" " ( e ) d 2 + L f . u(e)dx ~ L if" { t d 2 + / a f 9u dx as e ---, 0. We thus infer from these relations that the remaining terms in the right-hand side of the inequality converge to
A "- - { ~ Afi(ft) " fi(ft) d2+ faA~ " ~dx} + L )'.ftdFc+ L f .udx. Using the special form of the vector field u = (u{) found in part (iii) (u~ = ~ - xaO~a and ua = ~a), the special form of the tensor field ~ = (~{j) found in part (v) (n~ = %~(u), ec~a = O, a ~aa -- x+~ coo(u)), and expressing that the variational equations of parts (viii) and (ix) are satisfied in particular by (~, ~a) = ({t, ~a) and rl~/ = (H, respectively, we then easily verify that the limit A vanishes. Therefore the proof is complete.
Remark. The proof given here of part (v) is simpler than that originally given in Ciarlet, Le Dret & Nzengwa [1989]. Raoult [1992,
162
Junctions in linearly elastic multi-structures
[Ch. 2
Lemma 2.3] has also found a generalization of part (vi), which in turn significantly shortens the proofs of parts (vii) and (viii). II 2.4.
THE LIMIT SCALED PROBLEM: EXISTENCE A N D U N I Q U E N E S S OF A S O L U T I O N ; F O R M U L A T I O N AS A B O U N D A R Y V A L U E PROBLEM
The limit problem solved by ~H = (~-) (Thm. 2.3-1(d)) coincides with the scaled two-dimensional membrane equations found for a single plate, already considered in Thin. 1.5-2. It thus remains to study the limit problem solved by (~t,~3) (Thin. 2.3-1(c)); we recall that the problems solved by e H and by (~t, ~3) are independent. More specifically, we first give a "direct" proof of the existence of (~t, ~3) (one such proof, again as "highly improbable" as for a "single plate", is already de facto provided by the asymptotic analysis). We then describe the boundary value problem that is, at least formally, solved by (~z, ~3). To this end, we henceforth identify the sets f~ and O. This implies in particular that c9~ now denotes the partial derivative with respect to the i-th coordinate of a generic point in the set O, which is now denoted 2 = ( ~ ) , that notations such as ~i = ( ~ ) , ~ = (~) now represent functions in the space H i ( O ) , and that the set &d is identified with the set eva. We then define the open set (Fig. 2.4-1) Od -- 0 --~d, which is thus a three-dimensional open set with a two-dimensional slit, and we let cz~ and a~d denote the upper and lower "faces" of the slit. When viewed as sets, these faces are fictitiously distinguished, since they coincide with the set a~d; on the other hand, the introduction of different notations is convenient for distinguishing between traces "from above" and "from below" of a function defined over the set Od, as in the partial differential equation in a~ found in the next theorem. The unit outer normal vector along the set O O d - ~d is denoted 7i - ( ~ ) ; the notations ~ , 0., T~, 0~, 71 have the same meaning as in Thin. 1.5-1. Finally, if 0 denote a function defined
Sect. 2.4]
The limit scaled problem
-':'~ 9 .....
tL-_L- ~ "
....~2_1_--~ .!_--_-~' ."/9
%
- - T
.9' ~ - - - : b
.....
3"
- -/,
163
',
I
'
l'~"
',
SS SS SS SS SS eS S |
|
:
.
a
!
a
d
'
Fig. 2.4-1: Mathematical modeling of an elastic multi-structure. T h e limit p r o b l e m is a coupled, multi-dimensional, b o u n d a r y value p r o b l e m s i m u l t a n e o u s l y posed over t h e t h r e e - d i m e n s i o n a l o p e n set Oa and the two-dimensional open set w. T h e t h r e e - d i m e n s i o n a l set Od has a two-dimensional slit into which the p o r t i o n wa of t h e t w o - d i m e n s i o n a l set w is inserted; the n o t a t i o n s w + a n d w d for the u p p e r a n d lower "faces" of the slit allow to distinguish b e t w e e n traces "from above" and traces "from below" on the set Wd.
over the set We, we let Ext ~ 0 denotes the function defined over the set ~ by
ExtO0_{0 0
in in
w-~d, Wa.
164
[Ch. 2
Junctions in linearly elastic multi-structures
Theorem 2.4-1. (a)Assume that ~ E L2(O) and p3, q~ E L2(w). The variational problem satisfied by (~t, r viz.,
(li, r
E [nl(o) • V3(w)]d
-
-
T]3 = 0uT]3 -- 0
{(V, r/3) 6 Hi(O) • H2(w); on
70,
531&d --
7131Wd, 'Vz]c2d
-- 0 } ,
/O
{iCpp(~-$)Cqq(V) ~- 2~eij(~_L)Cij(~)) } dJc
4A# ACaAr/3+ ~0~z(3O~zT?3 } dw + ~ { 3 ( I + 2#)
for all (v,?~3)E [H~(O) • V3(w)]a,
has one and only one solution.
(b) Let
m~
- -{
3(~ + 2~)
A smooth enough solution equations:
(~i,(a) satisfies the following coupled
Sect. 2.4]
165
The limit scaled problem
-oja~j(a) - ]~ in Od,
~-~j(,a)% -
0 on OOd -- -gd,
--Oc~mc~ -- P3 + O~q~
+ Ext~ ~3
-
-
Ov~
-
-533(~)1~-; } in w,
0 0 n 70,
-
real3 b'a 1,"~3 -- 0 on 71,
,~31w+
-
~,~1~ -
-
'~,31w2 '5,~1%7
-
-
-
~3[w,t , 0.
(i) To establish that the variational problem satisfied by (~i, ~3) has one and only one solution, it suffices to show t h a t the semi-norm Pro@
I l (~, ~ ) - - "
{la(~)10~,o + 1~31~,~}1/~
is a norm over the space [HI(O) x Va(w)]e, equivalent to the product norm
II I1(~ ~3)~ {11~1121,o-Jr ll~3ll~ }1/2 L~t (~, ~a) c [HI(O) • V,(~)]~ b~ ~uch that I(~, ~ a ) l - 0. Since I" [2,~ is a norm over the space V3(w) - {~3 E H2(w); ~3 - 0 ~ 3 0 on 70} (Thm. 1.5-1, part (i) of the proof) and ~3 C V3(w), we first infer t h a t ~3 - 0. Since I~(~)10,o - 0, ~ - o on ad, and area 9 > O, we conclude t h a t v - 0 (Thm. 1.1-2). Hence [" I is a norm over [HI(O) x V3(w)] d. Assume t h a t I 9l is not equivalent to I1" II. Then there exist functions (~k, r/ak) e [HI(O) x V3(w)] d such that [l( ~ k , ~ ] ~ ) l l - 1
for all k and
1(~ k , ~ ) [ ~ 0 a s k - - ~ o c .
Since I" 12,~ is equivalent to II" 1[2,~ over Va(w) (Thm. 1.5-1, part (i) of the proof), I1~11~,~ --' 0, hence 5akin,, - r/akin,, ~ 0 in H1/2(Wa) as
166
Junctions in linearly elastic multi-structures
[Ch. 2
k --+ oc. Since 5~1~,, - 0 for all k by definition of [HI(O) • Va(cO)]d, we conclude that ~l~,,--+0
in
H1/2(COd).
The relation I~.(~)10,o --+ 0 as k --+ cc implies (Ex. 2.2) that there exist vectors g:~, d~ ~ R a and functions ~
~ Hi(O) such that
~k (5:) -- ~.~ + dk A off: + ~k (S:) for almost all S: ~ O, II"uIjk ll l ,O ---+ 0
aN k - - + oo.
Since ~kl~o,, --+ 0 in H1/2(cOe), we thus infer that ~k --+ 0 and ~/k --+ 0 in R a (it is no coincidence that the same argument was used in part (i) of the proof of Thm. 2.3-1). Consequently, ~k --+ 0 in H I ( O ) ; but this contradicts I1(~k, ~)11- 1 for all k (recall that ~Ta k --+ 0 in H2(co)). Hence part (a) is proved. (ii) Let 773 = 0 and let ~ vary in the space T~(Od), then in the space {~ C C ~ ( O ) ; ~ - 0 in a neighborhood of COd}, in the variational equations solved by (~i, ~a); this shows that the partial differential equations in the set Od and the boundary conditions on OOd--gd are satisfied. If (v,r/3) C [ H ~ ( O ) x V3(W)]d and ~i is smooth enough, Green's formula gives
- - ~ { e ~ ( , i ) ~ + - ~ ( , i ) , ~ : } ~ d~ d
= - ~ (Ext ~ { ~ ( , i ) ,~+ - ~ (,i)~%; } ) ~ d~, so that the variational equations reduce to
f m~O~r~3 da~ - ~ (p3 +
Ext ~ {53a(~i)l~+ - ~aa(~i),~ })rid da~
- f~ q~ c9~r/a dco
Sect. 2.5]
Modeling by a coupled, multi-dimensional problem
167
for all r/a E H2(co) that satisfy r/a = 0,r/a = 0 on 70; this shows that the partial differential equation in co and the boundary conditions on 7 are satisfied. The relations at the "junction" between the sets Oe and co are but another way of writing the conditions ~2al~,, = C31~, and g~[~,, = 0, satisfied by the particular function (g, (a) in the space [ H I ( O ) • V3(co)]d. Hence part ( b ) i s proved. II
Remarks. (1) It is easily seen that the variational problem satisfied by (ft, ~a) may be equivalently expressed as a minimization problem; the analogous minimization problem that the de-scaled unknown satisfies is described in the next section. (2) As for a "single plate" (Thin. 1.5-1), the partial differential equation in co may also be written as m
2.5.
MATHEMATICAL MODELING OF AN ELASTIC MULTI-STRUCTURE BY A COUPLED, MULTI-DIMENSIONAL BOUNDARY VALUE PROBLEM; JUNCTION CONDITIONS
We next "de-scale" the equations found in Thm. 2.4-1, in order that the "de-scaled" unknowns ~i~ and (~ (as defined below) be physically meaningful. First, with the "limit" vector field ~i = (ui) : O ---+ R a, we associate the vector field ~i~ - (g~) 9O ~ R a defined through the d e - s c a l i n g s : ~(S~) =: c~2~(5:) at all points ~ E O; next, with the "limit" vector field (~i) : ~ ~ R 3, we likewise associate the vector field (([) " ~ ---, R 3 defined through the de-scalings" (~-'e24~
and
{~-'r
168
[Ch. 2
Junctions in linearly elastic multi-structures
Naturally, these de-scalings are nothing but the "inverses" of the scalings made in Sect. 2.2; recall in this respect that the sets f / a n d O are identified.
Remark. The notation ~ ( 0 ) = (g~(0)), instead of the shorter ~ - ( ~ ) , would have been more consistent with that used for a "single plate" (Sect. 1.7); the same observation applies to the "descaled stresses" noted a~j(,i ~) in the next theorem. No confusion should however arise. II Naturally, the problems solved by (/K, (~) on the one hand and by ( ~ - ((a) on the other are independent, as their "scaled counterparts" (Thm. 2.3-1(e)). The first problem (see parts (a) and (b) in the next theorem) is simply a re-statement, modulo the de-scalings, of the problem described in Thm. 2.4-1; the second (see part (c)) coincides with the two-dimensional membrane equations of a "single plate", already described in Thm. 1.7-2 (for this reason, only its formulation as a boundary value problem is recalled here). Together, they constitute the limit e q u a t i o n s of t h e m u l t i - s t r u c t u r e . T h e o r e m 2.5-1. (a) The de-scaled unknown (~, r
belongs to
the space [H~(O) • Va(co)]d "-- {(V,r]3) E H i ( O )
• H2(w);
?~3 -- ~uT]3 -~- 0 on 70, v3[wa : ?']3[cod, Vc~lwa
=o},
and it satisfies the variational equations: fo {~,~e~,p ~ ('h~)eqq ~ ( ~ ) + 2 y e ~ ( ~ ) e ~ j ( ~ ) } d:K d
--~--s3
3( k~ -~- 2 P e) A~3,/~T]3 -}- ----~C~c~/3(3 C~a/31"13 dco Vi
P3~3 dw -
q~ ~113dco
for all ('u,?]3) ~ [H'(O) • V3(co)]~,
Sect. 2.5]
Modeling by a coupled, multi-dimensional problem
169
where
p~ "-
f~ dx~ ,
q~
.__
s
s
~ xaf ~ dx~ , C
(f[) E L2(O U gt ~) is the applied body force multi-structure, and ~ , fit~ and M, #~ are the "three-dimensional" and "thin" substructures, This variational problem has one and only (b) L~t
~(~)
density acting on the Lamd constants of its respectively. one solution.
. - ~ ~ . ( ~ )~j + 2f~%(~),
m2 ~ ._ _ e a {
4A~# ~ ~ 4y ~} 3(:~, + 2~,) Ar 5~, + - - ~ 0 ~ , r .
A smooth enough solution (ft ~, ~ ) satisfies the following coupled equations:
-O~cr~(~t *) - f~ in On,
+ Ext~
r
- &~
- cr~a(~K)l%; } in a~,
- o on 7o,
m~fl ar'fl -- 0 on ~/1~ m~
rn ~ L,~7~) - -q~r,~ on ")/1, U3lw+ -- U3lw S -- ~'3lw,z~
(c) The de-scaled unknown eH - - ( ~ ) satisfies, at least formally, the two-dimensional membrane equations of a linearly elastic plate,
170
[Ch. 2
Junctions in linearly elastic multi-structures
viz.,
- 0 ~ ha3 ~ - p ~ c~ i n w (~ - 0 on %, n~3~,z - 0 on ")/1,
where
p~ n:,
-
-~ ff~ dx~ , e~z(~ ~H) - - ~ ( O ~ + 0~(~), c
+ 2,
+
.
The problems described in (b) and (c) are independent.
I
Remark. T h e space [Hi(O)X V3(a))] d m a y be replaced with the space [Hl(Od) X V3(~)] d "- {('v, r]3)(E H l ( O d ) X H2(a;); r/a - &,r/a - 0 on %, ~?al~o~ - ~at~o,7 - Val~,,, ~1~o3 - ~ 1 ~
- 0},
in the variational formulation (a); for, if a function v E H~(Od) satisfies vl%+ -- vl%- , it is in the space H 1(O). I
A major conclusion is thus t h a t the function (,i ~, ~ ) solves a coupled, multi-dimensional, p r o b l e m : It is "multi-dimensional" because it is simultaneously posed over the three-dimensional set Od and the two-dimensional set co. It is "coupled" since the u n k n o w n s m u s t satisfy the j u n c t i o n c o n d i t i o n s ual~+ - Ual~oT, - (31~, and g~
- fi~
found in b o t h formulations (a) and (b); the coupling also a p p e a r s in the second formulation, via the additive t e r m E x t ~ -
Commentary; refinements and generalizations
Sect. 2 . 6 ]
171
cr~a(ft~)l~; } in the right-side of the partial differential equation satisfied by r The junction conditions are imposed along a two-dimensional slit of the three-dimensional set Oa, into which the portion CZd of the two-dimensional set a; is inserted. Note in passing that the boundary of the open set Oa is not Lipschitz-continuous! The variational problem solved by (ft ~, r may be equivalently written as a minimization problem: Find (~t~, ~ ) such that
fl~(~, r
(~, r [H~(O) • V3(~)]d, - inf{fl~(6, ~/3); (~, r/3) E [HI(O) x V3(W)]d} ,
where -
1/o {~
% (v) + 2/2~ei~J(V)ei~j(v)} d2~ (v)~
It is thus remarkable (though not quite unexpected) that the functional J~, which is the limit e n e r g y of the multi-structure, is nothing but the sum of the energy of the "three-dimensional" substructure and of the two-dimensional flexural energy of the plate; likewise, the space where the minimum is sought is nothing but that subspace of the product space associated with both substructures whose functions precisely satisfy the junctions conditions. 2.6.
COMMENTARY; REFINEMENTS AND GENERALIZATIONS
"Stiff" c h a r a c t e r of t h e limit p r o b l e m . The variational problem described in Thin. 2.5-1 (a) provides an example of a "stiff" elliptic problem: Different powers of the "small" parameter (c o and c 3, respectively) factorize the two bilinear forms appearing in the
Junctions in linearly elastic multi-structures
172
[Ch. 2
left-hand side of the variational equations. The asymptotic behavior of the solutions of such problems is studied for its own sake in Lions [1973, 1981 (Chap. 5), 1985], Panasenko [1980],and Sanchez-Palencia [1980 (Chap. 13), 1989a, 1989b, 1992]. R e m a r k s on the j u n c t i o n conditions. While the junction ~C ~C conditions nail+' - u31~; - ~al~,, express the continuity of the vertical displacement along the inserted portion of the plate, the other junction conditions ~ ~1~ - ~ ~1%7 - 0 do not involve the functions ~ 1 ~ ' which "independently" satisfy the two-dimensional membrane equations of a plate, and thus do not vanish in general (except in some special cases, as when f~ = 0 in ft~). This is only an apparent paradox, for the convergence result obtained in Thm. 2.3-1 implies that (U~l~)l~o,,- c~3 + o(c) in H1/2(Wd), (u~l~)l~o,, -- ~2r + o(~ 2) in H1/2 (COd); hence the first-order term (with respect to c) of the horizontal components of the displacement of the three-dimensional structure should also vanish in H1/2(Wd) if these "limit" horizontal components are to be continuous ; but this is exactly what is implied by the relations ~< ~1~,7 0 and the de-scalings u~(x ~) ~(,~(#c) The function -
~
_
-~
_
.
h~ "- Ext ~ {a~3(5~)1~+ - ~3(~)1~7 } appearing in the right-hand side of the otherwise familiar two-dimensional flexural equations of a plate (Thin. 2.5-1(b)) is nothing but a Lagrange multiplier, in the sense of optimization theory. Such Lagrange multipliers arise when a minimizer of a function (here the "limit" functional J~) is sought over a subspace (here the space [Hi(O) • Va(cO)]d) of a space (here the space H~(O) x Va(co)) whose elements satisfy "equality constraints" (here the junction conditions). For details about Lagrange multipliers, see, e.g., Ciarlet [1989, Thin. 7.2-2]. M e c h a n i c a l i n t e r p r e t a t i o n of the limit problem. Within the framework of linearized elasticity, the mechanical interpretation
Commentary; refinements and generalizations
Sect. 2 . 6 ]
173
of the solution of the limit problem is natural: The function ~t~ sat-
isfies the familiar equations of three-dimensional elasticity, while the functions ~ and ( ~ ) satisfy the familiar two-dimensional flexural and membrane equations of a plate (Thins. 1.7-1. and 1.7-2). As expected, the function h~ that is added to the right-hand side of the two-dimensional flexural equation "balances" the vertical resultant of the forces that act on the three-dimensional part Oe, in the sense that d
To see this, note t h a t +u~;
/o
,,
/o
,,
by definition of h~, by Green's formula, and by T h m . 2.5-1(b). W i t h the "limit" vector field u = (u~) : gt ~ R 3 found in Thin. 2.3-1 (a), we may also associate the "limit" vector field u~(0) = (u~(0)) : f2~ ~ R a defined by the same de-scalings as for a "single plate", viz.,
u~(O)(x ~) "- e2u~(x) and u~(O)(x ~) "- eu3(x) for all x ~ - 7r~x E ~. Then u~(0)=~-x~0~
and u ~ ( 0 ) = s
i.e, u ~(0) is a Kirchhoff-Love displacement field inside the plate. Assumptions o n t h e L a m ~ c o n s t a n t s in e a c h s u b s t r u c t u r e . T h e assumptions on the Lain5 constants (A~, /2~ are independent of e and A~, p~ vary as e -3) express t h a t the elastic material
constituting the plate must be "more rigid" than that constituting the three-dimensional substructure. Likewise, the assumptions on the applied b o d y force densities (they vary as c in the three-dimensional substructure; their vertical component is independent of c and their horizontal c o m p o n e n t s vary as c -1 in the plate) express t h a t the mass
density of the material constituting the plate must be higher than that constituting the three-dimensional substructure, at least for an applied force density t h a t is proportional to the mass density (e.g., the gravity or a centrifugal force).
174
Junctions in linearly elastic multi-structures
[Ch. 2
It is crucial to observe, however, that these "relative" asymptotic orders need to be assumed on the data only if the limit behavior of both parts of the structure is expected to be elastic. Indeed: Other ratios between the asymptotic orders in each substructure can lead to strikingly different limit behaviors. To illustrate this assertion, assume that the two substructures are made of the same material, and that a boundary condition of place is satisfied along a portion of the boundary of the three-dimensional part, in which case no boundary condition of place is needed along the boundary of the plate. Then, as shown by Ciarlet & Le Dret [1989], the threedimensional structure, as well as the inserted portion CUd o f the middle surface of the plate, become rigid as ~ ~ 0; besides, the junction conditions found in the limit problem represent the genuine boundary conditions that a two-dimensional clamped plate model should satisfy at the junction with the three-dimensional support; see Sect. 2.7 for further details. M i s c e l l a n e o u s e x t e n s i o n s . While the "full" three-dimensional problem is well defined for any r > 0 if d = 0, i.e., if there is no insertion, the present approach does not yield a coupled limit problem in this case: Even if a boundary condition of place is satisfied along a portion of the boundary of the three-dimensional part (in order to "hold" this part), the limit problem consists of two unrelated problems, i.e., there is no longer any junction condition in the limit problem when d = 0. More generally, Aufranc [1990] has shown that the same conclusion holds if d is a function of E that approaches 0 as c~0. As shown by Bourquin & Ciarlet [1989] and Raoult [1992], one can likewise identify and justify by an analogous asymptotic analysis the eigenvalue, and time-dependent problems, modeling the same elastic multi-structure; see Sects. 2.8 and 2.9. The limit stresses inside the plate have been studied by Aufranc [1990]. There remains however the challenging, and of major importance in practice, problem of identifying the "corner singularities" at the junction between the two substructures, singularities which are in turn responsible for the stress concentrations that are likely to occur there; in this direction, see Nicaise [1992].
Sect. 2 . 6 ]
Commentary; refinements and generalizations
175
The
asymptotic analysis described in this chapter is in fact of wide applicability, since it can be also used for modeling folded plates, possibly with corners (Le Dret [1989a, 1990a, 1990b, 1994]), junctions between plates and rods, plates with stiffeners (Aufranc [1990, 1991], Gruais [1993a], Conca & Zuazua [1995]), junctions between rods (Le Dret [1989b], Panasenko [1993]), and "thin-walled" rods (Rodriguez & Viafio [1997]). See in particular the monograph of Le Dret [1991], where these and other applications are treated in detail. Other extensions have been investigated, in particular the identification of the limit problem for nonlinearly elastic multi-structures (Aufranc [1990, 1991], Gruais [1993b]) by the method of asymptotic expansions described in Chap. 4 for a "single plate", and for junctions between three-dimensional structures and shallow shells (Sect.
3.8).
In each instance, at least one part of the whole three-dimensional elastic multi-structure has a "small" thickness, or diameter, deemed proportional to a dimensionless parameter e. If the various data (Lam~ constants and applied body or surface force densities) behave as specific powers of c as e ~ 0, the HI-convergence of the appropriately scaled components of the displacement vector field toward the solution of a limit variational problem can be established. Each such problem is "multi-dimensional" and "coupled", in that it is posed simultaneously over an open subset of R "~ and an open subset of R ~, with 1 _< m, n _< 3, and its solution must satisfy appropriate junction conditions at the "junctions" between the various "limit" substructures. Observe however that, stricto sensu, the modeling of junctions between plates (rn - n - 2), or of junctions between rods (rn n - 1), does not yield problems that are "multi"-dimensional. Such problems nevertheless share all the features of the "genuinely multidimensional" problem described here. Structures comprising "many" junctions between plates, or between rods, are also amenable to a completely different approach, based on the techniques of homogenization theory. The limit, "homogenized", problems obtained in this fashion are thus models of structures with "infinitely many" junctions. In this direction, see
176
Junctions in linearly elastic multi-structures
[Ch. 2
m .
.
.
.
Fig. 2.6-1: An H-shaped beam inserted into an elastic foundation. Two kinds of junctions are found in this multi-structure: Junctions between plates and junctions between plates and a three-dimensional substructure.
notably the works of Cioranescu & Saint Jean Paulin [1986, 1987, 1988] and Charpentier & Saint Jean Paulin [1996]. While the present approach essentially relies on a "Hi-setting '', a more refined asymptotic analysis, where "infinite energies" are allowed in the limit problems, has been advocated by Sanchez-Palencia [1988, 1994] (see also Leguillon & Sanchez-Palencia [1990], g a m p a s s i [1992], and Mampassi & Sanchez-Palencia [1992]); it encompasses in particular multi-structures where the depth d of the insertion vanishes. In the same spirit, a "multi-scaled" asymptotic analysis allows to model junctions between a three-dimensional structure and onedimensional substructures (rods); in this direction, see Argatov & Nazarov [1993] and Kozlov, Maz'ya & Movchan [1994, 1995]. M o d e l i n g a n d n u m e r i c a l a n a l y s i s of j u n c t i o n s . The modeling of junctions is indeed a problem of outstanding practical importance, since these are very commonly found in actual elastic multistructures, such as an H-shaped beam inserted into an elastic foun-
Sect. 2 . 6 ]
Commentary; refinements and generalizations
177
Fig. 2.6-2: A multi-structure from aerospace engineering. The solar panels of a satellite are two-dimensional substructures (plates), which are held together, and connected to the central structure, by one-dimensional substructures (rods). This sketch of the satellite "TDFI" is drawn by courtesy of the Centre National d'Etudes Spatiales (C.N.E.S.), Paris. dation (Fig. 2.6-1), the solar panels of a satellite (Fig 2.6-2), or the blades of a rotor (Fig. 2.6-3). Examples of multi-structures comprising shells are given in Vol. III. However, we know of few other works, prior to Ciarlet, Le Dret & Nzengwa [1989] and Ciarlet & Le Dret [1989], where the elastic equilibrium of a body is studied together with that of the interacting surrounding elastic bodies; see however Batra [1972], Feng Kang [1979], B h a r a t h a & Levinson [1980], Caillerie [1980], Feng Kang & Shi Zhong-ci [1981], Rigolot [1982], acerbi & Buttazzo [1986], PodioGuidugli, Vergara-Caffarelli & Virga [1987], and Acerbi, Buttazzo & Percivale [1988].
178
Junctions in linearly elastic multi-structures
[ch. 2
.I
-9 " -
_:
9
" " ~
. . . . o ' " ' D ~
9
9 9-
Fig. 2.6-3" A rotor and its blades. This multi-structure is composed of a "threedimensional" substructure (the rotor) and "two-dimensional" substrucures (the blades). The blades are often modeled as nonlinearly elastic shallow shells (Sect. 4.14).
Mention must also be made of the closely related asymptotic analysis of linearly or nonlinearly elastic adhesives, which has recently received particular attention; see Klarbring [1991], Geymonat & Krasucki [1996], Geymonat, Krasucki & Lenci [1996], Ganghoffer & Schultz [1996], and Licht & Michaille [1996].
Commentary; refinements and generalizations
Sect. 2 . 6 ]
~-~'/"-----~i 9
9
/
~
J ~
;---.__ ,,,, _....~ ~
!
~
,--~_
,, ___..........
~-----~ ~
..
179
~
~.~a&zTy
~
\
.
Fig. 2.6-4: Computation o/ the displacement vector field o/ a linearly elastic multi-structure comprising a "thin" substructure (a plate) inserted into a "threedimensional" substructure. The body force density is such that the "horizontal" components of the applied body force vanish and the "vertical" component is <: 0. The limit problem found in Thm. 2.5-1 has been used in this computation, performed at the INRIA with the MODULEF code. These results are due to Aufranc [1989].
180
Junctions in linearly elastic multi-structures
[Ch. 2
Once a mathematical model of an elastic multi-structure is available, its numerical analysis becomes possible. In this direction, Aufranc [1989] has obtained satisfactory numerical results using the limit model found in Thin. 2.5-1 (Fig. 2.6-4); see also Wang Lie-heng [1992, 1993, 1995] and Lods [1992]. Bernadou, Fayolle & L~n~ [1989] and Bernardou [1989] have made numerical experiments on various models of folded plates and junctions between shells; however the two-dimensional models used in these experiments are a priori constructed without reference to an asymptotic analysis. A challenging program consists in numerically approximating the mathematical models of elastic multi-structures that comprise "many" substructures, such as that of Fig. 2.6-2. For such multi-structures, domain decomposition methods (see, e.g., Bjorstad & Widlund [1986], Bramble, Pasciak & Schatz [1986], James & Plemmons [1990], and Plemmons & White [1990]) seem ideally adapted, since the "subdomains" are clearly identified! In this direction, see d'Hennezel [1993]. Optimization methods that are particularly well-suited for minimization problems with "many" equality constraints, such as the junction conditions considered here, have also been proposed by Cohen & Miara [1990]. 2.7 ~.
J U S T I F I C A T I O N OF T H E B O U N D A R Y C O N D I T I O N S OF A C L A M P E D P L A T E
So far, we have considered an elastic multi-structure consisting of a plate of thickness 2c whose Lam~ constants vary as ~-3, inserted into a "three-dimensional" substructure whose Lam~ constants are independent of c. We then established the HI-convergence of the (appropriately scaled) components of the displacement vector field towards the solution of a coupled, pluri-dimensional problem, posed simultaneously over a three-dimensional open set with a slit and a two-dimensional set (the middle surface of the plate). Tacit in our analysis was however the objective that both substructures should still have an elastic behavior "at the limit". A completely different limit behavior occurs if the Lamd constants of the three-dimensional substructure also approach +oc as e -2-~, for
Sect. 2.7]
Justification of the boundary conditions of a clamped plate
181
some s > 0 (in particular, the whole multi-structure may be made of the same elastic material): As shown by Ciarlet & Le Dret [1989], the three-dimensional substructure becomes rigid as e ~ 0; besides, the asymptotic analysis simultaneously provides a rigorous justification of the classical boundary conditions of a clamped plate. More specifically, let a, b, c, d, e, f denote constants that are all > 0, and assume that d < a. For each e > 0, we let (Fig. 2.7-1): - {(z,, ~)
e R ~., 0 < ~, < a, I~l < b},
(-Ud -- {(Zl,X2) e ]~2; 0 < X 1 < d, Ix=l < b},
0-
~-~• a5-
~[, ~•
c,c[,
{(Xl,X2,X3) ~ ]I~3; - c < x 1 < d, Ix~l < b, - e < ~ < f}, o,~ -
o -~,
s~-
oufl
~,
and we denote by x ~ - (x~) a generic point in the set S ~ and by 0[ the partial derivative O/Ox~. The set S ~ is the reference configuration of an elastic multi-structure that comprises two substructures, "perfectly bonded" together along their common boundary: A "thin" part ~ , otherwise called a plate, occupied by a linearly elastic body whose Lam~ constants are assumed to be of the form A~ - c - 3 A
and
#~-c-3#,
where /~ > 0 and # > 0 are two constants independent of e; and a "three-dimensional" part Od, which is occupied by a linearly elastic body whose Lam~ constants are assumed to be of the form A~ - r
and
/~ _ r
where A > 0,/2 > 0, and s > 0 are three constants independent of c. Note that, by letting A - A, # - / 2 and s - 1, we can handle the important special case where the whole structure is made of the same material. Of course, the distinction between the two substructures
182
Junctions in linearly elastic multi-structures
[iIil filet [,tl I
lil~'
it1,1
i,,,,
"F-~
2c
,j,,,
,2_~J.,:~_J ,a~!i
/
,.,; :;
,aiF~ . '" I, ! , ,j...,--,.
,,!
l.,~a'a
[Ch. 2
i
i
." ".'" .-" ."
II / /.":.-" I/ ." ---
f ~ ...... /
/
--J
', ;
,
:
Fig. 2.7-1: Another three-dimensional elastic multi-structure. The set ~ is the reference configuration of an elastic plate, inserted into a three-dimensional b o d y whose reference configuration is O~. A b o u n d a r y condition of place u ~ - 0 is imposed on the portion F0 of the b o u n d a r y of the three-dimensional s u b s t r u c t u r e (compare with the multi-structure of Fig. 2.1-1).
~ and O~ becomes somewhat artificial in this case; in particular, the constant d becomes undetermined. The unknown d i s p l a c e m e n t v e c t o r f i e l d u ~ - (u~) " S~ --~ l~ 3 is assumed to satisfy a b o u n d a r y c o n d i t i o n o f p l a c e u ~ - 0 on F0. In linearized elasticity, u ~ is the solution of the v a r i a t i o n a l p r o b l e m
Justificationof the boundary conditions of a clamped plate
Sect. 2.7] ~(s~)
183
9
u e E V ( S ~) " - {v ~ -
/o
{a ~
~ , (u)~~ ( ~ )
(v~) E H I ( S s ) ;
+ 2p ~ j ( ~ ) ~ j ( ~ ) }
v e-Oonl-'o), d~ ~ _
/o
f : ~ d~ ~
+ J~ {/~eepp(Ue)eqq(Ve)-t- 21.teeij(ue)e~j(ve)} dxe - Jr2 f.~v: - O for all v ~ C V(S~). We thus no longer assume t h a t a b o u n d a r y condition of place u ~ = 0 holds along the "right" edge of the plate, as in Sect. 2.1 (see Fig. 2.1-1). We assume instead t h a t such a b o u n d a r y condition holds on a portion F0 of the b o u n d a r y of the three-dimensional substructure: In other words, the plate ~ is only "held" by this "supporting" structure. This in turn requires some care in the mathematical analysis, where one has to show, among other things, t h a t Korn's inequality "passes" in a sense from the supporting structure to the entire structure, uniformly with respect to e (cf. L e m m a 1 of Ciarlet & Le Dret [1989]). W i t h the sets ft ~ and O, we then associate two disjoint sets gt and f~ as in Sect. 2.2 (see Fig. 2.2-1). W i t h the restriction (still denoted) u ~ " - (u~) 9~ ~ R 3 of the unknown u ~ to the set ~ , we associate the function u(e) - (u~(e))" ~ ~ R a defined by the s c a l i n g s (here and subsequently, all the undefined notations are as in Sect. 2.2):
~(~)
- ~~(~)(x)
and
?A~(Xe) -- ~U3(C)(X) for all x ~ - 7r~x E ~ ,
and with the restriction (still denoted) u ~ - (u~)" O --~ ]~3 o f the unknown u ~ to the set O, we associate the function ~t(e) - ( ~ ( e ) ) 9 {~)}- ~ R 3 defined by the scalings
u~(z ~) - c2~2i(e)
for all x * - ( ~ - t) C O.
Note that, while the scalings inside the plate are the same as in Sect. 2.2, t h e y are different inside the three-dimensional substructure.
Junctions in linearly elastic multi-structures
184
[Ch. 2
The unknown u ~ E V ( ~ ~) is thus mapped into a scaled displacement (s u(e)) which belongs to the space H I ( ~ ) x Hl(fl), which satisfies the boundary condition u(e) - 0 on f'0 "- F0 + t, and which satisfies the junction conditions for the scaled three-dimensional problem: g(e)(2) = u~(c)(x) and r = u3(e)(x), at all corresponding points 2 E f~ "-- f~ + t and x E 9td "-- a~dX ]-- 1, 1[, i.e., that correspond to the same point x ~ E ~5 (Fig. 2.2-1). Finally, we assume that there exist functions fi E L2(ft) and j~ E L2(f)) independent of r such that
f~(x ~) - e - ~ f , ( x ) and f~(x ~) - f3(x) for all x ~ - 7r~m E a ~, f [ ( x ~) - j~(S:) for all x ~ - (a?- t) E 0~. Note that, while the assumptions inside the plate are the same as in Sect. 2.2, they are different inside the three-dimensional substructure. Using the assumptions on the data and the scalings of the unknowns, we can reformulate the variational problem P ( S ~) in the following equivalent form: The scaled displacement (~t(r u(r solves the variational problem P(c; ~,Ft) (compare with Thm. 2.2-1)" (U(C), U(C)) E V ( c ; ~ , ~ ) " -
o
on
r0,
{(v,v) E HI(~) • HI(~); -
-
at all corresponding points ~ E ~
+
+ + ~
and x E ~d},
dx
{Aeoo(u(e))e33(v)+Xe33(u(e))e,,(v)+4#e~3(u(e))e~3(v)}dx (a + 2#)e33(u(e))e33(v)dx =
X(O~a)f~ dye +
f~v~ dx
for all (~, v) E V(e; a, f~).
Justification of the boundary conditions of a clamped plate
Sect. 2 . 7 ]
I
_ .=...L
(
,- . . . .
_L. . . . . ~ _ . _ _ ~
#
, ............
_
.......................
T .... i---1 ...................... $,::...;:.
/
i.
#
l ...i.i..:!,"
,'-<>a / ' / /
r ..... /I :. . . . . / t
i .... ]
I
/
............................................
I"'*':'.--',,-" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
:" .............
,
. . . .
4- . . . . .,,...I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
t ---i ...........
9. . . . .
A
..!
~-J . . . . . . . . . . . . 3___'
/
185
/
/ /
/
_.
/
r F i g . 2.7-2: The vector field u = l i m u ( e ) inside the set gt = w x ] 1,1[. T h e limit u = (ui) C nl(fl) v a n i s h e s in ~td = W d X ] - 1,1[, h e n c e 9o F* = ~ a F1Ft*,
and is a scaled Kirchhoff-Love displacement field in ~t* - g t - fld = w*X] -- 1, 1[" There exist ~ C Hl(w *) and @ e H2(w *) with ~ = 0~3 = 0 on ~,* such that us = ~ - x30~3 and u3 = ~3 in ~*.
w h e r e O~ - O~ + t. O n e c a n t h e n prove (cf. T h m . 1 in C i a r l e t & Le D r e t [1989]): 2 . 7 - 1 . Let (Fig. 2.7-2)
Theorem ~"~* -
~
__ --
~-~d,
CO* - -
__
CO - - 0 J d ,
F*
--
--
~'-~d ~'1
-~,,
.~,
-- ~d
~-~
~*
,
and let O, denote the outer normal derivative operator along the boundary of the set w*. (~) A~ ~ --~ o, th~ faintly ((~(~), ~(~)))~>o c o ~ v ~ m ~ ~t~o~gly ~n the space H ~ ( ~ ) • H~(~t) towards an element (/t, u) that satisfies"
--0in~
and
ula~--0inf~d.
Furthermore, the function ula. e H I ( ~ *) vanishes on F* and is a scaled Kirchhoff-Love field in ~*, i.e., there exist functions ~ C H l(w*) and ~ C g l(w*), satisfying in addition ~i = 0,~3 = 0 on
186
[Ch. 2
Junctions in linearly elastic multi-structures
7*, such that U, la* - ~ - x30~3
and
u31n* - (3 in ft*.
(b) Define the space Y3(o.)*) " - {?]3 e H2(02");
?]3 - 0u?]3 - 0 on ")'*}.
Then the function ~3, which belongs to V3(oz*) by (a), satisfies the variational equations {
4A# A43Ar/3 + -4-~cg~z~3cg~zr/3} d~* 3(A + 2p)
-
,l~fParl3dw*-J,~f
q,/),r/3 dw* for all ~73 E V3(w*),
where p3 "- f1_1 f3 dx3 and q~ " - f l 1 x3f(~ d x 3 . (c) Define the space VH(W*) -- {~TH -- (r/,) e H~(w*); ~TH -- 0 on ~/*}.
Then the function r "-- ( ~ ) , which belongs to VH (w*) by (a), satisfies the variational equations 4A#
eoo(~H)er~-(~TH) + 4pe,~z(r
} dw*
-- ~ . p,r/, dw* for all ~H --(z?~) e VH(~*),
where p~ "- f l 1 f~ dx3.
1
It remains to "de-scale" the unknowns and the equations that they satisfy: With the "limit" vector field ~ i - (~2i) " {f~}- ~ R a, we
Sect. 2 . 7 ]
Justificationof the boundary conditions of a clamped plate
187
associate the "limit" vector field ~t~ 9O --~ R 3 by letting
e~(x ~) . - ~ e , ( ~ ) , at all corresponding points x ~ C 0 and ~ E {f~}- and with the vector field r - (~) 9~* ~ R a, we associate the "limit" vector field (:~ - ( ~ [ ) " ~ ~ R 3 by letting
~2<~ and <~ " -
~ " - 0 in wa, ~ " -
e~s in ~*.
Using these de-scalings, we find t h a t the vector fields ~ satisfy the following boundary value problems and relations"
~ i ~ = O in O, ~ -0
in a~d,
- O ~ m ~ 9 -- p~ + O~q~ in w*, r
:/)~r
= 0 on 3'*,
m~.~
- 0 on ( 0 ~ - 7"),
(O~m;~)v~ + O,.(m;~v~T~) -- -q;v~ in ( O w - 7"), r
- 0 in CUd,
- O ~ n ~ - 0 in w*, r
= 0 on 7", *
nc~gv~--Oon7 , where m:~
--c~{ 9
n : z 9- e p~ " -
4~~
3(A~ + 2#~)
A<~.~+ 4~
~~ ( r
s f: dx;,
q; "-
~}
--~-O.z< 3 ,
+ 4~.~(r ~
~ dz~ ,
,
and ~
Junctions in linearly elastic multi-structures
188
[Ch. 2
)V, #~ are the Lamd constants of the material constituting the plate, and (f[) is the density of applied body force acting on the multistructure. The main conclusions of our analysis are thus the following: The three-dimensional supporting structure becomes rigid in the limit; the inserted portion of the plate also becomes rigid in the limit, i.e., the plate is "clamped" in the literal sense; and finally, the "limit" displacement field of the middle surface of the plate solves the classical two-dimensional plate equations in w*, together with the classical twodimensional boundary conditions of clamping along 7* (Fig. 2.7-3). In addition, the limit displacement field across the thickness of the plate is a Kirchhoff-Love field.
y/+
]-'0 1 I ,
I I ,
,
:, I
, ,, 0 d ,,,
. I
I
I
I
:
I I
I I
I I
I I
, ~ I
II !I s FI.d "~ I
lJ Sl
Ii
I t
I i
'
"
I
~
..."I- . . . . . . . . . .
,,
~
_
I i~
I-
. . . . .
""
" "-
~
I
i_
l
t I
:
,
+
:
,
i
9
.... "
/
I
I
I
.
I~
I
|
"_~--_-y---
..,
I I I I'
Is" l z. ,
...?_:~F.~. . ~ ~ ! .,:#-] ........
Fig. 2.7-3: Justification of the boundary conditions of a clamped plate. The threedimensional "limit" supporting structure Od is rigid and the inserted portion Wd of the middle surface of the plate is also rigid "in the limit". The "limit" displacement field ~ of the remaining portion w* of the middle surface of the plate solves the classical two-dimensional plate equations in w*, together with the classical two-dimensional boundary conditions of clamping along "7*.
Sect. 2.8]
189
Eigenvalue problems
To conclude, we mention that Blanchard & Xiang [1992] have also considered a multi-structure analogous to that of Fig. 2.7-1, i.e., where the plate is actually inserted in a surrounding body that is however considered to be rigid at the outset of the analysis. Their results nevertheless usefully complement those presented here, as they include a convergence analysis of the stresses, in a situation that is more realistic than that considered in Sect. 1.6; see also Gregory & Wan [1994]. Finally, a n t m a n & Lanza de Cristoforis [1996] have given an interesting discussion regarding the "real nature" of clamping in general settings, a notion that sometimes encompasses unsuspected subtleties.
2.8 ~.
EIGENVALUE
PROBLEMS
We consider the same three-dimensional elastic multi-structure as in Sect. 2.1, consisting of a plate inserted into a three-dimensional structure, and we use the same notations. The set O~ is the reference configuration of an elastic body with Lain6 constants A~,/~ and mass density fi~; the set ~ is the reference configuration of an elastic plate with Lain6 constants A~, #~ and mass density y . We assume that there are no applied forces. Let t denote the time. In linearized elastodynarnics, the displacement field _
.
,
e ~ •215
[__~
R
3
satisfies the following partial differential equations: 02 W ie
j 02w~ Ot 2
Oj { A~epp( ~ w ~)Sij + 2# ~e i j ( w e ) }
-- 0 in f~x]0, +oc[.
We assume as in Sect. 2.1 that the displacement vector field satisfies a boundary condition of place w ~ - 0 on F~ for all t >_ 0. The problem of finding stationary solutions of these equations, i.e.,
Junctions in linearly elastic multi-structures
190
[Ch. 2
s o l u t i o n s of t h e p a r t i c u l a r forms:
w ~ (x ~, t) - u ~ (x ~) cos v/-A-Tt a n d w ~ (x ~, t) - u ( x ~) sin v/A~t, ( ~ , t) e s ~ • [0, + ~ [ ,
for s o m e A ~ > O, t h u s r e d u c e s to f i n d i n g numbers A ~ > 0 a n d nonzero vector fields u ~ 9S~ ~ R 3 t h a t satisfy:
-div~{A
y A ~ u ~ in Od,
e~(u~)}-
- d i v ~ { A ~e ~ ( u ~) } = p~ A ~u ~ in ft ~ , u ~ - 0 on F~), h~
~(~),v
- 0 on a o ~ - a ~ ~,
A ~e ~ ( u ~ ) n ~ U~o ~ -
h
E
0 on 0 ~ ~ - 0 0 ~ , u~
on OO~ n 0 ~ ~,
e ~ ( u ~ ) ~ ~ + A ~ e ~ ( u ~ ) n ~ = 0 on 0 0 ~ N Oft ~.
As e x p e c t e d , A ~: a n d u ~ are t h e eigenvalues a n d eigenfunctions of t h e eigenval~e problem t h a t is n a t u r a l l y a s s o c i a t e d w i t h t h e " t r a n s m i s sion'" p r o b l e m of t h r e e - d i m e n s i o n a l l i n e a r i z e d e l a s t i c i t y d e s c r i b e d in Sect. 2 1. Each
p(s~):
eigensolution (A ~, u ~) also solves t h e variational problem
A ~ c R and u ~ E V ( S ~) " - { v ~ - (v~) E H I ( s ~ ) ; B ~ ( ~ ~, ~ ) -
v ~ - 0 on F~},
A ~ ( u ~, v~) ~ for ~11 v ~ C V ( S ~ ) ,
191
Eigenvalue problems
Sect. 2.8] where
B~(u ~,v ~).-
{ , X ~%p( u ~ )eqq( ~ v ~ ) + 2 f t ~ e i ~j ( u ~ ) e i j ( v ~ ) } d x
J~o5 "~
+
~
{a ~,,(~ ) % ( ~ ) + 2S~j(~)~j(~)} d~~,
(u ~,
p ui vi dx ~ +
P ui vi dx ~.
5 The V(S~)-ellipticity of the bilinear form B ~, the positivity of the mass densities f5~ and p~, and the compactness of the imbedding from V ( S ~) into L2(S ~) together imply t h a t the symmetric mapping G ~ : u ~ e V ( S ~) ---+ G~u ~ e V ( S ~) defined by
B~(G~u ~, v ~) - ( u ~, v~) ~ for all v ~ E V(Sr is compact and positive definite. Hence all the eigenvalues of problem 7) ( S ~) can be arranged as a sequence (Ae'~)~=1 satisfying 0 < A l'e < A 2'~ < . . .
< A e'~ < A e+~'~ < ... and lim A e'~ - +oc,
and there exists an associated sequence of eigenfunctions u e'~ C V(S~), g >_ 1, i.e., satisfying
B * ( u e'~, v ~) - Ae'~(ue'~, v ~) for all v ~ E V(S~), g >_ 1, t h a t form a complete orthogonal set in both Hilbert spaces V ( S ~) and L2(S~), and which can be orthonormalized in such a way t h a t
B~(u k'~, u e'~) -e2Ak'~Ske, and thus (u k'~, ue'~) ~ - e2~ke, 1 <_ k,g. In order to define a problem equivalent to 7)(S~), but now posed over two domains f~ and ~ t h a t are independent of e, we proceed exactly as in Sect. 2.2" First, we let ft - co x l - 1, 1[, F0 - % x [-1, 1],
192
[Ch. 2
Junctions in linearly elastic multi-structures
and we define the translated set ~ - O + t, the vector t being such that {~t}-N ~0 (Fig. 2.2-1). T h e n with the restriction of the unknown u ~ - (u~) to ~ , with the restriction of u ~ - (u~) to O, and with the unknown A ~, we associate the scaled unknowns u(e) (u~(e))" ~ --+ R 3, ,i(e) - (g~(e)) 9{~t}- --+ R 3, and A(e) C R defined by the s c a l i n g s (the notations are those of Sect. 2.2)"
e
u~(z ~) - e2u~(e)(z) and u3(z ~) - eu3(g) for all z ~ - rr~:c C u~(oc~) -eric(e)(2) for all z ~ - ( ~ : - t ) E O, A ~ -
~e
A(e).
Finally, we make the following a s s u m p t i o n s o n t h e d a t a : There exist constants ~ > 0, /2 > 0, t5 > 0, A > 0, # > 0, and p > 0 such t h a t the Lamfi constants and mass densities satisfy"
~-~ &~-e-3X
and and
/2~-/2, #~-e-3#,
y-fi, p ~ - c -1 p
.
The function tt ~ E V ( S ~) is m a p p e d through the above scalings into a s c a l e d u n k n o w n (~i(e),u(e)), which belongs to the space HI(~)) x H i ( a ) , which satisfies the b o u n d a r y condition u(e) - 0 on F0 - 3'0 x ] - 1, 1[, and which satisfies the junction conditions for the
three-dimensional problem: -
and
-
u3(c)(x),
at each corresponding points 2 E ft} - ft} + t and z C fte - cod x ] - 1, 1[, i.e., t h a t correspond to the same point z ~ C a~ (Fig. 2.2-1). Using the scalings of the unknowns and the assumptions on the data, we reformulate the variational problem P ( f t ~) in the following equivalent form (compare with T h m . 2.2-1, whose notations are used here):
Sect. 2.8]
193
Eigenvalue problems
T h e o r e m 2.S-1. The scaled unknowns ('~(c),u(c)) and A(c) satisfy the variational problem P(c; (~, ~t)"
A(c) > 0 and (5(c),u(e)) CV(c;~,~)'-{('b,v)
C HI((~) • HI(~);
v -- 0 on r0, v~(x) -- cv,~(x) and v3(Y:) -- v3(x) at all corresponding points 2 E ~ta and x E ~d}, x(O~){Aepp(~(e))eqq(~) + 2/2e~j(~i(e))qj(~)} d~
+ L{Ae~o(u(c))e~.(v) + 2 # e , ~ ( u ( e ) ) e ~ ( v ) } dx
+
(ae~(~(~))e~(~) + ae~(u(~))e.~(~) +4~e~(~(~))~(v)}d~
e L (A + 2#)e33(u(e))eaa(v)dx
+~-g
-a(e) {/~ x(O~)~(e)~ d~+~~/ap~(~)v~dz+ s p~a(e)vadz} for all ('b, v) C V(e; h, t~).
To each eigensolution (A ~'*, ue,*), g _> 1, of problem P(S*), there corresponds a scaled e i g e n s o l u t i o n (Ae(e), (s ue(c))) of problem P(c; ~, 9t), the scaled e i g e n f u n c t i o n s (~te(e), ue(e)) and scaled e i g e n v a l u e s Ae(e) satisfying"
g u~g,e (x e) - s %(e)(x) and u g3'e (x ~) - euf(e)(x) for all x ~ E -~e
uf'e(x e) -- c'~f (c) (Y:) for all x ~ E (), A e'~ - if(e).
The
scaled
eigenfunctions
also satisfy
the
orthonormalization
194
[Ch. 2
J u n c t i o n s in linearly elastic m u l t i - s t r u c t u r e s
condition"
~ X(0~)tSg)(e)gf(e)d2 + e2 fa pu~(e)u~(e)dx + fa PUk3(e)ue3(c)dx- 5ke. k,g > 1.
m
The next convergence theorem is due to Bourquin & Ciarlet [1989]" It shows that, for each g >_ 1, the family (Ae(e), (~ie(e), ue(e)))~>0 (or perhaps only a subsequence) converges in the space 10. +oc[xHl(f~) x
Hl(f~) to
a limit that can be recovered from the g-th eigensolution of the "expected" eigenvalue problem. The notations used here are the
same as in Thm. 2.3-1. Theorem
(a) Define the space (the same as in Thin.
2.8-2.
2.3-1)-
[HI(~) x g3(W)]d "-- {('v,/]3 ) C H I ( ~ ) x H2(w); 7]3 -- Ou~3 --
0
on 70.
'V3I&,, -- ~31wa,
and consider the eigenvalue problem:
V~l~. --
0}.
Find all eigensolutions
(A, (~i, {3)) E]0, +c~[x[Hl(D) • Vz(w)]d of the variational equations"
+
+
f~{
4Art A~3A'r/3-l- 4, 0a,6~'30a.,5,T]3} dw 3(A + 2p) V
= A { ~/5g~?~ as: + 2 ~ pr
dw }
for all (~, r/3)r [Hi(a) x V3(w)]e. This problem has an infinite sequence of eigenvalues
Ae, g _> 1,
Sect. 2.8]
195
Eigenvalue problems
which can be arranged so as to satisfy 0 < A ~ < A2 < . . .
< A e < A e+l < . . .
and lim A e - +oc. s
oo
(b) For each integer g > 1, the family (Ae(e))~>o converges to A e as ~ ---+O. (c) If A e is a simple eigenvalue, there exists Co(g) > 0 such that Ae(c) is also a simple eigenvalue of problem 7)(~;~,ft) for all c <_ ~o(~), and there exists an eigenfunction (~i~(s), ue(e)) of problem P(c; f~,ft) associated with Ae(c), and normalized as in Thm. 2.8-1, that converges strongly in the space H I ( ~ ) x Hl(f~) as e --~ 0 to a limit (~te, u e) - (((re), (u e)), possessing the following properties: There exists a function ~3 E H2(a;) satisfying ~e _ O,;ea _ 0 on ~/o such that
X30c~3 and
ue3 - ~3e in a ,
and the pair (ft e, ~ea) is an eigenfunction of the eigenvalue problem defined in (a), associated with the eigenvalue A e. (d) /f A e is not a simple eigenvalue, any eigenfunction (~te, ~e3) of the eigenvalue problem defined in (a) associated with A e and satisfying f puiui - e e d ~ : + 2 f ~ P~3~3 e e dw - 1 (no summation with respect to t~) is the limit of an ad hoc subsequence of eigenfunctions of 79(e; ft, ft) associated with Ae(e).
(e) The e genf nction
obtained
(c)
(d)
a
complete set in both spaces [H~((~) x V3(c~)]d and L2(~) x L2(a;); furthermore, they satisfy ~ft~ui-e d~c + 2 ~ p(k3 ~e3 da; - 5ke, k, f. _> 1.
m
Note that each function u e - (uf) is a scaled Kirchhoff-Love displacement field inside Ft, of the same special form (the functions u e vanish for x3 = 0) as for the eigenproblem of a "single plate" (Thin. 1.13-2). We then define the sets Od, a; +, and a;d- as in Sect. 2.4 (see notably Fig. 2.4-1); we also define the d e - s c a l e d u n k n o w n s ~ = (~2~) :
196
[Ch. 2
Junctions in linearly elastic multi-structures
Od ~ R 3, (~ " ~ --+ R, and A~(0) through the de-scalings" u~-~ - - ~2~ in
Od,
r
e~3 in w,
A
~
(0)
-
A
We next describe the boundary value problem that is, at least formally, satisfied by the de-scaled unknowns (compare with Thm. 2.5-1(b), whose notations are used here). T h e o r e m 2.8-3. Let
m~z
9- - c 3 {
4.~a ~}
4A~tt~ 3()~~ + 2#~)A(~5~, + - ~
~z~3 9
The de-scaled unknown (A~(0), (~{, ~ ) ) satisfies the following coupled equations: - O ~ a ~ ( ~ { ) - A*(0)fY~ in Oa, -O~m~
- 2eh~(O)p~:~
+ Ext~
-a33(~t~)l~2 } in w,
(~ - 0 . ( ~ - 0 on 70, C
Trta/3(~)/Ya///3 -- 0 Oil ~/1,
(Oam~;3)u, + OT(ma~ aT~) -- 0 on ~1
-
~1%7
_
O,
where )~, rid are the Lamd constants and fi~ is the mass density of the material constituting the "three-dimensional" substructure, and )~, p~ are the Lamd constants and p~ is the mass density, of the material constituting the plate, m
Sect. 2.8]
Eigenvalue problems
197
A major conclusion is t h a t (A, (gt~, (~)) satisfies a coupled, multidimensional, eigenvalue problem posed over a subspace of HI(Oa) x H2(co), whose elements satisfy junction conditions along the twodimensional set COd. Furthermore, this problem is precisely the eigenvalue problem associated with the problem found in Thin. 2.5-1; in particular, the junction conditions are the same. The convergence obtained in Thm. 2.8-2 implies t h a t each limit vector field u = (u~) satisfies (for convenience, the superscript g is dropped) (~ := u~(.,0) = 0 in ~. Thus the de-scaled unknowns ~ 9 ~ --+ R defined by ~ " - e2(, in co (in accordance with the scalings u~(x ~) -- e2u~(x) for x ~ -- 7r~x C ~ ) satisfy
(~ - 0
in co,
to within the second order with respect to c. Therefore, the conditions ~ 1 ~ , - ~1~, - 0 may also be viewed as "true" junction conditions to within the first order with respect to e (since g~l~+ - ~l~Z - 0 by T h m . 2.8-2, the de-scaled functions %-~ vanish on cod to within this same order). Note in passing t h a t the conditions ~ = 0 in co, or their de-scaled counterparts ~ 0 in co, are in agreement with the conclusions reached in ~ Sectl 2.6; there, it was found t h a t applied forces with 1 horizontal components of order - were needed in the plate in order to produce non-zero limits {~ (here, the corresponding right-hand sides - p ~ A ~%~ are of order e) The b o u n d a r y value problem found in Thm. 2.8-3 may be equivalently formulated as a variational problem: Find all solutions (Ae(0), (~e, r E]0, q - o o [ • • Va(co)]d, w h e r e
[ H I ( O ) • V3(cO)]d "-- {('b, ?73) C H i ( O ) • H2(co); f/3 -- (~t,713 -- 0 on 70, ~)3[wa -- ~31wa, ~3c~[w,, -0},
198
Junctions in linearly elastic multi-structures
[Ch. 2
such that {)Vepp(g~)eqq(iJ) + 2ye~j(~)e~j(iJ)} dJc 3 ( ~ + 2p~)
---~--0c~'30c~T]3dw
for all (~, r/a)E [HI(O) x Va(w)]d. This de-scaled limit problem provides an example of a "stiff" (variational) eigenproblem, in the sense that different powers ore (respectively, 0 and 3) appear in front of the two bilinear forms found in the left-hand side, and that different powers of e (respectively, 0 and 1) appear in front of the two linear forms found in the right-hand side. Such stiff problems are studied in Panasenko [1980], SanchezPalencia [1980, Chap. 13], Sanchez-Hubert & Sanchez-Palencia [1989, Chap. 7], and Sanchez-Palencia [1992]. The numerical analysis of the eigenvalue problem found in Thm. 2.8-3 may be performed by methods adapted to its multi-dimensional character, such as modal synthesis by substructuring methods (see Destuynder [1989], Bourquin [1990, 1992], and Bourquin & d'Hennezel [1992]). An analogous asymptotic analysis has been performed by Lods [1996] on the same elastic multi-structure, under the same asymptotic assumptions inside the plate (A~ - e-3A, #~ _ c-3#, and p~ - c-lp), but under different assumptions inside the "three-dimensional" substructure, viz., ~
-
c - 2 - ~ and/2 ~ - c-2-~/2, ~ - ~-2-~r
for some 0 < s _< 1.
V. Lods then reaches the interesting conclusion that the eigenfunctions inside the "three-dimensional" substructure are in this case asymptotically negligible in comparison with those inside the plate. Note that this conclusion is in accordance with that reached in Sect. 2.7, where the assumptions on the Lam6 constants ~ and #~ were of the same form.
Time-dependent problems
Sect. 2.9]
199
This analysis is a first step towards a better understanding of "micro-vibrations", i.e., vibrations that are "localized" only in some parts of a large multi-structure (like a satellite for instance), and whose control is of paramount importance; see Ohayon [1992]. The present analysis has also been applied to eigenvalue problems arising in other multi-structures such as folded plates (Le Dret [1990b]), plates connected to a vibrating support (Campbell & Nazarov [1997]), and multi-structures comprising junctions between rods (Kerdid [1995]) or junctions between a three-dimensional structure and a one-dimensional string (Conca & Zuazua [1994]). 2.9 ~.
TIME-DEPENDENT
PROBLEMS
We consider again the same elastic multi-structure as in Sect. 2.1. The scalings of the unknowns and the assumptions on the data are the same as in Sect. 2.2, with obvious modifications (as in Sect. 1.14 for a "single plate") for taking into account their time-dependence. In addition, it is assumed that the mass densities of the three-dimensional substructure and of the plate respectively satisfy r ~ - fi and p~
-
c-1/9,
for some constants/5 > 0 and p > 0 that are independent of c. The next convergence theorem is due to Raoult [1992]. Its various statements should be self-explanatory as regards the notations eraployed; in particular, the notations are consistent with those of Thins. 1.14-2 and 2.3-1, with one exception: For notational conciseness, we have dropped the dependence on the variable t, which denotes the time, in the variational equations found in (c) and (d). Theorem L2(~•
2.9-1.
(a) Assume that for some time T > 0, j~i C
TD, fi e L2(ft•
T[), and ~Of, e L2(~ • ]0,T D. Then
(ft(c), u(~)) --, (ft, u) in L2(0, T; H~(~)
• Hl(~t))
a s r --~
0.
Junctions in linearly elastic multi-structures
200
[Ch. 2
(b) For all t E [0, T], the limit u(., t) E Hl(~t)is a scaled KirchhoffLove displacement field in ~, i.e., there exist functions ~ ( . , t ) C H i ( w ) and ~3(',t) C He(w), satisfying in addition ~ - 0~3 - 0 on ~0, such that ~ta(" , t) -- Ca(', t) -- X3(0a~3(" , t) and U3(" , t) -- r
(c) For all t E [0, T], the pair (~t(., t), r [Hl(fi) •
t).
t)) belongs to the space
V3(w)]d "-- {('o,/]3) E H I ( ~ ) • H2(w);
r/3 -
0~r/3 -
0 on
70,
v31~,, -
r/31~.,
v-I~.
-
0},
and (~t, ~3) satisfies the time-dependent variational equations:
d2 {fi ~ (tiOi dx + 2p ff ~3~73dw} -J-V
+ +
f {~,~(~)~.(~)+ 2~j(~)~,j(~)}d~
f~{
4Ap A~aA~a + 4# 0~30~r/3 } dw 3(A + 2#) --3
/~ ~,~,d~§ {/~11~ dx3)~3d~-/i {/~1 x3~ d~3}0~ d~ for all (~, r/a) C [Hi(a) x Va(w)]a,
0 < t < T,
where the initial data (~(. 0) ~~3("~0)) and (o~ -~(. ~o) ~~(., cot o)) a ~ ~xplicitly derived from the initial data of the original scaled three-dimensional problem. (d) For all t E [0, T], the function ~H(., t) "-- (~(-, t)) belongs to the space VH(~)
"-- {r/H -- (r/a) C H i ( w ) ;
r/H - - 0
on 7o},
Time-dependent problems
Sect. 2.9]
201
and it satisfies the variational equations:
L{ -- L
k + 2p e~~162
}
+ 4#e~9(r
d~
{I 11~ dx3 }r/~ da~ for all rlH -- (r]~) E Vg(a~)and 0 < t < T,
m We now write the time-dependent problem solved by the des c a l e d u n k n o w n ( ~ , ~ ) , both as a variational problem and as a boundary value problem. T h e o r e m 2.9-2. (a) For all t E [0, T], the de-scaled unknown (~K (., t), ~ (., t) ) belongs to the space [HI(0) • V3(a))]d "-- {(0,773) E H i ( 0 ) • H2(cd); r/a = O~,r/a = 0 on %, Oal~<, = r/al~, O~l~o,,= 0},
and (~t~, ~ ) satisfies the time-dependent variational equations: d2 { Y / 5 ~ d ~
+
+~s -
Io
+
L 4aria dw}
{A %p(u )%q(O)+ 2# eo(~K)eo(O)} dJc
4~~
4~~
~
}
3(~ + 2~)zx~zx~ + --~0~0~9~3 d~
f~O~ dJc +
}
f~ axe3 113da~ c
L{I
}
x; f; dx; Oc~T]3da~ c
for all (0, r/3) E [HI(O) x V3(W)]d, 0 < t < T,
whose initial data (~t~( 9, O) , ~ ( . , 0)) and ( -~(. ae~ , o) , -o~ 5 c (', 0 ) ) c a n b e e z plicitly obtained from the initial data of the original three-dimensional problem.
202
[Ch. 2
Junctions in linearly elastic multi-structures
(b) Let
/Tte
.__
__s
4M# ~
4# ~
~}
A smooth enough solution of the variational problem found in (a) is also a solution of the following coupled equations: 02~ e Ot 2
-* % ~ ( ~*~ ) - f~ in Od X]0, T[, Oj a~5(~)nj - 0 on {OOa - ~a} x [0, T],
2 e 0~3
~S ot~
e e O~em~e -
/__
e
I3e d.; +
+ Ext~
/.-: J__
e e x30~L dx;
- cr~3(,i~)1%7 } in ~•
T[,
~ - 0 . r - 0 on 70 x [0, T ] , m e,~ev,~v~ - 0 on 71 x [0, T], g
(0~m~)u~ + O,(m;~u~r~) - g31~+ -Ual~,7 - r
~;
- ~
g
g
x 3 f ~ dx~ u~ on 71 x [0, T],
for all t e [0, TI,
,~1~,7 _ 0 for all t E [0, T]
II The problem found in (b) is thus the time-dependent problem that is naturally associated with the coupled, pluri-dimensional, problem found in Thm. 2.5-1(b). It provides an example of a "stiff" timedependent problem, since different powers of c appear at different places in its equations (this is especially apparent in the variational formulation (a)). For a given e > 0, in which case all the data A~, ~ , t3~, M, >~, p~, and f[ are to be regarded as independent of e, this problem thus becomes amenable to the techniques developed by Lions [1973, 1981,
203
Exercises
1985] for expanding its solution as a power series with respect to ~. In this direction, see also Panasenko [1980] and Sanchez-Palencia [1989a, 1989b, 1992]. A similar analysis can be carried out for other elastic multistructures; see in this direction Le Dret [1994], who has studied the case of "multi-plate" structures. The controllability of multi-structures is a problem of outstanding practical interest, especially in aerospace engineering, where the stabilization of large multi-structures, such as space stations, is a crucial problem. The controllability of time-dependent problems such as that found in Thm. 2.9-2 can be studied by the Hilbert Uniqueness Method ("HUM") of Lions [1988a], already used for a "single plate" (Sect. 1.14); in this direction, see Nicaise [1990, 1993], Lagnese, Leugering & Schmidt [1994], Puel & Zuazua [19931, Burq [1994], and Glowinski & Lions [1995]. Note that in the present case, such analyses are further complicated by the "stilT' character of the limit time-dependent problem. EXERCISES
2.1. The result of this exercise is used in the proof (part (i)) of Thin. 2.3-1 and in the proof (part (i)) of Thm. 2.4-1. (1) Let ft be domain in R a, and let
W ' - - {W EHI(fl), e ( w ) - 0 } ={w;
w(x)-c+dAox,
zEft,
withc, dcR3};
cf. Vol. I, part (ii) of the proof of Thin. 6.3-4. Show that there exists a constant c > 0 such that inf I I v - wll~, ~ < cle(v)lo, ~ for all v C H~(ft).
wEW
This inequality implies that the semi-norm le(.)]0,a is a norm over the quotient space HI(~)/W, equivalent to the quotient norm (the other inequality clearly holds). Hint: One may proceed as in Duvaut-Lions [1972, Thin. 3.4, p. 117, or as in Ciarlet [1978, Thm. 3.1.1].
204
Junctions in linearly elastic multi-structures
[Ch. 2
(2) Let there be given a sequence of functions v k E Hl(f~) t h a t satisfies le(v~)10,~ ~ o ~s k ~ ~ . Using (1), show t h a t there exist vectors c k, d k E R a and functions w k E Hl(f~) such t h a t
v k(x)-c
k+d kAox+w
II~ ~111,~ ~ 0 ~
k
xEl2
k -~ ~ .
2.2. We follow here Le Dret [1991, Sect. 2.2], who has shown t h a t an a s y m p t o t i c analysis considerably simpler t h a n t h a t of Sect. 2.3, yet t h a t preserves most of its features, can be carried out on a "model multi-structure" formed by an L-shaped domain in the plane: For each c > 0, define the sets fie =]0, l[x]0, e[, F;-{1}
fie =]0, e[x]0, 1[,
x [0, e],
F0
S e - fie U fie,
[0, e] x {1}
let y~ - (y~) denote a generic point in S ~, and define the space
V ( S e) -- {v e E HX(Se);v e - 0 on C~ U f';}. Given f~ E L2(Se), the variational problem" Find u ~ E V ( S ~) such that
/~ cqy~ Ou~Ov~ Oy[~ dye - ] f ~~v
~ dy e for all v ~ E V(Se),
has a unique solution, which satisfies, at least formally, the b o u n d a r y value problem
-Au~-f~inf~ ~, u ~ - 0 o n G U ~ ; ,
G~-OonOS~-GUf';.
Remark. The notations are on purpose reminiscent of, but should not be confused with, those in the text; in particular, ft ~, f~, S ~, ~, and ~ now denote two-dimensional domains. (1) Let t be any number < - 1 (its only purpose is to insure for "visual confort" t h a t ~ N { ~ } - - 0). Define the sets
=]0, l[• Co-
t + 1[,
{1} • [0,1],
h =]t, t + l[x]O, I[, ro-
[o, 1] • {1}.
Exercises
205
With each point y - (yl, y2) c f~, associate the point rr~y - (yl, gy2) E ~ , and with each point ~) - (~)l,y2) E {~)}-, associate the point # ~ Y - (@1, Y2) C { ~ } - . With the functions u ~, v ~" S ~ ' R , associate the functions u(e), v" a --+ R and g(e), 5" fi ---, R defined by
u~(y ~) - u(c)(y) and v~(y ~) - v(y) for all y~ - 7r~y e ~ , ue(y ~) - ~2(e)(9) and ve(y ~) - 5(~)) for all ye - #e# E {~e}-. Finally, assume that, for c small enough, there exist functions f E L2(f~) and f C L2(~)) independent of e such that
f f ( y ~ ) - f ( y ) for all y~ - 7c~y E fV, -
for
-
(this implies that, for r small enough, f ( y ) f(~) for all points y, corresponding to the same points y~ E ~ Cl ~ ) . Write the variational problem satisfied by the "scaled unknown" (g(e), u(e)) over the space V(c; ~r.~,~,_~) . _ {(v,v) E HI(~'~) x H I ( ~ ) ; v - 0 on F0, v - 0 on F0,
ga(~l) - v(y) at all points ~ E ~, y E f~ such that #~) - rr~y E ~ Cqf~ }. (2) Show that the n o r m [[U(C)III,~ and Ilu(~)lll,~ are bounded independently of e. Consequently, there exists a subsequence, still indexed by e for convenience, such that u(c) --~ ~ in Hi(D) and u(c) --~ u in Hl(f~) as c ---+O. (3) Show that ~ is independent of Yl and can therefore be identified with a function ~ E H I ( 0 , 1) and that likewise, u is independent of Y2 and can therefore be identified with a function ( E Hi(0, 1). In addition, show that ~(1) - ~(1) - 0. (4) Show that ((, () satisfies the "junction condition" q(o).
206
[Ch. 2
Junctions in linearly elastic multi-structures
Consequently (~, ~) belongs to the space V "- {(~, r/) E HI(0, 1) x Hi(0, 1); ~(1) - ~7(1) - 0, ~(0) - r/(0)}. (5) Show that (~, C) C V satisfies the variational equations
/o
1 ~'~' dY2 +
dyl -
]i
/5~ d92 -+-
ill
pr/dyl
for all (~, r/) E V, where t5 "- f01 f dyl and p "- fl f dy2. (6) Show that the variational problem of (5) has a unique solution. (7) Show that the whole family (~(e), u(e))~>0 strongly converges to (g, u ) i n Hl(ft) x Hi(a). (8) Show that (~, () satisfies, at least formally, the following "coupled" boundary value problem" -~"-
io in ]0, 1[ and - ~ " - p
in ]0, 1[,
~ ( 1 ) - ~(1)--0, ~ ( 0 ) - C(O) and ~ ' ( 0 ) - -~'(0).
CHAPTER 3
L I N E A R L Y E L A S T I C S H A L L O W S H E L L S IN CARTESIAN COORDINATES
INTRODUCTION In this chapter, we mathematically justify the two-dimensional theory of a linearly elastic shallow shell in Cartesian coordinates, by establishing as in Chap. 1 the convergence, as the thickness of the shell approaches zero, of the three-dimensional displacements (once appropriately scaled) to a limit obtained by solving two-dimensional equations that are classically those of a shallow shell. A noteworthy feature of our approach is that it also provides a rigorous definition of a "shallow" shell. Since shells constitute the central theme of Vol. III, it may seem peculiar that we already study some of these in this volume, devoted to plates. The reason is simple: It turns out that when the equations of shallow shells are expressed in terms of Cartesian coordinates, their asymptotic analysis closely follows that of a plate (the same remark applies to nonlinearly elastic shallow shells; cf. Sects. 4.14 and 5.12). It is only when their equations are expressed in terms of curvilinear coordinates that their analysis properly belongs to Vol. III. Although the asymptotic analysis that we use is indeed reminiscent of that used in Chap. 1 for a linearly elastic plate, its adaptation to the "geometry" of a shallow shell gives rise to a substantial amount of additional difficulties (cf. notably Sects. 3.3 and 3.4), which include in particular the proof of a crucial generalized Korn's inequality (Thin. 3.4-1). Let us now briefly outline the content of this 3.1, we consider a three-dimensional linearly elastic its reference configuration the set { ~ } - , where ~ a J x ] - c, c[, a~ is a domain in R 2, and the mapping
chapter. In Sect. shell occupying in - O~(fV), f~ = O~: {ft~} - ~ R 3
Linearly elastic shallow shells in Cartesian coordinates
208
[Ch. 3
is given by O e ( x e) -- ( X l , X 2 , 0 e ( X l , X 2 ) ) - t - x ~ a ~ ( x l , X 2 )
for all z ~ - (ml, m2, z3) E ~ , where a ~3 is a unit normal vector to the middle surface O~(~) of the shell (Fig. 3.1-1). The shell is subjected to applied body forces of density ( f I ) " ~ -+ R 3 in its interior ~ ; for simplicity, we assume in this introduction t h a t the applied surface forces acting on the "upper" and "lower" faces of the shell vanish. The shell is clamped along a portion F~ of its lateral face, where F~ - O~(F~), F~ - "Y0 • [-c, c], and 70 C " y - &o. Note t h a t if the function 0 ~ vanishes in ~, the shell becomes a plate. Let ~ and #~ denote the Lam~ constants of the elastic material constituting the shell. Then in linearized elasticity, the displacement vector field/t ~ - ( ~ ) " { ~ } - ~ R 3 satisfies ( ~ j ( / t ~) denote the linearized strains): ^
{t s E H I ( ~ e ) ,
li e - - 0 on I'0,
epp(~te)eqq(~ e) + 2#e~ij(~te)eij(~e)} d2 ~
L
f[v~ d2 ~
for all 9~ - ( 9 C H ~ ( h ~) t h a t vanish on f';. We then make the following scalings on the unknowns and assumptions on the data" We let f~ - a ; x ] - 1, 1[; we define the scaled displacement u(e) - (u~(e)) 9 f~ ---, R a by letting ~2~(a?~) -- c2u~(c)(w)
and
~(~e)
__ CU3(C)(X)
for all 5< - Oe(Trex), x E f t , where 7re(Xl,X2, X3) -- ( X l , X 2 , CX3); w e then assume t h a t there exist constants ~ > 0 and # > 0, functions f~ C L2(ft) independent of e, and a function 0 E Ca(~) independent of c such t h a t m
~-~ -
2f~(x)
and
and
#~-#,
f~(~?~) - c3fa(x) for all ~ - O~(Tc~x) E D ~,
O~(xl, x2) - cO(Xl, x2)
for all (Xl, x2) E ~.
209
Introduction
In this fashion, the scaled displacement u(c) satisfies a set of variational equations of the form" 1 1 e--~B_4(u(e), v) + -%B_2(u(e), v) + Bo(e; u(e), v) - L(e; v) for all v E V(ft), where the bilinear forms B_4 and B_2 are independent of c, the bilinear form Bo and linear form L are "of order 0 with respect to e", and the space V(ft) is given by V(f~)-
{V -- (Vi) E HI(f~); v - 0
on ro},
where F0 = % x [-1, 1]. Our main result (Thm. 3.5-1) then consists in showing that the scaled displacements u(e) converge in the space Hl(f~) as e + 0 to a limit u that can be computed from the solution of a two-dimensional problem; more specifically: (i) The limit vector field u = (u~) E H I(Q) is a (scaled) KirchhoffLove displacement field: The function u3 is independent of x3 and as such can be identified with a function r E H2(w) satisfying ~3 = (0.(3 = 0 on %; the functions u~ are of the form u~ = (~ - x30~3 with functions (~ E HI(w) satisfying r = 0 on %. (ii) The vector field (~ = (~) E Hi(w) x H~(w) x H2(w) satisfies the variational problem:
r
(r
~ V(CU)- {T~- (7]i) ~ sl(cd) X HI(cu) X H2(CU); 1]i- 0u713 - - 0 on
%},
1
- ~ pi rli dw - ~ q~oq~~73 d w for all rl E V(w), where a~T=
41# ~5~96,,I+2#
+ 2~t(5~,5~_ + 5~.5~,),
210
/_1
/1
Linearly elastic shallow shells in Cartesian coordinates
P~ -
1
fidx3,
q~ -
1
[Ch. 3
xaf~ dx3.
Equivalently~ the field ~ satisfies, at least formally, the following twodimensional boundary value problem:
-O~m~
- O~(ft~O~O) - P3 + O~q~ -O~g~
-
p~
~-O~a-O
in w,
i n a~,
on 70,
m ~ z p ~ z - 0 on 71, ~ZpZ where 71 - 7 - 7o and
- - 0 o n 3'1.
{
4.
3(~ + 2~) 4A# % , = ~ + 21----5~(r
-2
"~ - -
1
~(~) - ~(o~
+ o~)
} '
+ 4~~(~), 1 + ~(o~0o~
+ o~0o~3).
We then show that, after appropriate de-scalings, the problem satisfied by ~ is a known two-dimensional model for linearly elastic shallow shells (Thins. 3.7-1 and 3.7-2). As shown in the concluding discussion (Sect. 3.8), a noteworthy virtue of the present approach is that it clearly delineates the conditions under which a three-dimensional shell problem may be replaced by the limit two-dimensional problem found here. In particular, a major conclusion is that the function 0 ~ that describes the shape of the middle surface of the shell in its reference configuration should be of the order of e, i.e., of the thickness of the shell" This condition thus provides a rigorous definition of "shallowness". We shall see that it is precisely the same definition that gives rise to two-dimensional shallow shell equations in the nonlinear case (Sects. 4.14 and 5.12), or when the equations are expressed in terms of curvilinear coordinates (Vol. III).
The three-dimensional equations
Sect. 3.1]
211
THE THREE-DIMENSIONAL E Q U A T I O N S OF A L I N E A R L Y E L A S T I C C L A M P E D S H E L L IN CARTESIAN COORDINATES
3.1.
Let (e,) denote the basis of the Euclidean space R a, and let co be a d o m a i n in ttie plane s p a n n e d by the vectors e~. For each c > O, we define the sets
~ .- ~•
~, ~[,
r+
-
~
•
{~}
r ~ .-
~
•
{-~}
denote the generic point of the set ~ , and we let 0~ - 0~ "- 0 / 0 x ~ and c~ " - O/Ox~. We assume t h a t for each c > 0, we are giv/m a function 0 ~ C C 3 (~); we t h e n define the surface (Fig. 3.1-1)"
we let x e "-
(x~) -
~ -
(Xl, x2, x~)
{(~1, x~, 0~(x~, ~ ) ) e R~; (x~, ~ ) e ~ }
At each point of the surface {&~}-, the vector
a~ "- { ~ e } - l / 2 ( - a x 0 r
r 1),
where a ~ .-1010~l 2 + 1020~12 + 1, satisfies [ a ~ ] - 1 and is normal to {&~}-. For each e > 0, we define the m a p p i n g O ~ - (O~) 9~ ---, R a by letting
Or
e
e
e) "-- (Xl,X2,0~(Xl,X2))+ x3a3(xl,x2)
for all x e E
~e
.
We shall assume that, for all values of e > 0 subsequently considered, the mapping 0 ~ 9 --~ 0 ~(-~) is a Cl-diffeomorphism, i.e., t h a t O ~ is an injective m a p p i n g of class ~1 with an inverse m a p p i n g also of class (j1. For the class of m a p p i n g 0 ~ t h a t we shall allow later (Sect. 3.2), this a s s u m p t i o n can be rigorously justified if c is small enough; cf. Ex. 3.1. This assumption implies in particular t h a t the set (Fig. 3.1-1)
~ ._o~(~)
[Ch. 3
Linearly elastic shallow shells in Cartesian coordinates
212
I . .~176 ..
2
,: .... _~
J "~~
2 9g0
-"I--"
"~ .-,,.D ,~,.~
s -g I "~
P
Fig. 3.1-1" A three-dimensional shell problem. T h e set {fi~}- = O ~ ( ~ ) , w h e r e f2 ~ = ~ x ] - e , c[ a n d co C • 2, is the reference configuration of a shell,^ w i t h thickness 2~ a n d m i d d l e surface {&~}- - O ~ ( ~ ) , c l a m p e d on the p o r t i o n F~ = O~(F~)) of its lateral face, w h e r e F~) = 70 x [ - e , e ] a n d 70 C 0 ~ . Each point 2~ - (27) of {(2~} - is the image O ~ ( x ~) of the point x ~ - (x~) C f2 ~, which is itself t h e image 7Wx = (Xl,X2,Cx3) of the point x = (x~) C f2. In this fashion,_ a bijection is e s t a b l i s h e d for each c > 0 b e t w e e n t h e set { f ~ } - and the set f~. T h e set f~ does n o t d e p e n d on c (for a b e t t e r r e p r e s e n t a t i o n , a "cut" has b e e n m a d e in t h e sets (2 ~ , ft ~ , a n d f2).
The three-dimensional equations
Sect. 3.1]
213
is open and t h a t {~)~}- - O ~ ( ~ ) . We let 2~ - (2~) denote the generic point of the set {~)~}-, and we let c9[ - 0/02~. For each s > 0, the set {fi~}- is the r e f e r e n c e c o n f i g u r a t i o n of an elastic b o d y with Lam6 constants 1 ~ > 0 and #~ > 0. Because the p a r a m e t e r s is t h o u g h t of as being "small" compared to the dimensions of the set co, the elastic b o d y is called a shell, with thickness 28 and middle surface {&~}- := O ~(~).
Remarks. (1) It is only later (Sect. 3.8) t h a t we shall be able to give the definition of a "shallow" shell. (2) Since s is a dimensionless parameter, the thickness of the shell (as t h a t of a plate) should be written as 2sh, where h is the unit length. To save a notation, we let h = 1. m Let % be a subset of the b o u n d a r y "y of co, with
length % > 0 , and let F~) " - O~(P~),
where
F~ "- % x [ - s , s ] .
T h e unknown is the displacement f i e l d / { - ( ~ ) 9{ ~ } - + IRa, ^~ where the functions u i 9 { ~ } + R represent the Cartesian corn^~ (2~ )e~ is the displacement of the ponents of the displacement, i.e. , u~ point :~ - 0 ~(x ~) e {fi~ }-. T h e displacement is further assumed to satisfy a b o u n d a r y c o n d i t i o n o f p l a c e / { - 0 on F~. T h e n in linearized elasticity, /t ~ is the solution of the variational equations:
d}
-
f~ v~ d2 ~ +
9~ v~ dF ~ for a l l / ~ E V(~)~),
214
Linearly elastic shallow shells in Cartesian coordinates
where the space V ( ~ )
[Ch. 3
is defined by
V ( ~ c) - {/J~ - (~)~) E Hl(~c); /~ - 0 on F~)}, ^e 1 ^~ ^e ^~ ^~ where e~j(iJ ~) - -~(~ vj + 0jv~) denote the components of the linearized strain tensor, and where, for each c > O, the vector field (]~) e L2(~ ~) represents the given applied body force density acting in the interior ~ of the shell, and the vector field (t~) C L2(F~_ U F~) represents the given applied surface force density acting on the upper and lower faces of the shell, respectively defined by
.-
o
.-
o
(r
,
and dF ~ denotes the area element along the boundary of ~ . Note that the applied forces are also defined by their Cartesian components. These equations form a variational problem 7")(~), which has one and only one solution/t ~ (by Korn's inequality with boundary conditions applied to the functions/~ E V(~t~); cf. Thin. 1.1-2).
Remark. This solution can also be characterized as the unique solution of the minimization problem: Find / t ~ e V ( ~ ~)
such that
J~(/t~) -
inf J~(O~), ~v(a~)
where 1
{A~pp(/J~)~qq(~) ~) + 2#~a~j (/I)a~j (/~)} d2 ~ -
fi vi dk~ +
$u~
gi v~
dF ~
The function/t ~ is, at least formally, solution of a classical boundary value problem of three-dimensional linearized elasticity, which takes here the form
Sect. 3.2]
215
Fundamental scalings and assumptions epp(~te)(3ij + 2/te(~ij(~e)} -- f [ in ~ ,
--
u~ - 0 ~^
^~
on
~;,
I" gi on F+ U F_,
where ( ~ ) denotes the unit outer normal vector along the b o u n d a r y of the set ~t~. I
3.2.
TRANSFORMATION INTO A PROBLEM POSED OVER A DOMAIN INDEPENDENT OF r THE FUNDAMENTAL SCALINGS OF THE UNKNOWNS AND ASSUMPTIONS ON THE DATA
We describe in this and the next sections the basic preliminaries of the asymptotic analysis of an elastic "shallow" shell, as set forth in Ciarlet & P a u m i e r [1986] in the nonlinear case and Ciarlet & Miara [1992] in the linear case. To begin with, we let (Fig. 3.1-1)
ft - a~z] - 1, 1[, F+ - co x {1}, F_ - co • { - 1 } , Fo - 70 x [ - c , c ] ,
and with each point x C ~t, we associate the point x ~ E the bijection
~'x-
(x~) c f~
,
~-(~)-
through
( ~ , x~, ~ 3 ) z
W i t h the f u n c t i o n s / t ~,/J~ C V ( ~ ) , we then associate the s c a l e d displacement field u(c) = (u~(c)) and t h e s c a l e d f u n c t i o n s
216
Linearly elastic shallow shells in Cartesian coordinates
[Ch. 3
v = (v~) defined by the s c a l i n g s
~t~(Jc~) - e2u~(e)(x) and 5~(2~) - e2v~(x), ft~(~c~) = eua(e)(x) and ~)3(5:~) = eva(~c), for all 5:~ - O~(Tr~x) e { ~ } - . Finally, we make the following crucial a s s u m p t i o n s on the d a t a : There exist constants A > 0 and # > 0, and functions f~ C L2(~), g~ c L2(F+ u r_), and 0 C C3(~) t h a t are all independent of e, such t h a t
A~=A / ~ ( ~ ) - e~/~(x) a n d / ~ ( ~ )
and
#~=#,
- ea/s(x) for all ~ - O ~ ( ~ x ) 6 fi~,
gc, (a~e)^e -- esgc~(X) and g3(:~ e)^e
__ e 4 g 3 ( X )
for all a?~ - O~(Tr~x) C f'+ U F~., O~(Xl,X2) = eO(xl,x2) for all (Xl,X2) E ~. While other assumptions are possible on the L a m d constants and the applied force densities as in the case of a plate, we shall see in Sect. 3.8 that, by contrast, the assumption t h a t "the function 0 ~ is O(e)", which plays a crucial rhle in the definition of "shallowness", is ne varietur. Taking these scalings and assumptions into account, we next wish to transform the variational problem 7)(~ ~) of Sect. 3.1 into an equivalent variational problem posed over the set ft. To this end, we first transform P ( ~ ) into a problem posed over the set ft ~ (Thm. 3.2-1). Since the mappings O ~ 9~ ~ { ~ } - are assumed to be ( ] 1 _ diffeomorphisms, the correspondence t h a t associates with any function /J~ defined over the set { ~ } - the function 9~ 9~ --, R defined by
~(~)
- ~ ( x ~)
for ~u
induces a bijection between the
~ - O ( x ~) e
{fi~}-
spaces HI(~ ~) and HI(~ e) ( A d a m s
Fundamental scalings and assumptions
Sect. 3.2]
217
[1975, T h m . 3.35]), hence also between the spaces V ( f i ~) and V(f~ ~) := {9~ = (9~) E H~(f~); ~ = 0 on P~}. For each e > 0 and each x ~ E ft ~, let V ~O ~(x ~) denote the Jacobian m a t r i x ((~O~(x~)) and let
bi3(x ~) . - ( { V e O r 1 6 2
j
5~(x ~) := det {V~O~(x~)}
for all x ~ E
,
for all x ~ E ~ .
We also assume that, for all values of e > 0 considered, the mappings O ~ are orientation preserving, i.e., t h a t 5~(x ~ ) > 0
for all
x~E~.
Again, this is not a restriction in the present case (Thm. Using the formulas
3.3-1).
-
and the formula
df.~
_
~e{b~ib~i}l/2 d r ~
t h a t relates the area elements dF ~ along 0 ~ ~ and d r ~ along Oft ~, we easily obtain"
Let there be given an orientation-preserving C 1diffeomorphism 0 ~ " ~ ---+{(~ }-. Then the field s - (g~) E V(f~ ~) defined by Theorem
3.2-1.
g~(x ~) " - g ~ ( ~ )
for all
~-O~(w
~) C {f)~}-,
satisfies the variational equations: {)Vb~vO~ft;Sij + >~ (b~yOkft~ + b{iOkft~) }b~jS~Okg~ dx ~ i vi o {b3ib3i ~ ~ }1/2 d [ , ~ - fa f [ o~5~ dx~ + fr ~ur~ ~:~c~ for all ~ E V(ft~),
218
Linearly elastic shallow shells in Cartesian coordinates
[Ch. 3
where the f u n c t i o n s bi~ and 5 ~ are defined supra, and the f u n c t i o n s f [ " f ~ ~ R and (]~ " F+ U F~ ~ R are defined by
f:(x
. - ]:
for
~ ( x ~)'-t)~(:~ ~)
for all
-
~-O~(x
o
e
~) EF~_UF~ . i
Using the scalings on the displacements and the assumptions on the data, we can thus reformulate the variational problem P ( ~ ) as a variational problem P(c; f~) posed over the set f~. This problem takes the form" u(s) E V ( f ~ ) " - (v - (v~) E Hl(f~); v - 0 on F0),
~B-41(U(C),v ) +
1 7 B_2(u(s), v ) + B0(s; u(s), v) - L(s; v) for all v E V(ft),
B-4
where the bilinear forms and B_2 are independent of s and the bilinear form B0(s;., .) and linear form L(s; .) are "of order zero with respect to s", i.e., they do not contain any negative power of ~. R e m a r k . We postpone until part (iv) of the proof of the convergence theorem (Thm. 3.5-1) the explicit display of the somewhat complicated variational equations of problem 7)(s, ft), for they are not needed before. I ^~
.
A w o r d of c a u t i o n : We emphasize that the functions u~ -~ . ~: { ~ } - --~ R and u~ --~ ] ~ 3 represent here the Cartesian com^~ --~ ponents of the displacement, i.e., u~ (Jc~)e~ - u~ (x~)e~ is the displacement of the point 5:~ - O~(x ~) E { ~ } - ; the functions f/~" ~ ---+ R, t)~" f'~-tAI'~ ---, R and ] : " f~ --~ R, ~ " F~_tAF~ ---, R likewise represent the Cartesian components of the applied body and surface forces. As such, these are to be carefully distinguished from the covariant components u~ ~ R of the displacement and the contravariant components fi,~ . f~ __, R and g~'~ 9F~_ t2 F~ ---, R of the applied body and
Sect. 3.3]
219
Technical preliminaries
surface forces used in Vol. III, where shell equations are expressed in curvilinear coordinates. There, the displacement g~(2~)e~- g~(z~)e~ of the point 2~ - O~(x ~) C {~)~}- is expressed as u~(x~)g~'~(x~), the vectors 9~,~(z ~) forming the contravariant basis at the point 2~. The notations %, ^~ f[, gi ^~ and %, -~ f[, ~ have been chosen precisely in order to avoid possible ambiguities arising from these two essentially distinct choices of coordinates. II ^
3.3.
TECHNICAL
PRELIMINARIES
We needed
gather in the next two theorems various results that will be in the proof of convergence. In what follows, x - (xi) denotes a generic point in the set ~, and we let c9i - O/Oxi, c9~ - 02/Ox~Ox~.
For notational conciseness, we also suppress any explicit dependence on the function 0, but it should be clear however that remainders such as b~(e), a#(e), etc., or constants such as Co, C1 (in the next theorem), etc., do depend on O. Theorem
3.3-1. Let the function 0 ~ be such that
Oe(Xl: X2) -- gO(Xl, X2)
for all
(Xl, X2) e ~,
where 0 E C2(~) is independent of e. Then there exists eo - eo(O) > 0 such that the Jacobian matrix V~O~(x ~) is invertible for all x ~ C - ~ and all e <_ eo. For e <_ go, let the functions b i j ( e ) " f~ --+ R and 5 (e) 9ft ---, R be defined by:
for all
bij(c)(x) "-- bi~(x e)
5(c)(x) "-- 5e(x ~) where
b{~ -- ( { v e o e } - l ) i
for all
Xe Xe
-
-
-
-
7reX C 7reX E
j and 5 ~ -- det V~O ~. Then
b ~ ( e ) - 6 ~ + e2b~#(e), bao(e) - - e { 0 ~ 0 + e2b#~(e)},
-
+
baa(e) - 1 + e2b#a(e),
5(e) -- 1 + e25#(e),
220
Linearly elastic shallow shells in Cartesian coordinates
[Ch. 3
and there exists a c o n s t a n t Co such that sup maxma_x]b~(e)(x)l _< Co, 0<e_<eo z,3 xEf~ sup m ~ 0<e<_eo xE~
[5# (e) (x) l < Co.
P r o @ Define the matrix F(e)(x)
- (F~j(e)(x))'-
V~O~(x ~) for all x ~ - 7r~x E ~ .
Then a simple computation gives (for notational brevity, the variables xl and x2 are omitted)" 1 {ct (c) }_3/20ql oL(c) 010), -- 1 - c 2 x 3 ( { c t ( E ) } - l / 2 0 1 1 0 - -~
fll(C)(x)
c2 F~2(e)(x) - -~xa{c~(e)}-a/2(2c~(e)0120 - 02c~(e)010), _ ~ 12
F13(C)(x) -- --g{Ct(C)} " 010 , E-2 F21(C)(x) ---~-X3{(Y(E')}-3/2(2OL(~')0120 -
-
-
-
01OL(~')020),
C2 F22(e)(x) - 1 -- -~Xa{C~(e) }-a/Z(2c~(e)0220 - 02o~(e)020),
F z a ( e ) ( x ) - -e{c~(e)}-1/2020 , C F3l ( e ) ( x ) - eO, O - -~x3{o~(e) }-3/20,c~(e), -
F33(e)(x)-
co
o-
e
--3/
{c~(a)} -~/2,
where o<(e) "-- 1 +
(IOlOl + IO O1 ).
For a fixed (Xl, X2) ~ ~, an application of the mean value theorem ( f ( e ) - f(O) + e f ' ( t e ) for some 0 < t < 1) to the function f defined by f(e) - {c~(e)}-a, a > 0, shows that, for some t El0, 1[,
221
Technical preliminaries
Sect. 3.3]
{c~(e)} -~ - 1 - c 2
2at(]OlOI 2 + 10~01~) {1 + t2c2(101012 + 102012)}a+l
= 1 + e2r~(e), with sup m a x Ira (e) [ < +oc. O<e xCf~
Combining this last result with the expressions of the functions Fij(e), we infer t h a t for all e > 0 and all x C ft,
with F 1 --
/00 0 o1:/ 0
010
-02
02 0
F2 -
0
supmax O<e
,
xEf~
IIF#(c)(x)ll
<
•
•
0
0
0/ 0
x
+~,
where [[. [I denotes a fixed (but otherwise arbitrary) m a t r i x norm satisfying IIABII _< IIAII IIBII. Not~ t h a t only the location of the nonzero elements (simply indicated by "x" for this reason) of the m a t r i x F2 needs to be known for the ensuing analysis. We next show t h a t there exists e0 = e0(0) > 0 such t h a t the m a t r i x F(e)(x) is invertible for all e <_ e0 and all x E f t , t h a t {F(c)(x)}
-1 -
I - gFl(X) + ~2({F1(x)}2 -
F2(x)) + eaR#(c)(x),
where the matrices F1 and F2 are those appearing in the expansion of F ( e ) , and t h a t sup m a x ]]R # (c)(x)[[ < -t-oc,
0<~<_eo xEft
sup m a x [ ~ # ( g ) ( x ) l < +co.
O<e<_eo xEft
To this end, we recall t h a t if a m a t r i x A satisfies IIAll < 1, the m a t r i x ( I + A) is invertible, and
II(I + A ) -~ - { I -
A +
A2}II _<
IIAll ~ 1-
IIall "
222
Linearly elastic shallow shells in Cartesian coordinates
[Ch. 3
Let r be given such that 0 < r < 1. The expansion of F ( c ) ( x ) shows that there exists e0 > 0 such t h a t sup max IleFl(X)+
O<e<_eo xEgt
e2F2(x)+e3F#
(e) (x) [[ < r.
Hence for all e < e0 and all x E ~, the matrix F ( e ) ( x ) is invertible, and II{F(e)(x)} -1 - I + { C F l ( X ) +
c2F2(x) + e a F # ( e ) ( x ) }
- { ~ f l (~) + ~ F ~ ( x ) + ~ F # (~)(~)}~11
_< (1 - r)-l[IgFl(X)nu g2F2(x ) ~- eaF#(e)(x)ll ~ The asserted properties of the expansion of {F(c)(x)} -1 follow from this last inequality. It is then easily verified t h a t the location of the nonzero elements of the matrix {Fl(X)} 2 is the same as that of those of the m a t r i x F2(x). Hence the terms of order e 2 in the expansions of the functions b~a(e) and bao(e) vanish, and therefore all relations are proved. II We recall t h a t [ . [0,a and II" [ll,fl denote the L2(f/)-norm and H l ( f / ) - n o r m respectively, for both scalar-valued and vector-valued functions. T h e o r e m 3.3-2. Let the assumptions and notations be as in Thm. 3.3-1, and let the f u n c t i o n s / ~ - (~?~) C HI(~) ~) be related to the functions v - (v~) e Hl(f~) through the relations
9
e) - ~2v~(x) and 9
for all ~ - O~(Tr~x) e { ~ } - .
_ CV3(X)
Then
^~ ^~
"E ^~
0 ~ ( ~ ~) o ~ ( ~ ~) -
c{0~
- 0~00~
~{O~v~ + ~
o~,~(~ ~) - {o~v~ +
~(~;
+ e~#~(c;
~)}(~),
A generalized Korn inequality
Sect. 3.4]
223
at all corresponding points Jc~ E {D~} - and x E ~, and there exists a constant C~ such that sup
0<e_<eo
max lr~(c;v)10,~ ~< CIlIVlII,~ for all v C Hl(f~). z,3
Consequently,
a ~ , ( , ~ ) ( ~ ) - ~ : { ~ , ( v ) + ~ : ~#, (~; . ) ) ( ~ ) ~ ; 3 ( ~ ) ( ~ ~) - ~ { ~ 3 ( - ) + ~:~3(~; .)}(x), ~(~)(~)
- ~(~3~(v)+ ~3#~(~; v))(x),
~;3(~)(~ ~) - {~33(-)+ ~:~3~3(~; ~)}(x) = {g33(v)+ c2(0~0c3~v3 + be33(c)O3v3) +
c4e~33(c; V)}(X),
where
~ , ( v ) . - -1~ ( o ~ + o ~ )
1 (O, O03v~ + c3~OOav,), - -~
1 1 ~ 3 ( v ) - ~3~(v) "- -~(O~v3 + 03v~) - -~0~003v3,
~(~)
. - O~v~,
a n d there exists a constant C2 such that sup max le~(g; v)10, a _< c211Vlll,ft for all v c Hl(f~),
0<e_<eo
z,y
sup le~33(c;v)lo,~ ___C211Vlll,~e for all v c H I ( a ) .
0<e<__eo
Pro@ It suffices to combine the relations
o ~ ( ~ ) - o ~ (x ~ with the definition of the functions v~ and the asymptotic behavior of the functions bij(e) described in Thm. 3.3-1. m
3.4.
A GENERALIZED
KORN INEQUALITY
Following Ciarlet & Miara [1992, Lemma 3], we establish an inequality that plays a crucial r61e in establishing the uniform (with
224
Linearly elastic shallow shells in Cartesian coordinates
[Ch. 3
respect to c > 0) boundedness of the norms Ilu(g)lll,fl in the proof of the convergence theorem (Thm. 3.5-1). Note that when 0 = 0, the functions a~j(v) appearing in this inequality reduce to the functions e~j(v) appearing in the "classical" Korn inequality with boundary condition (Thm. 1.1-2). Theorem that
3.4-1.
Let f~ = a ~ •
1, 1[ and let 7o C Oa~ be such
length 7o > 0 Given v -- (vi) E H 1(~)~ let 1
~ j ( v ) . - -~ ( oj ~ + o ~ ~ ) , let the functions ~ j ( v ) = ~3~(v) be defined as in Thm. 3.3-2, i.e., 1 1 ~ ( ~ ) - ~ ( v ) - ~o~0o~,
~ ( v ) := ~ ( ~ ) , let the space V ( f t ) be defined by
V ( ~ ) = {v = (vi) E n l ( ~ ) ; v - - 0 on Fo}, and assume that 0 E C3(-~). Then there exists a constant C such that the following g e n e r a l i z e d K o r n i n e q u a l i t y holds:
1/2
[[V[]l,~ ~
C{~-~.. [eij(v)]2o,~}
for all v C V ( ~ ) .
Sect. 3.4]
A generalized Korn
225
inequality
Proof. For clarity, the proof is divided into four parts.
(i) Let the space E(f~) be defined as E(f~) "- { v -
(v~) E L2(f~); g~j(v) E L2(f~)}.
Then
E(gt) - H i ( a ) . Note that in the definition of E(f~), the relations "g~j(v) E L2(f~) '' are to be understood in the sense of distributions. For instance, the relations "g~z(v) C L2(a) '' mean that for each (c~, ~), there exists a function in L2(ft), denoted g~z(v), such that g~z(v)p dx - ~
v~ { - 0 z p + 03(~0z0) } dx + -~
vg{-O~ ~ + cga(~0~0)} dx
for all qp E D(g2) (only the assumption "0 E C2(-g) '' is needed here.) Let then v = (v~) be a function in the space E ( a ) . We show that each one of the second partial derivatives Ojkv~ (in the sense of distributions) can be written as a linear combination of distributions in H-l(f~) (the assumption "0 E Ca(~) '' is needed here, to guarantee that products such as O~OOav~ remain in H - l ( f t ) ) . To this end, we generalize the well-known relations
as follows (for brevity, we let g~j "- g~j(v))"
0c~3V3- 0c~e33, 033v~ - 203g~3 +
O~O0393a - 0~g33,
O~v3 - 2 0 ~ 3 + O~Oe3a + 0~00~33 - 0 3 ~
- O~OOaav~
(no summation), (Oc~3 V c~ - - (~3 ~ c~c~ + (~c~0 0 3 3 V ~
(no summation),
O~,v, - 0 ~ ~
(no summation),
C~33V3 -- 03e33,
+ O~,O03v~ + O~O0,3v~
226
Linearly elastic shallow shells in Cartesian coordinates
1 1
0~33)
(~13
[Ch. 3
1
~
1
--03e. 12 -- ~C~10C~33V2 -- ~020033Vl, 1
(~23V1- (~2 (C31-1-~010e33)- Oql(e--23-1- 21--C~20e33) -t- 03e12 -~- ~I o 10033V2 + ~1 o 20033Vl , 1
013V2- Oql(e32-1- ~020e'33)- Oq2(e13-t- ~OqlOe33) 1
1
-~- 03C21 -~- ~020C~33Vl "~- ~010033V2, 012Vl = Oq2ell + Oq1200q3Vl nt- OqlOO23Vl,
012V2 -- 01e22 -Jr-012003V2 -t- Oq20013V2, 022VI : 202r + 012003V2 -~- 010023V2 -Jr-022003VI + 020023VI -- Oqle22 -- Ol20O3V 2 -- 020013V2, C~llV2 : 201e12 -+- (~llOOq3V2-~- OlOOq13V2-~ 0~12003Vl -~- 020013Vl -- 02e-ll -- (~12003Vl -- 010023Vl 9 T h e n a simple induction on these relations, ordered as indicated, indeed shows t h a t Ojkv~ E H-l(f~). By the lemma of J. L. Lions (Thm. 1.1-1), if a distribution w C H-l(f~) is such t h a t Okw E H-~(f~), then w E L2(ft). To apply this result, we note that, by definition of the space E(Ft), v - (v~) E E(ft) ~ v~ e L2(ft) ~ Ojv~ e H-l(f~), and t h a t we also established the implication v - (v~) E E(ft) =~ Ojkvi = Ok(Ojvi) e H - l ( f t ) . Hence 0jv~ e L2(f~) and thus E ( f t ) = HI(Ft). (ii) The mapping I1" II defined by
1/2 i,j
Sect. 3.4]
A generalized K o r n inequality
Hl(f~),
is a n o r m o v e r the space
227
a n d there exists a c o n s t a n t C1 s u c h
that
Ilvll,,~ <__ CI[[V[[
for all v E H i ( a ) .
There clearly exists a constant c2 such that Ilvll
c211v111,~ for all v C Hi(f2).
Hence the identity mapping from the space Hl(f~) equipped with the norm I1" [[,,~ into the space E(f~) equipped with the norm I1" }} is continous, and it is also surjective by (i). Since the space E ( a ) H i ( a ) is a Hilbert space when it is equipped with the norm I1" II, the closed g r a p h t h e o r e m (see, e.g., Brezis [1983, p.19]) shows that the inverse of the identity mapping is also continuous, which is exactly what was to be proved. (iii) T h e s e m i - n o r m
l" I d e f i n e d by
1/2
is a n o r m o v e r the space V ( ft ) .
The only property that remains to be checked is that v E V ( a ) and I v l -
0
- 0.
So let v e V(ft) be such that g # ( v ) - O. Since g~3(v) - 0 implies e~3(v) - 0, there exist (Thm. 1.4-4) functions ~ such that (0~ denotes as usual the normal derivative operator along 7)" v~ - ~
- x30~3
and v3 - r/3,
r/~ C HI(w), r/~ - 0 on 70 and r/3 E H2(a~), U3 - 0,r/3 - 0 on 3'0. The relations g~z(v) - 0 then imply that
0aT]~ -~- 0r
-~- 0a00~T~3 -Jr-0~00c~?]3
-
-
2x30~f/3 in w.
Consequently 0~r/3 - 0 in w, since the left-hand side of the above equality is only a function of (x l, x2). By a standard result from
228
Linearly elastic shallow shells in Cartesian coordinates
[Ch. 3
distribution theory (cf. Schwartz [1966, p. 60]; the a s s u m p t i o n t h a t a~ is connected is used here), the function r/a is thus affine with respect to (xl, x2); hence it vanishes since it must also satisfy the b o u n d a r y conditions r/a = 0,r/3 = 0 on 3'o, and length ~/0 > 0 by assumption. T h e functions r/~ therefore satisfy 0~r/~ + 0~r/~ = 0 in a~, and t h e y are thus of the form (a similar a r g u m e n t is used in the proof of T h m . 1.1-2) 7]1 :
al
--
bx2,
r]2 - - a2
+
bXl,
for some constants al,a2, and b. Therefore they necessarily vanish since they must also satisfy the b o u n d a r y condition r/~ - 0 on 7o. Hence v - 0. (iv) There exists a constant C such that
Ilvlll,
_< Girl
for all v E V(ft).
Assume this p r o p e r t y is false. T h e n there exists a sequence (vk)~=l of functions v k E V(Ft) such t h a t
Itvk l l l , f t
--
1 for all k > 1 and Ivk l ~ 0 as k --, oc.
Since the sequence is bounded in the space I-Ii(f~), there exists a subsequence (vz)~l t h a t strongly converges in the space L2(f~) by the Rellich-Kondragov theorem (Vol. I, T h m . 6.1-5). Since IvZl + 0 as 1 + oc, this subsequence is a Cauchy sequence with respect to the norm II" II. Since this norm is equivalent to the norm II" IIl,a by (ii), and since the space Hl(f~) is complete, the subsequence (vl)~=l converges in the space Hl(f~). Let v "- lim v l. T h e n l--*o~
Ivl- l--,cx~ lim IvZl- 0, and thus v - 0 by (iii). But this contradicts the equalities IIv for all 1 _> 1, and the proof is complete.
lli, - 1 1
Remark. Another generalized Korn's inequality, also involving ad hoc generalizations of the functions e~j(v), will be likewise needed when we s t u d y shallow shells in curvilinear coordinates (Vol. III). 1
Sect. 3.5]
3.5.
229
Convergence of the scaled displacements
CONVERGENCE DISPLACEMENTS
OF THE
SCALED
A S c --~ 0
We are now in a position to prove the main result of this chapter, which consists in establishing that the family (u(e))~>0 strongly converges in H I ( ~ ) &s e ---* 0 and in identifying the "limit" variational problem that the limit of this family solves. We recall that the scaled displacement u(e) solves a variational problem 7)(e; f~), described in Sect. 3.2. The following theorem is due to Ciarlet & Miara [1992, Thm. 1]. T h e o r e m 3.5-1. Assume that f~ E L~(f~), g~ C L2(F+ U r _ ) , and that 0 C Cs (-g). (a) As c ---, O, the family (u(c))~>0 converges strongly in the space V(f~) - {v C Hl(f~); v - 0 on r0}. (b) Define the space V(CU) "-- { f ~ - (7]i) E Hi(co) x Hi(co) x H2(co);
r/~- O~,r/3 - 0
on 70 }.
J
Then u -
(ui) "- lim u(e) is such that 6---*0
us -- ~ -x3c9~3 and u3 - ~3 in f~, with ~ - (~) E V(w). (c) The vector field r (~) solves the following l i m i t s c a l e d t w o - d i m e n s i o n a l p r o b l e m 7)(aJ) 9
(~ 6 V(a;) and
- L P~Widcu- L q~O~W3 for all O -
(rh)E V(cu),
230
Linearly elastic shallow shells in Cartesian coordinates
[Ch. 3
where
-
=
a(a +
-2
'
1
p~'q~ " -
/1 /1
1
1
f~dxa+g ++g(
, 9~'-9~(', +1),
x a f ~ dxa + g + - g2
9
Proof. The proof follows essentially the same pattern as in the case of a plate (which corresponds to 0 = 0; cf. Thm. 1.4-1); it is however significantly more involved. For clarity, the proof is broken into six parts. Throughout the proof, Cl,... , cs denote various constants that are all > 0 and solely dependent on the function 0 (but for brevity, this dependence is not displayed).
(i) The n o r m s Ilu(e)lll,n are bounded i n d e p e n d e n t l y o f t . Expressing that the variational equations of problem P ( ~ ) (Sect. 3.1) are satisfied in particular by/J~ - / { , using the relations M = A > 0, >~ - > > 0, Thm. 3.2-1, the assumptions on the data, and Thm. 3.3-1, we obtain" / ,
e # ( i ~ ) e # ( i ~ ~) dJc~ < j a f~ u~ dJc~ + j p -
f:-~5~dx~+iu
~ur2
~]~Ui
g~ u~ dF ~
~_uD2
{be3ib3i} 1/2
= eS{L(u(e)) + e2L#(e; u(e))} , where
Sect. 3.5]
Convergence of the scaled displacements
L(v) "- L fivi dx + fr sup IL#(r
0
+UF_
givi d r
< clllvlll,~
231
for all v C V(f~) ,
for all v e V ( ~ ) .
Using Thm. 3.3-2, we next have I
c
e~ (/t~)e~ (it ~) as: ~ ~-~
{gaa(U(e)) + e2eaea(e; u(e))}2{1 + e25#(e)} dx
2 /a ~--~{g~a(U(e)) + e 2e~3 +)-7 # (e"' u(e))}2{1 + e25#(e)} dx O~
+~
~-~.{~.n (u(e))+,'%,#(,;u(e))}'{l+ ,'5#(,)}d~;). c~,~
and thus fi~5(/K)qj (/t ~) d~ ~ >_ ~-
~--~{~j (u(c)).
+ e2e~(e; u(c))} 2 dx
z,J
for c _< rain{l, (2C0)-1/2} (Thm. 3.3-1). The estimates on I~(~;v)10,~ obtained in Thm. 3.3-2 and the generalized Korn inequality established in Thm. 3.4-1 together imply
1
>- -2 Z.. I% (~(.)) 12~,0- (3.c.)' I1~(.) I1',,o -> ~
-
1,a-~
[[u(E)II1,~
for e _< min{e0, (6C2C)-1}. Hence the announced assertion follows by combining the above inequalities. (ii) Define the symmetric tensor ~(e) - ( ~ j ( c ) )
ting"
~(~)
.- ~(~(~)),
~(~)
1
e L~(ft) by let-
.- -~3(~(~)), E
232
Linearly elastic shallow shells in Cartesian coordinates
[Ch. a
1
~ ( ~ ) .- 7~<~(~(e)) + a ~ e a ~ ( e ) . Then the norms
l~(~)lo,~ a ~ bounded independently of ~.
To see this, we simply combine the triangular inequalities
I~e(~)lo,~ _< I ~ , ( u ( ~ ) ) +
# (c", u(c))lo,~ + e~ le~e(~; # ~ e~, u(~))lo,~,
I~(c)lo,~ ___-C1I ~ ( u ( ~ ) ) +
c 2e #~ (~; u(~))lo,~ + ~I e~3 # (~; u(~))lo,~
1
I~(c)lo,~ _< ~ l ~ ( u ( = ) ) +
~eg~(=; u(=))lo,~ + la~ea~(c)lo,~
+ le~(~; u(~))lo,~, with the relations established in the proof of part (i), the boundedness of the family (u(e))~>0 in the space Hl(~t) also established in part (i), and the estimates on lea(c; v)10,a established in Thm. 3.3-2. (iii) By part (i), there exists a subsequence, still indexed by e for notational convenience, and there exists an element u E V ( ~ ) such that (as usual --~ denotes weak convergence)" u(c)--u
inV(~)
Then there exist functions ~ ~ - O~a - 0 on % such that
E
asc-+0.
HI(aJ) and
u~-~-xaO~(a
and ~a E H2(w) satisfying
ua-~a.
Since the sequence (u(e))~>o is bounded in H I ( ~ ) (part (i)) and the sequence (s is bounded in L2(~) (part (ii)), there exists a constant c2 such that IC~3(u(c))]o,~ < c2c
and
l~33(u(c))[o,~ ~ c2~2,
by definition of the functions gi3(c). Hence ~i3(u(c)) --~ 0 in L2(f~) and consequently e~3(u(a))--~ 0 in L2(f~). As u(~) ~ u in H I ( ~ ) implies ~3(u(c)) ~ ~ 3 ( u ) i n L2(~), it follows that ~3(u) - 0. This in turn implies that e~a(u) - O, and the
Sect. 3.5]
Convergence of the scaled displacements
233
usual argument (Thin. 1.4-4) shows that the components u~ of the limit u are indeed of the announced form. (iv) By (ii), there exists a sequence, still indexed by c for notational convenience, and there exists an element ~ - (g~j) E L~(gt) such that
~(c) ~ ~
inL~(~) asa---+O
(we may assume that the subsequences found in (iii) and (iv) have the same indices). Then
~
-
~(~),
~
- ~
- 0,
~
-
A k+2#
-~e.~(~).
Since ~ ( c ) - g~(u(c)) and u ( c ) ~ u in H i ( f ] ) , we first infer that g ~ ( c ) - ~ ( u ) i n L2(ft). We next transform the variational equations found in Thin. 3.2-1 over the set ft ~ into an equivalent set of equations, but now posed over the set ft, thus forming the variational problem 79(c, ft) announced in Sect. 3.2. To this end, we use Thins. 3.3-1 and 3.3-2; we also make an essential use of the functions ~j(e). This gives:
+ 0~003v9) ) dx L {Agpp(e)5~ + 2#g~(e)}{O~v~ - -~(O~O03v~ 1 + L{Ag~(e)(O~OO~v3 + b#3(e)O3v3) + A%~(e; # u(e))03 v3 } dx + L 2#%#(c; u(e)){O3v~ + O~v3 - 0~003v3} dx + .f(A + 2#)e~a3(c; u(e))O3v3 dx + L(~
+ 2~)~(~)(o~oo~
+ b~#~(~)O~v~)d~
+ L ( A + 2#)b#a3(e)O3u3(e)(O~OO~v3 + b#3(e)O3v3)dx
+ L Ab#(e)O3ua(e)(O~v9 - O~O03v~)dx + -s
2pg~3(e)(O3v~ + O~v3 - 0~003v3) dx
234
[Ch. 3
Linearly elastic shallow shells in Cartesian coordinates
-~-)--~ {/~;~r
-4- ()~ -~- 2#)~;33(C)}03V3{1 -~- C2(~#(C)} dx
l f a (A + 2p)b#33(C)OaU3(C)OqaV3(1 + C25#(e))dx -I---~ +eB1# (e; k,(e), v) + e2B2# (e; u(e), v) - L(v) + e2n # (e; v) for all v E V(ft), where sup IBI#(a; ~,, v)[ < c3[~lo,~llvlll,~ for all (k, v) E L ] ( a ) x V(ft),
0<e<_eo
sup IB2#(g; u, v)[ _< c41,l/,[1,f~llvlll,ft for all (u, v) E V(ft) x V(ft),
0<e_<eo
and the linear forms L and L#(e; .) have been defined in part (i). Letting v3 - 0 in these variational equations and multiplying by e, we find that
with sup 174(a; s u, v)l < cs(l~lo,a + IluIIl,f~ -Jr-1)llvlll,a
O<e<_eo
for all ~ E L~(ft) and v - (vi) E V(ft) with V3 -- 0. F o r s u c h v E V(ft), the left-hand side converges to fn2pg~303v~dx as c ~ 0 by definition of weak convergence, and the right-hand side converges to 0 since a weakly convergent sequence is bounded. Hence fn g~303v~ d x - 0 and thus by letting the function v~ vary, we infer from Thm. 1.4-3 that g ~ 3 - 0. Letting v~ - 0 in the same variational equations, using 03u3(c) C 2 ( ~ ; 3 3 ( C ) - OaOOau3(s , and multiplying by c 2, we likewise find that
/f {)~;aa(C)-JI-()~ -4- 2~)~;33(E)}O3V3 dx -- s
~,(E), U(s
V),
with
sup IS( ; O
u,,,)l-
c6(1 [o, + [[Ulll, + 1)lIV[ll,n
,
Convergence of the scaled displacements
Sect. 3.5]
235
for all s E L2~(f~) and v - (v~) E V(f~) with v~ - 0. Hence passing to the limit as e --+ 0 gives
s
+ (~ + 21*) fCaa} Oav3 d~ - 0,
and we likewise conclude from Thm. 1.4-3 that As163
= 0.
(v) The vector field g = (~) found in (iii) satisfies the limit twodimensional problem announced in the theorem. To see this, restrict the functions v = (v~) E V(ft) appearing in the variational equations of problem P(e; ft) to be of the form
v~=rl~-x30~rla
and
va=r/3,
with r I = (~h) C V(co). A simple computation, which uses the relation O3u3(e) = e 2 ( ~ a a ( e ) - 0~00~u3(e)), then shows that these equations reduce for such functions to {A~,p(e)5~ + 2 # ~ ( e ) } { O ~ v ~ - -~(O~OOav~ + O~OO3v~)} dx (A + 2#)r%3(e)}O~OO~v3 dx
+ s
+ ~B1# (~; ~(~), ~ ) + ~B~# (~;.(~), ~) - t(~) + ~c#(~; ~). Passing to the limit as c -+ 0 and taking into account the relation X~
+ (~ + 2tt)~aa = 0
found in part (iv), we are left with {~,,~
+ 2.~}{0~
- 1 (0~003v~ + O~OOav~)} dx - L(v)
for all v = (vi) E V(ft) of the specific form considered here. It then suffices to replace the components u~ of u by their expressions in terms of the functions ~ and the components v~ of v by their expressions in terms of the functions ~h. (vi) Let u be the weak Hl(ft)-limit found in (iii). Then in fact the whole family (u(~))~>o strongly converges to u in the space H I ( f t).
236
Linearly elastic shallow shells in Cartesian coordinates
[Ch. 3
Since the solution ~ E V(w) of the limit two-dimensional problem is unique (this can be independently proved; cf. Thm. 3.6-1), the weak limit u is unique; likewise, the weak limit s is unique by (iv). Hence the whole families (u(c))~>0 and (s weakly converge, to u and ~ respectively. To show that the family (u(c))~>0 converges strongly to u in Hl(ft), it suffices to show that the family (~(u(c)))~>0, where ~(u(e)) "- (g#(u(c))), converges strongly to ~(u) in L2(a), as the conclusion will then follow from the generalized Korn inequality established in Thm. 3.4-1. As we shall see, this property is an easy consequence of the strong L2(f~)-convergence of the family (?~(c))~>0, which accordingly we establish first. Given two symmetric matrices S - (s~j) and T - (t~j) in L2(f~), let
AS" T
" - )~Spptqq + 2 p s i j t i j .
Then we have"
<-L A(k(e)
-
- f
dn
- ~)" (~,(e)- ~)dx Ag
(b-,,, 9 - 2~(e))dx + f
dft
Ag(e) " ~(e)dx.
Let v - u(e) in the variational equations of problem 7)(e; 12); it is then easily checked that they take the form
L
As
9s
+ eB#(e; s
u(e)) - L(u(e)) + e2L#(e; u(e)),
where the linear forms L and L#(c; .) are those defined in (i), and sup
O<e<eo
v)l
+ Ilvll , ).
Note that in order to establish this last estimate, we also use the inequality
which itself follows from the definition of the functions ~3a(c).
Convergence of the scaled displacements
Sect. 3.5]
237
Using this estimate on IB3#(e;~, v)l, the estimate on IL#(e; ~)1 established in part (iv), and the weak convergences of (u(e))~>0 and (K(e))~>0, we infer from the last equations that
L A~(e)"~(e)dz
L(u) as e ---, 0,
and thus
lim~f~Ag'(g- 27~(e))dz+
e---~0 I,
L
A~(c)"~(c)dz} - {-/a
A~" ~ d z + L(u)
o
Using the relations ~ 3 - 0 and A ~ + (A + 2 ~ ) ~ ; 3 3 - - 0 (part (iv)), and letting v - u in the variational equations found in part (v), we obtain
aArr #r dx - / a {~PP~qq ~- 2#~ij~ij } dx -- L { / ~ p p ~
"3L 2 ~ o ~ , c ~ Jr- (~pp Jr- 2 / - t ~ 3 3 ) ~ 3 3 } - f{;~r~,,e9z + 2 ~ ~ 9 }
dz
dx - L(u).
Hence it follows that (&(c))~>0 converges strongly to ?~ in L2(Ft). To show that this convergence implies that (~(u(c)))~>0 strongly converges to ~(u) in L2(ft), we note that ~3(u) - 0 (cf. part (iii)). Hence
l e ( u ( ~ ) ) - ~(~)1 0,~ ~ -- Z
I~ , (~) - ~ , l g , ~ + 2c2 ~
0~,~
and the proof is complete.
1~3 (c) 12o,~+
C4[,~'33(C) [O,f~,2
(It
I
As a complement to Thm. 3.5-1, note that u - limu(e) sat@~--~0 ties, and is the unique solution of, the following l i m i t s c a l e d t h r e e -
238
Linearly elastic shallow shells in Cartesian coordinates
dimensional problem
[Ch. 3
T'KL(fl)"
u e VKL(ft)"-- {V E V(t2); e ~ 3 ( v ) - 0 in ft}, 2A#
e~(u)~,,(v) + 2pe~,(u)~,(v)
)
dx
L(v)
for all v C VKL(ft). This result, established in part (v) of the proof, is thus reminiscent of that found for a plate (Thm. 1.4-1(b)), the functions e~z(v) there being replaced here by the more general functions g~z(v) (which reduce to e~z(v) when 0 = 0). However, the limit two-dimensional problem P(w) solved by ~ = (C~) can no longer be broken into two independent problems, one solved by ~3 and the other by ( ~ ) , by contrast with that found for a plate (Thm. 1.4-1(d)). 3.6.
THE LIMIT SCALED T W O - D I M E N S I O N A L PROBLEM: EXISTENCE AND UNIQUENESS A S O L U T I O N ; F O R M U L A T I O N AS A BOUNDARY VALUE PROBLEM
OF
We next give a "direct" proof of existence and uniqueness of the solution of the limit two-dimensional problem, and we also write the two-dimensional boundary value problem that is, at least formally, equivalent to this variational problem. T h e o r e m 3.6-1. (a) Assume that k e L2(ft), gi e L2(r+ u p_), and 0 E Ca(F). Then the limit two-dimensional problem 79(w) found in Tam. 3.5-1 possesses a unique solution r - (~) in the space V(w). (b) If the boundary 7 is smooth enough, a smooth enough solution of this problem is also a solution of the following two-dimensional boundary value problem (the functions m~z, fi~o, Pi, q~ are defined as in Thm. 3.5-1, the set ~/1, the vectors (u~), (T~) and the tangential
Sect. 3.6]
The limit scaled two-dimensional problem
239
derivative operator O. are defined as in Thm. 1.5-1)" -Gzrn~z - 0~(~0~0) -0~~ ~ -
0~3
171, o~fl lY o~ lYfl
~u~
- Pa + O~q~ in a;, -
p.
-
0 o n ~0,
-
-
-
0
in w,
o n ")/1,
0 on 51"
Let b(r denote the left-hand side of the variational equations of problem P(w). A simple computation, based on the expressions of the functions rn~ 9 and g,z, shows that: Proof.
b(f~, T~) -- / i 1 k4AP + 2 t t { ~(n?']3) 2 -~-(~acr(,)) 2 } dw
{1
}
By the generalized Korn's inequality established in Thm. 3.4-1, there exists a constant C > 0 such that
c~
i,j
[~j(v)[g,~ _ Iv[~,~ for all v E V(K2).
Given an arbitrary element r I = (~h) E V(w), the function v = (v~) defined by v~ = r/~ - xaO~rl3 and v3 = r/a belongs to the space V(ft). It is then easily verified that, for such functions v, the last inequality reduces to (note that g~a(v)= 0): 2C 2
c~ Z I~(vDl~.a - 2c ~ ~ I~.~(~) 10~.~+ --5- ~ IO~,~l~.~ c~,~
c~,~
> Ivl ~1,f~ - 2 ~
a,fl
10~1 0,w ~ + g2 ~ 10~,~1 ~
Hence there exists a constant c > 0 such that
b(,,rl) ->c{~]Oafl~)3]20,~ + ~ 10zr/~10,~ 2 }
for all rl E V(co ).
240
[Ch. 3
Linearly elastic shallow shells in Cartesian coordinates
Since the semi-norms (the spaces Va(w) and VH(W) are defined in Thms. 1.5-1 and 1.5-2)
1/2 ,3 E V3(02)--* 171312,w-{ E I(0af371312 O,w}1/2 c~,13 are respectively equivalent to the norms [[. IIl,w and I]" [[2,~ over the spaces VH(W) and Va(w) (see the proofs of Thms. 1.5-1 and 1.5-2), the bilinear form b(., .) is V ( w ) - elliptic; hence assertion (a) is proved. The proof of (b), which relies on successive uses of Green's formulas, is analogous to that used for a plate (see again the proofs of Thms. 1.5-1 and 1.5-2). II Remark.
in Ex. 3.2. 3.7.
Another proof of existence and uniqueness is proposed II
J U S T I F I C A T I O N OF T H E T W O - D I M E N S I O N A L E Q U A T I O N S OF A L I N E A R L Y E L A S T I C SHALLOW SHELL IN CARTESIAN COORDINATES
In order to get physically meaningful formulas, it remains to "descale" the functions {~ and u~ found in Thm. 3.6-1. In view of the scalings made in Sect. 3.2, we are naturally led to defining functions (~[ 9 ~ ~ R and ~ ( 0 ) 9 {t)}- ~ R a through the following descalings:
4~._~24~
and
~'-c~3
inw,
~ ( 0 ) ( ~ ~) . - c~,~(~) ~nd ~ ( 0 ) ( ~ ~) - c ~ ( ~ ) for ~n ~ - 0 ~ ( ~ )
e {~}-,
Sect. 3.7]
241
Justification of the two-dimensional equations
where the mappings r~~ 9~ ---+ ~ and O ~ 9~ + { ~ } - are those defined in Sects. 3.1 and 3.2. In this fashion, we obtain the following corollaries of Thms. 3.5-1 and 3.6-1: T h e o r e m 3.7-1. (a) The de-scaled vector field ~ := (~) satisfies the following two-dimensional variational problem 7)~ (co):
~e E V ( w ) - { ~ - (r]i) 6 Hi(w) x Hi(w) x H2(w); % = 0~r/3 = 0 on %},
-
l / ~ : % d w - ~ q-:0a% dw for all r / 6 V(w),
where
m ~
~z
.__
~3{
4M# ~ A(~5~Z + 4# ~ ~} 3(A~ + 2# ~) - ~ O~z(3 ,
{ 4~~ ~r ~ - ~ a~ + 2~ ~ ; ( ) ~ + 4~;~(r
~
}
'
)V and #~ are the Lamd constants of the material constituting the shell, 0 ~ :-~ ---+IR is the mapping that defines its middle surface, and
P[ "-
f
k(O~(., x~))dx~ + 0~(O~(., s)) + t)[(O~( ., - s ) ) , c
where the functions ]~ 9 ~ ~ R and ~ " F~+ U F~ ~ I~ are the Cartesian components of the applied body and surface force densities
242
Linearly elastic shallow shells in Cartesian coordinates
[Ch. 3
m
acting on the shell, and 0 ~ 9ft ~ ---, R a is the mapping that defines the reference configuration of the shell. (b) The de-scaled functions g~(0)" { ~ } - --~ R are then given by U~(0)(:~ r
-- ~a~ (Xl, X 2 ) -
a n d ~2~(0)(~~) -- ~(x~, x2),
X~c~(Xl,X2)
II
at all points Jc - O~((Xx,X2, X~)) e { ~ } - .
T h e o r e m 3.7-2. (a) Assume that f / E L2(f~),g~ C 0 ~ E C3(-~). Then the variational problem 70~(w) of Thm. 3.7-1(a) possesses a unique solution ~ in the space V(w). (b) If the boundary 7 is smooth enough, a smooth enough solution of 79~(w) is also a solution of the following two-dimensional boundary value problem (the functions m,z,~ n~,-~ p~,-~ q~-~ are defined as in Thm 3.7-1, and 71 = 7 - ~o):
-O~rn~
- O~(ft~O~O ~) - ~ + O~q-~ in a;, -0~ ~ (~
-
0~,(~
p~ ina;, -
0 on
70,
m e~v,~t,,~ -- 0 on V1
- ~ L 2 /3 -- 0 o n Tta~ ")/1
II This boundary value problem, like its variational counterpart, constitutes the t w o - d i m e n s i o n a l e q u a t i o n s of a l i n e a r l y elastic s h a l l o w shell in Cartesian coordinates. The adjective "shallow" is defined in the next section. The adjective "Cartesian" reminds that the unknowns ([ in either limit problem represent the Cartesian components of the displacement of the middle surface of the shell; this means that the vector ~/(Xl, x2)ei is the (limit) displacement of the point (x~,x2,0~(Xx,X2)), for any (x~,x2) E &. A w o r d of c a u t i o n . Different two-dimensional equations of a linearly elastic "shallow" shell, whose unknows are the covariant
Sect. 3.7]
Justification of the two-dimensional equations
243
components of the displacement of its middle surface (i.e., in a particular basis that "follows the geometry" of the shell), can be also derived by an asymptotic analysis; cf. Vol. III. m Note that the variational problem 72~(a2) found in Thm. 3.7-1(a) may be equivalently expressed as a minimization problem, viz., find r such that
(2~ C V(co) and j ~ ( ~ ) =
inf j~(rl), where neV(~)
1 9fw{c3 J~(~7) "- -~ -~a~arOarr/3Oc~r]3+ e a ~ , g ; ~ ( r l ) ~ ( r l ) p~ r]i daJ -
} daJ
% c~/]3 da~
A major conclusion is thus that we have been able to rigorously justify two-dimensional equations for linearly elastic shells by showing that (up to appropriate scalings) their solution can be identified (in the sense of Thms. 3.5-1(b) or 3.7-1(b)) with the Hl(f~)-limit of the three-dimensional solution as the thickness of the shell approaches zero.
The two-dimensional problem found here does indeed coincide with one found in the literature on shallow shell theory: It is the linearized version of the equations found, e.g., in Washizu [1975, p.173]. In addition, we have simultaneously justified an a priori assumption of a geometrical nature, by showing that the "limit" displacement field is a Kirchhoff-Love field, in the sense of Thin. 3.7-1(b). In this respect, no confusion should arise here between the variables x ~ C f~ and 2~ C {~t~}, which are carefully distinguished in the de-scaling process; this distinction is sometimes vague in the literature.
244
Linearly elastic shallow shells in Cartesian coordinates
3.8.
D E F I N I T I O N OF A " S H A L L O W " SHELL; COMMENTARY
[Ch. 3
Another major conclusion is that the existence of a function 0 E C3(-g) independent of c such that
O~(Xl, X2) -- gO(Xl, X2)
for all (Xl, X2) e ~,
provides a rigorous criterion for defining a s h a l l o w shell, and consequently, for deciding whether a linearly elastic shell may be modeled by the equations found in Thms. 3.7-1 or 3.7-2: Up to an additive constant, the mapping 0 ~ : ~ ~ R that measures the deviation of the middle surface of the reference configuration of the shell from a plane, should be of the order of the thickness of the shell (Fig. 3.8-1 (a)) in order that the three-dimensional equations be asymptotically equivalent to the two-dimensional model found here. This definition was first proposed by Ciarlet & Paumier [1986] in the nonlinear case; cf. Sects. 4.14 and 5.12. This assumption likewise implies that all partial derivatives 0~0 ~, 0 ~ 0 ~, etc., are also of order of c. In particular then, the radii of curvature of the shell must be of the order of c -1 in order that the shell may be deemed "shallow". Such a definition should be compared with more traditional definitions of "shallowness": For instance, Green s Zerna [1968, p. 400] define a shallow shell "to be one in which the amount of deviation from a plane, measured normally to the plane, is small compared with a maximum length of an edge of the shell, which in turn is small compared with a minimum radius of curvature of the middle surface"; Dikmen [1982, p. 158] states that "the shallowness of the shell is understood in the sense that the smallest radius of curvature is so large that the shell is nearly flat locally"; etc. As in the case of a plate (Sect. 1.8), asymptotic assumptions on the Lamd constants and applied forces are possible that are more general than those made in Sect. 3.2. More specifically, it is readily verified that the variational problem solved by the scaled displacement u(c)
Definition of a "shallow" shell; commentary
Sect. 3.8]
C~
245
~o(s)
(b) 2g~ii_
-
.........
-
Ce)
2 a~::-:
",o (e)
Fig. 3.8-1: Definition of a "shallow" shell. A shell is "shallow" if, in its reference configuration, the deviation of the middle surface from a plane is (up to an additive constant) of the order of the thickness of the shell (a). Special cases of interest include a junction between a plate and a shallow shell (b), and a "moderately slanted" plate (c).
Linearly elastic shallow shells in Cartesian coordinates
246
[Oh. a
is left unaltered if we assume t h a t
M-a
tk
and
# ~ - a t#,
f ~ ( x ~) - et+~f~(x) a n d / ~ ( x ~ ) - ~t+af~(x) for an x e ~, fl:(x ~) - ct+ag~(x) and f/~(x ~)
-
s
)
for all x E F+ tO F_
where the constants ~ > 0 , # > 0, the functions f~ E L2(~), the functions g~ C L2(F+ U F _ ) are independent ofe, and t is an arbitrary real number. A word of caution. By contrast, the "shallowness assumption" t h a t the functions 0 ~ be of the form 0 ~ - ~0 for some function 0 c (Ta(~) independent of c, cannot be replaced by a more general a s s u m p t i o n such as 0 ~ - e*0 for some constant s :/: 1 (Ex. 3.a). m Note t h a t our analysis includes cases where some portions of the middle surface of the shell are flat, such as junctions between plates and shallow shells (Fig. 3.8-1 (b)) or " moderately slanted" plates (Fig. 3.8-1 (c); see also Ex. 3.4). Combining the techniques used in this chapter and in Chap. 2, Rodrfguez [1997] has studied a multi-structure composed of a threedimensional s u b s t r u c t u r e and a shallow shell; one objective is to model a rotor and its blades rotating at high angular velocity (Fig. 2.6-3). An analogous analysis has also been applied to "shallow" rods by Alvarez-Dios & Viafio [1995].
EXERCISES 3.1. Let the m a p p i n g O ~ 9~
--+ R a be defined as in Sect. 3.1,
i.e., O e ( x e) -- (Xl,X2,s
~-
x~3a~(xl,x2)
e E ~e , where 0 C C3 (~) is a given function. for all m~ - (x~,x2, ma) Show t h a t there exists e0(0) > 0 such that, for all 0 < e _< e0(0), the m a p p i n g O ~ 9~ -+ O ~( ~ ) is a (71 - diffeomorphism. Hint: Use the relations
6(e) -- 1 + c26 # (g) and
sup m a x ]5# (g)(x)l < Co 0<e<_eo xE~
247
Exercises
established in Thm. 3.3-1, in conjunction with Thms. 1.2-6, 1.2-8, and 5.5-1 in Vol. I; this is the idea of the proof given in Ciarlet Paumier [1986, Prop. 3.2]. 3.2. Give another proof (i.e., different from that of Thm. 3.6-1) of the existence and uniqueness of the solution 4 E V(a~) of the limit two-dimensional problem found in Thm. 3.5-1, by showing directly the existence and uniqueness of a solution to the limit three-dimensional problem 7)Kr(ft), then by resorting to an argument similar to that proposed in Ex. 1.4(2). 3.3. Carry out the whole asymptotic analysis of this chapter when the "shallowness assumption" (Sect. 3.8) is replaced by Oe(Xl,X2) -- cSO(x,,z2)
for all (XI,X2) E ~,
with a function 0 E C3(~) again independent of s, but with a constant s =/= 1, all the other asymptotic assumptions made in Sect. 3.2 on the Lam6 constants and the applied forces being identical. 3.4. Consider a "moderately slanted" plate (Fig. 3.8-1 (c)), i.e., corresponding to a function 0 ~ of the form 0 ~ = e0, where 0 : ~ ~ R is a nonzero linear function of (x l, x2). (1) Write the associated variational and boundary value problems (these are mere special cases of those found in Thins. 3.7-1 and 3.7-2). (2) Compare the displacement field of the middle surface & of the plate found in this fashion with that found by solving the "usual" plate equations, i.e., with respect to a Euclidean frame (ei) where & is in the plane spanned by the vectors ~ .
This Page Intentionally Left Blank
PART B
NONLINEAR PLATE THEORY
This Page Intentionally Left Blank
CHAPTER 4
NONLINEARLY
ELASTIC
PLATES
INTRODUCTION The purpose of this chapter I is to describe the application of an asymptotic method, with the thickness as the "small" parameter, for justifying the two-dimensional equations of the classical nonlinear
Kirchhoff-Love theory of plates. The "preliminaries", viz., the passage to a fixed domain accompanied by fundamental scalings of the unknowns and assumptions on the data, are the same as in the linear case. As however no convergence result comparable to t h a t of Thm. 1.4-1 is as yet available for the nonlinear Kirchhoff-Love theory, we resort to a different approach, based on the method of formal asymptotic expansions. To begin with, we consider the displacement approach. We first pose (Sect. 4.1) the clamped plate equations as a problem in nonlinear three-dimensional elasticity. Let e > 0, let a~ be a domain in R 2 with b o u n d a r y ~/, and consider a plate occupying the set ~ - - w x I - e , e] subjected to applied body forces of density ( f I) : ft ~ ~ R a acting inside f~ (we assume in this introduction t h a t there are no applied surface forces on the upper and lower faces F~_ - w x {e} and F ~_ = cJ x { - c } ) , and clamped on a portion F; = % x I - e , e] of its lateral face ~, x I - e , el, where % C ~/. Let A~ and #~ be the Lamd constants of the elastic material constituting the plate. The unknown displacement field u ~ = (u~) then satisfies the following nonlinear b o u n d a r y value problem, where (n~) denotes the unit outer normal vector along the b o u n d a r y of the set f~ and "~1 = ~ / - ~0: 1To a
chapters.
large extent, this chapter can be read independently of the preceding
Nonlinearly elastic plates
252
[Oh. 4
- 0 ) (crij + crkjOkUi ) - f i in ~t e i
Oo n
F e 0
(a~5 + Cr;yi)~.u:)n~ - 0 on F+ U F5 U {')It
X
[--s s
,
where
+ 2, <5(u
s __ A ~
&a(u ~) -- -~(O~uj + 0;< + o~ UmO)Um). For pedagogical purposes, we a s s u m e in this c h a p t e r t h a t the elastic m a t e r i a l is a St Venant-Kirchhoff material, i.e., t h a t its constitutire equation is a linear relation between the Green-St Venant strain tensor (E~(u~)) and the second Piola-Kirchhoff stress tensor (a~). We shall see however (Sect. 4.10) t h a t the present analysis can be e x t e n d e d to the most general elastic materials t h a t are isotropic, homogeneous, and whose reference configuration is a n a t u r a l state. T h e nonlinear b o u n d a r y value p r o b l e m defined above is t h e n p u t in variational form, the u n k n o w n u ~ being sought in the space V(~-~ e) --- { V ~ --- (V~) e w l ' 4 ( ~ - ~ e ) ;
v e :
0
on F;}.
In Sect. 4.2, we t r a n s f o r m this p r o b l e m into an equivalent variational problem, b u t now posed over the set ft - & x [ - 1 , 1], which is independent of ~. This t r a n s f o r m a t i o n relies in a crucial v:ay on app r o p r i a t e scali~gs of the unknowns u~ and on a d e q u a t e assumptions on the data (the Lam~ c o n s t a n t s and applied force densities), which are the same as in the linear case (Sect. 1.3). More specifically, we let (as usual, Greek indices take their values in { 1, 2}) :
u~(x ~) -- ~2u~(c)(x)
u~(x ~) -- su3(x)
and
for all x ~ - 7r~x C , where TrY(x1, x2, xa) - (Xl, x2, cx3), and we assume t h a t there exist c o n s t a n t s A > 0, # > 0 and functions f~ E L2(f/) independent of e such t h a t A~ = A
and
#~=#,
f~(x ~) = e e L ( x ) and f~(x ~) = e a f 3 ( x )
for all x ~ = rffx C a ~
Introduction
253
In this fashion, the scaled unknown u(c) = (u~(s)) becomes the solution of a variational problem of the form (Thin. 4.2-1): U(C) ~ V(~'-~) ~--- {V =
B(E(c), v) + T ~
(Vi) ~ w l ' 4 ( a ) ;
v = 0 on F0 = 70 x [-1, 1]},
u(c), v) + c2T 2(E(c), u(~), v) - L(v) for all v E V(ft),
where -
1 S_ 4
+
1 8_2(u(s
) +
sO(,.u,(E) ) --~ E2S2(,U(E)),
and where the linear form L, the bilinear form B, the trilinear forms T O and T 2, and the tensor-valued mappings S -4, S -2, S ~ S 2 are all independent of e. The specific form of this variational problem suggests that we use the method of formal asymptotic expansions, i.e., we let U ( e ) -- U 0 + e U 1 + s
AVs
3 AV E4U 4 + . . .
in the variational equations, and then we equate to zero the factors of r q > - 4 , in all the equations found in this fashion. In so doing, we first find (Thin. 4.4-1) the rather surprising result that all the terms u ~ u ~, u 2, u a, U 4 must be considered before the leading term u ~ can be fully identified. We then establish the main results of this chapter (Thins. 4.5-2 and 4.6-2) by showing that the leading term is obtained by solving a two-dimensional problem; more specifically: (i) The vector field u ~ - (u ~ is a (scaled) Kirchhoff-Love displacement field" The function u ~ is independent of the variable x3 and it can be identified with a function ~3 C H2(a~) satisfying ~3 = 0.~3 = 0 on 70; the functions u~0 are of the form u~0 - ~ - x 3 0 ~ a with functions ~a C HI(w) satisfying ~ = 0 on 70. (ii) The vector field t; = (~), which represents the (scaled) displacement of the middle surface a~ of the plate, satisfies, at least formally, a two-dimensional boundary value problem, which (up to appropriate de-scalings; of. Sect. 4.9) coincides with the two-dimensional equations of the classical Kirchhoff-Love theory of a nonlinearly elastic clamped plate, viz. ,
254
[Ch. 4
Nonlinearly elastic plates --O~zm~z -- Oz(N~zO~a) = P3 + O~q~ in co, -OzNc~z = Pc~ in w,
~
=
0.~3
= 0 o n ~'o,
m~zu~uZ = 0 on 71, (O~m~z + N~zO~a)uz + O,.(m~zu~T~) = -q~u~ on "/1, N ~ u ~ = 0 on '~'1,
where
- -
"~
N~ -
{ a(~, + 2 , )
4.
}'
-5-
{ 4~# ~ } A + 2 , E o o ( ~ ) 6 ~ + 4pE~z(~) ,
E~
1
- ~(o~r
+ o~(~ + o~<~o~(~),
1
p~-/_
1
/__1
f~dxa,
q~ -
1
xaf~ dxa.
Observe that, starting with a three-dimensional boundary value problem that is quasilinear, we now obtain a two-dimensional boundary value problem that is semilinear. In other words, the method followed here has a "partial linearization effect" on the original systern of partial differential equations (see the discussion given in Sect. 4.10). We also show (Thin. 4.6-1) that particular solutions may be obtained by solving the following minimization problem: Find
--(~i) C V ( w ) - {~7- (rh)E HI(w) • HI(w) • H2(~); ~i - - 0u~73
such that j(~)= where
inf j(rl), n~v(~o)
"~ 0 on 70 },
255
Introduction ac~ar()ar?73C~a~r]3&z +
J(~) -- 5
a~no~Eo~(rl)E~n(rl) da~ -(~p+rl+daJ-fq~O~rlada~),
then we prove the existence of solutions to this minimization problem (Thin. 4.6-1); and finally, we establish the regularity of the solutions of the associated nonlinear boundary value problem when 70 = 7 (Thm. 4.6-3). In the displacement-stress approach (Sects. 4.7 and 4.8), the clamped plate problem is posed as in the displacement approach, but the stresses ai~ are also considered as unknowns (related to the unknowns u~ via the constitutive equation), and as such are also scaled. The scalings on the stresses (which can be fully justified through the consideration of the displacement approach; cf. Thm. 4.2-1(b)) are given by: ~ : , ( x +) - + ~ , ( + ) ( x )
+ +) - +40-33(s , ~+c~3(x+) - +3 O-c~3(+)(x), 033(x
'
for all x ~ - 7r~x C In this fashion, the scaled unknowns u(~) (ui(c)) and E(~) = (ai3(~)) satisfy a problem of the form (Thm. 4.7-1):
u(~) e v ( ~ ) ,
~(+) e L~(~) - { ( ~ )
B(E(c), v) + T~
e L~(a); ~+j -
~+},
u(c), v) + c2T2(E(c), u(~), v) - L(v) for all v E V(ft),
E~
+ +2E2(u(+)) - (B ~ + +2B2 + +4B4)~](+),
where the linear form L, the bilinear form/3, the trilinear forms 7 -0 and 7 -2 (the same as in the displacement approach), the matrix-valued mappings E ~ and E 2, and the fourth-order tensors B~ B2, B4 are all independent of c. Note that the second relation is obtained after the "original" constitutive equation has been inverted. We explain in Sect. 4.8 that this inversion, which cannot be fully understood
256
Nonlznearly elastic plates
[Ch. 4
within the displacement-stress approach alone, has its root in the displacement approach. The specific form of the above problem suggests that we use again the method o f f o r m a l asymptotic expansions. To this end, we let
in the equations of this problem and then we equate to zero the factors of E P , p 2 0, in all the relations found in this fashion. In so doing, we find that only the leadzng terms u0and C0 need to be conszdered in these asymptotic expansions, and that they satisfy the equations (Thm. 4.7-2): f3(C0,v )
+ 'T0(z0, uO,v ) = L ( v ) for all v E V(R), EO(UO)
= BOZO.
Our main result (Thm. 4.8-1) then consists in establishing the exzstence of such leadzng terms when yo = y (when yo # y , these equations have no solution in general): The leading term u0 = (up) is first computed from the solution of a two-dimensional nonlinear boundary value problem as in the displacement approach; the limit (scaled) "stresses" a:] are then given by explicit formulas involving the functions 5,and the densities of the applied forces. For instance, if we assume (for simplicity) that f, = 0, g, = 0, and f3 is independent of x3, these formulas reduce to:
We then show (Sect. 4.9) that, after the unknowns u: and a$ have been appropriately de-scaled, we have simultaneously found: (i) the standard two-dimensional equations of the nonlinear Kirchhoff-Love plate theory (see in particular the discussion in Sect. 4.10);
The three-dimensional equations
Sect. 4.1]
257
(ii) the s t a n d a r d a priori assumptions as a by-product, namely t h a t the "limit displacement" is necessarily a Kirchhoff-Love field, and t h a t the " limit stresses" necessarily take the special forms that are generally assumed a priori in the literature. We conclude this chapter by a detailed discussion about the m e t h o d followed here, regarding notably the scalings on the unknowns and the necessity of appropriate assumptions on the data (Sects. 4.10 and 4.11). We also describe the recent justifications of other nonlinear, "large deformation", "flame-indifferent", twodimensional theories, notably by means of F-convergence theory (Sects. 4.12 and 4.13). Finally, we show how the two-dimensional equations of a nonlinearly elastic shallow shell may be likewise justified by the m e t h o d of formal asymptotic expansions (Sect. 4.14). 4.1.
THE THREE-DIMENSIONAL E Q U A T I O N S OF A NONLINEARLY ELASTIC CLAMPED PLATE
We consider in this chapter the same clamped plate problem as in Chap. 1. The only (but essential!) difference is that the material constituting the plate is now assumed to be nonlinearly elastic. Let co be a domain in the plane spanned by the vectors e~ and let ?0 denote a subset of the boundary ? of the set co such that
length ?o > O. For each c > 0, we let (Fig. 1.2-1).
f~ "- aJx] - e,c[,
r;.-
x
r;.-
x
r
9- c o x
{-c}.
Again, we may have "70 - "7, in which case the set F~ coincides with the whole lateral face. For each e > 0, the set f~ is the reference configuration occupied by an elastic body in the absence of applied forces. The body under consideration is thus a plate, with thickness 2c, and middle surface ~.
258
Nonlinearly elastic plates
[Ch. 4
The plate is subjected to applied body forces acting in its interior, of density (f[) 9ft ~ ~ R 3 per unit volume, and to applied surface forces acting on its upper and lower faces, of density (g~) 9F~_ U F~ R 3 per unit area. Since these densities do not depend on the unknown, the applied forces considered here are dead loads (Vol. 11, Sect. 2.7). For definiteness, we assume t h a t f [ e L 2 ( f ~ ~)
and
g~EL2(F~_UF~).
The u n k n o w n of the problem is the d i s p l a c e m e n t vector field u ~ = (u~)" ft ~ --~ R 3, i.e., the vector u ~ ( x ~) represents the displacement t h a t each point x ~ C f~ undergoes under the action of the applied forces (see again Fig. 1.2-1, which represents indifferently a linearly elastic plate as in Chap. 1, or a nonlinearly elastic one as here); we also call d i s p l a c e m e n t s the components u~ 9ft ~ -~ R of the displacement vector field. The set q0~( ~ ) , where ~o~ " - i d + u ~ " x ~ e ~ e ~
(x ~ + Ue (x~)) E R3 ,
is called a d e f o r m e d c o n f i g u r a t i o n and the mapping ~o~ is called a deformation. Since the approach in this chapter is formal, we blithely assume t h a t the requirements t h a t the deformation ~o~ should satisfy in order to be physically admissible (orientation-preserving character and injectivity; cf. Vol. I, Sect. 1.4) are satisfied for all the values of c t h a t are considered. We assume t h a t the plate is clamped on the p o r t i o n F~ of its lateral face, in the sense t h a t the b o u n d a r y c o n d i t i o n o f p l a c e u~-
0 on F~)
is imposed to the displacement. Note t h a t the remaining portion (~Y-'Y0) x I-e, c] of the lateral face of the plate is free from all external actions. 1We recall that "Vol. I" stands for "Ciarlet, P. G. [1988]" Mathematical Elasticity, Volume I: Three-Dimensional Elasticity, North-Holland, Amsterdam".
Sect. 4.1]
259
The three-dimensional equations
Let (Xl, x2) and x ~ - (Xl, X2, X~) denote the generic points in the sets c~ and ~ , and let
0~-0~-
0
Ox~ and
0~=
The following e q u a t i o n s o f e q u i l i b r i u m tion (Vol. I, Sect. 2.6) are satisfied"
0
Ox~
in the reference configura-
in fY, on F+ U r ~ o n 71 x [ - ~ , e],
where ~E~ - (ai~j) 9~ --, g3 denotes the s e c o n d P i o l a - K i r c h h o f f s t r e s s t e n s o r field (cf. Vol. I, Sect. 2.5; S 3 denotes the set of all symmetric matrices of order 3), (n~) : OfF ~ R 3 denotes the unit outer normal vector along the b o u n d a r y of the set ~ , and 7~ = - 7 0 . For commodity, we shall simply call s t r e s s e s the components cri~j 9f~ ~ R of the second Piola-Kirchhoff stress tensor field. Note that, according to our rules governing Latin indices and summations, the equations in fF read in fact as 3
3
j=l
k=l
in ft ~~ i
-
1, 2 , 3 .
The b o u n d a r y conditions on c 0 f F - F ~ - F~_ U F ~_ U {71 X [--C, C]} constitute a b o u n d a r y c o n d i t i o n of t r a c t i o n . Let V(f~ ~) denote a space of sufficiently smooth vector-valued functions v ~ - (v~)" ~ ~ R 3 t h a t vanish on F~ (their smoothness will be specified later). Then, if the unknown displacement u ~ belongs to the space V ( f ~ ) , it can be easily established (by integration by parts; cf. Vol. I, Thm. 2.6-2) t h a t the equations of equilibrium are formally equivalent to the (three-dimensional) p r i n c i p l e of v i r t u a l w o r k , which states t h a t (dF ~ denotes the area element along 0fV):
(a j + kjuku )aj
dx
- L f ~v; dx~ + fr
g~v~ d r ~ for all (v~) C V(VY).
Nonlinearly elastic plates
260
[Ch. 4
Note that the principle of virtual work is nothing but the weak, or variational, form of the equations of equilibrium and t h a t the functions v ~ E V(f~ ~) that enter it are indeed "variations" around the actual deformation q0~ = i d + u ~ (Vol. I, Sect. 2.6). We finally assume that, for each c > 0, the material constituting the plate is e l a s t i c , h o m o g e n e o u s , i s o t r o p i c , and t h a t the reference configuration ~ is "stress-free, i.e., is a n a t u r a l s t a t e (these notions are defined in Vol. I, Chap. 3). For pedagogical purposes, we postpone until Sect. 4.10 the consideration of the most general n o n l i n e a r l y e l a s t i c m a t e r i a l s that satisfy these assumptions. We instead begin by considering a restricted class of such nonlinearly elastic materials, which permit an easier exposition, while retaining all the essential features of the method we wish to describe in this chapter. More specifically, let the s t r a i n s
E~(~ ~) .- -~(o~ + o;< + denote the components of the G r e e n - S t (Vol. I, Sect. 1.8)
E~(u~)- (E~j(~)).-
o[
strain tensor
~1 ( { V ~ ~}T + V~U~ + { V ~ ~}WV ~U ~),
where the matrix
Ve U r :--
is the d i s p l a c e m e n t g r a d i e n t . We assume that, for each ~ > 0, the material constituting the plate is a St V e n a n t - K i r c h h o f f m a t e r i a l . This means (Vol. I, Sect. 3.9) that, for each c > 0, there exist two constants t ~ and #~ such that the second Piola-Kirchhoff stress tensor is expressed in terms of the Green St-Venant strain tensor through a c o n s t i t u t i v e e q u a t i o n of the form: ~E~ -- E~ ( V ~ u ~) - A~(tr E ~ ( u ~ ) ) I + 2#~E~(u~),
Sect. 4.1]
The three-dimensional equations
261
where the mapping E~ - (d~5), which acts from the space of matrices of order 3 into the space of symmetric matrices of order 3, is called a r e s p o n s e f u n c t i o n . Componentwise, the constitutive equation thus reads -
+
-
where A~jkl
are the components of the (three-dimensional) e l a s t i c i t y t e n s o r . The two constants /V and #~ are the L a m d c o n s t a n t s of the material; in accordance to experimental evidence, we assume as usual that they satisfy k~>0
and
#~>0.
The constitutive equation may be also expressed in terms of the Poisson ratio S and the Youn 9 modulus E ~ of the same material (Sect. 1.2). We thus realize, simply by inspecting the equations, that for a St Venant-Kirchhoff material the minimum regularity needed on the components v~ of any function v~E V(f~ ~) in order that all integrals appearing in the left-hand sides of the principle of virtual work make sense is that they belong to the SoboIev space w l ' 4 ( ~ e) " - {v r E L4(f~e); oq[ve E L4(f~e)}.
Hence the space V(Q ~) may be defined in the present case as: V ( ~ ~) "- {v ~ - (v~) e w l ' 4 ( ~ ) ; v ~ - 0 on Pg}. If we assume that u ~ E Wl'4(~e), we also have X ~ -- A~(tr E ~ ( u ~ ) ) I + 2#~E~(u ~) C L ~ ( ~ ) , where L~(~ ~) "- {(7i~ ) E L 2 ( ~ ) ; 7i~ - r j ~ } .
Nonlinearly elastic plates
262
[Ch. 4
To sum up, the displacement field u ~ = (u~) satisfies the following nonlinear d i s p l a c e m e n t - t r a c t i o n p r o b l e m : S S g - 0 ) S (aijS + akjOkUi) - k S in
u~ i -Oon ~ ~)n;. -
(O'i~ ~-O'kjOkUi
{
fig
r~,
g~ o n r ; u r ~- ' O o n ")/1 X [--E, E],
which is in turn equivalent to the variational problem P(FY): U s e V ( ~ s) -- {V s -- (V~) e w l ' 4 ( ~ s ) ;
fa
s
V s -- 0 on F;},
s s ) O~v; dx ~
e
f~ur~
E
S
gi vi dF~
for all v ~ e V(fY),
where
1
E,j(~ ~) - ~(0~; + 0;< + a~r Problem P(fY) constitutes the t h r e e - d i m e n s i o n a l e q u a t i o n s of a n o n l i n e a r l y elastic c l a m p e d p l a t e made of a St Venant-Kirchhoff material. Because a St Venant-Kirchhoff material is h y p e r e l a s t i c (Vol. I, Thm. 4.4-3), solving the variational problem P(fY) is formally equivalent to finding the stationary points of an associated functional J~ (defined below), i.e., those points where the derivative of J~ vanishes. Particular stationary points are thus obtained by solving a minimization problem, viz., find u s such that u ~ e V(f~ ~) and J~(u ~) -
inf J~(v~), ,,~v(a~)
Sect. 4.1]
263
The three-dimensional equations
where the e n e r g y J~ :V(f~ ~) --+ R is defined by
1s
J~(v ~) "- -~ -
2
2
~{A~[tr E~(v~)] + 2# ~ tr[E~(v~)] } dx ~ fi vi dx~
+
g~v~
,
;urt
where
E~(v ~) - ( E i ~ ( v ~ ) )
and
1
E{5(v ~) "- -~(O[vj + O~v[ + O[v~O~v~).
Note however that there is no available result guaranteeing the existence of a solution u ~ to problem 7)(ft~), nor of a solution to its associated minimization problem. The only available existence result valid for St Venant-Kirchhoff materials is based on the implicit function theorem, and for this reason, is restricted to smooth boundaries and to special classes of boundary conditions, which do not include those considered here (see the discussion given in Vol. I, Sect. 6.7). The more powerful existence theory developed by Ball [1977] for minimizing energies of nonlinearly elastic materials does include boundary conditions of the type considered here. However, even within the class of elastic materials to which it applies, which does not include St Venant-Kirchhoff materials, it neither provides the existence of a solution to the corresponding problem 7)(ft ~) (Vol. I, Sect. 7.10), because the energy is not differentiable at the minimizers found in this fashion. Detailed expositions of the modeling of three-dimensional nonlinear elasticity are found in Truesdell & Noll [1965], Germain [1972], Wang & Truesdell [1973], Gurtin [1981], Marsden & Hughes [1983, Chaps. 1-5], and Vol. I, Chaps. 1-5. Its mathematical theory is exposed in Ball [1977], Marsden & Hughes [1983], Valent [1988], and Vol. I, Chaps. 6 and 7. A w o r d of c a u t i o n : For notational convenience, we use the ~ j~ same notations 9 %,~ a#, as in Chap. 1. It should be kept in mind however that in Chap. 1 these notations merely represented "approximations" of the "genuinely nonlinearly elastic" functions that they now represent, m
264
4.2.
Nonlinearly elastic plates
[Ch. 4
TRANSFORMATION INTO A PROBLEM POSED OVER A DOMAIN INDEPENDENT O F c; THE FUNDAMENTAL SCALINGS OF THE UNKNOWNS AND ASSUMPTIONS ON THE DATA
We describe in this section the basic preliminaries of the asymptotic analysis of a nonlinearly elastic plate, as set forth in Ciarlet D e s t u y n d e r [1979b]. As it t u r n s out, t h e y coincide w i t h those described in Chap. 1 for a linearly elastic plate; we neverthless briefly r e c a p i t u l a t e t h e m here for convenience. As in Sect. 1.3, we first transform problem 7)(~ ~) into a problem posed over a set that does not depend on c. Accordingly, we let (see Fig. 1.3-1, which applies as well here as in Chap. 1):
~ "- w •
1,1[,
r0 . - % • [ - 1 , 1], F+ . - w • {1}, F_ " - co • { - 1 } ,
through
a n d w i t h each point x E ft, we associate the point x ~ E t h e bijection
7r~ x -
(Xl,X2, X3)E ~ ~ x ~ -
( x ~ ) - (Xl,X2, Cx3) e
.
We t h e n set the following correspondences between the displace-
ment fields" W i t h the fields u ~ - (u~) and the s c a l e d d i s p l a c e m e n t field s c a l e d f u n c t i o n s v - (vi) 9~ ~ t h a t G r e e k indices vary in the set
u ~ (x ~)
v - (v~)" ~ --~ R 3 , we associate u(e)(ui(s))" ~ ~ R 3 and the R 3 defined by the s c a l i n g s (recall {1, 2})"
e2u~(c)(x) and u3(x ~) - eu3(c)(x) for all x ~ - ~ x
%C( x ~) -- s2v~(x) and %E( x ~) - sv3(x) for all x ~ - ~ x
C
C ~
.
Sect. 4.2]
Fundamental scalings and assumptions
265
We call s c a l e d d i s p l a c e m e n t s the functions ui(s) " f~ ~ R. Hence the components of the scaled displacement u(e) and scaled functions v belong to the space wl'4(~)
" - {v E L 4 ( ~ ) ;
0iv E L 4 ( ~ ) } .
Finally, we make the following a s s u m p t i o n s on t h e d a t a : We assume that the Lamd constants, the applied body force density, and the applied surface force density are of the following form: A~ = A
and
#~=#,
f ~ ( z e) -- e2f~(z) and f~(z e) - c3f3(z) for all z e - 7tea E ft e, g e ( x e) __ e 3g~(x) and g3(x e e)
-
-
C4g3 (X) for all Z e T l . e x E F +e U r e- ,
where the constants ~ > 0 and # > O, the functions f~ E L2(fi), and the functions gi E L2(F+ U F_) are independent of c. Note that, as in the linear case, other "equivalent" assumptions on the data are possible, and that, remarkably, the resulting "class of assumptions" can be fully justified through a careful analysis of the method of formal asymptotic expansions (Sect. 4.11). Using the scMings of the displacements and the assumptions on the data, we reformulate in the next theorem the variational problem 7)(~ ~) of Sect. 4.1 as a problem 7)(e; Ft) now posed over the set ~. Note that problem 7)(E; Ft) is not defined for e = 0, since negative powers of e appear in the expressions of a~j(e) in terms of u(e). In what follows, dF denote the area element along the boundary of the set Ft. T h e o r e m 4.2-1. Assume that u ~ E Wl'4(~-~e). (3,) The scaled displacement field u(e) = (u~(c)) satisfies the following variational problem 7)(e; Ft), called the s c a l e d t h r e e - d i m e n s i o n a l p r o b l e m of
266
[Ch. 4
Nonlinearly elastic plates
a nonlinearly elastic clamped plate: U(C) E V(~'~)"-- {V -- (72i) e w l ' 4 ( ~ ) ;
V -- 0 o n Fo},
f ~j(~)Ojv,dx+ f ~(~)O,~(~)Ojv~dx + ~ f ~j(~)O~o(~)Ojv~dx - f f, v~dx + f~+.~_g~v~ dF for all v E V(~), (7ij(s
--
1
1
_ -~Sij4(U(C)) -~- --~Sij2(U(s
o Sij(U(C)) -~- ~: 8ij: (u(~)) ,
where the mappings s~ - s~ " V(f~) --, L2(f~), p independent of ~; more specifically,
- 4 , - 2 , 0,2, are
1
~ , ( ~ ) . - ~{~EOz(u(~))5~,}
+ ~(E~ E~3(~(~)))5~, + 2 . E ~ + ~{~ES(~(~))~ + 2.E.~(~(~))}, ~.3(~) . - j 1 {2.EO (u(~)) } +
2.ES(u(~)),
1 a33(~) . - ~ { ( ~ + 2p)E~ 1 + j{~(E~
E~3(~(~)))+ 2,E~3(~(~))}
+ ~ E 5 (u(~)), where E~
-
1
~(0~j(~) + %~{(~) + 0{~(~)%~(~)),
1 E~j(u(~)) . - ~ ( 0 ~ ( ~ ) 0 ~ ( ~ ) ) . (b) The functions aij(r
e L2(~t) defined in (a) are also related to the components ai~ E L2(~ ~) of the second Piola-Kirchhoff stress
Sect. 4.2]
267
Fundamental scalings and assumptions
tensor by
<~(x~)_~~(~)(x),
~ 3 ( x ~) ~3~3(~)(~), ~33(x~)=~4~3(~)(~), -
for all x ~ - 7r~x E FY.
Proof. The proof of (a) reduces to simple computations, based on the scalings on the displacement, the assumptions on the data, and the formulas
s
O(x ~) dx ~ - e
s
O(~ff x) dx,
c9~ - 0~,
0~ - eOa.
The formulas in part (b) follow from the definition of the functions crij(e) given in part (a). m
Remark. The components Ei~(u ~) of the Green-St Venant strain tensor E ~(u ~) satisfy:
E ~z ~ ( ~ ) ( ~ ) - ~ {E~9(~(~)) o + ~ < ~~(~(~))}(x) , E~~ 3 ( ~ ) ( ~ ~) ~{E~3 o (~(~))+ ~ ~E~3 ~ (~(~))}(x) , E ~ z ( ~ ) ( x ~) - {E%(~(~))+ ~E~(~(~))}(x), -
for all x ~ - 7r~x E fY.
m
As a consequence of the "passage from fY to ft", combined with the scalings of the unknowns and the assumptions on the data, the
dependence on the parameter e is now "explicit", by means of the powers e -4, e -2, e ~ e2; more specifically, problem 7)(e; ft) is of the form (L~(ft) denotes the space of all symmetric tensors (T~j) E L2(ft)) 9 B(E(c), v) + T ~
u(c), v) + c2T2(E(c), u(c), v) - L(v) for all v E V(fl), 1 S_ 4
1
2
+ S~
~s~(~(~)),
Nonlinearly elastic plates
268
[Ch. 4
where the linear form L 9V(f~) - , R, the bilinear form B 9L~(f~) x V(f~) --, R, the trilinear forms T ~ T2 9L~(f~) x V(f~) x V(f~) --, R, and the tensor-valued mappings S p "- (s~Pj) 9V(f~) --, L~(Ft), p - 4 , - 2 , 0, 2, are all independent of c. For instance,
lvl
+UF_ s
B(E, v) - f~ cr~303v~dz,
0 0 0
T~
u, v) - ./o cr~30~uaO3v3dz,
0
0 (a + 2,)E~
/
, etc.
Remarks. (1) While for a fixed v E V(ft), B(., v) is indeed linear with respect to E(e) C L2(t2) it is no longer linear with respect to 8
u(c) E V(ft) when E(e) is replaced by its expression in terms of u(e). (2) If all the terms that are nonlinear with respect to u(c) are canceled in its formulation, problem P(c; ft) naturally reduces to that found in the linear case (Thm. 1.3-1). II 4.3.
THE METHOD EXPANSIONS: APPROACH
OF F O R M A L A S Y M P T O T I C THE DISPLACEMENT
The specific dependence on e of the equations of problem 79(E; ft) observed in Thm. 4.2-1 and the idea that c is a "small" parameter naturally lead us to apply the m e t h o d of f o r m a l a s y m p t o t i c e x p a n s i o n s , briefly described infra. For general presentations, see Lions [1973] and Eckhaus [1979]. As only the displacements are scaled here, only they will be formally expanded. For this reason, this application of the method is called the d i s p l a c e m e n t s a p p r o a c h , as opposed to the displacement-stress approach (Sect. 4.7), where the stresses are also scaled. It is the displacement-stress approach that was first applied to the nonlinear clamped plate problem by Ciarlet & Destuynder [1979b]. It is only later that Raoult [1988] showed that the displacement approach constitutes in fact the proper justification of the displacement-
Sect. 4.3]
The method of formal asymptotic expansions
269
stress approach (see in this respect the discussion given in Sect. 4.7). Accordingly, we present the displacement approach first, thus reversing the "historical" order. While the plates considered by Ciarlet & Destuynder [1979b] and Raoult [1988] were "completely" clamped (~/0 = ~, i.e., the displacement is assumed to vanish over the entire lateral face of the plate), we consider here the more general case of "partially" clamped plates, where the displacement is assumed to vanish only over a subset of the form ~0 x I-e, el, "~0 C ~/with length y0 > 0, of the lateral face. The m e t h o d of f o r m a l a s y m p t o t i c e x p a n s i o n s applied to problem 7)(e; ft) consists in using the following basic A n s a t z : (i) Write a priori u(e) as a f o r m a l e x p a n s i o n U ( g ) --- U 0 -11-~ U 1 + C2U 2 -Jr-
h.o.t.,
where u ~ is called the l e a d i n g t e r m , and more generally u p, p >_ 0, is called the t e r m of o r d e r p, of the formal expansion; "h.o.t." is an abbreviation for "higher-order terms", which accounts in particular for the fact that the number of successive terms u ~ u ~, u 2 , . . . , that will be eventually needed is left unspecified at this stage; the expansion is "formal" in that it is not required to prove that the successive terms u 1, u 2, etc., do exist in the space V(ft), let alone that the above "series" converges! (ii) Equate to zero the factors of the successive powers c q, arranged by increasing values of q > - 4 , found in problem 79(c; f~) when u(c) is replaced by its formal expansion; (iii) Assuming ad hoc properties on whichever successive terms u ~ u 1, u 2, etc., are needed, pursue this procedure until the problem that the leading term u ~ should satisfy can be fully identified. In the present case, it turns out (Thins. 4.4-1 and 4.5-1) that carrying out step (iii) necessitates that the scaled displacement u(e) 4
be formally expanded as ( ~ cPu p + h.o.t.) with u ~ e V(f~), Oau~ e p=0
C~ and u p E wl'4(f~), 1 < p _< 4; in particular then, only the leading term is required to satisfy the boundary conditions found in the definition of the space V(f~).
270
Nonlinearly elastic plates
[Ch. 4
It is not the least paradoxical virtue of this m e t h o d that crucial information can be drawn about the leading term u ~ from the assumed existence of such a formal expansion (Thms. 4.4-1 and 4.5-1) even though the terms u ; of order p >_ 1 cannot usually satisfy the b o u n d a r y condition of place u" - 0 on F0; hence they cannot belong to the "original" space V ( ~ ) (the same restriction already holds in the linear case; cf. Sect. 1.12). There are even cases where already the leading term u ~ - (u ~ cannot fulfill the boundary condition of place! This occurs for instance in the osymptotic analysis of linearly elastic "membrane" shells (Vol. IIIX), where the "transverse" component u ~ only belongs to the space L2(~) and as such, cannot be required to satisfy the expected boundary condition u ~ - 0 on F0.
4.4.
CANCELLATION
OF THE FACTORS
OF
Cq, --4 < q _< 0, I N T H E S C A L E D
THREE-DIMENSIONAL
PROBLEM
The identification of the leading term u ~ in the formal expansion of u(e) will be carried out in two stages. To begin with, we gather all the information that can be derived from the cancellation of the factors of e -q, - 4 _< q _< 0, in the variational equations of problem More specifically, we first show that the cancellation of the factors of e q, - 4 < q _< - 1 , implies that the formal expansion of the tensor E(e) induced by that of the scaled displacement does not contain any negative power of e; this is a particularly striking simplification, since an inspection reveals that the expansion of E(e) is a priori of the form {g-4~]-4 _~_g-3y]-3 -4- h.o.t.}. Secondly, we show t h a t the cancellation of the factor of e ~ provides variational equations, which will play a key r61e in the sequel. The next result is due to Raoult [1988, Chap. 2, Sect. 2.2]. 1We recall that "Vol. III" stands for "Ciarlet, P.G. [1998]" Mathematical Elasticity, Volume III: Theory of Shells, North-Holland, Amsterdam".
Sect. 4.4]
C a n c e l l a t i o n of the f a c t o r s of e q, - 4 <_ q < 0
Theorem written as U(s
4.4-1.
271
A s s u m e that the scaled displacement can be
: U 0 -[- s
1 -~- C2U 2 -Jr- C3U 3 -~- C4U 4 -~- h.o.t.,
with U 0 -- (U 0) e V(~-~) -- {V -- (Vi) ~ w l ' 4 ( a ) ;
v -- 0 on F0},
03 uO e C ~
u p --(u~) E W~'4(f~), p >_ 1. Let ~-'~(e) "-- s
-~- e - - 3 E -3 -}- s
-~- s
-t-" E 0 -Jr- h.o.t.,
where the tensor fields Eq - ( a ] ) , q > - 4 , are independent of e, denote the induced formal expansion of the tensor ~E(c):= (aij(e)) found in Thin. 4.2-1 when u(e) is replaced by its formal expansion in the functions sij (u(e)) , p - - 4 , - 2 , 0 , 2; see again Thm. 4.2-1. Then the cancellation of the factors of c q, - 4 < q < O, in problem 7)(s; f/) successively implies: ~--]-4
E-a
-
0 and Oau ~ - O,
-
0 and Oau 1 - O, ~ +
E-2
-
0
and
~ = o,
Oaul - -
~(0o u~ + -~ 1~
o~ u o
-2 03 U a 03 1 +
E-1 - 0 and
a
1 ~ O~
Oauaa- - A ~ 2 # (O~176 + O~u~176 t!3
uo ~ u 1 a t!3
a
~
Nonlinearly elastic plates
272
[Ch. 4
1 o ~ OOouOa-Jr-~1 o ~ o 03uO)Sa~ ~o~ _ ~ ( o ~ + o ~ ~ + -2 + ,(o~~ + o ~ ~ + o ~ ~ 1 7 6 ~ 0 _ ~o _ , ( o ~ ~ + o ~
+ o ~ 0 03 ~0) ,
+ o~~
~o _ (~ + 2 ~ ) ( o ~ + ~103U203U2 + 03 ~ 0o ~ 2 + ~1 o ~ o1~ ) 1
+ ~(o~2 + o~~ fa a~
2 + -1~ o ~ 1 o ~ + ~o~ 1 uzO~uz), o o
dx + fa aij~176
- [
+[
J~
J r +UF_
dr for
v/a/.
Proof. (i) We recall the following simple result (Thm. 1.4-3)" Let w C L2(fl) be a function such that ~ w O a v d x - 0 for all
v
- 0 on "7 x [-1, 1].
c C ~ ( ~ ) that satisfy
Then w - O. In our applications of this result, we shall use the fact that, if F 0 - ~0 x [-1, 1] with ~/0 C ~/, then {v e C~(t)); v - 0 on 7 x [-1, 1]} c {v e W"4; V -- 0 o n r0}. (ii) Cancellation of the factor of C -4" W e have (Thm. 4.2-1)"
~ ( c ) - e-~-~ ~9 +
h.o.t.,
cry3 + h.o.t., 0"33(s ) -- s
-}- s
2
-~
h.o.t.,
with 0-3- 4 -- (/~-~-
2p)O3u~
+ 1 03u0),
cr3-aa - (A + 2#)(1 + 03u~
1.
Since then
If ~
dx + / f crij(g)Oiua(c)Ojv3 dx = e -4 f c~-a4(1 + 03u~ da
dx + h.o.t.,
Sect. 4.4]
Cancellation of the factors of c q, - 4 < q < 0
273
it follows that ~ ~r334(1 + C~3uO)c~3V3d x -
0 for all v C V(f~),
and thus, by (i), 0aU3~ + ~10au~
+ Osu~ - 0
Since we have assumed that 0~3u0 ~ C0(~=~), U 0 -- 0 on F0, and area F0 > 0, we conclude that the only possible solution to this cubic equation is Oau~ -
Hence or24 - 0, and thus
O.
E - 4 -- 0.
(iii) Cancellation of the factor of
g -3"
Since
C~3u0 -- 0,
~ cr{j(s)Ojv{ dx + ~ O'ij(C)OiU3(g)~V 3 dx ms
-3 /
0-3303V3 d x -Jr- h.o.t.,
and thus
j~~3303
V3 a~
- 0 for ~n
~ v(~).
V
Therefore, by (i), (733 -- 0. Hence E - a _
0 and OaU1 - 0 by (ii).
(iv) Cancellation of the factor of c -2" The expressions of the -1 _ 0 since functions cr~3(c) found in Thin. 4.2-1 show that (cr{~ - a~z E~ s - h.o.t, by (ii)and (iii))" ~(c)-
~o~ + h.o.t.,
0"c~3(C ) -- s
~(~)-
-2
-2 -1 -1 0 O'c~3 -Jr- s O'c~3 -Jr- O'a3 -Jr- h.o.t.,
~ ~ 3 3 + ~-1~;31 + ~33 + h.o.t., -
0
Nonlinearlyelasticplates
274
[Ch. 4
with 1
o
o
cr~ - A(/)au~ + / ) o u o + E 010Qo.UOoa,.U,0 -Jl- -~ O3Uo.O3Ua)(~o~ fl
+ ~(o~G + o ~ ~ + o~~176 ~ d - , ( o ~ ~ + o~~ o - ~(o~ ~3
+ o~ ~ . + 0 . ~ o 0 ~
+ o . ~oo ~ ) o.
~;~ - (:~ + 2 ~ ) ( o ~ + 2 o ~ ~
lo~176176176 ~ + ~ ( o ~ ~ + -~ 1
+ ~(Oa ~ 1 + o ~ o ~ 1 ) ,
~;1 _ (~ + 2 , ) ( o . ~
+ o~~
~o _ (~ + 2 , ) ( o ~
10au~Oau~ + Oa~O~u~ o 2 + -1~ o ~ o1~ 1 ~ ) + -~ 1
Since
~ crij(e)Ojvidxnt- ~ crij(e)Oiu3(g)Ojv3dx dx +
O ~247
hot,
and since Oau~ -O, we m u s t now have
fa ~-s
+ O~va + O~u~
+ fa a~Oava dx - 0 for all v E V(f~).
By considering functions v - (vi) first with v3 - 0, secondly with v~ - 0 , we find, again by applying (i), t h a t a~
- O,
hence t h a t O~u~ + Oau~ - O, then t h a t a3-a2 - O, hence t h a t E - 2 03U~ --
0 and A A+2#
1 ~o~)(o~ ~ + ~o~
~1 o ~ oo ~ . o
(v)
275
Cancellation of the factors of eq,-4 <_q <_0
Sect. 4.4]
Cancellation of the factor of g-1. We now have
~ a~j(e)OyV~dx
+ J['aa~y(e)O~u3(e)Ojvadx -- s
1 O'ijl - Ojvi dx + e 1 f a:~lcOiu~
dx + h.o.t.,
and thus we conclude t h a t -1 Oa3
hence t h a t
__ O~
O.u~ + Oau1 -O; likewise, cra-d - 0,
hence E - 1 -
0 and
c03u~ = - ~
A+2p
(vi) Cancellation of the E -1 - 0, we are left with
(Oo'Ula -J'--(~o"
factor of e ~ Since
0
E -4
1
--
__
E -3
--
E -2
=
fa o~j(e)Ojv~ dx + fa o~j(e)O~ua(e)Ojva dx - fa a~ Ojvi dx + fa crij ~ O~u~
+ h.o.t.
Hence the announced variational equations are satisfied for all v E v(a). "
Remarks. (1) Note the crucial r61es played in (ii) by the regularity assumption "0au ~ E C~ '' and by the assumption "area F0 > 0". (2) In order to avoid cumbersome statements, we have assumed t h a t u ~ belongs to V(Ft) and t h a t all the remaining relevant terms of the formal expansion belong to the same space Wl'4(f~). However, an inspection of the proof reveals that the same conclusions could have been drawn under weaker assumptions, since not all partial derivatives Oju~ occur for a given order p; for instance, the terms u a and u 4 occur only through their partial derivatives 0au~ and Oau4. Likewise, only u ~ is required to satisfy the boundary condition on F0. (3) The integral fa o~j(e)O~u~(e)Ojv~ dx that factorizes e 2 in problem 7)(e; f~) plays no rSle here; it would play a rSle only if the formal
Nonlinearly elastic plates
276
[Ch. 4
expansion of u(e) were pursued until it includes the term eau 6, and if the factors of c and e 2 were also cancelled. (4) The subsequent analysis is not altered if the asymptotic expansion of u(c) is a priori assumed to contain only terms of even order; cf. Ex. 4.1. II 4.5.
IDENTIFICATION OF THE LEADING IN THE DISPLACEMENT APPROACH
TERM
u~
Among other things, we have established in Thm. 4.4-1 that c~u ~
~
int2.
o is a scaled Kirchhoff-Love In other words, the leading term u ~ - (ui) displacement field (Sect. 1.4); as such, it belongs to the space 9KL(~'~ ) "-- {V -- (Yi) E wl'4(~"~)" V -- 0 on r o , Oiv 3 -qt-O3vi -- 0 in f~}. We are now in a position to make another crucial observation. We also saw in Thin. 4.4-1 that the following variational equations should be satisfied:
fa a~
dx+/~
cr~176
a m - / ~ fiv~ d x + f r
+OF_
9ivi dF
for ~n (~,)c v(~), or, more explicitly (recall that a~a~ _ cro).
j f c~~ +
dx + /~ ~,,..o z v~~ ~oaC,Zv3 dx
0 (03 v~ + c9~v3) dx + 0-~3
dx+/
/ oa~3 c9~u oa03 v3 dx +
cr~ 03 u ~ O~v3 dx
+UF_
We established in the same theorem that c93u~ - 0 (as already noted) and that the function OaU~ found in the expression of 0~/3 0 0 Hence if in the is in fact a known function of the functions &Uy.
above equations, we restrict the functions v = (v~) to belong to the
Sect. 4.5]
277
Identification of the leading term u~
space Vi
Theorem
the leading term u ~ - (u ~ of the formal asymptotic expansion of the scaled displacements satisfies the following l i m i t s c a l e d t h r e e dimensional
problem/)KL(Ft)"
u ~ C VKL(f~)"-- { V -
(vi) E HI(f~); v -
Oonr0,
cgiv3 + 0ira - 0 in ft },
fa cr~z~
~
u~
Y3 d x - / a f i v i d x + J r
for all v -
+UF_
gividF
(v~) E VKL(f~),
whe~'e
0 ._ .
E~
2~# +
-
~ .-
1
0
+
+ 2 # E O (u ~ , ~+
o
0
Pro@ As already observed, the variational equations found in problem 7)Kr(ft) directly follow from Thin. 4.4-1 and the definition of the space V K r ( f t ) . It thus remains to verify t h a t this problem is still well defined if we assume t h a t u ~ and the functions v belong to the space V K t ( f t ) , instead of the smaller space V K r ( f t ) . To see this, it sumces to observe t h a t the imbeddings Hi(a;) ~-, Lq(a~) hold for all q >_ 1 since a~ is a two-dimensional domain (cf. Vol. I, Thm. 6.1-3; the notation X ~-, Y means t h a t the normed vector space X is continuously imbedded in the normed vector space Y). T h a t u~ can
Nonlinearly elastic plates
278
[Ch. 4
be identified with a function in the space H2(a~) by Thm. 1.4-4 then implies that c)~u3i)~u 0 a0 E L ~(fl) for all r >_ 1, hence that a ~0 E L 2(ft); for the same reason, O~u~ E L~(ft) for all r > 1. Since all the multilinear forms found in problem 7)KL(ft) remain continuous over the larger space VKL(f~) once the functions a ~0 are replaced by their expressions in terms of the functions E ~ 1 7 6 and since Wm'4(~) is a dense subspace of g ' ~ ( ~ ) , any solution of these equations obtained by letting v vary in the space 9 K L ( ~ ) is also a solution of the same variational equations when v vary in the space
VKL(~t).
m
We first observe that, as expected, problem T)KL(ft) reduces to that found in the linear case (Thin. 1.4-1(b)) if all the terms that are nonlinear with respect to u ~ are canceled in its formulation. Noting t h a t both the unknown u ~ and the functions v appearing in the formulation of problem 7)KL(ft) belong to the space VKL(f~), we next show t h a t problem 7)KL(fl) is in fact a "two-dimensional problem in disguise" (as in the linear case; cf. Thm. 1.4-1), in the sense t h a t the three function u~0 are entirely determined by the solution = (~i) of a two-dimensional problem. The following result is due to Ciarlet & Destuynder [1979b]. Theorem
V(w) "- {~-
4.5-2. (a) Define the space
(rli) e H i ( w ) x H i ( w ) x H 2 ( w ) ; r h -
0 . ? ' / 3 - 0 on "Y0}.
The leading term u ~ - (u ~ is a s c a l e d K i r c h h o f f - L o v e d i s p l a c e m e n t field, in the sense that O~u~ + 03u ~ - 0 in f~. Hence (Thm. 1.4-4) there exists (~ - (~) E V(w) such that
0~__ ~a ~ X36~a~3 'U,O
and
u ~ - ~3
279
Identification of the leading term u ~
Sect. 4.5] (b) Let
1 "-- ~ ( 0 c ~ / 9 -3I- 6~/9
E~
1 p{'--/_
1
gi~ "-- g i ( ' , - 4 - 1 ) ,
/_l f { d x 3 + g + + g[,
q~ "-
1
x3f~ dx3 + g + - 9 2 .
The leading term u ~ satisfies problem T)KL(f~) if and only if r satisfies the following l i m i t s c a l e d t w o - d i m e n s i o n a l e q u a t i o n s 7)(aJ) of a nonlinearly elastic clamped plate:
= (~)e v(~),
- ~ Pi~i dw - ~ q~O~3 ace for all r; - ( r I i ) E V(co), where m~n "- ~
{
4 )t #
3(A + 2#)
/~ ~ 3 (~o~~ nt-
4)~# o ~ + 2~ E ~ ( r
.-
+
Conversely,
ma~
_
/1 1
1
0 X3Cra~
dx3
0 ~ ~ ~3
}
'
4pEOn
(c) The functions a~n~ defined in Thm. functions n~z and m~ n by 0
-3-
(r
4.5-1 are related to the
3
and
n~z --
1
0 dx3 . a~Z
Nonlinearly elastic plates
280
[Ch. 4
Proof. As c~u ~ + Oau0i _ 0, assertion (a) is simply a re-statement of Thm. 1.4-4. To prove (b), let first the functions v E VKL(FZ)in problem 7)KL(f~) be of the particular form V --- (--X3C~IT]3,--X3C~2713, T]3),
where T]3 E H2(w) and/13
s
=
0uT]3
Z3 (70c~/3Oa/3T]3 dx + f O'03C~ar Ja
--
0 on 70; this gives dx - s {--X3fc~0c~T]3 -~- f3/13} d x
q- ~F
+UF_
{--x3gaOa~3 -Jr-937"/3 } d F .
Next, let the functions v E V~cL(f~) in problem IZKL(f~) be of the particular form V --- (771,/'/2, 0), where r/, E H 1 (co) and r/~ = 0 on 70; this gives
facr~163163
+UF_
dr.
Writing each integral f a p dx as f~ ( f l I ~gdx3} do.) yields
s s
o X30"c~/3C~al3f]3 dx f3r]3 d x +
JF
+uF_
1
o x3crc~/3 dx3
93?73 dF -
C~c~3/13dx,
1 f3dx3 1
-1t- g+ -~- 93 } ?']3 dco, etc.
Using the definitions of the functions m~z, n ~ , p~, q~, we then obtain the variational equations indicated in the theorem. If, conversely, ~ = (~,) satisfies the scaled two-dimensional problem 7)(w), it is easily verified that u ~ - ( u ~ with u~0 __ ~a - X30c~ 3 and u~ -- ~3, satisfies problem T)KL(f~), and thus part (b)is proved. Part (c) immediately follows from the expressions of the functions ~ 0 given in Thm. 4.5-1. II
Remarks. (1) The limit two-dimensional problem 79(w) can no longer be broken into two independent problems as in the linear case
Sect. 4.5]
281
Identification of the leading term u ~
(their unknowns were respectively (r
1.5).
and (a; cf. Sects.
1.4 and
(2) If (a, r/3 r D(w), two applications of the fundamental Green formula give
s Oqc~/3r
dcu - - f~
0c~(OQc~r
dcd -- o~w 0c~c~r
d(~.
Since D(c~) is dense in H2(w), we thus conclude that, if 70 = 7, the first integral appearing in the left-hand side of the variational equations may be also written as -- fw mo,~0o~73 doa -
8/z(A -+- p) f Ar
3 daa for all r r]3 c
3(), + 2,)
(3) If p~ = 0, the functions n~ z C L2(co) satisfy
~ n~90~7~ da~ - 0
for M1 r]~ E Hol(W).
We show that this property implies that ~ T~c~/3C~cxr
dc~ -- - / ~ {n~90~9r
}7]3 dc~ for all ~3, ?]3 E /-/02((,0 ) 9
Again it suffices to establish this relation when 4a, r/3 E 2)(co), since both sides are continuous when 4a and r/a vary in the space Hg(co); but in this case, we can write
f~ Tbc~/30c~30/3113dw - s Tto~/30/3{(~c~r
} d(.,d -- s {T/,c~/30c~/3r }713d~,
and f~ n~O;3{(O~r = 0 since (0c~r E ~)(a2). Hence we conclude that, if % = 7 and p~ = O, the second integral appearing in the left-hand side of the variational equations may be also written as /wnc~/3Oc~r
dcu - - ~
{nc~0c~r 3}7]3 dcd for all ~3,7]3 ~ H3(a)).
(4) The notation E ~ ( r used in Thm. 4.5-2 is consistent with the notation E~ ~ used in Thin. 4.5-1. II
282 4.6.
Nonlinearly elastic plates
[Ch. 4
THE LIMIT SCALED TWO-DIMENSIONAL PROBLEM: EXISTENCE AND REGULARITY OF S O L U T I O N S , F O R M U L A T I O N AS A B O U N D A R Y VALUE PROBLEM
It has been shown in Thm. 4.5-1 that the leading term of the formal asymptotic expansion of the scaled displacement should satisfy the limit scaled three-dimensional problem 7:'KL(ft). By Thm. 4.5-2, examining questions of existence, multiplicity, or regularity, relative to solutions of 7)KL (f t) is equivalent to examining the same questions, but relative to the limit scaled two-dimensional problem P(w) found in this theorem. Remark. As the present asymptotic analysis is formal, it cannot be expected to provide "as a by-product" any information regarding the existence of a solution, by contrast with the linear case (Thm. 1.4-1). I Keeping this observation in mind, we begin by examining the question of existence of solutions to problem/)(co). The result of part (b) is an extension to the case where 70 C 7 of Ciarlet & Destuynder [1979b, Thm. 4.2] and the result of part (c) is due to Rabier [1979]; for related results, see Neeas & Naumann [1974] and Bielski & Telega [19881 9 We refer to Vol. I, Sects. 1.2 and 1.3, for a brief review of differential calculus in normed vector spaces. Other properties needed here, such as the sequential weak lower semi-continuity of functionals, are reviewed in Vol. I, Sects. 7.1-7.3. T h e o r e m 4.6-1. Assume that p~, q~ E L2(w) and length 70 > 0. Consider the scaled two-dimensional problem T)(w) of a nonlinearly
The limit scaled two-dimensionalproblem
Sect. 4.6]
283
elastic clamped plate, viz., find ~ such that
(- ((i)C V(w)"-- {~- (~7i)C Hi(w) •
HI((M) X
H2((.~);
r / i - O,,r/3 - - 0 on 70},
-~rn~O~rl3dw+jf
N~O~(30~rl3dw+ f N~O~ri~dw
-fpirl~do:-~q~O~ado:
for all rl E V(o:),
where
"~'
-
-
4A# 3(~ + 2 ~ ) ~ r
4A# o N ~ := A + 2p E~176
4p + -5 -~ +
4#EO
}
1
-
-5 a~€176162
o (r - a~#or162162
a~zor "- A + 2p 1 e~,(r
1
1 . - [ ( o ~ ( , + o,(~).
(a) Let the sc a l e d t w o - d i m e n s i o n a l e n e r g y (of a nonlinearly elastic plate) be the functional j 9V(w) -~ R defined by
J(~7) "- -~
{-~a~or162
+ a~or176162176
} dw
- ( j f pirlidw-~q~O~rl3dw), for all rl - ('r/~) E V(w). Then solving problem T)(w) is equivalent to finding all the stationary points of the functional j, i.e., those ~ that satisfy r c v(~)
~nd
j'(r
- o,
Nonlinearly elastic plates
284
[Ch. 4
where j' denotes the Frdchet derivative of j. (b) If the norms IP~lo,~ are small enough, there exists at least one such that (2 < V(w)
and
j((:)=
inf j(rl). .cv(~)
Hence any such minimizer ~ is a solution of 7)(a;). (c) If % = 7, the same conclusion holds without any restriction on the magnitude of the norms IP~lo,,~. Proof. (i) The functional j is differentiable over the space V(a;), and solving problem T)(a;) is equivalent to finding the stationary points of this functional. Since the continuous imbeddings Hi(w) ~ Lq(w) hold for all q >_ 1, the functional j is well defined and differentiable (in fact, infinitely) over the space V(oo), as a sum of continuous k-linear forms, k = 1, 2, 3, 4. Finding the expression j'((:)r/ for arbitrary functions (:, r/E V(w) thus amounts to identifying the linear part with respect to r / i n the difference (j((: + r / ) - j((:)). This gives
-(of
P~rl~d~-ofq~O~rl3dc~ ) 9
Hence r satisfies/)(co) if and only if j'(r if and only if j'((:) = O.
= 0 for all rl C V(co), i.e.,
(ii) The functional j is sequentially weakly lower-semicontinuous over the space V(a;). Given a sequence of functions rl k E V(co) such that (as usual, - denotes weak convergence): rlk~rl
in
V(co),
consider the behavior of the various terms found in j(rt k). The linear terms converge by definition of weak convergence (the corresponding
Sect. 4.6]
285
The limit scaled two-dimensional problem
linear functionals are continuous). The quadratic part of j, viz.,
12{1
}
is sequentially weakly lower-semicontinuous as it is continuous with respect to the strong topology of V(a~) and convex ( a ~ 9 ~ t ~ t ~ ~ > 4 # t ~ t ~ 9 for all symmetric matrices (t~9)). The compact imbedding Hl(cu) ~ L4(a;)implies that r/a 0~r/a Oor/a dw ---,
aac~arO~a/]a6~r~aOqa~]30~713 dw;
hence the quartic terms converge. Together with the weak convergences e~9(rl~ ) - - e~9(rlH ) in L2(a;), the same compact imbedding implies that
and thus the cubic terms also converge. Hence j is sequentially weakly lower semi-continuous. (iii) I f the n o r m s ercive on V(w), i.e., rl E V(w)
Ip~10,~ a ~ and
~o~gh, the
~all
Iinllv<~> ~
~
functional j is co-
~ j(rl) ~ +oc,
where
II/~llV(w) "--)l/~HIIl,co -~ 11713112,w. An inspection of the functional j shows that there exists a constant Cl - Cl (ql, q2, Pa) >_ 0 such that
-
c111~3111,~
-
~21n.10,~
Nonlinearly elastic plates
286
for all T / - (7"/H,~3)e Hi(w) • Hi(w) x
He(w), where
1/2 IT]312'w --
[0a/3T]3[0, w
[Ch. 4
~ IT~H[O,w - -
1/2
{ E
}
1/2
[T]al20,w
We have shown (proof of Thin. 1.5-2, part ( i ) ) t h a t there exists c3 > 0 such that ct ,~9
for all T/H = (r]~) that vanish on 7o (length 70 > 0 is needed here). Combining this inequality with the continuous imbedding HI(w) ~-, L4(a~), we infer that there exists c4 such that (recall that E~ -
10~30~)"
_
C3 'llrtH[[1,~ <
1
~ [E~
+ ~ ~ [lOo~T]3llL4(w)llOl3T]3llL4(w)
--< E [E~ (T~)[0,w j- c411T13[t22,w c~,O
for all r; E V(w). We have also shown (proof of Thm. 1.5-1, part (i)) that there exists c5 > 0 such that C5]]T]3]]2,w <_ ]T]312,w for all ?73 ~ H2(a-') that satisfy 713 -- Ov?]3 -- 0 on '~o (length 7o > 0 is again needed here). The conjunction of the above inequalities implies that
2p 2
)
2
j(T~) _> v C 5 -- C2C3C4 Ilr/3l]2,~- cxllr/3ll2,~
+ 2~ ~ IEo~[3(~)[0,r ~ ~
-
-
C2C3E
for all r/C V(w). Hence if c2 satisfies
2~c~
0
0 IE~z(r;)l~
Sect. 4.6]
The limit scaled two-dimensional problem
287
there exist Ca > 0, c7 > 0, and c8 such that
j(~) >_ c611~3112 ~,~+c~l
E ~9 ~ (~)12o,~+c~
for all rl E V(a~). Consequently,
=v j(r;) ~ +oc.
(iv) If
the n o r m s
Ip~[0,~ a<~ ~maZZ ~no~gh, th~ f~n~tionaZ j ha~ at
least one m i n i m i z e r over the space V(w), and any such m i n i m i z e r is a solution of 7)(a~).
Let ((:k) be a an i n f i m i z i n g sequence of j, i.e., Ck C V(w)
and
j(r
--+ inf j(r r
As j is coercive on V(w) if the norms IP~10,~ are small enough (part (iii)), the sequence (4k) is bounded; thus there exists a subsequence (~l) that weakly converges to some ~ E V ( a ) , since the space V(aJ) is reflexive and complete. As j is sequentially weakly lower semicontinuous (part (ii)), j((:) _< l i m i n f j ( ( t ) - inf j(~). l~oc r Hence ~ is a minimizer of j over V(co). That it satisfies j'(~) - 0 a paradigmatic property of differentiable functionals. (v) The proof of (c) is more delicate; see Rabier [1979].
is m
The existence (established in part (iv))of a minimizer of a coercive and sequentially weakly lower semi-continuous functional over a
288
Nonlinearly elastic plates
[Ch. 4
reflexive Banach space is a fundamental result, which pervades the direct methods of the calculus of variations; as such it is proved (in the same manner as above) in many texts; see, e.g., C~a [1971, p.62], Dacorogna [1989, p. 48], and Struwe [1990, p.4] (other applications of this result will occur in the proofs of Thins. 4.12-3 and 5.8-3). When % = -y, another approach to existence theory is possible, via the implicit function theorem (Ex. 4.2). II As an immediate corollary to Thms. 4.5-2 and 4.6-1, we have: A s s u m e that f~ E L2(ft) and g~ C L2(F+ U F_). Then if the norms rf~10,~ and ]g~10,~ are small enough, or if 70 - ~/, problem 7)KL(ft) found in Thin. 4.5-1 has at least one solution u ~ - (u ~ E VKL(ft); besides, u ~ C H 2 ( f~ ) . A w o r d of c a u t i o n " No mention is made here about the possible uniqueness of solutions, precisely because nonlinear problems often possess several solutions; however, this non-uniqueness is not of the kind observed in linear problems, where multiple solutions automatically occur in an infinite number as they lie in vector subspaces (a paradigm is the pure traction problem in linearized elasticity; cf. Vol. I, Ex. 6.3). By contrast, a nonlinear problem such as that considered here may have a finite number _> 2 of solutions. This kind of non-uniqueness is perhaps best illustrated by the yon Kdrrndn equations (Chap. 5), which may possess one, three, or more, solutions according to the nature of the linear form in their right-hand side (see Thms. 5.9-2, 5.11-1, and 5.11-2). II Observe that the minimizers of the associated energy (as those found in the previous proof) are only particular stationary points, and as such, provide only particular solutions to problem/9(co). Be that as it may, the multiplicity of solutions u ~ to problem PKL(f~) is "governed" by the multiplicity of solutions ~ -- (~) to problem 7)(co). in the following sense: To each function ~a, there correspond unique functions ~, and thus a unique u ~ We next write the boundary value problem that is, at least forreally, equivalent to the variational problem ?(co). In what follows, (r,~) denotes the unit outer normal vector along ~/; (r~) de-
Sect. 4.6]
The limit scaled two-dimensional problem
289
notes the unit tangential vector along 7 oriented in such a way t h a t the angle from (t,~) to (~-~) is 7r/2; 0" and & denote the associated outer normal and tangential derivative operators along 7, reC~4
spectively; A 2 . . . .
C~4
~ - 2 ~
04
+
= 0 ~ ( 0 9 9 ) denotes the
biharmonic operator;, finally, we let ~/1 " - - ~ - ~0" Theorem
4 . 6 - 2 . (a) Assume that the boundary 7, the functions
Pi, %, and the solution ~ of problem 79(c~) are smooth enough. Then - ( ~ ) satisfies the following boundary value problem: -o~~
- o~(x~9o~r -0zN~z
t~
= p~ + O~q~ i~ ~. = p~ in co,
= 0.C3 = 0 on ~0,
mc~ztzc~tzZ = 0 on ")'1,
(o~.~9 + N ~ O ~ ) . 9
+ 0 ~ ( ~ 9 . ~ 9 ) : -q~.~
on ~.
N~gu 9 = 0 on 71,
where the functions m ~ and N~ 9 are defined in T h m . 4.6-1. (b) If 7 - 70 and p~ - O, this boundary value problem takes the f o?~m:
3(A + 2#) A (a - N~z0~9~a - P3 + 0.q~ in w,
~
0oNto
- 0 in w,
-
-
O~a
0 on 7.
Proof. In the proof of T h m . 1.5-1, we have established the Green formula
+ Jj{(O~m~9)tJ ~ + g)~(m~tJ~r~)}rl3 d7 j.y
Nonlmearly elastic plates
290
[Ch. 4
We likewise have
L
Nc~/30~<3~/~T]3 dco-- -- L
{0~(Nc~13G~c~3)}/73dw
+
L(N~/90~<3)///3/73
d~/,
LN~zOzrl{~dw - -L(cgzN~z)rh~dw+/iN~z~'zr#~d'7, -L%cg{~r]3 dw-~(cg{~q~)rl3dw-Lq{~'{~rl3d"/.
Combining these various Green formulas with the b o u n d a r y conditions r]~ = 0"r/3 = 0 on 7o then yield the b o u n d a r y value problem given in (a). The special form given in (b) follows from the rela+
tion -O~zm{~z - ~-)~ T 89 A ~3 and from the equations ,gnNom - 0 obtained from (a) when p~ = 0.
II
Remarks. (1) As in the linear case, we always have -O~nm~ n = +
3(A + 2#) A ~3, irrespective of whether % - ~/or not. (2) If f~ - 0 and g~ - 0, then p~ - q~ - 0.
II
Finally we establish a regularity result that holds when % - 3'. The proof given here is due to Ciarlet & Destuynder [1979b, T h m . 4-2]; it crucially hinges on an idea of Lions [1969, p.56]. T h e o r e m 4.6-3. Assume that P3 E L2(w),p~ E Hl(w),q~ E H i ( w ) , the boundary ~/ is smooth enough, and % - "7. Let ~ ((~H, ~3) e H~(w) x Hg(w) denote any solution of problem 7)(w). Then
(~H --(r
E H ~ ( w ) A Ha(w)
and
r E H~)(w)CI H4((M).
Proof. By T h m . 4.6-2, any solution ((s
{3) E H~(w) x H2(w) of problem 79(w) satisfies the following equations in the sense of distributions (the notations are as in T h m . 4.6-1)"
291
The limit scaled two-dimensional problem
Sect. 4.6]
1
--Oz{ac~z~,~eo~(~H)} -- -~O~{aa[3a~-O(7~30T~3} -nt- Pa, 8,(/~ -~- ~)A2~3 __ O~{aa~c, rear(r162 3(,~ + 2#) + ~10 e{aaeaTO,~3OT~3Oa;3} + P3 + Oc~qa. Since c%~3 E HI(w)r
Lq(w) for all q _> 1 and 0z~a E L2(w),
OB{@aBcrTOcr<3Orr --(aaBaTO/3~<3OT-r
) E Lr(o3)for all 1 < r < 2,
and thus the equations satisfied by ~H show that
--O~{a~e~(~g)} E U(w) for all 1 < r < 2. Consequently, O {aa crre rr( H)Oa 3} =
3 + aaf3arear( H)Oa
For the same
3) E
LI(w).
reasons,
= (aa~aTOBa~3OT~3Oa< 3 - ~ . . . ) E LV(a)) for all 1 <_ r < 2,
and thus the equation satisfied by ~3 shows that A2s E L l(w). As a consequence, we show that ~'3 ~ H 3-6 (w)
for 5 > 0 small enough,
by resorting to an argument due to Lions [1969, p.56] (the definitions and properties needed here of the Sobolev spaces H~(w) for nonintegral values of s are found in Lions & Magenes [1968, Chap. 1] and Adams [1975, Chap. 7]): It is easily seen that the imbedding Li(w) ~-~ H-2(w) holds, as a consequence of the imbedding H2(w) ~-~ C~ But in fact the imbedding H~+~(w) ~-* L~ already holds for 5 > 0, where H0~+~(w) denotes the closure of D(w) in Hl+e(w) (Peetre [1966]). Hence we likewise have L~(~) ~-+ H-~-~(w),
Nonlinearly elastic plates
292
[Ch. 4
where H-1-6(CU) denotes the dual space of H~)+e(w). Consequently, save for some "exceptional" values of 6 > 0 that are automatically excluded if 6 is small enough, the implication
<3 ~ H3(co)
and
/~2<3 ~
H-1-5(co)~ r
E
H3-6(c0)
holds (Lions & Magenes [1968]), and the assertion is proved. The continuous imbedding H1-6(co) ~-+ L2/e(co) for 6 > 0 small enough (Peetre [1966]) then shows that 0~r C L2/e(co) for 6 > 0 small enough, hence that
O~{a~oTOo~30,-~3} C Lq(w)
for all q _> 1.
Considering the particular value q - 2, we thus have, by Thm. 1.5-2(c), ~u E Hi(co) and - 0 ~ { a ~ o . e o . ( ( u ) } E L2(co) :v ~ . C H2(co). Since 0~r C L2/~(co) for 6 > 0 small enough and a~ore~r(~I-i) E HI(w) ~-+ Lq(w) for all q >_ 1 imply
aa~a~-ear(~H)Oa~3 E L2(w), since Oz{a~zore~,T(~H)} E L2(w) and 0a~3 C H~ +5 ~ L~(w) for 6 > 0 imply Os{a~so,eo,(~H)}O,~a C L2(a~), and since 0 ~ 3 6 L2/5(co) for 6 > 0 small enough and 0 ~ 3 E Lq(w) for all q >_ 1 imply
O~{a~-Oo~30~
L2(a~); by Thin. 1.5-1(c),
~a 6 H3(w) Therefore 0 ~ 3
and
/xr2
L2
H4
E H2(a~) and 0 ~ 3 C Ha(a~) ~-* CI(~), and thus
O~{ac~-Oa~3Or~3} E H 1(co).
The displacement-stress approach
Sect. 4.7]
293
Another application of Thin. 1.5-2(c) then shows that (~u C H3(w), and the proof is complete. II
Remark. The same assumed regularities led to exactly the same regularity on the solution of the corresponding problem 79(w) in the linear case (cf. Thms. 1.5-1(c) and 1.5-2(c)with m = 1). As an immediate corollary to Thms. 4.5-2 and 4.6-3, we thus have: Assume that f~ E Hl(f~), f3 E L2(t2), g~ E /-/l(w), 9i3 E L2(w),
the boundary "7 is smooth enough, and "7o = "7. Then any solution u ~ - ( u ~ E VKL(ft) of problem T~Kn(ft)found in Thin. 4.5-1 has the following regularity: 0
H3
i,d
H 4(a).
Incidentally, this provides an instance where all the assumptions made about the leading term in Thm. 4.4-1 are satisfied. 4.7.
T H E M E T H O D OF F O R M A L A S Y M P T O T I C EXPANSIONS: THE DISPLACEMENT-STRESS APPROACH
In Thm. 4.2-1, we saw in part (b) that the functions chj(g) appearing in the scaled three-dimensional problem 7)(s; ft) found in part (a) could be also viewed as the components of an appropriately scaled second Piola-Kirchhoff stress tensor E ( e ) : = (Cho(e)). In Thm. 4.4-1, we saw that the formal expansion of the tensor E(s) induced by that of the scaled displacement does not contain any factor of e q, --4 _< q < - 1 ; we also showed that the pair (u ~ El~ with u ~ " - ( u ~0) and ~0 "-(oh~ satisfies
fa ~ ~vi dz + /a ~io~176
faf~v~dz+/; for
+OF_
v =
9iv~dF V(a).
Hence a simple inspection reveals that we would have obtained precisely the same equations had we required at the outset not only
294
[Ch. 4
Nonlinearly elastic plates
that u(e) be of the form u(e) = u ~ h.o.t., but also that the tensor E(e) be of the form E(e) - E ~ h.o.t., in problem 7)(e; Ft) found in Thm. 4.2-1. This simple, yet crucial, observation is the basis for another way of applying the method of formal asymptotic expansions, called the d i s p l a c e m e n t - s t r e s s a p p r o a c h . This approach, which we describe now. was originally proposed by Ciarlet & Destuynder [1979b]. To begin with, we reformulate the three-dimensional problem T)(~ ~) of a nonlinear elastic clamped plate made of a St Venant-Kirchhoff material (Sect. 4.1) as an equivalent, but different, problem Q ( ~ ) : First, the stress tensor field is considered as an unknown per se; secondly, the constitutive equation that relates the displacement vector field u ~ - (u~) and the stress tensor field E ~ - (a~j) is inverted, i.e., the Green St-Venant strain tensor E~(u ~) - (E~5(u~)) is expressed as a function of the stress tensor. Note that this is possible because the Lamd constants M and p~ are > 0 by assumption. More explicitly, we now consider that the displacement field u ~ and the second Piola-Kirchhoff tensor field E ~ satisfy the following problem Q ( ~ ) , consisting of the p r i n c i p l e of v i r t u a l w o r k , together with the i n v e r t e d c o n s t i t u t i v e e q u a t i o n :
U e -- (U~) ~ V(~"~ e) "-- { V e -- (V~) e w l ' 4 ( ~ ' ~ e ) ;
v -- 0
on P~},
E ~ - (a~j)E L~(~ ~) "- {(rib ) E L 2 ( ~ ) ; r~5 - r j ~ } ,
(% + kjOku " ~)O)v
dx"
--
fi
v~ dx" +
~
~urt
g~ v~ dP"
for all v ~ E V(~t ~), where E
j(u
.-
+
+
A
~
Note that the last relation may be also written as
1
O"e
The displacement-stress approach
Sect. 4.7]
295
where
W i t h the fields u ~ - (u~) and v ~ - (v~)" ~ ~ R 3, we then associate the s c a l e d d i s p l a c e m e n t field u(e) - (u~(e))" ft --, R a and the s c a l e d v e c t o r fields v - (v~) 9fi ~ R 3 defined by the scalings
u~(z ~) - e2u~(c)(z) and u3(z ~) - cu3(c)(z) for all z ~ - 7r~z E
~(~)
- ~(~)
~nd ~(~)
- ~(~)
for ~11 ~
- ~
c
,
as in the displacement approach (Sect. 4.2). In addition, with the stress tensor field E ~ - (cr~) 9f~ ~ ~3, we associate the s c a l e d s t r e s s t e n s o r field E(c) - (cr#(c))" ft ~ ~3 defined by the s c a l i n g s
~;~(~)
- ~~(~)(x),
~;~(~)-
~
~(~)(z), ~ ( x ~) _ ~4~(~)(~), for all x ~ = 7r~x E f t ~,
and we call s c a l e d s t r e s s e s the functions oij (C) : ~ ~ R. Of course, the scalings on the stresses are entirely motivated by the observation m a d e in T h m . 4.2-1(b)! Finally, we make the same assumptions on the data (the Lam6 constants and the applied force densities) as in Sect. 4.2. C o m b i n i n g the scalings on the displacements and stresses with the a s s u m p t i o n s on the data, we then reformulate problem Q(t2 ~) as a problem Q(e; ft) posed over the set ft, called the s c a l e d t h r e e d i m e n s i o n a l p r o b l e m in t h e d i s p l a c e m e n t - s t r e s s approach. As shown in the following theorem, problem Q(e; ft) consists of a s c a l e d p r i n c i p l e o f v i r t u a l w o r k (already found in T h m . 4.2-1(a)) and of a s c a l e d i n v e r t e d c o n s t i t u t i v e e q u a t i o n . Since this result is simply an alter ego of T h m . 4.2-1, its proof is omitted.
296
[Ch. 4
Nonlinearly elastic plates
4.%1. Assume that u ~ E W~'4(ft~). The scaled displacement field u(e) = (u~(e)) and the scaled stress tensor field E(e) = (cr~j(c)) satisfy the following problem Q(c; ~): Theorem
U(@) ~ V ( ~ ) " - E(c) E L~e(f~)"-
{'O -- (Vi) ~ w l ' 4 ( ~ ) ; v -- 0 o n Fo},
{(T~j)E L2(f~); T{j- Tj~},
L aij(c)Ojvi dx +/ua~j(c)Oiu3(c)Ojv3 dx +e2/aa{j(e)O{u~(e)Ojv~dz- ~ f i v { d z + fr + giv~ dE tAF_ for all v E V(f~),
E~ (u(E'))4- a2E~2j(u(c)) -- B~ (][](~'))+ c2B[j (][](~))+ a'4B4j (~(a')), where the mappings E~j - Ej~ 9V(fi) ~ L2(fi), p - 0, 2, and Bqj = Bq~" L~(ft) --~ L2(ft), q - 0,2,4, which are independent of e, are defined by 1
l o u~,(e)Oju~(e)
.(~(~)) +
<, )~
_ _
E~
1
~(~)}<~ + ~~(~),
+ ~E~(~(~)) - ~1 r ~ ( ~ ) , /~
= -2.(3~ + 2.){~"(~) +
s
~(~)} +
1 4
~ ~(~)
II
Remark. Thin. 4.7-1 is essentially an adaptation of Thm. 3.1 from Ciarlet & Destuynder [1979b]; there, the scaled inverted constitutive equation was instead expressed in a "weak" form, by taking
The displacement-stress approach
Sect. 4.7]
297
inner products against arbitrary "test-functions" Tij ~ L2(f~).
[]
As in the displacement approach, the dependence on the parameter e is now "explicit", by means of the powers c ~ e2, r more specifically, problem Q(e; ft) is of the form"
B(E(c), v) + T ~
u(e), v) + e2T2(E(e), u(c), v) - L(v) for all v E V(f~), + c2E2(u(c)) - (B ~ + E2B2 + c4B4)S(G),
E~
where the linear form L 9V ( ~ ) ~ R, the bilinear form B 9L~(~) • V ( ~ ) ~ R, the trilinear forms 7 "~ T2 9L~(~) • V(ft) • V ( ~ ) ~ R (the same as in Sect. 4.2), the matrix-valued mappings E ~ E 2 9 V(ft) --+ L~(f~), and the fourth-order tensors B~ B 2, B 4 9L~(f~)2 L~(~), are all independent of c. The notation used here is selfexplanatory; for instance,
BoE
.--
/
{ 2-~ O-11 -- 2tt(3~+2tt) O"7-'1-} 1 2#O"21
1 2pgrl2 1 A { 2-#~O-22 -- 2#(3A+2tt)
0
CTTT}
0
i/
for an arbitrary matrix E - (aij), etc. The polynomial dependence of these relations with respect to the parameter e again leads us to apply the basic A n s a t z of the m e t h o d of f o r m a l a s y m p t o t i c e x p a n s i o n s (Sect. 4.3), in the following way:
(i) Write a priori u(c) and E(c) as f o r m a l e x p a n s i o n s : U ( E ) -- U 0 -t- EU 1 Jr- E2U 2 -atE(C)
-- E 0 - ~ - C E 1 - t - C 2 E 2 -t-
h.o.t.,
h.o.t.;
(ii) Equate to zero the factors of the successive powers of s, arranged by increasing values of the exponents, found in problem Q(c; ~2) when u(s) and E(s) are replaced by their formal expansions;
Nonlinearly elastic plates
298
[Ch. 4
(iii) Assuming ad hoc properties on the successive terms found in both formal expansions, pursue this procedure until the leading terms u ~ and E ~ can be fully identified. While all the t e r m s u ~ and u 4 were needed in the formal expansion of u(c) in order t h a t u ~ could be identified in the displacement approach (Thins. 4.4-1 and 4.5-1), it is remarkable t h a t the identification of the leading terms u ~ and E ~ in the displacementstress approach does not require t h a t the formal expansions of u(e) and ]E(e) contain any further term, i.e., the leading terms u ~ and ]E~ are simply obtained by solving the equations:
g ( E ~ v) + T ~
for all v C V(f~), ~ ,u~ E ~ (u ~ - BoE o,
as we now prove. T h e o r e m 4.7-2. Assume that the scaled displacement and stress tensor fields can be written as
u(e) = u ~ +
h.o.t.
and
E(e) - E ~
h.o.t.,
and that the leading terms of these formal expansions satisfy
U 0 -- (U 0) ~ V ( f ~ )
r~~
-- { v ~ wl'4(~"~);
v -- 0 on Fo}, O3u ~ E C ~
(~?) ~ L~(~)- {(~j)e L~(~); ~ j - ~}.
Then the cancellation of the factors of e ~ in problem Q(c; ~) implies that the leading terms u ~ and E ~ satisfy the following problem
Sect. 4.7]
299
The displacement-stress approach
QKL(f~): u ~ e VKL(f~)"-- {V -- (Vi) E H I ( f ~ ) ; v -- 0 on Vo, OiVa + Oavi = 0 inf'},
X ~ e L~(a), O'ijOj v i d x +
aij
+UF_
g~v~dF
for all v E V(f~), where
~ o . = ~ 2+~ #2 E ~o( u o ) ~
+ 2#EO9 (uO) ,
o o ) . - -~(o~~ 1 e~(~ + o9~o + O~uOOzu o ) .
Proof. Cancelling the factor of c o in the scaled principle of virtual
work (Thm. 4.7-1) immediately gives the equations that should hold for all v E V(f~). Cancelling the factor of e ~ in the scaled inverted constitutive equation (cf. again Thin. 4.7-1) likewise gives: Eo
)~
o
1
o
~9(u~ - - 2 p ( 3 A + 2p)o-~_~_~z + ~-~o~9,
Eo~3(u ~ - ~1 (Oc~uO -Jr-03 ~ o + o ~ o o 3 ~ ~ - o,
Eo~(uo) - o ~ o + ~1 ( 0 3 u ~ ) 2 - 0 Inverting the first set of equations then yields the equations expressing a~~ in terms of the functions E~176 1 The equation E~ ~ - 0 is equivalent to 03u~ + -~03u ~ - O; hence 03u ~ -
0
Nonlinearly elastic plates
300
is the only possible solution to this equation, since Oau~ E C~ assumption, u ~ - 0 on F0 - % • [-1, 1], and area F0 > 0. equations E ~ ~ - 0 thus reduce to
~ +
~ -
[Ch. 4 by The
0.
Consequently, ,t o belongs to the space VKL(t2) defined in Sect. 4.5, hence a fortiori to the space VKL (ft). II A w o r d of c a u t i o n " The displacement-stress approach is thus seemingly simpler than the displacement approach, since the equations that constitute problem QKL(f~) are obtained "at a simpler cost" than the equations obtained in Thin. 4.4-1" It now suffices to cancel the factors of ~0 in the equations that constitute problem Q(c;Ft), while all factors of e q, - 4 _< q _< 0, had to be cancelled in problem 7)(c; Ft) in order that the same equations be obtained in the displacement approach (cf. again Thin. 4.4-1). This observation must be however amended in two ways: (i) It must first be proved that all the factors of c q, - 4 _< q _< - 1 , in the formal expansion of the scaled stress tensor E(e) do indeed vanish. As showed in Thin. 4.4-1, this is a consequence of the assumption that the scaled displacement vector u(c) can be formally 4 expanded as ~-s h.o.t.; it is thus the displacement approach p=0
that provides this unvaluable information. (ii) Once it is known that the formal expansion of E(c) starts with a zero-order term. it is clear that it is easier to use the inverted scaled constitutive equation; which only involves factors of c p, p _> 0. If the scaled constitutive equation were used instead as in Thin. 4.2-1, all factors of c q, - 4 < q < 0, would have again to be cancelled in order to provide the same information as in Thm. 4.4-1. II
Sect. 4.8] 4.8.
Identification of the leading term E ~
301
IDENTIFICATION OF THE LEADING TERM E ~ IN THE DISPLACEMENT-STRESS APPROACH; EXPLICIT FORMS OF THE LIMIT SCALED STRESSES
It remains to examine whether, and how, we can solve problem QKL(~) found in Thin. 4.7-2. Since u ~ E VKL(~2) as in the displacement approach, it follows (by restricting the functions v to belong to the space VKL(9) as in Sect. 4.5) that u ~ solves the same problem 7)Kr(Vt) that was found in Thin. 4.5-1. Since the functions (7~ are simultaneously obtained in this process (Thin. 4.5-1), it remains in effect to see whether, and how, we can also prove an existence result concerning the functions (7~a. 0 This is the object of the next theorein, where it is shown that, while explicit formulas can be found for o when 70 - "Y, which thus establish their existence, the functions 0"i3
problem ~KL(~'~) cannot be solved in general when % ~ ~/. 0 found in the next theorem The expressions of the function a~3 were first given by Ciarlet L: Destuynder [1979b, Thin. 4.1]; the observation of part (a) was first made by Blanchard [1981] and de 01iveira [1981]. We recall that, if 0 denotes a function defined almost everywhere and integrable over ~, f l 1 0 dy3 denotes the function defined for almost all (XI, X2) E COby f l 1 0(Xl, x2, Y3)dy3, and fx~ 0 dy3 denotes the function defined for almost all (Xl, x2, x3) E Vt by fx~ O(Xl, x2, Y3)dy3. Theorem
4.8-1. (a) I f % r 7, problem QKL(Vt) has no solution
in general. (b) If 7 is smooth enough, % - - y ,
and
fa E H I ( ~ ) , f3 E L2(~), g+ E Hi(w) C~
~
g3~ E L2(w)
problem QKL (Vt) has at least one solution. All its solutions (u ~ E ~ 0 ((u~ ((7~j)) are obtained first by solving problem 7)(co) (Thm. 4.6-1), whose solutions satisfy (Thm. 4.6-3)" ~'c~ ~ H1 (co) ~ H 3 (02) &nd
<3 ~ H3 (02) ~ H 4 (~),
Nonlinearly elastic plates
302
[Ch. 4
0 then by defining the functions u~o and the l i m i t s c a l e d s t r e s s e s O'ij by 0 Uc~ "-- ('c~ -- X3Oqa('3
1
u ~ -- ~-3,
3
c roz " = - 2N ~ + (7.0c~3
and
~ x 3mc~/~,
0 -- f~ } dy3, 9-- --gc~- -]- J : l a {__Oq/3 (TaZ
a~ "- -g3 +
{-OZcr~ - f3 } dy3 + g2 0 ~ 3 O ~ : a ,~~~
-
u 3) ~ dy3,
where the functions m , z and N~Z are defined as in T h m . 4.6-1 and satisfy: m , z E H2(w)
and
N , z C Hi(w).
These solutions possess the following regularities: ~.o e H 3 (a) , ~o ~ H 4 (a) , ~o~ c (c) The functions
H 2 ( f ~ ) ,~ o a3 e H 1( ~ ) , 0"303 e L 2 ( ~ ) .
0 ave also given by: 0"i3
~3 - ~(1 - ~3)0~,~.~ + ~(1 + ~3) 1
+ ~(1 + 1 :3 - -~x3(1
X 3)g~+
1
- ~(1 -
2 - ~3)0~,~
+ ~(1 + xa)
1
1
1
f . @3 -
f . @3
x3)g2,
3 + ~(1 - ~ ) m ~ O ~ 3
fa @a -
f3 dy3
1
.
- {-~(l + x3) f l f~dy3 - /:(~ f~dy3}O~3
1
+ ~(1 + x3)
+ x3
J__l
1 yaO~f~
cg~f~ dy3 -
1
d y 3 - ~ (1 + xa) 2
yaO~f~ dy3
/1
I cg~f~ dya
Sect. 4.8]
Identification of the leading term 1
+
E~
303
1
+ ~(1 + x~)g~ - ~(1 - x~)g; -
{1
+
~(1 + x3)ga
I
-
-
}
I
~(1 - x3)g-s 0ar
-~- ~(1
-
2
X3)0a(g + + g2).
Pro@ (i) Preliminary remarks. The only remaining possibility 0 is clearly to let the functions v in for computing the functions cry3 the variational equations of problem QKL(f~) vary in the whole space V(ft). In other words, we have to see whether we can find functions (703 E L2(f~) that satisfy the variational equations: ./o Oc~30(0c~V3 -~- 03Vc~ -~- 0c~U003V3)dx+ ./i Cr30303V3dx
- - / a c r a/3 ~ (Ozv~+O~u~
dx+/f~vidx+
f~ +uF_ gividF
for all v - (v~) E Wl'4(f~) that vanish on ro - "y x [-1, 1] (we have taken into account that 03u ~ - 0 ) . We first note that these equations are linear with respect to the unknowns ~ri3~and linear with respect to the functions v, and that, since the functions u~ and a~z 0 are already known, we may first solve
with respect to erda~by specializing the functions v to be of the special form (Vl, v2, 0), and secondly with respect to or~ by specializing the functions v to be of the special form (0, O, va). 0 then these We next note that, if these equations have solutions a~a, solutions are unique for, if cr E L2(ft) is such that ffl aOavdx - 0 for all v E C~ that satisfy v - 0 on ~ x [-1, 1], then cr - 0 (Thm. 1.4-3). (ii) Computation of the functions a~3 , necessity of the condition "Y0 - "Y. W h e n va - 0, the variational equations of part (i) reduce to 0
~ a~ Oav~ dx - - fa a~
o
g~v~ dF
for all functions v~ E WI'4(Q) that vanish on Fo, hence for all functions v~ E H~(ft) that vanish on Fo (the space WI'4(ft) is dense in Hl(f~), and all the linear forms appearing in the equations are continuous with respect to the norm of Hl(f~); hence we may replace the space Wl'4(f~) by the larger space H I ( Q ) ) .
Nonlinearly elastic plates
304
[Ch. 4
The basic idea for solving these equations consists in using a Green formula so as to transform, at least formally, this variational problem into a boundary value problem in f~, together with a set of variational equations posed over the remaining portion ")/1 X [--1, 1] of the lateral surface of the set ft, where 71 "- 7 - 70. We then show that this variational problem posed over 71 x [-1, 1] has no solution in general, ezcept if To - 7, while the boundary value problem in ft can always be solved explicitly. This is quite unexpected in view of the special nature of this boundary value problem, which turns out to be, for each (Xx,X2) E w, a two-point boundary value problem of the first order with respect to the variable x3. A consequence of this existence result is that the necessary condition for existence, viz.,
/~cr ~~O z v ~ d x - f f ~ v ~ d x + f r
+uF_
g~v~dF
for functions v~ C H l(fl) satisfying v~ - 0 on F0 and O~v~ - 0 in ft is automatically satisfied. More specifically, using Green's formula and taking into account the cancellations that occur in various boundary integrals, we obtain (the unit outer normal vector along 71 x [-1, 1] is denoted (nl, n2, 0))" /aa~
_
-
crzo Ozv,~ dx -
--
~f(03 O-a3 0 )Va d x + J~F O'a3 0 Vc~ dF o ~ +
(Oza~
dx
-
--
O'c~3 o Vc~ dF,
fr -
lx[-1,1] a~zv~n z dF.
Hence the variational equations of part (i) reduce when v3 - 0 to -- ~ ( 0 3 O-c~3)Vc~ d x qL fF 0
~ ) v ~ dx +
+
; +
g~v~ dF +
+
O-a3Vc 0 ~ dF __
f~v~ dx
/; _
g~v~ dF -
/
fr
O'a3Vc o ~ dF
0
, x[-1,1]
cr~zv~n z dF,
i.e., they are equivalent, at least formally, to the boundary value problem (~3
0 cry3 -- c 9 ~ c0~ + f~ in gt,
Sect. 4.8]
Identification of the leading term E ~ 0
305
+
(7c~3 -- gc~ on F + (70c~3 - - -g~ on F_
together with the variational equations: ~1X[--1,1]
o cr~zv~nz dF-0
for all v~
E
Hi(Q) that vanish on Fo.
Since (7~ 0 _ -~ 1 N a ~ + -~x3maZ, 3 and since each function ( X l , X2) ~ ")'1 --+ f l 1 X3Vc~ dy3 may equate the restriction to 7~ of any function ~ E Hi(w) that vanishes on 70 (to see this, consider the function v, E Hl(f~) defined by v~(xl,x2, x3) - 53X 3 ~ ( X l x2)), we conclude that, if the above variational equations are satisfied, the functions m ~ must satisfy the boundary conditions m ~ v ~ = 0 on 71. But these boundary conditions cannot be satisfied in general, since the functions m ~ are already entirely determined (together with the functions N~9 ) by solving problem 7)(w) (Sect. 4.6). Hence we are led to henceforth assume that 70=7. The remaining boundary value problem can be solved if and only if the compatibility conditions /-1 1 0/3 (70c~
@3_ / 11 f~ @3 + g+ + g2
(which simply expresses t h a t satisfied. But 0Z~~
1 =
N
f l 1 03 (70c~3 dY 3 _
(70c~3[F + -- (70c~31F-) a r e
3x309m~
l{j/_'1 } 3 0,ma,, f~dy3+g ++g2 +~x3 2 1
by Thm. 4.6-2, and thus these compatibility conditions are satisfied. Using this expression of the functions 09or~ we then obtain
306
[Ch. 4
Nonlinearly elastic plates 0.0 a3 -- - - g 2 -~-
0 {--0~O'c~ ~ -- f~} dy3
_ 3 (1 - x~)Ozm~z +
(1 + x3)
f~ dy3 -
1
f~ dy3
1
+ ~(1 + xa)g + - ~(1 - x3)g 2. The functions oa3~ and ~30"O3 obtained in this fashion belong to the space Hl(f~). Hence all the integrations by parts that led to the b o u n d a r y value problem are a posteriori justified, and we can conclude that a~3 0 as given supra indeed solves the variational equations of part (i) when v3 = 0. (iii) Computation of the function 003 . T h e proof follows essentially the same pattern as in part (ii). W h e n v~ - 0, the variational equations of part (i) reduce to 0"30303V3 d x -- - ~ o'O~C~c~U03C~qV3d x - ~ o'O3(0c~V3 -3t- C~c~U003V3)dx
+j['af3v3dx+fr
+UF_
g3v3 dF
for all functions V3 E Wl'4(~'-~) that vanish on Fo. Using the Green formulas (recall that we assume that 7o - 7) '
-
-
~/ /o
/o(O3o~
~O~u~
/o
dx -
o 3 O~v3 d x 0"o~
Cr~aO~U~Oav3 dx -
O~(a~176 (0~%3)v3
/o
03 ( ~ 3oo ~ u ~
-}-
~ +
dx, dx,
dz -
/o
~3 (O~u~
dr
+
0 O'a3
dr,
we infer that solving these equations when v~ - 0 is equivalent, at least formally, to solving the boundary value problem"
Sect. 4.8]
307
Identification of the leading term IEO
-a~~
- a~(~~
~ + a~~
+ o3( cr~a0~u o ~ + fa in ft,
o ~ g3+ - g +a0 ~ r on F + 0-33 a~ - -g~ + g2 0~r on r_,
which can be solved if and only if the compatibility condition j_l j~_l /_1 1
O,(cr~
~ @3 +
(which simply expresses isfied. Let us check this Since o ~o 1Na z + -~ _
/_~
O~cr~ @3 +
1
f3 dya + g+ + g; - 0
that f_l I c93a~3dy3 - a~ property. 3 -~x3rnc~z, we first have
-a~
-) is sat-
(o-0 D u 0
1
-
1
-O~,m~, -
{;
1
f3 @a + g+ + g;
) {? - O~
1
)
xaf~ dya + g+ - g2 ,
by Thm. 4.6-2. From part (ii), we next infer that 0~
~o
3
-
/x~
fl
1
+ ~(1 + x~) j_ o~f~ dy~
- ~(1 - ~ ) o ~ , . ~ ,
1
1
O~f~ dy3 -+- -~oqa(g +
1
- 9 2 ) + -~x3~
+ + 92),
and thus, 0~cr~3~dy3 - O~zm~z + 1
O~f~ dy3 1
- S l l { / ~ ~ O ~ f ~ d z 3 } d y 3 + O ~ ( g + -g-s
Hence the compatibility condition reduces to the relation
O~ (-/_11 Y3f~ my3+/_11 f~ dy3-j~_l 1{j_~a fa dz3 } my3) - 0, which certainly holds, since an integration by parts shows that: j~_l1 Yaf~ d y a - - SII {/_'~3 f~ dza } d y a +
f_~lf~ dya.
9
Nonlinearly elastic plates
308
[Ch. 4
Since the compatibility condition is satisfied, the function a~3 is given by
0
(733 -- - g ; - +
J_Xl't{ - 0 ~ ~
-- f3} dy3 + g20~(3 --
0 ~) dy3. J71a 0~ (crc~iOqc~u
It thus remains in principle to replace the functions cr~9~and era0 by their previously computed expressions in order to write or~a as a function of m,~, N~a, f~,g+, and 9~-. It turns out however t h a t carrying these computations through to their completion in a finite time is by no means obvious! To this end, we resort to an ingenious observation of Raoult [1988, Chap. 2, Sect. 2.4], who noticed that the functions ~r~a~may be altogether eliminated from the b o u n d a r y value problem t h a t is used for computing the function (,~a. To this end, we return to the variational equation t h a t we used for computing a~ viz.,
s 0"03Oq3v3dx - --/f2 crc~/30OqaU0Oq/3v3dx -- s crc~3 ~ ( Oa Va -l- oqc~u ~oq3v3 ) dx q- s f ava dx -i- j fr
+UF_
g3v3 dF
and we henceforth assume t h a t the functions vs are such t h a t va E Wl'4(ft) and va(',xs) E H~(a~) for all xa E [-1,1]; this is licit, since the space formed by such functions is dense in the space {v E wl'n(f~); v = 0 on F0}. W i t h each such function vs, we associate the functions
vc~ "--
c9c~v3dy3 + (oqc,u~
which satisfy (recall that u ~ E Ha(co) since Oat ~ - 0 and (3 C H4(co)Cq v~ E H1 (~)
O~v~ -
and
v. - 0 on Fo,
O~va dya + (O~u~
+ cg~u~
03v~ - O~va + O~u~
thus, for such functions va and v~" Oa3
~t 3
~
__
O'a3
Identification of the leading term ]Eo
Sect. 4.8]
309
Letting as in part (ii) v3 - 0 in the variational equations of part (i), we obtain -
a~
dx -
-
dx f~v~ dx ii cr~zOgv~ ~ o o %z(O~9~a)v3dz+ ~o +
cr~90~u309v3 d x -
+UF_
g~v~ dF
O~nva dya
f~v~ dx -
+UF_
In this fashion, the variational equations used for computing come 9
+UF_
dx g~v~ (IF.
o~3 be-
+UF_
+ fafavadx- faf~{i~~O~vadya} dx + t..JF_
for all v3 e W1'4(~'-~) such t h a t v3(',xa) E H{(w) for all x3 E [-1, 1]. If p and ~ are two smooth enough real-valued functions defined over the set ft, a simple application of Fubini's theorem shows t h a t
Nonlinearlyelasticplates
310
[Ch. 4
Combining this relation with several applications of Green's formulas (note that the functions w3 - f ~ v3 dy3 and O~w3 vanish along 7), we obtain
jf~~176
dx - ~ (0~ (70~){~3v3dy3} dx ~{t3
;
1 0~Z0 ~~ dy3 iv3 dx,
- f~ f ~ {~/~i'~'O~v3dy3 } dx - - /a f ~O~{ s '~v3 dy3 } dx
- fn{fiO~f~dy3}v3 dx" We also have, again by Green's formula, 0~
dx-
-fa(O3~,g3)v3 d x +
J/r+ O'03v3d F - fr_ O'03v3dF,
- ~+ 9a { tll OC~vady3 } dP - - ~ +O~ 3 dx - fa (0~9~+)v3 dx. The variational equations used for computing o~ may thus eventually be written as --~(030"03)U3 d x + fr O'03v3 d F - f r O'03v3 dP + 1 ~ 0~ a~3 dy3}vadx : / ; ( 0.0,~;~o~aUa)Vadx + / n { t O~aa~
+~favadx+~{JxiO~f~dy3}vadx-~(f~O~u~ -[-
-
+OF_
g3V3dr +
L (9'~O~u~ dl +
dF
( c~ga)V3 dx -
f
dl"
(9~O~u~
dF.
Accordingly, the boundary value problem used for computing a~ may be also written as (note that the functions a~3 0 no longer appear,
Identification of the leading term E ~
Sect. 4.8]
311
as was desired!)"
-
0~o - cr~OO~ou~ 0 0 dy3 + + ~ O~gcr~9 - f~O~u ~
3
in ft,
+
~~3 - g + - 9 + 0 ~ 3
O~f~ dy3
f3 +
3
onr+
or~ - - 9 3 + g20~(3
on F_,
and thus the function cr~ is given by"
a~ - - /~i~ a~176
- /~~ { fyi O~gcr~
dya
-f~_] f a d y a - /-7 {Jjl O~f~dza} dya+/_~(~ f~O~u~ - J~~~ 0ag + dy3 - g2 + 92 C~a(3 (we have already checked the validity of the compatibility condition fl 1 03003 dy3 -- 0303 IF+ -- 003 IF_ ). Let us transform the first two integrals appearing in the expression of or~ 9Using the relations o z0 _ -~ 1N~ 9 + -~x3m~o 3 , we first obtain
z__.[cr~ Oo~zU0 dy3 -
1 2 + ~3(1 -za)m~zO~zi3,
-
/j~~ {~1 a
}
O~za c~~ dza dya-- -
1
(1 + x a ) { ( 3 - xa)O~oN~z
2 -t- (2 -}- X3 -- X3)0c~/~?/tc~e}.
By repeated applications of Thm. 4.6-2, we next obtain:
= -O~m~9 O~N~
-
-O~p~
- pa -
O~s~ + p~O~(a,
.
Hence, upon combining these expressions, we find that
312 __ i ~ a
[Ch. 4
Nonlinearly elastic plates O.0 ,a
0
.
1
1
2
3
2
- ~ xa (1 - x a ) O ~ m ~ + ~ (1 - x a ) m ~ 9 0 ~ a 1
1
1
1
+ ~(1 + xa)O~q~ + ~(1 + x a ) ( 3 - xa)O~p~. An integration by parts also shows that
i;l~{LiO~f~dz3}dy.a-i~l~Yacg~f~dya + xa
a
O~f~ dya +
]_1 1
cg~f~ dxa.
It is then a simple computational matter to obtain the expression of the function a~a given in the statement of the theorem. 88 A w o r d of c a u t i o n . A close scrutiny of part (ii) of the above proof shows that the impossibility of solving problem QKL(f~) when 7o =/= 7 arises from the impossibility of satisfying the boundary condition c r0~ n ~ - 0 on 71 x [-1, 1]. This does not conflict with the linear case, where we showed (Thm. 1.6-1) that the scaled stresses cr~j(c) always converge, irrespective of whether 70 = 7 or not, to limits given by the linear part of the expressions found in Thin. 4.8-1(b). But the convergence of ao(c ) as e + 0 was then established in spaces with such "feeble" norms that the boundary conditions need not be satisfied by the limits; in fact, they do not even make sense in these spaces! m
Remark. The terms involving functions m~o in the expressions of the functions or~a may be expressed in terms of the function ~a, as"
8~(~++2t~) ~)0.Ar 8~(~+ ~)/x~
O, rn~, - - 3 ( ~
Sect. 4.9]
The two-dimensional equations; nonlinear Kirchhoff-Love theory 313
4
4A#
II
-
Note that the multiplicity of solutions (u ~ E ~ to problem QKC(ft) is "governed" by the multiplicity of solutions ( = (~) to problem P(a~), in the following sense: To each function G, there correspond unique functions (~, m~ 9, and n ~ , hence a unique solution (u ~ E~ While problem QKL(f~) cannot be solved in general if ~/0 r 7 (Thm. 4.8-1(a)), its "subproblem" T)KC(f~) can always be solved (this observation is due to Blanchard [1981]); more specifically, even 0 found in its when QKC(~) has no solution, the unknowns u 0i a n d cr,~ formulation can always be computed and given a meaning by solving problem 7)KC(t2).
This fact alone would justify the consideration of the displacement approach, which precisely leads to solving problem 7)Kr(t2). Otherwise, i.e., if only the displacement-stress approach were considered, no plausible reason could be found explaining this "only partial" solvability of problem QKL(ft), observed when the displacement-stress approach is applied to a partially clamped plate. 4.9.
T H E T W O - D I M E N S I O N A L E Q U A T I O N S OF A NONLINEARLY ELASTIC CLAMPED PLATE; NONLINEAR KIRCHHOFF-LOVE THEORY
In view of the scalings made in Sect. 4.2, we define functions ([ "~ ~ R and u~(0) 9ft ~ R through the de-scalings: ( ~r ' - s
2
(~ andCac . - s C a i n ~ ,
s u~(z) and us(O)(z ~) "- su~
for all z ~ = 7c~z E f~
As in the linear theory (Sect. 1.7), the functions (~ are called the d i s p l a c e m e n t s of t h e m i d d l e s u r f a c e (of the plate), the functions (~ are called the in-plane displacements and the function (~ is called the transverse displacement, and the functions u~(0) are called the l i m i t d i s p l a c e m e n t s (inside the plate).
314
[Ch. 4
Nonlinearly elastic plates
These de-scalings, combined with the assumptions on the data made in Sect. 4.2 yield the following corollary of Thm. 4.5-2, where (f[) C L2(Ft ~) and (g[) E L2(F~_ U F~) are the actual densities of the applied body and surface forces and M and #~ are the actual Lamfi constants of the material constituting the plate: T h e o r e m 4.9-1. (a) The displacements ~ of the middle surface satisfy the following variational problem 7)~(w): ~e ._ (~;) e V ( w ) - { r / - (~i) E Hl(w)•
• H2(w); V~ = 0~Va on % }
-/,., m:~O~arl3 d w + f~ N~oO~(~ O~rl3d w + f~ N~zO~7~ dw -
/p~rli d w - [ q:O,~rl3dco for all r t - (rh)E V(co), J,o J,,
where
m~z
9- -c3{
N~--c
4M# ~ 4# ~ ~} 3(A~ + 2p~)A(~5~z + - ~ O ~ ( 3 ,
A~+2# E ~
~
,
1
E~162~) -- ~(0~r + 0,r + 0~r162 p~ "-
f: dx~ + g+~ + g;~, 9 +~ . -
g:(.,
~),
~ .__
~ ~ x3f~dx ~ + c,( g + ,~ - g~-~ ),
g:-~ .- g~(., -c).
(b) The vector field u~(0) = (u~(0)) is a Kirchhoff-Love disp l a c e m e n t field, in that the limit displacements u~(O) satisfy O~u~3(O) + O~u~(O) = 0 in ~ . Consequently, the functions u~(O) are of the form (Thm. 1.4-4)
Sect. 4.9]
The two-dimensional equations; nonlinear Kirchhoff-Love theory 315 u~(O)--~-x30~3
and
u3(0)-~3
in
,
and the functions ~ are those found in (a).
I
Remarks. (1) T h e relations ma~
_
c4maz,
N aZ ~
-
p ~a --
,
c3N~9
c3Pa,
hold between the functions defined in T h m . analogs defined in T h m . 4.5-2.
qa
--
c4qa,
4.9-1 and their scaled
(2) T h e notation E ~z( ~ r ) is consistent with the notation E ~ 1 6 2 introduced in T h m . 4.5-2, since only partial derivatives with respect to x~c - x~ occur in both cases. I Arguing as in T h m . 4.6-2 (and using the same notations), we next write the two-dimensional boundary value problem t h a t is, at least formally, equivalent to the variational problem 7'~(w). T h e o r e m 4 . 9 - 2 . (a) Assume that the data and a solution ~ of problem 7)~(~) ( T a m . 4.9-1) are smooth enough. Then ~ = ( ~ ) satisfies the following boundary value problem:
-O~zm~z - O z ( N ~ z O ~ ) - p~ + O~q~ in ~, - O z N C~/3 ~ - p~(It i n w r
-
0.(~
-
0
on 70,
s
m,~Vo~V~ -- 0 on 71, (O~m~ + N~D~)v~
+ 0~-(m~u~T~) - -q~u~ on ")/17
N ~ u 9 = 0 on 717
where the functions m~z,~ N ~9, P~,~ q~ are defined as in Thin. 4.9-1. (b) If 7 = 70 and p~ = 0, this boundary value problem may be equivalently written as:
[Ch. 4
Nonlinearly elastic plates
316
8/re(/~c -+-/ff)C3/k2~3
3(A~ + 2#~)
~
_ N ~ ~ _ c c~0c~f~3 P3 +
O~q~ in ~,
0 z N ~ , = 0 in ~, ~
-
0.~i
-
0 on
"~ I
Note that one always has
__Oc~m~/3
---
8/-re( A~ -~- ]-re) C3
3(A~ + 2# ~)
2 e A ~'3"
A w o r d of c a u t i o n . The two-dimensional equations found in either theorem can no longer be split into two separate problems, one solved by ~ and the other by ( ~ ) , as in the linear case (Sect. 1.7). II The existence and regularity of solutions to problem 7)(w) established in Thins. 4.6-1 and 4.6-3 apply verbatim to the solutions of problem 7)~(w). In particular, solving T)~(w) is equivalent to finding all the stationary points of the t w o - d i m e n s i o n a l e n e r g y j~ : V(w) --+ R (of a nonlinearly elastic plate) defined by
-(fp:r/~da~-fq~O~rladaJ
)
where a,~o~
A~ + 2# ~
EOc~/3(T~) -- ~1 (Oo~T]/3 + O/3T]o~ + Oo~T]SOI3T]3) 9
for all rl E V(w),
Sect. 4.9] The two-dimensional equations; nonlinear Kirchhoff-Love theory 317 The two-dimensional energy is the sum of the same flezural energy as in the linear case and of a membrane energy that differs from that of the linear theory, through the nonlinear terms 0~r/309~]3 appearing in the expressions E~ The variational, or boundary value, problem found in Thins. 4.91 and 4.9-2, together with the specific form of the field u~(0) (Thin. 4.9-1 (b)), constitutes the n o n l i n e a r K i r c h h o f f - L o v e t h e o r y of an elastic clamped plate: First, it comprises the t w o - d i m e n s i o n a l principle of virtual work"
-jf
rn~90~r/ada~+J2N~~9 0 ~3c99 ~ rla da~ + s - ~ P:Wi da~- f~ q:0~r/a d~
~c9~7~ dcz ~
for all r / - (r/i)E g(a~),
or the equivalent (at least formally) t w o - d i m e n s i o n a l equations of equilibrum:
- 0 ~ 9 m ~ 9 - 0 9 ( N ~ 0 ~ ( 3 ) - P3 + O~q~ in cz, C
C
C
C
- O z N ~ z - p~ in w,
m~;~u~u;~ -- 0 on ~/1, c
N~9~ z = 0 on 71" It also comprises the t w o - d i m e n s i o n a l b o u n d a r y c o n d i t i o n s of clamping (i.e., those boundary conditions that are imposed on the displacements r of the middle surface)"
r
-
O~ r
-
0 o n "Yo.
Finally, it comprises the t w o - d i m e n s i o n a l constitutive equations
318
Nonlinearly elastic plates
[Ch. 4
of a plate:
m~
Ve ~ -
N~z(V4
-e a{
~)
4M#~ 4M#
~
4#~
~}
zxC~9 + --~0~zCa ,
E~162 ) ~ + 4-~E , ~(; ~ ~)},
where
2e
V ~a-
011~ 012~
021;~ 022;~] and V r
/ 01~1~ 02~1~ ~ 01;~
02;~1
01;~ 0~r
"
The mappings (rh~ ~ z ) 9g2 ~ g2 and (fi~ ~ z ) 9Nia• ---, g2 defined in this fashion (g2 denotes the set of symmetric matrices of order 2; NIa• the set of matrices with three rows and two columns) are called t w o - d i m e n s i o n a l r e s p o n s e f u n c t i o n s . In this context, the tensor E~Z( )) is called the t w o - d i m e n s i o n a l s t r a i n t e n s o r . The two-dimensional constitutive equations may be written in a form more reminiscent of the three-dimensional constitutive equation used here. We recall (Sect. 4.1) that it takes the form a~j
~j(V~u - a^~
~)-x~
~ ~ Epp(u~)6~y + 2# ~ E~j(u ~)
_
~ ~ A~y~qEpq(U ~) ,
where
E~j(u~) - -~(o~ + o;~ + o ~ : a ; ~ : ) denote the components of the Green-St Venant strain tensor, and
denote the components of the three-dimensional elasticity tensor. In order to mimic these formulas, we write m~z
__
~ ~ - e 3 a~z~,~.oq~,~_~3 and
N ~ ~
- r
~
o
( ~ ),
Sect. 4.9] The two-dimensional equations; nonlinear Kirchhoff-Love theory 319 where the matrices (c9~;~) and (E~;~(~)) are the c h a n g e of c u r v a t u r e and c h a n g e of m e t r i c , or s t r a i n , t e n s o r s of the middle surface of the plate, and the constants
a~o~ ~ -
4)~# ~ ~ + 2/*~ 5 ~ 9 ~ + 2 # ~ ( ~ 5 ~
+ 5~59~ )
are the components of the t w o - d i m e n s i o n a l e l a s t i c i t y t e n s o r of the elastic plate under consideration; as in the linear case (Sect. 1.7), these definitions are best understood if the plate is viewed as a special case of a shell (Vol. III). We emphasize that the asymptotic method followed here thus auto2~# ~ matically justifies the "passage f r o m / ~ to " in the elasticity i~~ + 21,~ tensors (as in the linear case; cf. Sect. 1.8). Note that the two-dimensional constitutive equations of a plate are often expressed in terms of the Poisson ratio z~~ and Young roodulus E ~ (Sect. 1.2) as: m~ __ 2E~ c3 { z/eA~3( 5a~ l eif_ ~ 3(1 -(tJ~) 2)
N~e = (1 -2E~ (~)~)c
{ sE..(g~)e~e 0
- z/e) 0 c~/~~3, e}
+ (1 -
s
0~}) )&9(
The coefficient D ~ .__ 8# ~( ~ + #~) e3 _-
2E~
3(~ ~ + 2 ~ )
3(1 - ( s ) 2)
g3
that factorizes A2(~ in the boundary value problem found in Thm. 4.9-2 is called the f l e x u r a l r i g i d i t y of t h e p l a t e (as in the linear case; cf. Sect. 1.7). In view of the scalings made in Sect. 4.7 on the stresses, we are led to defining the l i m i t s t r e s s e s ~r~5(0) 9~ -+ IR through the following de-scalings:
Nonlinearly elastic plates
320
0.~/3 (0) (Xr
__ 6.20.0o~13(X), 0"o~3 ( ~(0)) -X-r
[Ch. 4
0 3 (X), 0"33 e (0) (Xr C 3 O'c~
-- ~40"13(X),
for all x ~ - 7r~x E
,
where the functions a0j are the limit scaled stresses found in Thm. 4.8-1. We then have the following corollary of this theorem: T h e o r e m 4.9-3. Let the assumptions be as in Thin. 4.8-1(b); in particular, 70 - 7. Then the limit stresses ch~(O) are given by (the functions m~m and N ~ are defined as in Thin. 4.9-1)" 1
3
<,(o) - ~ms + ~3~.~2s.
'{
09~T/~~z ~
O'ea3(O) 9 -- ~--~C 1 - -
/Ddy3+7 1+
-
e
(
+ ~1 1 + ~X 3
g~-7
e X3{ (__~)2 }
1
-4- 7
-
X3 1 -4- - -
( ')I_': x3 g
r e A dy3
e
=
y30afa~ dy~ -
c "~
2
1+
S;
e
cO.f; dy I
1 ( x1~+) - + x; . i "~ O~f OL~ dy; - S_'iE y;O~f; dy; + -~ C 1
2
1-
x;
c
.
f2dY/~- f x'= A d y l } o~162
{1( =)? 7 l+--xz 1
1---~ 92~,
e 2] 3 { (X3) ~r
e f~ dY3-
+ ~ 1 + --
f~r dy 3e
93- ~ _
2
1+
9~
-~
1---
s
g3+= 92 ~ 0~r 3
2
+-4
1-
O,(g,
+g,).
II
Sect. 4.10]
Justification of the nonlinear Kirchhoff-Love theory
321
Since the functions m ~ , N s0, ~ and cr~ (0) are also related by the formulas m~z -
/:
xao-~o(O) dx~ and N2~ -
f
(r~, (0) dx~ c
the functions m~z are called (as in the linear case; cf. Sect. 1.7) the s t r e s s c o u p l e s and the functions N~Z are called the s t r e s s result a n t s . The functions m~z are also called b e n d i n g m o m e n t s . The expressions giving the stress couples m~z and stress resultants N~z are justifiably called two-dimensional "constitutive equations", since they are defined in terms of partial derivatives of the functions ~~ through response functions m~z^ ~ and N~z ^~ that depend solely on the elastic material constituting the plate, through its Lam6 constants. We can likewise consider that the limit stresses a~9(0 ) are given through an "-~ x a- dependent" constitutive equation of the form
~
~
~:, - ~:,(x~, v r ~, v:C~).- 2--22~9(vr
~)
3x~
+~
~9(v
:r
A w o r d of c a u t i o n . By contrast, the expressions giving the limit stresses crib(0) cannot be considered as constitutive equations, since they explicitly depend on the applied force densities. II 4.10.
J U S T I F I C A T I O N OF T H E N O N L I N E A R KIRCHHOFF-LOVE THEORY; COMMENTARY REFINEMENTS AND GENERALIZATIONS
J u s t i f i c a t i o n of t h e "classical" n o n l i n e a r K i r c h h o f f - L o v e t h e o r y . A major conclusion that can be drawn from Thins. 4.9-1 and 4.9-2 is that they provide a complete mathematical justification of the "classical" nonlinear Kirchhoff-Love theory; indeed, the equations found in these theorems coincide with those found in the literature, where they are usually derived from the equations of threedimensional elasticity through a priori assumptions, of a geometrical or mechanical nature.
322
Nonlinearly elastic plates
[Ch. 4
Usually, these assumptions either specify how the components of the displacement vector and stress tensor should vary (as functions of x~) across the thickness of the plate, or they specify which terms should be "neglected" in the three-dimensional problem, in view of their hypothetical "smallness". More specifically, without any such a priori assumption, we have found the standard two-dimensional equations of equilibrium, the standard two-dimensional strain tensor, and the standard two-dimensional constitutive equations, of a nonlinearly elastic plate (see, e.g., Landau & Lifchitz [1967, eqs. (13,2), (14,1), (14,4), and (14,5)]). In the same fashion, we have shown that the de-scaled vector field u ~(0) = (u~ (0)) is a Kirchhoff-nove displacement field. This is usually obtained from an a priori assumption of a geometrical nature (Ex. 1.5), according to which the normals to the middle surface should remain normal to the deformed middle surface and the distances on these remain unchanged; see, e.g., Washizu [1975, eq. (8.60)]. Likewise, the expressions found in Thm. 4.9-3 for the limit stresses a~z(0 ) are standard in nonlinear plate theory (see, e.g., Stoker [1968, pp. 42-47], and Landau & Lifchitz [1967, eqs. (13.2)and (14.1)]), where they are usually derived after a priori assumptions have been made regarding which terms should be "neglected" in the components of the three-dimensional strain tensor; the expression found (again in Thm. 4.9-3) for the limit stresses a~a(0 ) are similar to those found in Green & Zerna [1968, eq. (7.7.3)], after they have been assumed to be quadratic in x~, etc. P a s s a g e from a quasilinear to a s e m i l i n e a r s y s t e m of partial differential e q u a t i o n s . A striking effect of the application of the method of asymptotic expansions is a "partial linearization" of the three-dimensional equations, in that a system of quasilinear partial differential equations, i.e., with nonlinearities in the higher order terms, is replaced as e ~ 0 by a system of semilinear partial differential equations, i.e., with nonlinearities only in the lower order terms. More specifically, the quasilinear equations
-0~ { ~ j ( V u ~) + ~r~j(Vu~)O~u~} - f~ in ft ~, (where the response function E~ -
(~j) could be in fact that of
Sect. 4.10]
Justification of the nonlinear Kirchhoff-Love theory
323
a more general elastic material; see below) are "replaced" by the semilinear equations (recall that rh~9(X72~) is a linear function of the matrix V 2~; cf. Sect. 4.9) ~
-
n~9
)0~3 }
- 0 z ~z(V~ ~)
_
P3 + O~q~ in co,
p~ in co.
Nevertheless, this "semilinearization" has its price: Contrary to the original three-dimensional equations, the limit two-dimensional equations are no longer frame-indifferent (Ex. 4.3). Another "linearization" affects the nature of the displacement field across the thickness of the plate: The limit displacements u~(O) are affine functions of x~, as (u~(0))is a Kirchhoff-Love displacement field (Thm. 4.9-1). As was already pointed out in Sect. 4.1, the only available existence result for the three-dimensional problem with "changing" boundary conditions of the type considered here, is that of Ball [1977] (also described and established in Vol. I, Chap. 7), who proved the existence of minimizers of the associated energy. This otherwise powerful and elegant approach suffers from one drawback, as the minimizers do not necessarily satisfy the associated quasilinear boundary value problem, even in the weak form of the principle of virtual work, which is the "prelude" to our use of the method of formal asymptotic expansions. This drawback is not encountered in the existence theory based on the implicit function theorem, but then this second approach is limited to smooth boundaries and to very special cases of boundary conditions, such as a boundary condition of place everywhere along the boundary of the reference configuration (see Ciarlet and Destuynder [1979b], Marsden and Hughes [1978], and Valent [1979, 1988]; see also Vol. I, Chap. 6), with one noticeable ezception, however: Paumier [1990] has shown that the implicit function theorem can similarly provide an existence theorem, the first of its kind, for a nonlinearly elastic rectangular plate either subjected to periodic boundary conditions, or with "sliding edges". By contrast, there exist satisfactory ezistence (Thm. 4.6-1), regularity (Thm. 4.6-3), uniqueness, multiplicity, and bifurcation results
324
Nonlinearly elastic plates
[Ch. 4
(see Sects. 5.8, 5.9, and 5.11 and the references quoted therein) for the two-dimensional nonlinear plate equations associated with various kinds of two-dimensional boundary conditions. That these results have at the present time no counterpart for the "original" threedimensional equations illustrates the fact that semilinear partial differential equations are usually much easier to study than quasilinear ones. A natural idea for establishing an existence result for a nonlinearly elastic plate consists in trying to construct a three-dimensional solution u(c) of problem 7J(~; f~) close to a "two-dimensional" solution u ~ of the limit problem J~KL(ft), by means of the Nash-Moser implicit function theorem (Moser [1961]). The first successful attempt in this direction is due to Paumier [1985, Chap. 5]. T h e nonlinear Kirchhoff-Love theory as a " s m a l l displacement t h e o r y " . Following Miara [1994b], we shall see in the next section that a rigorous justification can be given of the scalings u3(x e) -- cu3(c)(x) and u~(x ~) - c2u~(~)(x)
for all x e - 7rex C
performed on the displacements. Consequently, the displacement field engendered by the nonlinear Kirchhoff-Love theory necessarily satisfies, in some formal sense at least, ~ - cu3(., 0) - O(c) and r
- e2u~( ., 0) -- O(E2).
Anticipating this justification, we can therefore assert that the nonlinear Kirchhoff-Love theory is a "small displacement theory": It remains valid only if the transverse displacements remain of the order of the thickness of the plate. That the in-plane displacements are in turn of the order of the square of the thickness should then be intuitively clear, geometrically.
"Large displacement" nonlinear two-dimensional theories and convergence theorems. Following Fox, Raoult & Simo [1993] and Le Dret & Raoult [1995], we shall see later (Sects. 4.12 and
Justification of the nonlinear Kirchhoff-Love theory
Sect. 4.10]
325
4.13) that there do exist "large displacements", also called "large deformations", nonlinear two-dimensional theories, which may be likewise justified by an asymptotic analysis; one of these, called the "nonlinear membrane theory", is even justified by a convergence theorem. Note that such a convergence theorem that would likewise justify the nonlinear Kirchhoff-Love theory is still lacking at the present time. In this direction, some hope may reside in the novel and penetrating approach that Mielke [1988, 1990] has successfully applied for justifying one-dimensional equations of nonlinearly elastic rods; an illuminating introduction to this approach is given in the book by Antman [1995, pp. 559-564]; its application to linearly elastic plates is briefly discussed in Sect. 1.9. M o r e g e n e r a l a s s u m p t i o n s on t h e d a t a . As the Lain6 constants and the applied force densities enter linearly the three-dimensional equations, the conclusions drawn in Sect. 1.8 in the linear case hold verbatim: The same "limit" equations of the nonlinear KirchhoffLove theory are still obtained if the more general assumptions on the data:
M = eta and #~ = ct#, -
-
for
m
x
-
g~(x~)--e3+tg~(x) and g~(x~)-c4+tg3(x ) for all
hold for some arbitrary real number t. We shall see in the next section that these assumptions are in fact the only ones that give rise to this theory. E x t e n s i o n to m o r e g e n e r a l c o n s t i t u t i v e e q u a t i o n s . The most general constitutive equation of a n o n l i n e a r l y elastic m a t e rial that is homogeneous and isotropic takes the form (Vol. I, Thin. 3.6-2):
Nonlinearly elastic plates
326 E~ - E ~ ( V ~ u ~ )
[Ch. 4
- E~(E~(u~))
= 7~(~(C~))I + 7~(~(C~))C ~ + 7~(~(C~))(C~) 2, where C ~:=I+2E(u
~)
denotes the right Cauchy-Green strain tensor, E~(u ~) denotes the Green-St Venant strain tensor (already introduced in Sect. 4.1), and ,7~, 72 are real-valued functions of ~(C ~) := (tr C ~, t r C o f C ~, det C~), i.e., functions of the three principal invariants of the matrix C ~. If in addition the reference configuration is a natural state, we must also have (Vol. I, Thm. 3.8-1): ~]~(E) - A ~ ( t r E ) I + 2#~E + o(llEII), for all symmetric matrices E of order 3 close to 0, where A~ and #~ are the Lam6 constants of the material under consideration. Note that a St Venant-Kirchhoff material fulfills all these conditions (it corresponds to o(l[Ell ) = 0). If such a more general constitutive equation is used instead of the simplified equation of a St Venant-Kirchhoff material (as was so far assumed in this chapter), a striking conclusion, due to Davet [1986], is that the application of the method of asymptotic expansions still
yields the same nonlinear Kirchhoff-Love theory! More specifically, J.L. Davet has shown that the same limit twodimensional equations of a nonlinearly elastic clamped plate are found if the following assumptions on the d a t a are made:
f~(x ~) -- r
and f~(x ~) - a3+tf3(x) for all x ~ - 7r~x E fY, 4 + , ga(x) for all x ~ - ~ xeC+~UP ~_,
g~ (x ~) _ ~_ 3 + ~ g~(x) ~ ~ andga(x~) -~
70(~(C))I + 71(~(C))C + 7 2 ( ~ ( C ) ) 6 2 - )~(tr E ) I + 2 p E +o(l[E[[), C = I + 2E,
Sect. 4.10]
Justification of the nonlinear Kirchhoff-Love theory
327
where t is an arbitrary real number, the functions f~ : ft ~ R a, 9~ : F+ O F_ ~ R a are independent of e, the functions 7o, 71,72 are independent of e, and )~ > O, # > 0 are two constants independent of C.
Hence the two-dimensional equations of the nonlinear KirchhoffLove theory have a generic character. O t h e r e x t e n s i o n s . As shown in detail in the next chapter, the application of the method of asymptotic expansions to a nonlinearly elastic plate subjected to another specific class of boundary conditions yields the well-known yon Kdrmdn equations (Ciarlet [1980]). The three-dimensional boundary conditions may even be live loads (Blanchard & Ciarlet [1983]; see also Ex. 5.2); incidentally, this shows that different three-dimensional problems may be "asymptotically equivalent" to the same limit problem. In this respect, one of the merits of the present method is to clearly identify which twodimensional boundary conditions should correspond to a given set of three-dimensional boundary conditions. Time-dependent problems for nonlinearly elastic plates have been thoroughly studied by Raoult [1988, Chap. 2]. Adapting the method of formal asymptotic expansions followed here in the "static" case, she has provided a full justification of the two-dimensional equations of the time-dependent nonlinear Kirchhoff-Love theory; her discussion includes in particular the consideration of various sets of boundary conditions. Then Karwowski [1993] further extended the displacement-stress approach, by scaling the first Piola-Kirchhoff stress tensor (Vol. I, Sect. 2.5), rather than the second as here, then by investigating more general sets of possible scalings, in a manner reminiscent of that described in Sect. 4.12; in this fashion, timedependent two-dimensional nonlinear "membrane" theories are also recovered. Other extensions consist in applying the method of asymptotic expansions to nonlinearly elastic plates with rapidly varying thickness (Quintela-Estevez [1989], Alvarez-Vazquez & Quintela-Estevez [1992]), to more realistic boundary conditions of clamping (Blanchard & Xiang [1990]), to nonlinearly elastic anisotropic plates (Begehr, Gilbert & Lo [1991], and to nonlinearly elastic shallow shells (Ciarlet
328
Nonlinearly elastic plates
[Ch. 4
Paumier [1986]; cf. Sect. 4.14). The method of asymptotic expansions can be also adapted to the "one-dimensional" modeling of nonlinearly elastic rods. In this case, the reference configuration is of the form ~ - &~ • [-1, 1], where co~ := {(eXl,eX2) E IR2; (Xl,X2) E co} and co is a fixed domain in IR2 with (0, 0) as its centroid. Through appropriate scalings, the components of the displacement field are then transformed into functions defined over the fixed set f~ "- & • [-1, 1], and specific assumptions on the data are made. In this fashion, it is found that the leading term of a formal asymptotic expansion of the scaled displacement field is a Bernoulli-Navier displacement field that satisfies a nonlinear ordinary differential equation of the fourth-order along the "center line" of the rod. For details and various extensions, see the thorough analyses of Cimeti~re, Geymonat, Le Dret, Raoult &: Tutek [1988] who also investigated the nature of the limit stresses inside the rod, of Trabucho & Viafio [1996, Chaps. 9 and 10], and of Zarwowski [1990], who recover different nonlinear rod and string equations under various constitutive assumptions. Nonlinear one-dimensional rod theory has also been related to the three-dimensional theory by Mielke [1988, 1990], who justified St Venant's principle by a remarkable use of the center manifold theorem. Special mention must also be made of the pioneering contributions of Rigolot [1976, 1977a]. Two-dimensional nonlinear plate theories may be also found, first by integrating the three-dimensional equations across the thickness, and secondly by approximating the resulting equations by quadrature formulas; see Vashakmadze [1986]. A nonlinearly elastic plate may be also viewed "directly" as a
two-dimensional deformable body. This viewpoint leads notably to the Cosserat theory of plates, perhaps best understood as special case of the Cosserat theory of shells, briefly discussed in Vol. III (an illuminating introduction to this theory is given in Antman [1995, Chap. 14, Sects. 10 and 13]). A noticeable feature is the frameindifference of the two-dimensional equations found in this theory (in this respect, see also Sect. 4.12). In the same vein, the existence and uniqueness results obtained by Bielski gz Telega [1996] for a nonlinear Reissner-Mindlin theory
Sect. 4.11]
Justification of the scalings and assumptions
329
are worthy of interest.
4.11.
JUSTIFICATION OF T H E S C A L I N G S A N D ASSUMPTIONS IN THE NONLINEAR CASE
In Sect. 1.10, the scalings of the unknowns and assumptions on the d a t a were justified (after Miara [1994a]), but only up to a multiplication, inevitable in the linear case, by an arbitrary power of c (the same for all the components of the displacements and applied force densities). Miara [1994b] has further shown t h a t this "dangling factor" becomes "frozen" when the nonlinear case is considered (and specific, but natural, requirements are set), thus providing a rigorous justification of the scalings and assumptions considered so far. Let us describe her analysis. We first note that it is no loss of generality to assume at the outset that the Lamd constants are independent of e, i.e., t h a t l ~-I
and
#~-#,
as the Lam6 constants and applied force densities can be multiplied by a same power of c without altering the ensuing developments. Then functions u~(c)" ~ ---, R, f~(c)" ft ~ R, and g~(c) 9F+ U F_ --, N are defined by letting u~e (x e) - ~ 9(E)(x) for all x e - 7re~ E ~c , f.~(x ~) - L(E)(x) for all x e - 7rex E ~ e -
for all
--
C r;
v r
As a result of these definitions, the "new" scaled displacement u* (c) "(u~@)) solves the following variational problem 7)*(e; f t) (compare with problem P(e; f~) found in Thin. 4.2-1)" I/,*(E) ~ V(~"~) -- { v ~ W1'4(~"~);
v - 0 Oil F o } ~
330
Nonlinearly
,
1
elastic
[Ch. 4
plates
.
-~- 0"33(6) (O3V 3 -~- --O3tti (6)03Vi)} dx c
- e / ~ f,(c)v, dx +/c
+LJF_
gi(---c)v,dF
for all v E V(a),
where . 9 ~;~(~) .- ~ 0 ~ ; ( e ) + 1 0 o ~(~)0~(~)
1.5.,
+ -g o ~ ( c ) +
o~(e)o~(c)
o-*
1
.
.
1
.
}~
.
~ ( ~ ) .- a{o~;(c)+ ~ 0 ~ ( c ) 0 ~ ( ~ ) } 1 , .t)o3 '~ + ()~ @ 2"){~03U3(E)-1 t- ~ 1C2 o.t)j'-~3u*'e'~u*'e Assume next that u* (e) can be expanded as a formal series" U * ( s ) --
1 -~ U - I -Jr-...-Jr-
1 -- U - 1 C
-~- U 0 At- C U 1 -~- . . .
where the order -1 <_ 0 of the leading term is to be determined, the terms u p - (u~), p >_ -1, belong to w~'n(ft), and only the (eventually found) leading term is required to satisfy the b o u n d a r y condition of place on F0. A w o r d of c a u t i o n . There was no loss of generality in starting such a formal series by a term of order 0 in the linear case (Sect. 1.10). By contrast, this "freedom" is lost in the nonlinear case. I The smallest power of c found in the left-hand side of the variational equations in problem 7)*(~; f~) is ( - 3 / - 3); accordingly, we
Sect. 4.11]
331
Justification of the scalings and assumptions
first "try" f ( c ) - c31+4 1 f-3l-4
and
1 g -31-3 , g(c) - c3l+3
where, here and subsequently, fq - (fq), q _> - 3 1 - 4, and g~ = (g~), r >_ - 3 1 - 3, stand for vector fields in L2(ft) and L2(F+ U F_) respectively, t h a t are independent of c. E q u a t i n g to zero the coefficient of c -3t-3 shows t h a t u -I E V(f~) satisfies
s
a+2
~03U-~IO3U-~ZO3Ur~ LO3Vidx 2 g~-31-3v~ dF +UF_
for all v - (v~) C V(ft); hence (take v independent of xa)" t__f al-4 dxa + g - a l - a (., 1) + g - a l - a (., _ 1) - O. 1 A first requirement t h a t guides the analysis is that, as in the linear case, we do not wish to retain limit equations where restrictions (e.g., the ones found supra) m u s t be imposed on the applied force densities in order that these equations possess solutions. This does not m e a n t h a t such limit equations are b o u n d to oblivion; indeed, they can be studied for their own sake (Ciarlet & Miara [1997]). Using the first requirement, we are thus forced to conclude t h a t f-3t-4 0 and g-3Z-a = 0, which also shows t h a t Oau -I - O, and to next "try" =
f(e)-
1 -az-3 eaz+ a f
and
g(e)-
1 e3/+2g
-al-2
Successively equating to zero the coefficients of c -31-2 -3L-1 c -3l, and relying on the same requirement (and also using the relation 03u -z - 0), we find that, if I > 1, fq - O, - 3 1 - 3 < q < - 3 1 - 1 and g~ - O, -31 - 2 < r < -31, which also shows t h a t ~3u -1+1 - O. We are thus led to "try" f(e)-
1 -3l (c))-57f and g
1 1 cal_
g-
3/+1
.
Nonlinearly elastic plates
332
[Ch. 4
If 1 >_ 2, the cancellation of the factor of s then yields to solving (as 03u -I - O, it is licit to identify u -I with a function defined over w)"
_
f~-aldxa+g{al+l("' 1) + gi-3/+l ( "' - 1 )
{fl1
}r/i dw
for all (r/i)E H i ( w ) t h a t vanish on ~/0. At this stage, we need to resort to a second requirement:
By linearization with respect to the unknowns we should find the problem solved by the leading term of the linear theory; in other words, taking formal limits as ~ ~ 0 and linearizing should commute. Applying this second requirement shows t h a t for any 1 >_ 2, f-31 = 0 and g - a + ~ = 0 on the one hand, and u -I - 0 on the other. We thus conclude t h a t
1
-
- u -
1 _~_ U 0 _1_ s
1 -Jr 9 9 9 ,
s
and t h a t we must "try" f(s)_
1
_gf - 5
and
g(e)-~g
-4
But then the first requirement shows (as before; only the restriction 1 >_ 1 was then imposed) t h a t f - 5 _ f - 4 _ f-3 _ 0 and 9-4 _ g - a __ g-2 _ 0, then t h a t u -1 - 0 and finally, t h a t f - 2 _ 0 and g-~ - 0. We should therefore let u*(e) - u ~ + e u 1 + e 2 u 2 + . . . , and
"try"
f(e) -- I f _ , g
and
g ( e ) - gO.
From t h a t point on. the m e t h o d proceeds by carefully blending the first and second requirements, together with a r g u m e n t s similar to those used in the proof of Thin. 4.4-1. In so doing, it is successively found t h a t f - 1 - f0 _ 0 and g o - 91 - 0, u ~ - 0, f l __ 0 and
Justification of the scalings and assumptions
Sect. 4.11]
333
g2 _ 0 , u s1 __ 0 , f sl __ 0 and g~ -- 0, f~ -- 0 and g~ - 0; finally, the problem solved by the leading term is also identified. In this fashion, the following result was obtained by Miara [1994b]: 4.11-1. Define the space
Theorem
V(w)-
{ r / - (rh) E Hi(w) • Hi(w) • H2(~); ~]i- 0,r]a - 0
on ~0}.
Assume that the Lain6 constants are independent of c. In order that the leading terms in the formal asymptotic expansions of each component u~(c) of the scaled displacement u*(c) may be computed without any restriction on the applied forces and in order that taking such formal limits commute with linearization, we must have
9 us(s ) - s 2 u 2s + . . . f~(s)- 2 2
and
9 u3(s )-su~+..., c3 3
g~
and
g3
c g3.
Moreover, (u~, u~, u~) is a scaled Kirchhoff-Love displacement field, i.e.,
2 ~
its
~2 s
--
X30s~3
1
1
1
2
2
1
and u 3 - Ca with (4a, 42, C3) E V(~),
Nonlinearly elastic plates
334
[Ch. 4
and the functions r and ~ solve the variational equations"
li{
3(A + 2#)Ar
+
dw
a+2 1 1 + 0 ~ 2 + 0 ~r162
+ 2p(0~r -
1
1
1
faa dxa + g4( ., 1) + g4( ., - 1 )
/ {fl L{]_1x3f: -
+
}
+ --O~O~r/a
f : dxa + 9a~(., 1) + g 3 (. - 1 )
}
+ 0zr/~ } dw
r/3 dw
)
dxa + g~(., 1) - g a ( . , - 1 )
r/~ dw
}
0 ~ 3 dw
1
for
all
(r/~) e V(w). m
The variational problem satisfied by (r r r ) coincides with that of the nonlinear Kirchhoff-Love theory (Thin. 4.5-2). Under the two requirements enounced in its statement, Thm. 4.11-1 thus provides a full justification of the scalings and assumptions set forth in Sect. 4.2. More specifically, it shows that the displacement field ~ - (~) found after de-scalings by the nonlinear Kirchhoff-Love theory necessarily satisfies (Sect. 4.10) 4~ - O(e 2) and 4~ --O(c), and that the Lam~ constants and the components of the applied forces that produce such displacements necessarily satisfy
A~ - O ( c t)
and
f~ - O(c 2+t) and g; - O(c3+t) and
p~-O(ct), f~ - - O ( E 3 + t ) , g~ - O(c4+t),
for some arbitrary real number t. A major virtue of B. Miara's analysis is thus to provide a conclusive evidence that the nonlinear
Sect. 4.12]
Frame-indifferentnonlinear membrane and flexural theories
335
Kirchhoff-Love theory (and consequently the linear Kirchhoff-Love theory, as already noted in Sect. 1.8) is necessarily a "small displacement" theory. 4.12 ~.
FRAME-INDIFFERENT NONLINEAR MEMBRANE AND FLEXURAL THEORIES
Remarkably, other limit equations, corresponding to different scalings of the unknowns and orders on the applied forces, can also be ohtained by the method of formal asymptotic expansions if one no longer insists on recovering the linear Kirchhoff-Love theory by linearization. This key observation is due to Fox, Raoult & Simo [1993], who in fact achieved this greater generality by scaling the deformations instead of the displacements. In this fashion, they obtain other two-dimensional theories, the nonlinear membrane and nonlinear flexural ones, that possess the specific features of allowing "large" deformations of order O ( 1 ) w i t h respect to c, and of preserving the frame-indifference of the original three-dimensional model; for these reasons, they constitute "large d e f o r m a t i o n " , and f r a m e - i n d i f f e r e n t , theories, frame-indifferent theories being synonymously called properly invariant theories. Note that a similar analysis was conducted by Karwowski [1990] for modeling nonlinearly elastic rods. Let us outline this approach. Consider the same nonlinearly elastic clamped plate as in Sect. 4.1; in particular, the plate is made of a St Venant-Kirchhoff material. The deformation ~
- (~{)"-
id + u ~
thus satisfies the variational equations
r
dx ~ - ~
~
f~v~ dx~+ fr
f o r g l l V ~ --- (V~) ~ V(~'~ ~) -- { v ~ ~ W 1 ' 4 ( ~ c ) ;
S u r ~_
g~v~ d E
~
v ~ -- 0 o i l F~)}, w h e r e
Nonlinearly elastic plates
336
1 Ve(~ e TVr E ~ ( u ~-) - ~({ }
c
[Ch. 4
- I)~j
,
and the matrix .
-
is the d e f o r m a t i o n g r a d i e n t . Notice that, without loss of generality, we assume at the outset that the Lamd constants are independent oft. The associated e n e r g y I ~ is then defined by (Vol. I, Sects. 4.1 and 4.10)
I~(~b~) "- L~ 14r({V~b~}rv~b~) dx~
-{fa
f [ v : d x ~ + fr
~_urt
g~~vi dF ~} ,
where the s t o r e d e n e r g y f u n c t i o n l~ (of a St Venant-Kirchhoff material) is defined by
W(C)'-
-
3~+2# 4
trC+
8
tr
+
trCofC
for any symmetric positive definite matrix C, and the associated set of a d m i s s i b l e d e f o r m a t i o n s is ~hen defined as (Vol. I. Sect. 7.4)
(I)e(~ e) "-- {~2 e E w l ' 4 ( ~ e ) ;
~ e ( X e ) -- X e for z e C P~),
det V~b ~ > 0 in f~}. Note that the definition of the set O~(ft ~) incorporates the o r i e n t a t i o n p r e s e r v i n g c o n d i t i o n det V~p ~ > 0 in ft ~ (Vol. I, Sect. 1.4). Particular solutions of the variational equations are formally obtained by finding the m i n i m i z e r s of the energy I ~ over the set 9 ~(f~),
Frame-indifferentnonlinear membrane and flexural theories
Sect. 4.12] i.e., those ~
337
t h a t satisfy r
E Os (ft ~) and I ~(r
=
inf i ~(~e). ~ c ~ ( a ~)
As the above stored energy function lfV is a function of the right Cauchy-Green strain tensor { V ~ h ~ ) T V ~ r ~ (Vol. I, Sect. 1.8) associated with an a r b i t r a r y deformation ~ E O~(t2~), it is frameindifferent (Vol. I, Sects. 3.3 and 4.2; see also Ex. 4.3). A l t h o u g h a most desirable requirement in C o n t i n u u m Mechanics, frame-indifference is often violated by some of the most favorite models, such as linearized elasticity (Vol. I, Ex. 3.7), the nonlinear Kirchhoff-Love theory of plates (Ex. 4.3), or the von K s 1 6 3 equations (Chap. 5)! By contrast, the "first" and "second" two-
dimensional plate theories found below do retain this invariance property of the "original" three-dimensional model. Let the set f~ and the m a p p i n g 7r~ 9~ ~ -f~ be defined as in Sect. 4.2. Let then the s c a l e d d e f o r m a t i o n ~ ( e ) - ( ~ ( c ) ) " ~ ~ R 3 be defined by qp~(z ~) - ~ ( c ) ( z ) for all z ~ - 7r~x E ~ , and let the vector fields f ( c ) = (f~(c)): t2 F+ U F_ ~ R 3 be defined by
f ~ ( x ~) = f ( c ) ( x )
--,
]~3
aIld g(~)
---
(9i(C)) :
for all x ~ - 7r~x E f ~ ,
g~(* ") - g ( ~ ) ( * ) eo~ ,11 0: - ~ .
~ r ~ u r~_.
Observe t h a t no assumption is made at this stage regarding the orders with respect to s of the components of the applied force densities. It is found in this fashion t h a t the scaled deformation satisfies
:(s) E ~ ( s ; f t ) and I ( c ) ( q p ( c ) ) :
inf I(e)(@), #JE,I,(e;f~)
338
Nonlinearly elastic plates
[Ch. 4
where (I~(c;~) "-- { r C w l ' 4 ( ~ ) ; r
(Xl,X2, Cx3)
for x = (Xl,X2, x3) e F0, det V r > 0 in Ft},
I(e) ( r
~1fa {AEo~(e; r
(e; r + 2#E~z (c; r
+2-~e {2)~Eoo(e; ~b)Eaa(e;r + 4#E~a(e; r +2-~e2 -
(A + 2p)Eaa(e; r
{/a
f~(c)v~ dx + -l f r ~
r
dx
~)} dx
~b)dx +UF_
g~(e)v~ d r
},
and E~,(s; r
.-
1 -~(0~r162
-
~),
1 Eaa(c;~b) - ~ (1) - 71 0 a~b~0a~bi-1) Fox, Raoult & Simo [1993] apply the basic A n s a t z of the method to the variational equations that are formally equivalent to the above minimization problem, viz., of f o r m a l asymptotic expansions
dx
1L2#E~3(e;
+-
qO(e)){O~cp~(e)Oav~ + Oa~(e)O~v~} dx
C
1
+7~
f o(AEoo(c;~o(c)) + (A + 2#)Eaa(c; ~a(c))}O3~(e)O3v~dz --
f~(e)v~ dx + -
C
+uF_
9~(e)v~ dr'
Sect. 4.12]
Frame-indifferent nonlinear membrane and flexural theories
339
for all (v~) e V ( ~ ) , where V ( ~ ' ~ ) "-- {V - - (Vi) E w l ' 4 ( ~ ' ~ ) ;
v -- 0
on Fo}.
They then show that several choices of orders (with respect to ~) of the applied forces are possible that give rise to two distinct nonlinear two-dimensional theories. For conciseness, we express their results in the next theorems as minimization, rather that variational, problems, and we do not "de-scale" the limit equations; we refer to the original paper for a more detailed exposition, the proofs, and a thorough commentary. Let us consider the "first" set of possible assumptions on the forces. Theorem
4.12-1. Assume that
f(c)-
(fo)
and
g(e) - c(gr
where the functions f o e L2(~) and g] C of c, and that
L (r+ur_)
are independent
(.~(C) -- (~0 + C(~01 AV .. "
Then the leading term qO~ is independent of the "transverse" variable x3 and it satisfies the following minimization problem, where it is (justifiably) identified with a function qpo .-g ___.R 3.
~o e (I)M(W) and Ira(: ~ --
r
inf
(~)
IM(r
[Oh. 4
Nonlinearly elastic plates
340
where, ~ denoting the mapping
(Xl, X2) E ~ ---> (Zl, X2, 0) E ]I~3,
(I~M(Cd) "-- { r -- L q- ~; ~ e w l ' 4 ( O d ) ;
~ -- 0 on ~0,
01r • 02r r 0 in co},
s pi ~
-
EM
-
1
dw
-
1
= ~(a~z(~b) - 6~z) where a~z(r ) "- c9~r c9zr p0._[
1
j_ 1
fOdx3+gl(',1)+9~(',-1),
4Ap
m
This result has three important consequences: First, the de-scaling produces a deformation that is O(1) with respect to c; secondly, the stored energy function ~b ---+a ~ , E M ( ~ ) E ~ ( ~ b ) is frame-indifferent, as its value is not altered if r is replaced by 0 o r where 0 is any isometry of R 3 (see also Ex. 4.4); thirdly, only the first fundamental form (a~z(~b)) of the deformed middle surface r (Vol. III) appears in the expression of the stored energy function. For these reasons, this "first" theory is called a " l a r g e d e f o r m a t i o n " , f r a m e i n d i f f e r e n t , n o n l i n e a r m e m b r a n e t h e o r y . We also note that it coincides (once de-scaled and written as a boundary value problem) with the "nonlinear membrane equations" found in Green & Zerna [1968, eqs. (11.1.13)]. Another noteworthy characteristic is the quasilinearity of the (formally) equivalent boundary value problem (Ex. 4.4), as opposed to the semilinearity of that found in the nonlinear Kirchhoff-Love theory (Sect. 4.10). As a result, the existence theory for such quasilinear equations is a delicate question. Promising results have nevertheless been recently obtained by Coutand [1997b].
Sect. 4.12]
Frame-indifferent nonlinear membrane and flexural theories
341
A w o r d of c a u t i o n . Surprisingly, these equations, obtained by a formal approach, are "not always" identical to those obtained by a convergence theorem; see Sect. 4.13. m Remarks. (1) Once it is proved that the leading term q~0 is independent of x3, the orientation-preserving condition takes the form det(01q~ ~ 02qp~ Oq3~1) ~ 0, since it should be satisfied "at the lowest possible order". This is a useful relation, as it is used to derive the condition 01qp~ x 02q~~ ~ 0, found in turn in the definition of the set (2) Let q~0 = L+ (j so that (j = (~) may be understood as a scaled displacement of the middle surface. Then the functions 1 S ~ ( ~ o) - [(0~< 9 + 09(~ + O~<m09(~) found in the scaled energy IM differ from the functions 1 found in the "membrane part" of the scaled energy of the nonlinear Kirchhoff-Love theory (Thm. 4.6-1). More specifically, the functions E~(~a ~ retain all the terms 0~(,~09(,~ found in the original threedimensional strains (Sect. 4.1) <,(u
-
while the terms 0~(o0~(~ no longer appear in the functions E~ (3) As noted by Fox, Raoult & Simo [1993], the linearization of the nonlinear membrane theory about particular deformed configurations of the form ~ap 9 (Zl,Z2) E & + (pz~,pz2,0) E R a, p > 1, corresponding to states of "uniform tension", shows that the vertical component Ca of the resulting deformation satisfies the famed l i n e a r membrane equation -TpA(3 - p0
in co,
Nonlinearly elastic plates
342
[Ch. 4
where the number Tp (a function of p) measures the tension of the m e m b r a n e (the b o u n d a r y condition ~b = ~ on "70, which was chosen in Thin. 4.12-1 for simplicity, is replaced here by ~ = q0p on 3'0). This linearization does not conflict with B. Miara's (Sect. 4.11), who wished to recover instead the linear Kirchhoff-Love theory (besides, the reference configurations corresponding to such deformations (~:::~P a r e no longer natural states). (4) The linear membrane equation may be also recovered in an entirely different manner from an ad hoc limit analysis of the von Ks equations (Thm. 5.10-1). 1 Now we t u r n to a "second" admissible set of assumptions on the applied forces. Theorem
4 . 1 2 - 2 . Assume that
f ( c ) - c2(f~)
and
g(c) - c3(g3),
where the functions f~ E L2(f~) and g3 E oft, r
L(F+uF_)
are independent
- ~o + c~p~ + . . . ,
and the leading terms qo~ is independent of the transverse variable x3. Then the leading term, henceforth identified with a function ~o . --+ R a, solves the minimization problem"
qo~ E O~(w)
and
IF(qO ~ --
inf
IF(C),
where (the mapping t. and the constants a~z~ are defined in Thin.
Sect. 4.12]
Frame-indifferent nonlinear membrane and flexural theories
343
4.12-1)
@F(C~) := {r = ~ + rl; rl E H2(a~); rl = 0 on "7o,
0~r IF(C) " -
b~z(r
~l f ~ a~z~bo~(r162
:= n ( r
p~ "--
/1 -1
0~zr
n(r
09r = 8 ~ in co},
da~ :=
da~,
01r x 02r
101r x 02'~1'
f? dx3 + 9ia(", 1) + 9ia(., - 1 ) .
II As with the membrane theory (Thm. 4.12-1), the de-scaling produces a deformation that is O(1) with respect to e, and the stored energy function ~2 --~ a~o~.bo~(~b)b~(~b) is frame-indifferent. Besides, only the second fundamental form ( b ~ ( ~ ) ) of the deformed middle surface ~b(~) (Vol. III) appears in the expression of the stored energy function. For these reasons, this "second" theory is called a " l a r g e deformation",
frame-indifferent,
nonlinear flexural theory.
Note however that the first fundamental form (a~z(r of the deformed middle surface also appears in the formulation of this theory, via the definition of the set @F(a~), which consists of deformations satisfying ( a ~ ( r = (a~z(e)). For this reason, OF(CO) is called a set of inextensional deformations, and the nonlinear flexural theory is also called an inextensional theory. A w o r d o f c a u t i o n . Naturally, the "interesting" situations covered by the flexural theory found in Thin. 4.12-2 are those where the set OF(CO ) contains other deformations than @ = t. For instance, assume that ~ is a rectangle; then OF(W) = {t} if 70 = ~/, while OF(W) does not reduce to {t} if "7o is one side of the rectangle (Ex. 4.5). II
Remark. Body and surface forces of order 1 and 2 respectively may also contribute to the linear form found in the total energy of the flexural theory, provided however they are subjected to ad hoc
Nonlinearly elastic plates
344
[Ch. 4
restrictions: If f(c) - e f ' + c2f 2 and g(c) - c2g 2 + C393, then the fields f l and g2 must satisfy
/i fl
dx3 + g2(., 1) + g 2 ( . , - 1 ) - 0.
Incidentally, this is precisely the type of restrictions that was ruled out in B. Miara's analysis (Sect. 4.11). m The "third", and last, choice of assumptions on the applied forces consists in assuming that f(c) -- (c2f~, c2f~, c3f2) and g(c) - (c3913, e3923, r where the functions f~,f~ E L2(t2) and 9~,9~ E L2(F+ U r _ ) are independent of c. In this case, D. Fox, A. Raoult and J. C. Simo find that the components of the scaled deformation ~(e) = ( ~ ( e ) ) are necessarily of the form ~ ( c ) - x~ + c u~ + . . .
and V)3(c)
-
c ( x 3 + u 3) 1
+.-.,
where (u~, u~, u~)is a scaled Kirchhoff-Love displacement field, i.e., it 2 _ r
_ 2~3(~ar
and u I - r
and the vector field (el, r r precisely solves the equations of the (scaled) nonlinear Kirchhoff-Love theory (see., e.g., Thm. 4.11-1). By first scaling the deformations rather than the displacements, then by systematically searching assumptions that give rise to nonlinear limit behaviors, altogether without insisting on preserving any property by linearization, Fox, Raoult, & Simo [1993] have thus identified, and clearly delineated, all possible nonlinear plate theories. A remarkable feature of their membrane and flexural theories is the striking similarities (about the order of the applied forces, the expressions of the energies, the sets where the minimizers are sought, etc.) that they share with the "membrane" and "flexural" shell theories, both in the linear and nonlinear cases (Vol. III).
Sect. 4.12]
Frame-indifferent nonlinear membrane and flexural theories
345
To conclude, we give a proof, due to Coutand [1997a], of the existence of a solution to the minimization problem corresponding to the nonlinear flexural theory (Thm. 4.12-2). It relies on the clever observation that the associated energy reduces to a quadratic functional over the set "~F(a~) (part (iii) of the proof)! T h e o r e m 4 . 1 2 - 3 . Let functions pi ~ Le(a~) be given, and assume that length 7o > O. Then there exists at least one qp such that:
q~ E OF(W) and I y ( c p ) q'F(W) -- {r E H2(aJ); /)~r Is(C)
-
1 f~{ 4AP3(A + 2
inf IF(C), where ~eeF(~) 0Zr
8~Z in co, ~b - t on 7o},
p ) 4 p + -~b~z(r162 b~(~2)b~(r
} da~
- .f. pi~2i da~,
b~,(r
- n(e)
0~,r
n(r
-
01r • 02r Io1r • o r
Proof. (i) The integral IF(C) is well defined if r C '~F(a~).
Let r E ~s(a~). The relations 0 ~ r 1 6 2 = 8 ~ may be also written as 101r = 102r = 1 and 0 1 ~ . 02~b = 0; hence the vectors 01~ and 02~b are linearly independent, and consequently the vector n(~b) is well defined almost everywhere in co. In fact, the vector field n ( r is in L~ (since In(r = 1), and thus b~z(~) c L2(a~). (ii) The set Oy(a~) is weakly closed in H2(aJ).
Let ~bk E ~y(a~), k _> 1, be such that Ck ~ r in H2(a~). The compact imbedding H2(a~) e Hi(co) shows that Ck ~ r in Hl(a~); hence 0~r k. 0 z r k --~ 0 ~ r 0 z r in Ll(w) and thus 0 ~ r 1 6 2 - 8~z in co. The convergence r ---, r in HI(w) also implies tr Ck -+ tr in L2(7); hence r - ~ on 7o.
Nonlinearly elastic plates
346
[Ch. 4
(iii) Let the functional/~ 9H2(a~) --. R be defined by
-
+
for r = (r
H2(a~). Then
Differentiating the relations 0 ~ . 0 z r = 5~z in the sense of distributions yield the successive implications (it is easily verified t h a t such differentiations are licit): Oql~) 9Oqll/) -- 1 => 0 1 1 r
/)2r
01r = 012r
02~ = 1 ~ 0 2 2 r 1 6 2
- 0~ur
0 1 r --- 0,
0 u r = 0,
Oqlr " 0 2 r z 0 ==~ 0 1 1 r
0 2 r 4- 0 1 2 r
01r
0 2 ~ + 022~" 0 1 r --- 0 ~ 022~" 0 1 r = O,
02r = 0 ~ 012r
0 1 r -~- 0 ~ Oqllr 90 2 r --- 0,
which show that, if r E (I)y(aJ), the three vectors 0 ~ z r are colinear with the vector n ( r almost everywhere in ca. In order words, for almost every y E a~ and for each c~,/3 = 1, 2, there exists a constant C~z(y) such t h a t
O~zr
) = C~z(y)n(r
hence
b~9(~)(y ) = n ( r
. O~zr
= C~z(y).
Consequently (for brevity, the dependence on y is dropped),
likewise,
r E OF(CO) =~ b ~ ( ~ ) b ~ ( ~ b ) = C ~ C ~ Therefore,
= I0~r
] = oq~iO~~.
Sect. 4.12]
Frame-indifferentnonlinear membrane and flexural theories 4A# boo(r162 A+2# _
+ 4#b~z(r162 4~#
~+2#
347
)
0o~r162
+ 4#0~r
(iv) The functional f is weakly lower semi-continuous on H2(a;), and coercive on the set Or(w), i.e., r C Cr(w) and I1r
-~ + ~ ~ I(r
~ +~
Consequently, there exists at least one minimizer of the functional I over the
set ~F(Cd).
The quadratic part of the functional I is convex, as a sum of squares of Hilbertian semi-norms. Since it is also continuous over H2(cz), it is weakly lower semi-continuous over H2(cz). Let
Ir
1/2
1/2
{E , ]Oar.
Ir
a,~,i
By the generalized Poincar6 inequality (Vol. I, Thin. 6.1-8), there exists a constant Co such that (the assumption length 7o > 0 is crucially needed here)" 2
I1~11~1,w < c0{lr ~1,w -
-
Cd-y
+fo
Besides, the relations 01r OF(w) show that
~ ~(~)
01r
--
02r
for all r C Hi(a;). 02r
--
1 satisfied by r E
~ I~IY,~ - 2 areaw.
Hence
fo~d7 } 2
~ ~(~)
~ I1~11~l,w <- - C1 "-- Co{2area~ +
We thus obtain the implication
Nonlinearly elastic plates
348
r er
-
2A#
2
[Ch. 4 2#
s ( ~ ) - 3(~ + 2v)1~162 O,w + --3 Ir
~
f
L p. Cd~
> 2#
- -2,~-Iplo,~lr 3 Ir
> 2# -
3
I1r
2#
2,w - I p l o ,w Ig, lo,~ - - ~ C 1 ~
where p "- (p~); hence/~(~p) --+ +oc if r E OF(W) and 11r ~ +oc. As the set q)y(aJ) is weakly closed in H2(a~) by (ii), we conclude by a classical argument (already detailed in part (iv) of the proof of Thm. 4.6-1) that there exists at least one qO such that ~a E OF(W) and/7(qO) hence such that IF(qO) i n f r coincide over the set OF(a) ) by (iii). =
inf
/~(~p),
since the two functionals II
Remark. Thm. 4.12-3 immediately extends to the case where the set (I)F(aJ) is of the more general form ~ F ( W ) - {~ C H2(w); O ~ " o q ~ -
5 ~ in w, ~ p - qPo on 70},
provided the mapping qPo : 7o ~ R3 is chosen in such a way that An existence result for a nonlinear membrane theory akin to that found in Thm. 4.12-1 is given in the next section (Thm. 4.13-1) 4 . 1 3 ~.
FRAME-INDIFFERENT MEMBRANE THEORY
NONLINEAR AND F-CONVERGENCE
A remarkable progress in the asymptotic analysis of nonlinearly elastic plates was achieved when Le Dret & Raoult [1995] gave the first proof of convergence. Their approach was inspired by Acerbi, Buttazzo & Percivale [1991], who had successfully used F-convergence theory for justifying the equations of nonlinearly elastic strings. More specifically, H. Le Dret and A. Raoult show that a subsequence of the deformations that minimize, or rather "almost minimize" (in a sense explained below), the scaled three-dimensional en-
Sect. 4.13] Frame-indifferentnonlinear membrane theory and F-convergence 349 is dictated by the growth properties of the stored energy function). They show in addition that the weak limit minimizes a "nonlinear membrane energy" that is the F-limit of the scaled energies. V~renow give an abridged account of their analysis. Let M a denote the space of all real square matrices of order 3, and let I" I denote any norm on M 3. Consider a completely clamped plate (for simplicity; partially clamped plates are likewise covered by the subsequent analysis), made of an elastic material whose stored energy function W : NP ~ R is continuous and satisfies the following growth assumptions: There exist constants C > 0, c~ > 0, fl 6 R, and 1 < p < ec such that ^
II~(F)I _%
Remark. The stored energy function of a linearly elastic material, given by
tt A (tr(FT + F W ( F ) - ~IIF + F T - 21112 + N
2I) }z
where IIF[I "- {tr FTF} 1/2, satisfies the first inequality with p but not the second one.
2, It
The three-dimensional problem is then posed as a minimization
problem: Find qO~ such that :~ 6 (I)(f~~) and U(qp ~) =
9 ( ~ ) . - {r
e w',~(~);
r
inf
U(~b~), where
~ on ~ • [-c, d } ,
U(@) " - / a l ~ ( V ~ b ~ ) d z ~ - { / a f ~ . ~ b ~ d z ~ + jfr;ur~ 9 ~.
r
Nonlinearly elastic plates
350
[Ch. 4
Note that this problem may have no solution; it would have one (Vol. I, Thin. 7.3-2) if it were required in addition t h a t the stored energy function be convex with respect to its argument F C 1~ 3, but then this requirement would contradict frame-indifference (Vol. I, Ex. 3.7 and Thin. 4.8-1). This is not a shortcoming however, as only the existence of a "diagonal infimizing family", as defined in Thin. 4.13-1, is required in the ensuing analysis. This problem is then transformed as in Sect. 4.12 into an analogous problem over the set ft, i.e., the deformations are scaled, by letting qC~(x~) - ~ ( c ) ( x ) for all x ~ - 7r~x r ~ , and it is furthermore assumed that there exist functions f r L2(ft) and g E L2(F+ U F_) independent of e such that
f~(x ~) -- f ( x ) for all x ~ - 7r~x E f~, g~(x ~) - ~.g(x) for all x ~ - 7rex E F+ U F~. As a consequence of these scalings and assumptions, the scaled
deformation satisfies the minimization problem: qp(c) E (I)(c; Ft) and I ( c ) ( q p ) -
(I)(E'; ~ ) " - - {r E WI'P(~); r
inf
where
(~O0(E) on 7 X [-1, 1]},
(~0(C)(X) "-- (Xl, X2,CX3) for all x -
I(e)(O) .-
I(c)(r
(Xl,X2, x3)E ')' x [-1, 1],
w((o1r o2r lo3r - { /a f " ~b dx + fr+ur g " ~b dF } ,
where the notation (a l; a2; aa) stands for the matrix in NI 3 whose three column vectors are a l , a2, a3 (in this order). The scaled
Sect. 4.13]
Frame-indifferent nonlinear membrane theory and F-convergence 351
displacement : =
-
therefore solves the minimization problem (recall that {el, e2, e3} denotes the basis in R3): u(e) e V ( ~ ) and J ( e ) ( u ( c ) ) = V(~'-~) : = {V E W I ' p ( ~ ) ;
J(e)(v) .-
fa ~7((el
inf
J(e)(v), where
v -- 0 OIl ")' X [--1, 1]},
-~- 01v; e2 -~- 02v; e3 -~- -10 3 v ) ) d x C
- {L f "(q~o(e)+ v) dx + /r+ur_g 9(~0(c) + v) dr}. Central to the subsequent analysis is the notion of quasiconvexity, due to Morrey [1952, 1966] (an illuminating account of its importance in the calculus of variations is provided in Dacorogna [1989, Chap. 5]): Let NI"~x n denote the space of all real matrices with m rows and n columns; a measurable and locally integrable function l ~ " NIm• --~ R is q u a s i - c o n v e x if, for all bounded open subsets D C R ~, all F e M mx~, and all 0 - (0~)~ 1 e w l ' ~ ( D ) ,
I~(F) _<1TIES1 D L I~(F + V0(x))dx, where V 0 denote the matrix (OjO~) E NIm• Given any function 1~" NIm• + R, its q u a s i - c o n v e x e n v e l o p e is the function Q I ~ " NI m• + R defined by QI~ - sup{X; 2 " ~]~mxn ~ R is quasi-convex and 2 _< l~}. The scaled energies J ( e ) : V(fi) ~ R are then extended to the larger space LP(f~) by letting
{ J(a)(v) if v E V(f~), J(a)(v)+ oc if v E LP(f~) but v r V(f~).
352
Nonlinearly elastic plates
[Ch. 4
This ruse, customary in F-convergence theory, has inter alia the advantage of "incorporating" the boundary condition in the extended functional. In this fashion, Le Dret & Raoult [1995] establish that the family (3(~))~>0 F-converges as c ~ 0 in LP(~), and that its F-limit can be computed by means of quasi-convex envelopes (the definition of F-convergence and F-limits are given in Sect. 1.11). More precisely, their analysis leads to the following theorem, where the limit problems are directly posed as two-dimensional problems (part (c)); this is licit since their solutions do not depend on the "transverse" variable (part (b)). Note that while minimizers of J(c) over V(ft) need not exist, the existence of a "diagonal infimizing family", in the sense understood below, is always guaranteed. The notation (al; a2) stands for the matrix in N[3• that has a l , a 2 (in this order) as its column vectors. T h e o r e m 4.13-1. Let (v(c))~>0 be a "diagonal infimizing family" of the scaled energies, i.e., v(~) E V(f~) and J(e)(v(e)) <
inf J(c)(v) + oh(c) for all c > 0, vcv(f~)
where h :]0, oc[~]0, ~ [ is any function that satisfies h(c) ~ 0 as c ~ O. Then: (a) The family (v(c))~>0 lies in a weakly compact subset of W~'P(~). (b) The limit u as c ~ 0 of any weakly convergent subsequence of (v(~))~>0 satisfies u E V ( g t ) : = {v C V(f~); 03v = 0 in f~}. (c) The vector field
1/
u- - - ~
u dx3
solves the minimization problem (QI~Vo denotes the quasi-convex
Sect. 4.13] Frame-indifferent nonlinear membrane theory and F-convergence 353
of Wo)" ~i E W~'P(a~) and J ( ~ ) -
2 j~ ~ / ' 0 ( ( e l d(rt) "-
inf 9' (rt), where ~ewl.,(~)
-Jr-C01T~;e2 -~- 02T~))d(.d -
p . r / d w if rt E w l ' p ( w ) ,
+oc if rt E LP(w) but rt ~ w l ' p ( w ) ,
I/Yo((al; a2)) p "-
1
"--
inf
a3 EIR3
W ( ( a l ; a2; aa)),
f d x a + g(., 1) + g ( . , - 1 ) . II
A natural question immediately arises: How does this limit model compares with those found by ~bx, Raoult &: Simo [1993] via the method of formal asymptotic expansions (Sect. 4.12)? As we now explain, it is a " l a r g e d e f o r m a t i o n " , " f r a m e - i n d i f f e r e n t " , n o n l i n e a r m e m b r a n e t h e o r y , that presents similarities with, as well as subtle differences from, the nonlinear membrane theory previously found by formal means (Thm. 4.12-1). First, it is a "lar9e deformation theory", as the de-scaling produces a deformation that is O(1) with respect to e. Secondly, it is flame-indifferent in the following sense: Assume that the original stored energy function is flame-indifferent, in the sense that W(RF) - W(FR)
R e
F e M
where 9 denotes the set of all real orthogonal matrices R of order 3 with det R = 1. This relation is stronger than the usual one, which needs to hold only for F E NI a with det F > 0 (Vol. I, Thm. 4.2-1); it is however verified by the kinds of stored energy functions to which the present analysis applies, e.g., that of a St Venant-Kirchhoff material. Under this stronger assumption, Le Dret & Raoult [1995,
Nonlinearly elastic plates
354
[Ch. 4
Thm. 9] establish that the quasi-convex envelope QI/V0 " NI3x2 ~ R appearing in the functional J (Thin. 4.13-1(c)) satisfies
QIs
- Oleo(/~) for all R E 9
F e M3•
furthermore there exists a real-valued function I~0 defined over the set of all real symmetric positive definite matrices of order 2, such that QIs I~(/~T/~) for all/~ E l ~ 3x2. This last property is thus reminiscent of a relation satisfied by frameindifferent three-dimensional stored energy functions (Vol. I, Thm. 4.2-1). Thirdly, it is remarkable that both functions Is and QI~0 can be explicitly computed when the three-dimensional energy function ~ r . [~i[3 ---+ R is that of a St Venant-Kirchhoff material, i.e., when I ~ ( F ) - ~# t r ( F T F -
)~ { t r ( F T F - I)} 2 I ) : + -~
In this case, the function l~0 9M 3x2 ~ IF{ and its quasi-convex envelope Ql~0" NI 3x2 ~ R are given by (Ce Dret & Raoult [1995, Prop. 16])" ls
-
- ~P t r ( ~ ' T F _ ~)2 + +
{[Atr(F F -
+ 2.)
8(~ + # )
4 ( ~ + 2#)
W
-
{t
(F
-
I) + (~ + 2#)]+} 2 -
{[v2(F) - 1]+
-~- 2(~ -~- p)(/~ + 2p) {[(/~ + p)Vl(F)2 + 2 v2('~~)2 ~-
1
+ 2,)
3~ + 2# 2
{ [/~(Yi (F)2 + v2(1~)2)_ (3~ + 2#)1+} 2,
where I denotes the unit matrix of order 2, [x]+ "- max{0, x}, and V 1 ( g ) , v2(f ) denote the two singular values of the matrix F C M 3•
Sect. 4.13] Frame-indifferent nonlinear membrane theory and F-convergence 355 (i.e., the square roots of the two eigenvalues of the matrix /~TF), always arranged as 0 _< V l ( F ) < v2(F). If
and 2(A + ~)Vl(-P)~ + A ~ ( F ) ~ _> 3A + 2~, the quasi-convex envelope Ql~0 coincides with the stored energy function found in Thm. 4.12-1 , viz. , r ----+ -~a~9~,,-E~,,._.E~,._.. 1 The frame-indifferent membrane theory of Fox, Raoult & Simo [1993] is thus recovered in this case. A w o r d of c a u t i o n . These two approaches thus provide an intriguing instance where the limit equations found by a formal asymptotic analysis (Thm. 4.12-1) do no always coincide with those found by a convergence theorem. II Le Dret & Raoult [1995, Thm. 10] have further shown that, if the stored energy function is frame-indifferent and satisfies in addition l ~ ( I ) - 0 and I ~ ( F ) _> 0 for all F E NI3 (as does the stored energy function of a St Venant-Kirchhoff material), then QI~0(F) - 0 for all F E NP • such that 0 <_ v~(F) <_ v2(F) _< 1. This result has the striking consequence that "nonlinear membranes offer no resistance to crumpling. This is an empirical fact, witnessed by anyone who ever played with a deflated balloon" (to quote H. Le Dret and A. Raoult). Other behaviors occur depending on where the singular values Vl(F) and v2(F) lie in the set {(z~) E R2; z~ > 0}. The reader interested in further deciphering the arcana of such membrane theories should consult Genevey [1996]. A highly desirable requirement is that the stored energy function of an hyperelastic material, normally only defined for matrices F satisfying det F > 0, be such that I ~ ( F ) ~ +oc as det F ~ 0 +. Ben Belgacem [1997] has shown how to include this condition in the above analysis. A most valuable virtue of the nonlinear membrane theories found in this section and in Sect. 4.12 is their ability to answer the following
356
Nonlinearly elastic plates
[Ch. 4
question: Consider a family of plates, all made with the same nonlinearly elastic materials (e.g., a St Venant-Kirchhoff material), and only subjected to their own weight; as the mass density p and the gravitational constant g are independent of c, so is the function f E L2(ft) defined by f ( x ) = f ~ ( x ~) = (0, O , - p g ) for all x ~ = rr~z E f t ~. What is the asymptotic behavior of such plates as c -+ 07 This question cannot be resolved by the nonlinear Kirchhoff-Love theory, which requires that f~ = O(e a) if the Lamd constants do not depend on e as here (Sect. 4.11). But it can be resolved by the nonlinear membrane theories, as ttiey allow body forces that are O(1): Ultimately, i.e., for a "sufficiently small" thickness, "a thin nonIinearly elastic body submitted to its own weight does not behave like a plate, but indeed like a membrane" (Le Dret & Raoult [1995, p.552]). The same theories also provide the invaluable information that the corresponding transverse displacement is O(1), another reason why such limit behaviors cannot be recovered by the Kirchhoff-Love theory, which corresponds to transverse displacements that are O(c) (Sect. 4.10). 4 . 1 4 ~.
NONLINEARLY IN CARTESIAN
ELASTIC SHALLOW COORDINATES
SHELLS
The method of formal asymptotic expansions may be also used for deriving the two-dimensional equations of a nonlinearly elastic "shallow" shell in Cartesian coordinates (the "shallowness" being defined as in the linear case; cf. Sect. 3.8), by extending the analysis of Sects. 4.1-4.5 to the more general geometry of a "shallow" shell. Such an extension is due to Ciarlet & Paumier [1986], who used the displacement-stress approach and for this reason, were restricted to "completely clamped" shells (this restriction already holds for plates; cf. Thm. 4.8-1). We briefly describe here the application of the displacement approach to a "partially clamped" shallow shell. Let a~ be a domain in R 2 with boundary ~/, and let 0 ~ : ~ ~ R be a function of class C3. The reference configuration of the shell is
Sect. 4.14]
Nonlinearly elastic shallow shells in Cartesian coordinates
{~t~} -, where ~
357
- O~(ft~), ft ~ - c o x ] - c,c[,
O e ( x e) -- (Zl,X2 , 0 s ( x l , z 2 ) )
~- x ~ a ~ ( z l , x 2 )
for all x ~ - ( X l , X 2 , X 3 ) E , and a 3 is a unit vector normal to the middle surface O ~(~) of the shell. T h e "geometry" of the shell is thus defined as in the linear case; cf. Fig. 3.1-1. T h e shell is s u b j e c t e d to applied body forces of density ( ) ) " fi~ R a in its interior and to applied surface forces of density (t)~)" F~- U F~ + R a on its u p p e r and lower faces F~_ and ['~_ (defined as in Sect. 3.1), and it is clamped on a portion O~(% • I-e, el) of its lateral face, where % C 7 and length % > 0. Let k ~ - (k~) denote the generic point in the set {~)~}-, and o5~ . - O/Ok~. A s s u m e t h a t the shell is m a d e of a St Venant-Kirchhoff material, with Lamd c o n s t a n t s ~ and #~. T h e n the displacement field it ~ satisfies the variational problem 7 ) ( ~ ) 9 "s e -- ( ~ )
E V(~') e) "-- {.~e __ (/)~) E wl,4(~.~e);
~e _ 0 Oil r ; } , 9iYi
for an where ^C
C ^C
^C "E ^~"
Let f~ = co x ] - 1, 1[. We defined the s c a l e d d i s p l a c e m e n t u ( c ) = (u~(c)): f~ -+ R a by
field
gz~(k ~) - s2u~(s)(x) and ~2~(k~) - cu3(g)(x ) for all a~~ C O~(rcCx) C { ~ r where 7re(Xl,X2, Z3) -- (Zl,X2,~X3); cf. again Fig. 3.1-1. Assume t h a t there exist c o n s t a n t s A > 0 and
Nonlinearly elastic plates
358
p > 0 and functions f~ E L2(f~), 9~ E independent of e such t h a t A~ = A
and
L2(F+u
[Ch. 4 r _ ) , and 0 E C3(~)
#~=#,
~(Jc~)--c2f~(x)~ and f~ (~?~) - c3f3 (x) for all ~ ? ~ - O ~(Tr~x) e { ~ }-, g~(3c~)--E3g~(x) and g3($~) - s 4gz(x) for all 5 : ~ _ O ~(Tffx) E F+ ^~ U ~ _ ^~ ^~ O~(Xl,X2) = eO(Xl,X2) for all (x~,x2) E -~. In this fashion, the scaled unknown u(c) solves over gt a variational problem (which reduces to t h a t described in T h m . 4.2-1 when 0 = 0) posed over the space V(~-~) "-- {V -- (Vi) e w l ' 4 ( ~ ' ~ ) ; V -- 0 on F0},
and whose form again suggests to write U ( E ) -- U 0 Jr- CU 1 "-]- s
-~- E3U 3 + y 4 U 4 -~- h.o.t,
so as to apply the basic Ansatz of the method of formal asymptotic expansions. In so doing, we find as in Sect. 4.3 t h a t the leading t e r m u ~ should solve a limit scaled three-dimensional problem 7)KL(~), which reduces to t h a t found in Thin. 4.5-1 when 0 = 0. According to the following result, which reduces to T h m . 4.1 in Ciarlet A P a u m i e r [1986] when 7 = 70, solving this problem in turn a m o u n t s to solving
a two-dimensional problem: Theorem
4 . 1 4 - 1 . (a) Define the space
V(w) "- {v/-(rh) C Hi(w)
x Hl(cd) x
H2(w); r/~- O~r/3- 0 on "~0}.I
Then there exists ( ~ - (~) E V(w) such that the components u~0 of the leading term u ~ are of the form u 0~ - ~ - x 3 0 ~ 3
and
u~
Sect. 4.14]
Nonlinearly elastic shallow shells in Cartesian coordinates
359
(b) Let -o
1
/1 /1
1
q~ "-
z a f ~ d z 3 + g+ - g2 Ot
1
Then r = (r E V(a;) satisfies the following limit scaled two-dimensional problem 79(a;): -
.~9o~
d~ +
~:~0~((~ + 0)09~ d~ +
N ~ 9 0 , ~ d~
- f p i r l i d a j - o f q~O~rl3daJ f o r a l l r l - (rli) E V(aJ), where m~
-
N~9 -
-
{ 3 ( ~4A# A~35~9 + ~0~ZC3} + 2~)
4A# -o ~ + 2# E~(r
+
4#/)09 (r
(c) Assume that the boundary ~, the functions pi, q~ and the solution r = (r of 7)(a;) are smooth enough. Then r satisfies the following boundary value problem (the notations are defined as in,
e.g., Thin. 4.6-2): m
- O ~ r n ~ - O~{N~O~((3 + O)} - P3 + O~q~ in w, ~ = 0~,~3 = 0 o n 7o,
rn~gu~u 9 = 0 on 71,
360
[Ch. 4
Nonlinearly elastic plates
{0c~7~afl @ Nafl(~o~(~ 3 -nt.--0)}./3 -q--Or(me, ill/aTe) -- --qalJc~ on 71,
No~guo - 0 on ")'1. II Defining functions C[ "& --' R through the de-scalings 4~'-c24~
and
(~'-c~-ainco,
we then obtain the following immediate corollary of Thm. 4.14-1" Theorem 4.14-2. (a) The de-scaled vector field ~ V(w) satisfies the following variational equations:
- s ~:~ a~- s q-;Oo,~d~
for .11 ., - ( , ~ ) ~
(~) E
v(~).
where the space V(w) is defined as in Thin. 4.14-1, and
e3{ 'F~
"--
dg~ .
3
4A~#~ A ~ z 3(~-7 ; 2~e)
{ 4M#~
(~
+ 4#~
~}
-~Oc~fl~3
-o,~
,
}
- o,~ 1 0 ~ ~ ~ G~,(~ ~) - ~( ~ , ~ + o~,G + oo,0 ~ o~,~3 + o~,0~GC3~ + GGo,~3)
p~ .-
/
,
f;(O~(, x;)) dx; + 0~(0~(-, e)) + 0~(0~(., -e)), s
q-; . -
x~f;(O~(.,x;))dx~+e{O;(O~(.,c))-f];(O~(.,-e))}. E
(b) Assume that the boundary 7, the function ~ , Cfl~, and the solution ~ - ( ~ ) are smooth enough. Then ~ satisfies the following
Sect. 4.14]
Nonlinearly elastic shallow shells in Cartesian coordinates
361
boundary value problem: --Oo~m~
--
-- ~ { N ~ O o ~
s
--s
s
s
(r -+- OS) } -- P3 + Oc~q~ in w, _ 0 9 N ~9 ~ - p~ -~ in w,
r
-- Ov<~ - - 0
o n 70,
s
on ")/1,
Tl~a fl l/ c~ldfl - - 0
{O~rnt, + N ; , O ~ ( r a + 0~)}~9 + oq.(rn~,L,~v-,) - --q.LG on ")/1,
Nj~u~ -- 0 on ~/1" I
Either one of the above problems constitutes the t w o - d i m e n sional e q u a t i o n s of a n o n l i n e a r l y elastic shallow shell in C a r t e s i a n c o o r d i n a t e s . As in its linear counterpart (Thm. 3.7-2), the adjective "Cartesian" reflects that the unknowns ~[ are the Cartesian components of the displacement of the middle surface of the shell. It is easily seen that solving the variational equations found in Thin. 4.14-2(b) is equivalent to finding the stationary points of the t w o - d i m e n s i o n a l e n e r g y j ~ : V(w)--+ R (of a nonlinearly elastic shallow shell) defined by: 1L{a3
J s (,7) 9 ~ _
_
-0,e -2 a ~e ' ~ (~o...r./730c~~ T~3 + ~ 9s~ G , -0,e (,7)G9('7)
}
d~
-(f~/~:r/~ d w - L O:O~r/3 dw) for all r / - (r/~) E V(w), where
a~,~.
:= ~
+ 2/-*~ 8~( ~ + ~)sa
Remarks. (1) We also have -0~gma~ -- ~()~g ~ ~73
2
A r
(2) The limit stresses can also be computed when 70 = 7 by means of the displacement-stress approach; cf. Ciarlet & Paumier [1986, Thin. 4.11.
362
Nonlinearly elastic plates
[Ch. 4
~ ( ~ ) that is linear (3) As expected, , the part of the functions/)o,~ with respect to (~ coincides with the functions ~~((~ ~) found in the linear theory (Thin. 3.7-1). (4) The functions E0,~ ~ ( ~ ) reduce to the functions E~ ~) when 0 = 0, i.e., when the shell is a plate (Thm. 4.9-1). (5) It is likely that existence and regularity properties of solutions to problem/)(co) found in Thm. 1.14-1(b) can be derived by the same methods as in Sect. 4.6. However, such results do not seem to be available in the existing literature. (6) The method of asymptotic expansions can also be applied to nonlinearly elastic shallow shells with varying thickness; see de Figueiredo [1989]. m We have thus been able to justify two-dimensional nonlinear shallow shell equations in Cartesian coordinates from three-dimensional elasticity; moreover, these equations coincide with those found in the literature by means of a priori assumptions; see, e.g., Washizu [1975, p. 173] or Dikmen [1982, p. 160]. Two important conclusions have also arisen: First, the deviation of the middle surface of the shell from a plane should be of order of the thickness of the shell (as in the linear case) in order that the shell be deemed "shallow"; this is the essence of the assumption 0 ~ = O(c). Second, the (de-scaled) vertical deflection ~ := c~a of the points of the middle surface should remain of the order of the thickness of the shell (as in the nonlinear Kirchhoff-Love theory of an elastic plate) in order that the two-dimensional equations be safely used in lieu of the three-dimensional ones: The nonlinear shallow shell theory found here is thus a s m a l l d i s p l a c e m e n t t h e o r y . EXERCISES
4.1. Show that the conclusions of Thm.4.5-1 are unaltered if the scaled displacement is expanded in Thm. 4.4-1 as u(e) = u ~ +c2u 2 +e4u 4 +...
,
363
Exercises
i.e., the terms of odd order are a priori canceled in the formal asymptotic expansion of u(e) (this observation is due to Raoult [1988, Prop. r, p. ss]. 4.2. This problem offers a direct proof of the existence of a smooth solution ~ = (~i) of the boundary value problem:
-O~N~
= r~ in co,
4i = c9,4a = 0 on 7, where m~9 and N~z are the functions of (~ = (Ci) defined in Thin. 4.6-1. W h e n 3'o = 7, this problem thus provides an alternative to the existence theory of Sect. 4.6 (with a restriction however on the m a g n i t u d e of the applied forces; cf. (3)). (1) Define the nonlinear operator A by
A(r
- (-OzNI~,-O~N~,-0~.~
- Oz(N~O~r
for any smooth enough vector field (~ = (~) : & - , R a. Given any p > 2, show that A maps the space Wa'P(a~) x wa'p(a~) x W4,p(a~) into the space WI'p(a~) x Wl,p(cz) x LP(a~), and that A is infinitely differentiable between these two spaces. (2) Show that, if the b o u n d a r y 7 is smooth enough, the derivative of the operator A at the origin is an isomorphism from the space
v~(~) .- {.-
(,~) ~ w~.~(~)•
• w4.~(~); r]i = 0,r]3 = 0 on 7}
onto the space WP(aJ) " - wl'p(a~) x WI'p(a~) x LP(a~). (3) Show that, if the boundary 7 is smooth enough, there exist for each p > 2 a neighborhood F p of the origin in WP(a~) and a neighborhood U p of the origin in the space VP(a~) such that, for each r = (ri) E F p, the nonlinear equation A(r
=r
364
Nonlinearly elastic plates
[Ch. 4
has exactly one solution (2 in U p. Hint: Use the implicit function theorem as in three-dimensional elasticity; cf. Vol. I, Thm. 6.4-1. 4.3. The theme of this exercise is due to A. Raoult. A necessary and sufficient condition that the response function E for the second Piola-Kirchhoff be flame-indifferent is that there exists a mapping . ~;a> __~ ga (~;a denotes the set of all symmetric matrices of order 3 and ga> the subset of S 3 consisting of positive definitive matrices) such that E ( F ) - "~(FTF) for all matrices F of order 3 with det F > 0 (cf. Vol. I, Thm. 3.3-1; for simplicity, only homogeneous materials are considered here). (1) Show that, if E is frame-indifferent, the equations of threedimensional nonlinear elasticity are also flame-indifferent in the following sense" Let ft be a d o m a i n in R 3 and let qp 9 ft ---+ R 3 be a deformation of the reference configuration ft that satisfies the equations of equilibrium (Vol. I, Thin. 2.6-2)" - d i v { V q O E ( V q o ) } - f in ft. Let Q be an orthogonal matrix of order 3 with det Q = 1. Then Qq0 satisfies the same equilibrium equations but with Q f as their righthand side: In other words, "if the applied body force is rotated by Q, so is the deformation ~o" (naturally, the matrix Q is independent o f x E ~). (2) The two-dimensional equilibrium equations of the nonlinear Kirchhoff-Love theory (Sect. 4.9) may be written as follows, once self-explanatory notational simplifications have been performed to facilitate the comparison with the semilinear three-dimensional equilibrium equations of (1):
-O~{a~o~E~
-- r~ in co,
By means of a counter-example, show that these equations are
not frame-indifferent" Let q 0 - ~ + ~ where ~(xl, x 2 ) - (Xl,X2, 0), and let r - (r~); then there exists an orthogonal matrix Q of order 3 such
Exercises
365
t h a t Q ~ does not satisfy the same equilibrium equations but with Q r as their r i g h t - h a n d side. Remark. The semilinear yon Kdrmdn equations studied in the next chapter are neither frame-indifferent, since they correspond to the same two-dimensional equilibrium equations (Thin. 5.4-2(c)). There do exist however two-dimensional equilibrium equations t h a t are b o t h frame-indifferent and quasilinear in addition, as their threedimensional counterparts; cf. Ex. 4.4. 4.4. T h e notations are those of Thin. 4.12-1. (1) Show t h a t the "limit" scaled deformation ~o~ - (~~ 9c~ ~ R 3 (which m a y also be viewed as a de-scaled unknown, since the "original" u n k n o w n q~ is simply scaled as qa~(x ~) - qa(s)(x) for all x ~ = 7r~x E fY) obtained in the nonlinear membrane theory of Fox, Raoult A Simo [1993] satisfies, at least formally, the following quasilinear b o u n d a r y value problem:
_ O ~ { a ~ 9 ~ E M (qao)o~oo} _ pO in co, V,~
{aa~or176176 where p0 ._ (p0) and 0/1 - - " ) / (2) Show that, by contrast tions are frame-indifferent in thogonal m a t r i x of order 3. differential equations in co but
~ on 70,
-- 0 on 0/1, 0/0.
with those of Ex. 4.3 (2), these equathe following sense: Let Q be an orT h e n Q~o ~ satisfies the same partial with QpO as their right-hand side.
4.5. Given a domain co C R 2 and a portion 70 of its b o u n d a r y 7, define the set cI, r ( ~ ) " - {~p - ~+rl; r / C H2(a;); 0~p.c3~p - ~
in ~, r / -
0 on 7o},
as in T h m . 4.12-2. (1) Assume t h a t ~ is a rectangle and t h a t 7o is one of its sides. Show that (2) Assume t h a t ~ is a rectangle and t h a t 70 = 7. Show t h a t r = {~}.
366
Nonlinearly elastic plates
[Ch. 4
Remark. The conclusion of (2) is a special case of a general result in differential geometry, asserting that a planar domain fixed on its entire boundary cannot undergo any metric-preserving deformation other than ~ (such questions are discussed in Vol. III).
CHAPTER 5
THE VON K/~RM/i~N EQUATIONS
INTRODUCTION The two-dimensional yon Kdrmdn equations for nonlinearly elastic plates, originally proposed by T. von Kgrms in 1910 (see p. lxiii), play an almost mythical role in applied mathematics. While they have been abundantly, and satisfactorily, studied from the mathematical standpoint, as regards notably various questions of existence, regularity, and bifurcation of their solutions, their physical soundness has often been seriously questioned. For instance, Truesdell [1978, pp. 601-602] made the following statements : "An analyst may regard that theory (von Kgrmgn's theory of plates) as handed out by some higher power (a Hungarian wizard, say) and study it as a matter of pure analysis. To do so for von Kgrmgn theory is particularly tempting because nobody can make sense out of the "derivations" ... I asked an expert, Mr. Antman, what was wrong with it (von K~rm~n theory). I can do no better than paraphrase what he told me: It relies upon: (i) "approximate geometry", the validity of which is assessable only in terms of some other theory; (ii) assumptions about the way the stress varies over a crosssection, assumptions that could be justified only in terms of some other theory; (iii) commitment to some specific linear constitutive relation linear, that is, in some measure of strain, while such approximate linearity should be the outcome, not the basis, of a theory; (iv) neglect of some components of strain- again, something that should be proved mathematically from an overriding, self-consistent theory; (v) an apparent confusion of the referential and spatial descriptions - a confusion that is easily justified for classical linearized
368
The yon Kdrrndn equations
[Ch. 5
elasticity but here is carried over unquestioned, contrary to all recent studies of the elasticity of finite deformations." Using the same method as in Chap. 4, we show in this chapter t h a t the yon Kdrmdn equations may be given a full justification by means of the leading term of a formal asymptotic expansion (in terms of the thickness of the plate as the "small" parameter) of the exact three-dimensional equations of nonlinear elasticity associated with a specific class of boundary conditions that characterizes the "yon Kdrrndn plates" (Sect. 5.1). For ease of exposition, we again restrict ourselves to St Venant-Kirchhoff materials, but our conclusions apply as well to the most general elastic materials, by means of an extension identical to that discussed in Sect. 4.10. In this fashion, we are able to provide an effective strategy for imbedding the von K~rmfin equations in a rational approximation scheme that overcomes the five objections raised by S.S. Antman. More specifically, our development clearly delineates the validity of these equations, which should be used only under carefully circumscribed situations. First, the validity of these "limit" two-dimensional equations is definitely dependent on an appropriate relative behavior of the varions physical data involved when the thickness approaches zero. As shown by the analyses made in the previous chapters, this observation pervades in fact plate theory. Secondly, this approach clarifies the nature of admissible boundary conditions for the three-dimensional model from which these equations are obtained, and consequently for the von K~rm~n equations themselves. Let us outline the content of this chapter. In Sect. 5.1, we pose the three-dimensional problem of a yon Kdrmdn plate: We consider a plate occcupying the set ~ - a~ - x [-c, c], where a~ is a domain in R 2 and ~ is > 0, subjected to applied body forces (f~) = (0, 0, f~) in f~, to applied surface forces (g~) = (0, 0, g~) on the upper and lower faces F ~+ - a~ x {• and to applied surface forces on the entire lateral face 7 x I-c, c] whose only the resultant (h~, h~, 0) along the b o u n d a r y ~ of the set a~ is given. The boundary conditions involving
Introduction
369
the displacement (u~) are (as usual, Greek indices vary in {1,2}) u~ independent of x ~a and u ~3 - 0 o n T x
[-c,c].
Notice in passing the novelty of the conditions "u~ independent of x~" the special form of which plays an essential r61e in later devel3 opments; see the discussion given in Sect. 5.1. The problem then consists in finding the displacement field u ~ - (u~) and the second Piola-Kirchhoff stress tensor field E ~ - (cr~j) as solutions of the following nonlinear b o u n d a r y value problem ( ~ and #~ are the Lam(~ constants of the elastic material; (~,~) denotes the unit outer normal vector along 7):
-0~ (o~ q- crkjc9k % ) -- f[ in E
C
(~ + ~0~)
1/
~
- +g; o~ r+,
u~ i n d e p e n d e n t of x a and u a~ - 0 o n T x
e
(
E
g
~
E
, [ - e e, ] ,
E
~ cr~e + akeOk%)t,e dx~a- h~ on 7,
where _ ~E~(u
~
1
E~j(u ~) - -~(O~u~ + 0 ; < + O~u~O;U~m). In Sect. 5.2, we define an equivalent problem, but now posed over the set (~ - & • [-1, 1], which is independent of e. This transformation involves a p p r o p r i a t e scalings on the unknowns (u~) and (erda)), and a d e q u a t e assumptions on the data )~, P~, f~, g~, and h~ regarding their a s y m p t o t i c behavior as functions of c. In other words, we use the displacement-stress approach described in Chap. 4 for a clamped plate, i.e., we let:
u~(x ~) -- c2u~(c)(x), ~,(x~)-~~,(~)(x)
.
~ ( . ~ ) ~ ~ ~ _
u~(x ~) -- cu3(c)(x), (~)(~) . ~ (x~) - ~4 ~ ( ~ ) ( x )
for all x e - 7rex E , where rc~(xl,x2, xa) - (Xl,X2, cx3); WC then assume t h a t there exist constants A > 0 and > > 0 and functions
The von Kdrmdn equations
370
f3 C L2(f~),ga e that
L(F+ u F_) and A~ = A
g~(x ~) - e4g3(x) h:(y)-c
h~ E L2(~/) independent of ~ such
and
f~(x ~) - e3f3(x)
[Ch. 5
#~=#,
for all x ~ - 7r~x e f~,
for all x ~ - 7r~x e F+ U F~, 2h~(y)
for a l l y e ~ .
In this fashion, the scaled unknowns ( u ( e ) = (u~(c)) and E(g)) = (cr~j(e)) solve a problem of the form (Thin. 5.2-1) :
U(g) E V ( ~ ) - - {V = (V i) E wl'4(['~); Vc~ independent of xa andva=0onTx
[-1,1]},
E(e) e L ~ ( f ~ ) - {(T~j)E L2(f~); 7~j -Tj~},
u(c), v) + e2T2(E(c), u(e), v) - L(v)
B(E(e), v) + T~
for all v C V(f~),
E~
+ c2E2(u(~)) --(B ~ +
c2B 2 + e4B4)E(c),
where the linear form L, the bilinear form B, the trilinear forms T o and T 2, the tensor-valued mappings E ~ and E 2, and the fourth-order tensors B~ B2 B 4 a r e all independent of e The specific form of this problem again suggests that we use the method of formal asymptotic expansions, i.e., that we let _
+...,
-
~ +
.
.
.
.
In doing so, we find (Thm. 5.3-1) that the leading terms u ~ and E ~ should satisfy the equations
B(E ~ v) + T ~
~ u ~ v) - L(v) for all v e V(f~), E ~(u ~ - BOZ ~
Our main result then consists in identifying situations where the above equations are nothing but a disguised form of the yon Kdrmdn equations (up to appropriate de-scalings). More specifically, assume that the set a~ is simply connected (this assumption is essential), the
Introduction
371
data are sufficiently regular and the functions h~ satisfy the compatibility conditions (whose justifications are given at length in this chapter)
fhld~/-fh2dT-~(
Xl h2 -
x2hl ) d~/ =
O.
Then we prove the following (Thins. 5.4-2 and 5.6-1): (i) The vector field u ~ - (u ~ is a (scaled) Kirchhoff-Love displacement field" The function u ~ is independent of the variable xa, and it can be identified with a function ~a in the space Hg(co) NH4(aa); the functions u~0 are of the form u ~c~ - ~ - xaO~a, with ~ E Ha(w). (ii) In order to compute the vector field ~ = (C~), one first solves the (scaled) yon Kdrmdn equations: Find (~a, r &--* R 2 such that 8#()~ + #)A2~. a _ [qh,~3] + P3 in w
a(A + 2u)
A2q5 _ _#(3A + 2#)[(3, ~3] in w, k+> (a = O~,4a = 0 on ~/, r162
h2) oil ~,
Our = el(hi, h2) on ~/, where r is the (scaled) Airy stress function, O, is the normal derivative operator along 7, r and r are known functions of hi, h2, and [~, ~)] -- 011~022r -~- 022~011r -- 2012~012r
Pc-9 ++9;+
?
1
fadxa,
9~-ga(',4-1).
(iii) Next, one sets Nil : 022r
N12 = N21 = -012r
N22 = 011r in co.
Then, for a given function Ca, the functions ~1 and C2 are obtained as the solutions (unique up to an arbitrary infinitesimal rigid displacement in the plane of co) of two-dimensional (scaled) membrane
372
[Ch. 5
The yon Kdrmdn equations
equations, whose right-hand sides are known functions of hi, h2, and ~3. (iv) The limit scaled stresses a~j, o 1 < _ i , j _< 3 , are then given by explicit formulas involving the previously determined functions (Thin. 5.5-1): 0
1
0
3 ~(1
0"o~ 3 - -
3
-
1
o-~ - - ~ x ~ ( 1
1
2
x3)Oarrtaz
2
- x~)o~,.~,
+ ~(1 + x3)
/1
3
2
+ ~(1 - x ~ ) ~ , o ~ , ~
f3 dy3 -
f3 dy3
1 + 1 _ + ~(1 + x3)g3 - ~(1 - x3)g3, where 4Ap
m~
- -
3(~ + 2~) A I 3 5 ~
+ -~0~9~-3
9
The thread of this derivation is the equivalence between the yon K d r m d n equations and a two-dimensional "displacement" problem, posed only in terms of the components of the "limit displacements" along the middle surface w of the plate (Thin. 5.6-1). To sum up, we have simultaneously justified (after appropriate descalings; cf. Sect. 5.7) the two-dimensional yon K d r m d n equations of a plate, together with standard a priori assumptions, according to which the "displacement" u ~ is necessarily a Kirchhoff-Love field, and the "stresses" a~j 0 should take special forms. Perhaps the most noticable virtue of this method is that it clearly identifies those boundary conditions that are admissible for the corresponding three-dimensional problem, and from which the boundary conditions for the Airy stress function r must be in turn derived in a specific fashion. These aspects, which are often omitted in the literature, are further commented upon in Sect. 5.7. We then give (Sects. 5.8 to 5.11) a mathematical analysis of the von Ks163 equations. We notably study the questions of existence,
The three-dimensional equations
Sect. 5.1]
373
regularity, multiplicity, and bifurcation of its solutions; we also show how they m a y even degenerate into the famed Poisson equation of a linearly elastic membrane (Thin. 5.10-1)! We conclude this volume by showing how the celebrated Marguerreyon Kdrmdn equations of a nonlinearly elastic shallow shell m a y be likewise justified by a formal asymptotic analysis (Sect. 5.12). 5.1.
T H E T H R E E - D I M E N S I O N A L E Q U A T I O N S OF A N O N L I N E A R L Y E L A S T I C V O N K/~RM_&N PLATE
Let a2 be a d o m a i n in the plane spanned by the vectors e~. We denote by ( ~ ) and (~%) the unit outer normM vector and unit tangent vector along the b o u n d a r y 7 of a2, related by ~-1 - -~2, ~-2 - Lq. Given e > 0, let
a
r+
-
-
•
r
-
•
so t h a t the b o u n d a r y 0Q ~ of the set ~ is partitioned into the lateral face 7 • [ - e , a] and the upper and lower faces F~_ and F~. Finally, we let (n~) 9c9~ ~ --~ R a denote the unit outer normal vector along 0 ~ ; hence (n~) -- (Lq, L,2, 0) along the lateral face 7 x [-e, e]. We assume that, for each s > 0, the set ~ is the reference configuration of a nonlinearly elastic plate, subjected to three kinds of applied forces: (i) applied body forces acting in ft ~, of density ( f ~ ) : f~ + R3; (ii) applied surface forces acting on the upper and lower faces, of density (g~) 9r ~ U P t + Ra; (iii) applied surface forces parallel to the plane spanned by the vectors e~ acting on the lateral face -y x [ - s , s], whose only the resultant density (h~, h~, 0) 9"~ --~ R a per unit length, obtained by integration across the thickness, is known along the b o u n d a r y ~/ of the middle surface w of the plate.
The von Kdrmdn equations
374
[Ch. 5
For definiteness, we assume at this stage that f [ E L2(f~), g~ E
L~(F+u
F~),
hL ~
L2(7).
The boundary conditions involving the displacement field u ~ = (u~)" f~ ---, R 3 are: u~~ independent o f x a
and
u 3~ - 0 o n T x
[- e , e] .
In other words, if we think of the plane spanned by el and e2 as being "horizontal", any "vertical" segment along the lateral face can only undergo "horizontal" translations (Fig. 5.1-1). As in Chap. 4, we assume for ease of exposition that the plate is made of a St Venant-Kirchhoff material, but the present analysis carries over to more general nonlinearly elastic materials as for a clamped plate (Sect. 4.10). Let 3,~ and #~ denote the Lam~ constants of the elastic material. The three-dimensional problem then consists in finding the displacement vector field u ~ - (u~) 9~ --~ R a and the second Piola-Kirchhoff stress tensor field E ~ - (cr~) " - ~ ~ ga (g3 denotes the set of all symmetric matrices of order 3) t h a t satisfy the equilibrum equations:
(~,~ + e
%o~,)~;
- g~ on r+ u
_,
~ cr~ + ak~Oku~)v~dx ~ -- h~ on 7,
together with the constitutive equation
a~ - )~E~p(u~)6ij + 2p~Ei~j(u~), where
E~j(~ ~) .- -~(o;~j + o;~ + o ; ~ o ; ~ ) , and the boundary conditions u s~ independent of z a and u ~ 3-0onTx
[-c , r .
The three-dimensional equations
Sect. 5.1]
375
e5 ,/t
e~
f
. . . .
--
__
_--7_
-(..~ - ~
---x
~_..
__~ _ - ~
.......
k:
Fig. 5.1-1" A von Kdrrndn plate. The three-dimensional equations are characterized by specific boundary conditions on the whole lateral face ~y x I-c, c], where ~, = 0aJ. Applied surface forces parallel to the plane spanned by the vectors e~ are acting on the lateral face through their resultant (h~) ''~ --+ R 2 obtained by integration across the thickness of the plate. The admissible displacements us are independent of x~ and u~ - 0 along -~ x [-c,c]; in other words, any "vertical" segment along the lateral face can only undergo "horizontal" translations. Finally, all applied forces are "vertical", i.e., f~ = 0 and g~ - 0.
A s in S e c t .
4.7, we r e f o r m u l a t e
but different, problem:
this problem
as a n e q u i v a l e n t ,
F i r s t , t h e equilibrium equations a r e w r i t t e n
in t h e v a r i a t i o n a l f o r m of t h e principle of virtual work; s e c o n d l y , t h e
constitutive equation is inverted.
I n o t h e r w o r d s , we n o w c o n s i d e r
t h a t u ~ a n d E~ s a t i s f y t h e f o l l o w i n g p r o b l e m Q(gt ~) ( d ~ / d e n o t e s t h e
376
[Ch. 5
The von K d r m d n equations
arc length element along 7)" U s ~ V(~"~ c) "-- {V e -- (V~) ~ wl'4(~"~g); Vc~
independent of x~
and v~ - 0
on 7 • [-c,c]},
E ~ C L ] ( ~ ~) "- {(Ti~.) E L ~ ( ~ ) ; rio - r j < } , dz ~ -
+~
9~ vi dF~
Ev{
--C
v;dx;
hadTforallv ~ e
1
M
1
With these choices of spaces V ( ~ ~) and L~(~t~), all the integrals appearing in the left-hand sides of the principle of virtual work are well defined and both sides of the inverted constitutive equations are in L~(9~). Since we have assumed that f[ E L2(ft~), g~ E L2(F~_ U F~), and h~ 6 L2(7), the integrals appearing in the right-hand sides of the principle of virtual work are likewise well defined. Some comments are in order about the boundary conditions on 7 and on 7 x [-e,r the conjunction of which defines a (threedimensional) y o n K&rm&n plate" The boundary conditions '"'U,c~ ~ independent of x~ and u~ - 0 on 7 x [-c,e]" found in the definition of the space V ( ~ ~) were introduced by Ciarlet [1980]: Their effect is to precisely yield the other boundary conditions ,,_1
(~
+
~k~oku~)u~dx~ -- h~ on 7", as a result of an application of Green's formula to the principle of virtual work. Had we instead chosen more standard "pointwise" boundary conditions of the form:
~ + crkeOkU~)V'Z ~ ~'~ ~ ( cr~e -- H~~
and
it ~3 - 0 o n T x
[-e , e] ,
with functions H~ now defined on the lateral face 7 x I-e, e], serious difficulties would have arisen in later developments (see the discus-
The three-dimensional equations
Sect. 5.1]
377
sion in Sect. 5.5). Otherwise such pointwise boundary conditions are perfectly admissible for the three-dimensional problem (see, e.g., Duvaut & Lions [1972, p. 106] in the linear case); note that boundary conditions involving the components u~ of the displacement are no longer specified along ~/x [-e, e] in this case. Another worthwhile observation is that the applied surface forces along the lateral surface cannot be arbitrary. More specifically, after assuming that f~ - 0 and g~ - 0, we shall need to impose the
compatibility conditions
f~ h~l d7 - f~ h~ d7 - J~ { xlh~ - x2hel} d7 - O,
on the given functions h~ 9 7 ~ R. Whether such compatibility conditions may be needed depends upon the nature of the boundary conditions (in particular, no such conditions occur when the displacement is required to vanish on a portion of the boundary with strictly positive area). Their mathematical justification in the linear case is to allow for the definition of the energy in an appropriate quotient space, thereby providing an existence theory, the displacements being then defined only up to horizontal infinitesimal rigid displacements; see Ex. 1.9. In the nonlinear case, the situation is less clear on the mathematical side, except when such compatibility conditions can be related to an adequate existence theory, as in the work of Ball [1977]. We also note that these compatibility conditions are in agreement with the conclusions of the discussion given by Truesdell & Noll [1965, p. 127], who observe that these conditions can be written in the reference configuration (rather than in the defbrmed configuration) exactly as in the linear case. Such compatibility conditions also arise quite naturally in the proof of the existence of the leading term of the asymptotic expansion of the three_dimensional solution, as well as in the proof of its relation to the solution of the von Ks163 equations.
378 5.2.
The von Kdrrndn equations
[Ch. 5
TRANSFORMATION INTO A PROBLEM POSED OVER A DOMAIN INDEPENDENT O F c; T H E FUNDAMENTAL SCALINGS OF THE UNKNOWNS AND ASSUMPTIONS ON THE DATA
We follow here Ciarlet [1980]. As usual, our first task is to define a p r o b l e m equivalent to p r o b l e m Q ( f ~ ) , but now posed over a d o m a i n t h a t does not d e p e n d on c. Accordingly, we first let
ft = wx] - 1, 1[, P+=wx
{1}, F_ = w x
{-1},
and, with each point x E f t , we associate the point x ~ C the bijection 7r e
X-
(Xl,X2,
X3)
E ~-~ "'+ Z ~ -
(Z~)
-
(Xl,Z2,~Z3)
through
E
.
W i t h the fields u ~, v ~ e V(gt ~) and E ~ C L~(ft~), we t h e n associate as in Sect. 4.7 the s c a l e d d i s p l a c e m e n t field u (e ) = (u~(c)) : ft --. R a, the s c a l e d f u n c t i o n s v - (v~) 9ft ~ R 3, and the scaled s t r e s s t e n s o r field X:(c) = (a~j(c)): ft ~ ~3 defined by the seal-
ings"
u ~ ( x ~) - e 2 u , ( e ) ( x ) and u~a(x ~) - eu3(e)(x) v ; ( x ~) - e 2 v , ( x ) and v~(x ~) - eVa(X),
~(~)-
~9(~)(~), ~( ) -
~(~)(~),~
~(~)(~),
for all x ~ = 7r~x E ft ~. Naturally, these scalings on the stresses can be justified exactly as in C h a p . 4 from the p r e l i m i n a r y consideration of a displacement
approach.
Fundamental scalings and assumptions
Sect. 5.2]
379
Finally, we assume that the Lamd constants and the applied force densities satisfy the following a s s u m p t i o n s o n t h e d a t a : /V=~
f~--O -
0
and
and
and
#~=#,
f~(x*)-~3f3(x ) for
g3 ( x e )
--
allx *-TrsxE~t
*,
e 4 g4 (X) for all x ~ = 7r~x E F+~ U F~ ,
h~(y) - ~2h~(y) for all y E 7, where the constants )~ > 0 and # > O, the functions f3 E L2(f~), g3 E L2(F~ U F~), and h~ E L2(7) are independent of ~. A w o r d o f c a u t i o n : Had we replaced the assumptions f~ = 0 and g~ - 0 by the same assumptions as for a clamped plate (Sect. 4.2), viz.,
f ~ ( x ~) = s2f~(x) for all x ~ = 7r~x E ft ~, s 3 g~(x) for all
x
-
7r~x E F+ U F~,
with nonzero functions f~, g~ independent of e, the functions OzN~z introduced in Thm. 5.6-1 below would not vanish in general, and we would be led to equations more general than the von Ks163 equations. In other words, the powers of e characterizing the Lamd constants ~ and #~ and the functions fS, g~, and h~, together with the relations f~ - 0 and g~ - 0, represent precisely the kind of assumptions on the data that the von Kdrmdn equations are designed to handle. I
Remarks. (1) This being said, it should be clear that assumptions such as f ~ ( x ~) = caf~(x) and g~(x ~) - c4g~(x) are perfectly admissible; but such functions f~ and g~ do not contribute to the limit equations. (2) As for a clamped plate (Sect. 4.10), other sets of assumptions are possible, where each power of c is multiplied by the same power ct, t E I~. I Combining the scalings with the assumptions on the data, we then reformulate problem Q(f~) as a problem Q(e;f~) posed over
380
[Ch. 5
The von Kdrmdn equations
the set f~, called the s c a l e d t h r e e - d i m e n s i o n a l e q u a t i o n s of a v o n K&rm&n p l a t e in the d i s p l a c e m e n t - s t r e s s approach: It consists of a scaled principle of virtual work and of a scaled inverted constitutive equation (compare with Thin. 4.7-1) 9 T h e o r e m 5.2-1. Assume that u ~ E Wl'4(ft~). The scaled displacement field u(s) - (u~(s)) and the scaled stress tensor field E(c) (cr{3(c)) satisfy the following problem Q(E; f~)"
U(s e V(~)"-- { V - (Vi) E wl'4(~); ca independent of x3 and v3 --0 on 7 • [-1, 1]}, E(c) E L ~ ( ~ ) " - {(T{j)C L2(fl);f{j -Tj{},
/ cr~j(c)cgjv~dx + / a~j(c)O~Uz(e)cgjvz dx
+ e2 fa a~j(e)O~u~(e)Ojv~dx - /u fzvz dx + fr +UF_ gzvzdr l Va dx3
-[- -~
ha d7 for all v E V(f~),
E~ (~(~))+ ~: E~~,(u(~)) - B~ (r~(~))+ ~:W~ (z(~))+ ~4Bej (r~(~)), where the mappings E~ -
Ej~ 9 V(Ft) ~ L2(f~), p - 0,2, and 0, 2, 4, are independent of e', more
Bqj - Bq,~" L2(gt)~ ~ n2(gt), q -
specifically,
s~
1
~(o~j(~)+ %~(~) + o~3(~)%~3(~)), 1
E ~ (u(~))+ ~ E ~ (u(~)) A
1
381
The method of formal asymptotic expansions
Sect. 5.3' +
C2
-
+
= --
5.3.
~4 {g20"rr(g ) -~- g40"33(g)} -4- ~-~O'33(C).
/~
I
T H E M E T H O D OF F O R M A L A S Y M P T O T I C EXPANSIONS: THE DISPLACEMENT-STRESS APPROACH
As in Sect. 4.7, the polynomial dependence of problem Q(e; ft) with respect to ~ naturally leads us to apply the basic A n s a t z of the m e t h o d of f o r m a l a s y m p t o t i c e x p a n s i o n s . We define formal expansions U(C) -- U 0 -4- CU 1 -+- s
E(e) -- 7~-~0-4- g ~ l "4- s
-4- h . o . t . ,
_of_ h . o . t . ,
of both scaled unknowns, and we equate to zero the factors of the successive powers c p, p >_ 0, found in the equations of problem Q(c; ~) until the leading terms u ~ and E ~ can be fully identified. As in Sect. 4.7, the main virtue of the d i s p l a c e m e n t - s t r e s s a p p r o a c h followed here is that the "higher-order" terms u p and E p, p _> 1, are in fact not needed for this purpose: T h e o r e m 5.3-1. Assume that the scaled displacement and stress can be written as u(c)-u
~
h.o.t,
and
E(c)-E
~
h.o.t.,
and that the leading terms of these formal expansions satisfy U 0 -- (U/0) ~ V ( ~ ' - ~ ) - { V -
(Vi) ~ W1'4(~'-~); Va independent of Za
and v3 - 0 on 7 x [-1, 1]}, ~
5] o - ( a ~ ~
L ~ ( f t ) - {(r~j) E L2(ft); rij - r j i } .
The von Kdrmdn equations
382
[Ch. 5
Then the cancellation of the factors of c o in problem Q(e; ft) implies that the leading terms u ~ and E ~ should satisfy the following problem QKL(f~): (v i) E Hl(ft); v~ independent of x3 and
u ~ E VKL(f~)"-- { v -
va = 0 on 7 x [-1, 1], &va + Oav~ = 0 in ft}, E ~ e L2(f~) 8
dx + /cr~176 / a.~jOjv~ o
- /nfav3dx+fr
ljf~{f_ 1v ~ d x a + -~
} h~d'y
+UF_
93v3 dF
for all v C V(f~),
where cr0 ~
E~ Proof.
.=
0 ~ 2A# + 2~E~0 (u~200 + 2,E~,(u~
1 ~ .- ~(o~~ + o, uO~ + o~~176
The proof resembles that of Thm. reason, is omitted. 5.4.
4.7-2, and for this II
I D E N T I F I C A T I O N OF T H E L E A D I N G T E R M u~ THE LIMIT SCALED "DISPLACEMENT" TWO-DIMENSIONAL PROBLEM
Restricting in Thm. 5.3-1 the functions v E V(f~) appearing in the variational equations of problem QKL(~) to lie in the space VKL(f~) immediately yields the problem that the leading term u ~ should satisfy: T h e o r e m 5.4-1. Let the assumptions be as in Thm. 5.3-1. Then the leading term u ~ should satisfy the following l i m i t s c a l e d t h r e e -
Sect. 5.4]
Identification of the leading term u ~
383
d i m e n s i o n a l p r o b l e m "]')KL(~-~):
u ~ E VKL(a)-
{V -- (Vi) e Hl(f~); va independent of xa and Va=0onTx
/~ o Ozv~ d x + Ja 0o~~
[-1,1], 0 i v a + 0 a v i = 0 i n a } ,
~
f3v3dx+jfr+UF_ g3v3 dF
0 "0 aj3Oalt 3
lf~{f_ 1l v~ dxa }h~ d7 for all v E VKL(~')),
+ ~
where o Eo
2)~tt
=
(uo) -
+
1
o
+
o m
+
~ +
It is easily checked that 7)Kc(ft) is precisely the problem obtained if the displacement approach were instead applied to the scaled threedimensional problem, as in Sects. 4.3-4.5 for the clamped plate problt3m.
As a first step towards recognizing the von Ks equations in problem 7)KC(f~), we show in the next theorem, due to Ciarlet [1980, Thin. 4.1], that 7)Kc(f~) is in effect a two-dimensional problem, in the sense that its unknown u ~ - ( u ~ can be computed from the solution ~ = (~) of a two-dimensional problem 7)(a~). Because the three unknowns ~ are the components of the scaled "limit displacement field" along the middle surface ~ of the plate, 7)(a~) is called a two-dimensional "displacement" problem. The second step will consist in showing that this two-dimensional "displacement" problem is in turn equivalent to the yon Kdrmdn equations (Sect. 5.6) . The questions of existence, regularity, multiplicity, and bifurcation, of solutions for these problems will then be examined in the final part of this chapter (Sects. 5.8 to 5.11).
384
[Ch. 5
The yon K d r m d n equations
Theorem 5.4-2. (a) Define the space V(o2) - -
{~ -
(/]i) E H i ( a ) )
--_ H I ( a j )
• HI(a))
• HI(~d) • H2(~);/]3
- OqL,?]3 -- 0 o n ~/}
• H 3 ( w ).
The leading term u ~ - (u ~ is a scaled Kirchhoff-Love displacement field, in the sense that O~u~ + 03u ~ - 0 in ft. Hence (Thin. 1.4-4) there exists r = ({~) E V(w) such that uO -- r
-- Z3C~o~r
and
u~ -
r
(b) Let EO~ ( r
P3"-
Ji'
1 . - ~(0~r + 0~r + 0~r162
fadxa+g ++g;,
g ~ ' - g 3 ( ' , +1).
1
Then the leading term u ~ satisfies problem 7)KL(~) if and only if ~ =
(~) E V(w) satisfies the following limit scaled two-dimensional "displacement" problem 7)(w) of a yon K d r m d n plate:
= f~ p3r/3 dw + f~ h~r/~ d'7 for all r/E V(w), where
m~ N~--
-
-
3(~ + 2~)
o ~ 4~# + 2 E~(r
+
4#EO (r
Sect. 5.4]
Identification of the leading term u ~
385
(c) Assume that the boundary 7, the functions pa and h~, and th~ ~ol~t~on r of p~obl~,~ P(~), a~ ~,~ooth ~no~,gh. T h ~ r = (~) satisfies the following two-dimensional boundary value problem:
a(a + 2u)
A ~a - N~90~9('a - Pa in w 09N~ 9 = 0 in co, ~a = O ~ a = 0 o n 7,
N~gtJ9 = h~ on 7. Pro@ The proof of (b) resembles that of Thm. 4.5-2 and the proof of (c) that of Thm. 4.6-2; for this reason, they are left as exercises (Ex. 5.1). II Remarks. (1) As will be shown in Thm. 5.6-1, the assumed czistence of solutions to either problem considered in Thm. 5.4-2 (in parts (b) and (c)) automatically implies that the functions h~ satisfy certain compatibility conditions; for the sake of clarity, these have not been yet mentioned. (2) By virtue of the equations O~N~z = 0 in co, each vector-valued function (Nlz, N2z) E L2(co) belongs to the space H(div;co) "- { X -
(X~) E L2(w); d i v X -
O~X~ E L2(c0)}
and consequently (see Ladyzhenskaya [1969] or Temam [1977, p.9]), the boundary conditions N~gt, 9 = h~ on 7 make sense if we only assume that the functions h~ are in the space H-1/2(7) (which contains the space L2(7) ;recall that, for definiteness, we have so far assumed h~ E L2(7)). This is also reflected by the equivalence of these equations and boundary conditions with the variational equations N~O~7~ dco - f~ h~rl~ d7 for all (r/~) E Hi(co).
I
It remains to "de-scale" the boundary value probem found in part (c) of Thm. 5.4-2 (the effect of the "de-scaling of parts (a) and (b)" is similar to that the described in Thin. 4.9-1 and for this reason, is
386
[Ch. 5
The von Kdrmdn equations
omitted). As in Sect. 4.9, we define the d i s p l a c e m e n t s (~ of t h e m i d d l e s u r f a c e of the plate through the d e - s c a l i n g s {~'-c2~
and
~'-r
These de-scalings, combined with the assumptions on the data made in Sect. 5.2, lead to the following corollary of Thm. 5.4-2(c)" T h e o r e m 5.4-3. Assume that the data and a solution ~ = (~) of problem 7)(w) (Thin. 5.4-2(b)) are smooth enough. Then ~ = ( ~ ) satisfies the following boundary value problem, called the limit twod i m e n s i o n a l " d i s p l a c e m e n t " p r o b l e m of a yon Kdrmdn plate:
8"~(a ~ + ,~) 3(A~ + 2# ~)
e 3A 2~ - N ~ O ~
- p~ in w,
OzN~z - 0
~
-
0.~
-
in co, 0 on
7,
N ; ~ v ~ - e h ; on 7,
where
p~ .c
N;~.-~
f~ dx~ +
9~+~ +
a ~ + 2 . E~~
Eo~(~ ~) . - ~(o~5 1
g;~,
g~--
~+
.
9~(., i~), ~(r
,
+ o~; + o ~ o ~ ) . m
Note that the coefficient D ~ . _ 8 ~ ( ~ + S ) ~3
3(~ + 2~)
factorizing A 2~3 in the first equation in w is the flexural rigidity of the plate (already encountered for a linearly or nonlinearly elastic clampled plate; cf. Sects. 1.7 and 4.9).
Sect. 5.5] 5.5.
Identification of the leading term
IDENTIFICATION EXPLICIT FORMS STRESSES
387
IE O
OF THE LEADING TERM OF THE LIMIT SCALED
IE~
It remains to establish the existence of the leading term E ~ = (a~~ As in the case of the totally clamped plate problem (Thm. 4.8-1(b)), it is again possible to explicitly compute all the limit O. scaled s t r e s s e s a~j T h e o r e m 5.5-1. Assume that fa e L2(~t),
g~ E L2(w),
h~ C L2(7),
and that problem 7)(~) has at least one solution (~) satisfying
r E H3(w)
and
r e H3(w)N H4(w).
To such a solution there corresponds one solution ((u~ (a~~ problem QKL(ft) given by
0
and
Uc~-r162 o
1
of
u~162
3
o ~ - -~N~z + ~x3mo~, o cry3
-
-
3 ~(1
-
x~)Ozm~z,
1
1
+ ~(1 + X3) 1
3
/_l s dy3 -- f__X; f3 dy3 1 1
+ ~(1 + xa)g + - ~(1 - xa)g;, where the functions m~z and N~z are defined as in Thm. 5.4-2. Proof. The proof is analogous to that of Thm. 4.8-1, and for this reason, is left as a problem (Ex. 5.1). m Remark. The existence of solutions to problem 7)(a~) possessing the regularity assumed in the above theorem does indeed hold if the
388
The yon Kdrmdn equations
[Ch. 5
b o u n d a r y 7 is smooth enough, as in the case of a clamped plate (Thm. 4.6-3). It can also be deduced from the existence and regularity of solutions to the von Ks equations (Thm. 5.8-4). We are now in a position to explain why "pointwise " boundary conditions of the form
(cr;z + crkzOkU~). ~ -- H ;
and
U3
are not appropriate for the "original" three-dimensional problem (such b o u n d a r y conditions were already alluded to at the end of Sect. 5.1). Had we chosen these, we would have found that the functions a0 1 3 a[3 - -~Nae + -~x3ma3, where the functions m~z and N ~ are defined as in Thm. 5.4-2, but where the function ~3 is now in the space H2(co) N Hi(co) instead of the space Hg(a~), should satisfy boundary conditions of the form
0
c r ~ v ~ - Ha on 7 x [-1, 1], where H~ is a given function, defined over the entire lateral face x [-1, 1]. It is easily seen that it is not possible in general to satisfy such "pointwise" boundary conditions on 7 x [-1, 1]. By contrast, the functions a ,0~ need only satisfy the boundary conditions
{/_1 } a af~ ~ dx3 u ~ - h ~ o n 7 1 in the present case, which are indeed satisfied. 5.6.
E Q U I V A L E N C E OF T H E L I M I T S C A L E D "DISPLACEMENT" PROBLEM WITH THE SCALED VON K/~RMAN EQUATIONS
As a domain (according to the definition of Sect. 1.1), the set a~ C R 2 is a Nikodpm set, in the sense of Deny & Lions [1954, p.328]:
Sect. 5.6]
The scaled yon Kdrmdn equations
389
Whenever a distribution T E D'(~) is such that O~T C L2(~), then T E L2(~); see Amrouche & Girault [1994]. For the definitions and properties of the s p a c e Hm+l/2(,y), m _> 0, we refer to Lions & Magenes [1968, p.45] or Adams [1975, Chap. 7]. Without loss of generality, we also assume that the origin 0 belongs to the boundary 7 of the set cz, and we denote by "y(y) the arc, oriented in the usual manner, joining the origin 0 to the point y along the boundary g'. Notice that "y(0) = 7 since the set :v is assumed to be simply connected. We let u~ (y) and u2(y) denote the components of the unit outer normal vector at each point y E 7. We now establish the equivalence of the (scaled) two-dimensional "displacement" problem found in Thin. 5.4-2 with another two-dimensional problem, constituting the (scaled) yon Kdrrndn equations. In the former problem, the unknowns are the three components ~ of the displacement of the points of the middle surface ~, while in the latter, there are only two unknowns, one being the "transverse" component ~a of the displacement and the other a function ~b again defined on ~; it is remarkable that from their knowledge, one c~n also compute the other two components 4~. The following result is due to Ciarlet [1980, Thm. 5.1]. T h e o r e m 5.6-1. Assume that the domain ~ is simply connected and that its boundary "~ is smooth enough. (a) Consider the limit scaled two-dimensional "displacement" problem P ( ~ ) of a v o n Kdrmdn plate (Thm. 5.4-2(c))"
8#(.X + #) AZ~.s _ N ~ O ~ 3
s(), + 2#)
- Pa in w
O~N~ - 0
in u,,,
~3 -- 0~,~3 -- 0 on 7, N~v,,~ - h~ on 7,
390
[Ch. 5
The von Kdrmdn equations
where 4)~# o 4#E o , (r , N ~ := A + 2# G ~ (r G , + 1
E~162 - ~(Gr
+ o~G + o~r162
and let there be given a solution r - (~) of 7)(oz) with the following regularity: r E H 3(w)
and
r E H 4 (0d) A H 3 (CO).
Then the functions h, are in the space H3/2(~/), they satisfy the compatibility conditions /~ hi d'), - ~ h2 d'), - f~ (x~h2 - x2hl)d"/ - 0,
and there exists a function r E H4(w), called the scal ed A i r y s t r e s s f u n c t i o n , uniquely determined by the requirements that r = 01(~(0)- 02r 0, such that
011r N22, 012r
- N 1 2 - -N21,
022r Nil
in c~.
Furthermore, the pair (Ca, r E { (Hg(w)AH4(co) } x H4(02) satisfies the scaled v o n Kfirmfin e q u a t i o n s : 8p(A + p)A2~3 _ [r r 3(X + 2p)
+ Pa in 02
A2 r _ _p(3X + 2#)[~3, ~3] in w, A+# ~-3 -- c9~,~3- - 0 on ")', r
r
and 0 ~ , r
r
on 7,
Sect. 5.6]
where the
Monge-Amp~re form [., .] is [X, r
and the functions
r
defined by
"-- C~11X022r -}- C~22X011~/) -- 2G912X012r
r
r
(y)
r
391
The scaled yon K d r m d n equations
"~/ ~
]1~ are
h2dT+y2~
(y)
defined by
h~dT+~
(y)
(xlh2-x2hl)d~/,
"-- --ul(Y) f7 (y) h2 d~ + ~2(y)/~ (y) hide,
for all y - (yl, Y2) C 7. (b) Conversely, assume that the functions h~ are in the space H3/2(7), and let there be given a solution ((3, r of the scaled yon Kdrmdn equations with the following regularity"
Ho2(W) and r
(3 e H 4 ( w ) N
e H4(w).
Then the functions h~ necessarily satisfy the same compatibility conditions as above. If we define functions N~Z by N i l "-- ~22(~,
N12 - N21 " - - 0 1 2 r
N22 " - 011(~,
there exists a unique function (H -- ((~) in the space such that
4A# =
H3(~)/V~(~)
o
+ 2
+
where 1
V~(w) "- {rIH -- (~) e :D'(~);
1
e~Z(~TH) -- 0
in w}
-- {(?~a) e ~:)'(02); 711 -- a l -- bx2, ?72 -- a2 + b X l } .
In addition, the vector field ~-= (~H,<3) e H3(w)• {H~(w) AH4(~)} satisfies the two-dimensional "displacement" problem T)(a~).
The von Kdrrndn equations
392
[Ch. 5
Proof. For the sake of clarity, the proof is divided in several parts. Preliminary technical results are assembled in parts (i) and (ii). (i) For some integer m >_ 0, let there be given functions hi, h2 E Hm+l/2(~/) that satisfy
~hld"/-fh2d'7-f(xlh2-x2hl)d')'-O. Then the functions r
and r
defined by
r162
(y)
+[
J~ (y)
r
h2d~/+y2f
(y)
hide/
( x l h 2 - X2hl)dT,
" Y ~ ")/ ~ r (Y) "-- --l]l (Y) /
J-y(~)
h2 d7 + u2(y) [ hi d")/, J~(~)
belong to the spaces H'~+5/2(-y) and Hm+a/2('~), respectively. Using the definition of the space Hm+l/2(?), one easily establishes the following: If a function h E Hm+l/2('y) satisfies f~ h d'y - 0, and if ~ 9 -y ~ R is a sufficiently smooth function of the arc length parameter along 7, then the function r
E7 ~ r
r
f hd7 J7 (y)
is in the space H'~+3/2(7) (the compatibility condition f7 h d7 - 0 is needed to insure that the function r is unambiguously defined at y - 0). An application of this result shows that both functions r and r belong to the space Hm+3/2(~/). The assumption that ~/ is
smooth enough is thus crucially needed here. We next notice that
Co(u) - f
J~ (y)
0o aT,
where the function 0o is defined by
0o .y c
00(y)--
h2dT+Ul(y)]
f h~dT. J~ (y)
Sect. 5.6]
The scaled yon Kdrmdn equations
393
Since the function 00 is in the space H'~+3/2(7 ) and since f~ 00 d7 - 0, we conclude that r C H'~+5/2(7) (these properties are easily seen by introducing the arc length parameter along 7). (ii) For some integer m >_ O, let there be given functions f~ E Hm(w) and h~ E H'~+~/2(y) that satisfy the compatibility conditions (the space V~(w) is defined in the statement of the theorem)" j2 f~r]~ dw + L h~r/~ d7 - 0 for all (r/~) E V~ Then the boundary value problem -0~
n !a/~ - -
fa in
w,
/
rta~LJ/~ -- ha on "7, wheT"e
4A# %9 "= A + 2# ,
has a unique solution CH -- (C~) ~ the space H'~+~(~)/V~(~). The unknown (2H satisfies the variational equations 4A# - L f~r]~ dw + L h~r]~ d7 for all r/H = (r/~) C HI(w), and by assumption, the linear form appearing in the right-hand side of these equations vanishes for all ~/H E V~(w); hence the corresponding variational problem is well defined over the space H l ( w ) / V ~ Furthermore, its bilinear form is H l ( w ) / V ~ (Ex. 1.9), and thus it has a unique solution (~H in the sp a c e HI (w) / V ~ (w). It remains to prove a regularity-result for this problem. As its boundary conditions are of the Neumann type, this requires a special proof; the following one is due to Ciarlet Rabier [1980, Lemma 1.5-5]). We may assume without loss of generality that f~ = 0. To see this, it suffices to subtract the solution of the Dirichlet problem -O~n'~
-
f~
i n cv,
394
[Ch. 5
The von Kdrmdn equations
~H = 0 o n ' ) , which is in the space H'~+2(a~) if f~ E H'~(a~) (Thm. 1.5-2(c); recall t h a t 7 is smooth by assumption). T h e n the a r g u m e n t will rely on the following result: Given a s y m m e t r i c tensor ( F ~ ) E T f (w), a sufficient (and clearly necessary) condition that there exists an element X = (X~) E T f (co) such that -
1
+
is that
011F22 -~- 0221-'11 - 20121-'12 = 0. To prove our assertion, we write the above condition as 01(01F22 - 02F12) = 02(01F21 - 02F,1). Using a result in distribution theory (cf. Schwartz [1966, p. 59]; the assumption t h a t aJ is simply connected is crucially needed here), we infer t h a t there exists a distribution T E D'(a~), unique up to additive constants, such that:
02Fll,
OIT = 01F21 -
02T
-
-
01F22
-
02F12.
A n o t h e r application of the same result shows t h a t there exist two distributions X;1, X;2 E D'(a~) such t h a t C')lX1 z Fll, 01~2 =
and the assertion follows.
02)(.1 =
1-'12-~- T,
F21 - T ,
02;g2 = F22,
Notice t h a t the element X = (X~) is ,pac
Let us then assume t h a t we have established the existence of a s y m m e t r i c tensor (F~z) satisfying the following relations:
(F~fl) C Hm+l(w), 0 1 1 F 2 2 -]- 0 2 2 F 1 1
-
- O z ( a ~ z ~ . r ~ . ) = 0 in w,
2012F12 =
0
in co,
a ~ z ~ . r ~ . u z = h~ on 7,
Sect. 5.6]
395
The scaled von Kdrrndn equations
where a~,F~,
4A# F ~ 8 ~ + 4 # F ~ . := A + 2#
S i n c e t h e relation 011F22 q- 6922F11 - 2012F12 -- 0 is satisfied, we deduce from the previous assertion that there exists a unique element X/~ - ( X ~ ) C 79'(w)/V~ such that
and we in turn deduce
(~11Xl
--
that
01Fll E H'~(CO), 012Xl 022Xl
=
=
02Fll C Hm(CO),
(202F21- oqlF22) C Hm(w),
since ( F ~ z ) e Hm+l(CO). As COis a Nikodym set, this implies that XH E Hm+2(CO)/V~ But then the equations - 0 z ( a ~ z ~ , F ~ ) = 0 in w, a ~ z ~ , F ~ z = h~ on 7, together with the relations F~Z = e~Z(XH), show that XH coincides with the solution CH of the boundary value problem; hence CH possesses the required regularity. To complete the proof, it remains to show that there ezists a symmetric tensor (F~z)E Hm+l(w)satisfying OqllP22 -t- oq22F11 - 2012F12 = 0 in CO,
-0~(a~,Fo~)
= 0 in w and a ~ o ~ P ~ , ~ = h~ on 7.
To this end, we rely on regularity results for the Dirichlet problem: A20 0 = O0
and
=
0
in CO,
0~0
--
(~1 on 7,
where the functions 00 and ~bx are defined as in part (i). Since we showed there that r C Hm+5/2(7 ) and (]~1 E Hm+3/2(~/), infer that the unique solution of this problem satisfies (Lions & Magenes [1968]) w e
0 E H m+3(CO).
The yon Kdrmdn equations
396
[Ch. 5
We then proceed to show that the symmetric tensor (F~z) defined
by
1 V~9 = - 4 p ( 3 A + 2p)Z~g~z + ~-~pE~ 9
(note that a~z~,F~, - E~9), where 211 -- 0220,
E l 2 -- E21 -- --012 0,
E22 -- 0110,
satisfies all the desired requirements. First, (C~) E Hm+l(cd) since 0 E Hm+a(cv). Next, a simple computation shows that h+#
0111-'22 -1- 0 2 2 F l l - 2012F12 = 2#(3A + 2#) A 2 0 - -
O.
The equations --0a(a~oTF~.) -- 0 in a~ are likewise satisfied since -
Finally, we must verify that the boundary conditions a ~ . - P . T u 9 -h~ hold, or equivalently, that E~S//~- ha on ~/. Taking the arc length parameter along 7 as the variable, we readily infer from the boundary condition 0 = r on 7 and the definition of the function ~0 that the tangential derivative 0.0 of the function 0 along 7 is given by
OTO(y)
-
-
//1(~]) ~(y) hi d7 + u2(y) f(y) h2 d7 for all y E 7.
Combining this relation with the boundary condition 0~0 - r and the definition of the function r we find that
(y)
(u)
h~ d7 for all y E 7.
Consequently, E l l / / 1 -+- E12//2 -- / / 1 0 2 2 0 -
on 3'
//2012 0 -- OQr(02 0) -- h i ,
221//1 -~- 222//2 -- - / / 1 0 1 2 0 Jr-//20110 -- - 0 T ( 0 1 0 )
-- h2,
Sect. 5.6]
397
The scaled yon Kdrmdn equations
and the assertion is proved. (iii) Given an integer (m + 1) >_ O, let there be given functions N~ 9 E H'~+l(a~) that satisfy N12 - N21
and
O~N~ 9 - 0
in w.
Then there exists a function r E Hm+3(w), unique up to the addition of polynomials of degree < 1, such that (~11r
N22,
012r
-N12-
-N21,
022r
Nil.
Using a result from distribution theory (Schwartz [1966, Thm. VI, p.59]), we infer from the equations O~N~ 9 - 0 in a~ that there exist distributions ~ E D'(a~), unique up to additive constants, such that N i l -- 02if)l,
N21 - - 0 1 r
N12 -- 02r
N22 - - 0 1 r
Combining these relations with N12 -- N21, we in turn infer that 0~r - 0; hence the same result shows that there exists a distribution r E D'(~), unique up to the addition of polynomials of degree _< 1, such that r
02(~,
~)2 -- --01(/),
and consequently, the relations 0~1~ - N22, etc., hold. Since a~ is a Nikodym set, we deduce from these relations, combined with the assumptions N~Z E Hm+l(a~), that r E H "~+a(cJ). (iv) Given an integer m >_ 0 and functions N~Z E Hm+l(w) satisfying N12 - N21 and O~N~z - 0 in a~, let the function r E H'~+3(w) of (iii) be uniquely determined by the conditions (recall that 0 E 7 by assumption) r
-
-
0,
and define the functions h~ "- N~9. 9 e Hm+~/2(7).
398
[Ch. 5
The von Kdrmdn equations
Then the functions r and h~ are related along 7 as follows:
r
(y) 01r
h2dT+y2/s
-- -- f~
(y)
(y)
hld~/-+-f7
h2 d"/,
(y)
02~b(y) - ~
(Xlh2-x2hl)dT,
(y)
hid"/,
for all y = (Yl, Y2) E 7.
We observe that, along 7, hi =/11022(/)-//2021r
0r(02r
h2 -- -/]1c~12r162
: c~r(-01r
so that C~lr ) -- --f3'(y)h2d7 sequently, we find that 0~r
J~ (y)
and 02r
- f~(y)hi dT, y E 3'. Con-
h2dT+~'2(y) f hldTforallyE J~(y)
7,
but this is exactly the expression that we get by differentiating with respect to the arc length parameter along 7 the function YE~/---*--ylf~
(y)
h2d7 + y2 f~
(y)
hi d7 + j/
(y)
(Xlh2-x2hl)d"/.
In view of the relations r = 0~r = 0, we thus conclude that r is indeed equal to the above function along 7. (v) Given functions ~H E Hi(w) and ~3 C H3(w), define the functions , 4A# n ~ "= A + 2#
N"
9
2A#
1
N ~ "-- n ~' + N"a~, and assume that the functions N ~ (which belong to L2(w)) satisfy c3~N~ = 0 in w.
The scaled von Kdrrndn equations
Sect. 5.6]
399
Let r C H2(co) be a function determined as in (iii), or more generally, any function that satisfies 011r
N22
and
02~r = N l l in w.
Then A2 r _ _#(3A + 2#)[~3, ~3] in ~. A+# First, the definition of the functions N~Z, n~z, ' N~Z, " and the assumed relations between the functions r NI~, and N22, together iraply that (here and below, expressions such as A ( 0 ~ ) are to be understood in the sense of distributions) A2r
~ ( N ~ ) - 2/z(3A + 2 / z ) { 2 ~ ( 0 ~ ) + A+2p
A(0c~'3C')o~'3) }.
Using the definitions of the functions n~z, N~%, and the equations O~N~z = 0 in w, we next obtain 0 - Oz{O~n~z' + O~N~z}" _ 4#(A + # ) A ( 0 ~ ) A+2p
+ 0~zN"~z,
and consequently, by combining the last two relations, we have 3A+2# -
-
-5- 7)
+
2#(3A + 2 # ) A ( 0 ~ a 0 ~ a ) . A+2#
A straightforward computation, based on the above equation and on the definition of the functions N"Z, then yields the required expression for A2r (vi) Converse to (iii)-(v)" Given functions h~ e Ha/2(7) that sat-
i4y /hld~/-/h2d~/-J;(Xlh2-x2hl)d~-O, define the functions r (Y)
(/)1 on "~ by
(Y)
(Y)
400
[Ch. 5
The von Kdrmdn equations
r
Y ~ ")/ ---+r (Y) "-- --l]l(Y) !(Y) h2 d7 + u2(y) / (v) hi dT,
for all y C 7 (these functions belong to the spaces H7/2(7 ) and H5/2(7) respectively, by (i)). Given a function ~3 E Ha(w) (hence the function [~3, ~3] belongs to the space L2(w)), let next r E H4(w) be the unique solution of the boundary value problem
/%2q5 -- - tt(3A + 2it)[~3, ~3] in w, A+# r162
Our
on 7,
r
on
7,
and define the functions N~9 E H2(w) by letting
Nil := 0q22(~, N12 --- N21 :-- - 0 1 2 r
N22 = 011 r
in w.
Then these functions satisfy O~N~z = 0 in co, N ~ u z = ha on 7, and there exists a unique element CH--(~) in the space H 3 ( w ) / V ~ such that N~ ' N" where n'~ and N'~'z are defined as in (v) as functions of ~g and ~3, respectively.
First, it is clear that the functions N ~ defined by N l l = 022~), etc., satisfy O~N~ = 0 in a;. That they satisfy N ~ u ~ = h~ on 7 follows from the boundary conditions q5 = r and 0~r = ~1 on ")' by an argument already used at the end of the proof of part (ii) (notice that the equation A 2 r - u(a~+2.) x+. [~a, ~a] is not needed in this argument). To determine 4/~ - (C~) such that the relations N~z n'~z + N'~'z are satisfied, it is natural to solve the following boundary value problem:
Sect. 5.6]
The scaled yon Kdrmdn
a~e~(~ij)p
401
equations
~ - h~ on 7,
where the functions f~ and/t~ are defined by 1
f~ "- -~Oa{a~a~,~O~C30,.~3} in w, 1 2
{a~a~,~-O~3OT~3}va + ha on 7.
The functions f~ and h~ defined in this fashion belong to the spaces HI(w) and H3/2(7 ) respectively; besides, they satisfy the compatibility condition f L ~ dw + ~ / ~ r / ~ d 7 - 0
for all r t t / - ( ~ )
C V~
To see this, we observe that ~ f ~ u ~ dw +
/t~r/~d 7 - - ~
a~9,~_O~,~3OT~ae~o(rtH) dw
+ ~ h~r/~ d7 for all rtH -- (U~) E H I ( w ) , and thus, for all (r/~) - (al-bX2, a2+bXl) (the space V~ consists of such functions (r/~)):
f~rl~ dw +
h~rl~d7 - a~
h~ d7 + b
(x~h2
--
precisely
X2hl) d7 - 0.
We therefore infer from part (ii) that the above two-dimensional linear system has a unique solution CH -- (C~) in the space H3(w)/V~ Once the functions 4~ are obtained in this fashion, define the functions 1 + By construction, they belong to the space H2(w), and they satisfy cOrNea = 0 in w, NI*2 - _N~I in w,
402
[Ch. 5
The von Kdrrndn equations
N2~t9 = ha on 7Consequently, by (iii)-(v), there exists a function r that (~11r
---
N~2,
0~24)* = -N[2 = -N21,
(922r
C H4(w) such
=
N;1
in w,
and 0* satisfies the boundary value problem A2r r
0~r
_ _ p(3A + 2p)[~3, ~3] in w, A+# = 00 on 7,
= r
on 7-
Since the solution 0* of this problem is unique, we conclude that 0* = r and consequently that N~ = N~, as was to be proved.
U
Remarks. (1) The linear boundary value problem encountered several times in the above proof, viz., =
i n ca,
constitutes another instance of two-dimensional (scaled) m e m b r a n e equations, this time with boundary conditions of the N e u m a n n type along the entire boundary; the first instance occured in Thin. 1.5-2(b); there, boundary conditions of the Dirichlet type, viz., ~ = 0 on 70 C 7, were imposed. (2) The assumption that w is simply connected plays a crucial role in part (ii) of the proof of Thm. 5.6-1. The case where w is multiply connected is studied in Ciarlet & Rabier [1980, p. 61 ft.]. II
Sect. 5.7]
5.7.
Justification of the von Kdrmdn equations
403
J U S T I F I C A T I O N OF T H E V O N K / k R M / k N E Q U A T I O N S OF A N O N L I N E A R L Y E L A S T I C PLATE; COMMENTARY AND BIBLIOGRAPHICAL NOTES
It remains to "de-scale" the scaled yon Ks163 equation found in Thm. 5.6-1. To this end, we define the t r a n s v e r s e d i s p l a c e m e n t ~ " ~ ---+R of the middle surface of the plate and the A i r y s t r e s s f u n c t i o n r : ~ ~ R through the following d e - s c a l i n g s : 4~'-r
and
r162
Together with Thm. 5.6-1, these de-scalings immediately give: T h e o r e m 5.7-1. The de-scaled functions ~ and df satisfy the von K~irm~in equations: 81,~(~ ~ + ~ ) ea/x2
3(A~ + 2#~)
~
~
43 - c [ r ~, 431 + P3 in w,
/~2(/)~ __ m
r
r
= r
- 0~r
-0
and 0 , r ~ - r
#~(3)V + 2# ~) [~, ~] in a;, A~ + #~ o n 7,
on 7,
where [X, ~] -- OQllXOQ22~/J -iv (~22XO11~/J -- 2012~012~/J,
p~ "r
i;
f~ dx~ + g +~ + 93-~, g +~ "- g; (', e), g ; ~ "- g~ ( ' , - e ) , (y)
h~dT+y2 f~ (y)
(y)
h~dT+j~
(y)
(Xlh~ -x2h~l)dT, yET, (y)
404
[Ch. 5
The von Kdrmdn equations
Besides, the functions h~ must satisfy the compatibility conditions: h I d ~ / - ~ h~ d~
- ~(Zlh; -X2hl)d~
-0,
in order that the yon Kdrmdn equations possess a solution,
m
Note that the partial differential equations in the von Ks163 equations may be also written as
D S A 2 < a - c [ r 5,r 5
f a d x ~ + 9 +5+93
5
5
--5
inw,
5
zx
r
-
where D ~ -- 8#~(A~ + #~)c 3 and E ~ -- #~(3A~ + 2#~) respectively represent the flexural rigidity of the plate and the Young modulus of its constituting material. There is an abundant literature on the yon Kdrmdn equations, and the brief list given below is by no means exhaustive. The original reference is yon Ks163 [1910] (an excerpt is given p. lxiii). More recent treatments from a mechanical perspective are given in Novozhilov [1953], Timoshenko & Woinowsky-Krieger [1959], Stoker [1968], and Washizu [1975]. Mathematical treatments concerning existence and regularity theory can be found in Berger [1967, 1977], Knightly [1967], Berger & Fife [1968], Lions [1969], Hlavs & Naumann [1974, 1975], Duvaut & Lions [1974a, 1974b],. Ne~as & Naumann [1974], John & Ne~as [1975], Rabier [1979], Ciarlet & Rabier [1980], Cibula [1984], John, Kondratiev, Lekveichvili, Ne~as & Oleinik [1988]. References concerning the bifurcation of the solutions are given in Sect. 5.11. To complete the de-scaling, it remains to define the i n - p l a n e d i s p l a c e m e n t s ~ of t h e m i d d l e surface, the limit displacem e n t s u~(0), and the limit stresses cry5(0) through the following
Justification of the yon Kdrmdn equations
Sect. 5.7]
de-scalings
405
9
r
c ~(~ in co,
~ ( 0 ) ( . ~ ) . _ e ~u~(x) o ~ and Ua(0)(x ~) "-cu~ for all x ~ - r K x E ~ ~ ~ ( o ) ( ~ ) - ~~~e(~), o ~ ( o ) (~) - ~ ~~o ( . ) , ~ ~ ( o ) ( . = ) _ ~ o ( . )
for all x ~ = rr~x E ~ . In the following corollary to Thms. 5.5-1 and 5.6-1, we show how these de-scaled functions can be computed. T h e o r e m 5.7-2. Let (r r be a solution of the von Kdrmdn equations (Thin. 5.7-1) that possesses the following regularity: ~ E H3 (w) ('l H4 (w)
and
05s e H4(co).
Let the functions m~o E H2(co) and N ~ E H2(oa) be defined by 9- - c a {
m~e
NIl := 6022r e,
4MP ~ 4 # = ~ r=} 3(A= + 2M=)a(~&~e + --~u~es a in w,
N[2 = N~I : = --g012r s
N~9.= r162 s in w.
Then the limit stresses criS(O) are given by
~(o)
1
3
- U N:e + g/-~e~~ . ~ ,
~aa(0) g -
=) ..
xa 1 4e
={ (==)"} --s
(9=0 *= + 47g 1 -
+i
1+
+~
1+--g~
s
s
fldYl-
fldYl
-~
g;~,
1-
rn]0cg=0(i
406
[Ch. 5
The von Kdrrndn equations
and thus a~z(O) E H2(ft~), (7~3(0) C HI(~e), and O'~3(0) E L2(f~). The vector field (u~ (0)) is a K i r c h h o f f - L o v e d i s p l a c e m e n t field, in that the limit displacements u~(O) satisfy O~u~(O) + O~u~(O) - 0 in f~. Consequently (Thm. 1.4-4), the functions u~(O) are of the form Uc~(O)
and
-- Ca -- X3~c~r
U3 -- ~3'
where the in-plane displacements ~ of the middle surface are solutions of the following boundary value problem: 1
- 0 ~ { a ~ o ~ e o , ( ~ ) } - -~O~{a~o,Oo~O~} in w, 1
a:zo.eo.((5)~, z - --~a;zo.Oo~O.~L, z + eh 2 on 7, where 4A~# ~ l~+2# ~
a~zor'=e
+
/
1
For a given function ~ , the function ~H -- ( ~ ) is uniquely determined in the space H~(w)/V~ where the space V ~ is defined as in Thin. 5.6-1, and in fact is in the space Ha(w). Consequently, u~(0) E g3(f~ ~) and u~(O)e H4(f~e). I Remarks. (1) The functions N ~ automatically satisfy -
a
1
(r
}.
(2) Since the functions N ~ satisfy N
~
~
?
C
(as in the case of a clamped plate; cf. Sect. 4.9), they are also called stress resultants. That they are computed from the function r explains why r is called the Airy "stress" function. I We now list various conclusions that can be drawn from our analysis, and we mention several extensions. In addition, we also refer
Sect. 5.7]
Justification of the yon Kdrmdn equations
407
to Sect. 4.10: Most comments there apply verbatim to the present problem. The main conclusion is, of course, that we have been able to mathematically justify the derivation of the yon Kdrmdn equations in a rational manner from three-dimensional nonlinear elasticity, by identifying in particular specific boundary conditions along the lateral face that give rise to these equations. We have thus provided answers to the various objections mentioned in the introduction to this chapter, originally raised by S.S. Antman. In addition, we have established the equivalence of the yon Kdrmdn equations with a two-dimensional "displacement problem", which, consequently, can be also studied on its own sake; this is particularly worthwhile when the set w is not simply connected, since only the latter problem is well defined in this case. As in the case of a clamped plate (Sect. 4.10), the constitutive equation may be replaced by that of the most general elastic, homogeneous, and isotropic, material whose reference configuration is a natural state. This does not modify the definition of the "limit" two-dimensional equations found here, which thus exhibit a generic character. As in the case of a clamped plate (cf. again Sect. 4.10), it is noteworthy that a quasilinear, second-order problem has been replaced by a semilinear, fourth-order problem, whose mathematical properties may be therefore expected to exhibit crucial simplifications, as exemplified by the available existence and bifurcation theories for the von Ks163 equations, which have no comparable counterpart (as of now) for the original three-dimensional problem. As those of the nonlinear Kirchhoff-Love theory (Sect. 4.10), the yon Kdrmdn equations are not frame-indifferent (as is best seen on the equivalent "displacement" problem; cf. Ex. 4.3). A w o r d of c a u t i o n . Which boundary conditions are appropriate for the three-dimensional problem is a question of particular importance, inasmuch as the yon Kdrmdn equations are sometimes used when they should not be! Consider for instance a completely clamped
The von Kdrmdn equations
408
[Ch. 5
plate. Then, instead of u~~ independent of x 3 and u 3~ - 0 0 n T x
12
-
(
~
~
[-e,e]
dx~-h~onT,
as here, the b o u n d a r y conditions on the lateral face are u~-0onTx
[-e,c].
As shown in Chap. 4, an asymptotic analysis can be applied t h a t yields a scaled two-dimensional "displacement" problem over the set w of the following form (compare with Thm. 5.6-1(a)):
8~()~ -~- /_t)A2~3 _ NaOOc~3 -3(~ + 2 , )
1
f3 dx3 + 9 + + 93- in w,
O ~ N ~ = 0 in w, (3 -- (~.~3 = 0 o n
7,
~',~ = 0 o n
7,
where the functions N~a have the same expressions as in Thm. 5.6-1. Hence the partial differential equation in w and the b o u n d a r y conditions ~a = 0,~3 = 0 on 7 do coincide with those found in Thin. 5.6-1, but the b o u n d a r y conditions N~zv~ = hz on 7 do not: T h e y are replaced by the b o u n d a r y conditions ~ = 0 on 7. As a simple analysis shows, it is then impossible to compute boundary conditions for the Airy stress function, which still exists in view of the equations OzN~z = 0 in w, from the data of the problem. Consequently, the yon Kdrmdn equations are inappropriate in this case, and it is no surprise that they yield erroneous results if they are used for modeling a clamped plate! m The b o u n d a r y conditions 1
-
(
ae
+
e ~e
e
h;
on
correspond to an applied force that is a dead load, since the functions h~ are assumed to be independent of the unknown u ~. In Blanchard &
Sect. 5.8]
Existence and regularity of solutions
409
Ciarlet [1983], a more general boundary condition of pressure ( Vol. I, Sect. 2.7), which is no longer a dead load, has been instead considered on the lateral face. It is interesting to notice that, while these two kinds of three-dimensional boundary conditions are different, they nevertheless correspond to the same "limit" two-dimensional equations as those found here; see Ex. 5.2. 5.8.
THE VON K/kRM/kN EQUATIONS. EXISTENCE A N D R E G U L A R I T Y OF S O L U T I O N S
The existence and regularity theory described in this section is adapted from Ciarlet & Rabier [1980, Sect. 2.2], whose presentation was itself based on the method set forth by Berger [1967, 1977]. To begin with, we write the von Ks equations found in Thm. 5.7-1 in a simpler form, "where all constants are equal to 1". To this end, we associate with the unknowns ~ , r and the data p~, r r appearing in these equations the "new" unknowns ~, ~ and the "new" data f, r r defined by the relations ~
--
cD1/2E-1/2~ and r
P~ - e4Da/2E-1/2f, r
- e2D~,
- E2D~o, r - c 2 D r
where
D'-
8#(A+#) a(a + 2,)
and
E ' - #(3A+2#) a+
In this fashion, we find that the pair (~, ~b) solves the canonical von Kdrmdn equations: A2~ - [ ~ , ~] + f in a2, A2~ = -[~, ~] in a2, ~ = 0 ~ = 0 on 7, = ~0 and 0 ~
=
~D1 on
")/,
The yon Kdrmdn equations
410
[Ch. 5
where the Monge-Amp~re form [., .] is defined as before by [T], ~] -- 0117"1~22~ -]- 0227"1011~ -- 20127]Oq12~,
co is a domain in R 2, and f, ~b0, and ~bl are given functions. Since our objective is to establish the existence of (at least) one solution (~, ~) E H~(w) x H2(w) (Thm. 5.8-3) of these equations, we accordingly assume that the data have the following "minimal" regularities (H-2(w) is the dual space of H02(w); references about the spaces Hm+l/2(~/), m _> 0, have been already provided in Sect. 5.6)"
f e H-2(co), ~2o E H3/2(7 ), r
e H1/2(7).
In other words, we are studying here the canonical yon Kgrmgn equations for their own sake, momentarily forgetting that they were derived from a "displacement" problem (Thm. 5.6-1) under the assumptions that co was simply connected and the data were regular. We first transform the canonical von Kgrmgn equations into a more condensed form, by reducing their solutions to that of a single nonlinear equation in the unknown ~. Not only is this equation particularily convenient for proving the existence of a solution, but it also shows that the the nonlinearity in the yon Kdrmdn equations lies in the term C(~) = B(B(~, ~), ~),
which is "cubic" (Thm. 5.8-1). We let [1" 1[-2,~ a n d [ . [0,p,~ respectively denote the norms in the spaces H-2(w) and LP(w); we also define the semi-norm
O,p,w o~
Note that the biharmonic equations in the next theorem are to be understood in the sense of distributions. Theorem
5.8-1. Let the bilinear and symmetric operator
Sect. 5.8]
Existence and regularity of solutions
411
be defined as follows: Given (~, ~l) E H2(~) • H2(w), we let B(r ~l) denote the unique solution of B({, r/) C Hg(co)
and
A2B({, ~7) - [{, 'r/] in co.
Then define the operator
c . ~ e H~o (OO) --~ C(e) . - B ( B ( e , ~), ~) e H3(oo), which is "cubic", in that C(c~{) - c~aC({) for all c~ E I~. Assume that ~2o E Ha/2(7) and r E H1/2(7); let Oo be the unique solution of 0o E H2(co), A20o - 0 in co, 0o - ~bo and 0~0o - ~bl on 7, and define the linear operator A . g c Ho2(Co) --+ A(g) "- /3(00, r ~ Ho2(Co). Finally, assume that f C H-2(w) and let F be the unique solution
of FcHg(co)
and
A2F-finco.
Then (~,~2) E He(co ) • He(co) satisfies the canonical yon Kdrmdn equations if and only if { satisfies the r e d u c e d v o n K~irm~in e q u a tion
EHg(w)
and
and ~b is then given by
r
C({)+(I-A){-F-0,
412
[Ch. 5
The von Kdrrndn equations
Proof. By assumption, ~0 E Ha/2(w), ~1 ~ H1/2(w), and f E H-2(w); hence the definitions of 00 and F show that these functions are uniquely determined in the spaces H2(w) and Hg(w). If (r/,x)-E H2(w) x H2(w), the bracket It/, X] belongs to Ll(w); hence B(r/, X)is likewise uniquely determined since Ll(w) ~-+ H-2(w), as we now show. Let g E L ~(co); since H2(w) ~-+ C~ there exists a constant c such that ( < . , > denotes the duality between D'(w) and z~(~)) l < g, ~ > 1 _ < Ig[o,l,~l~lo,oo,~ < clglO,l,~ll~ll2,~
for all 7) C D(w), hence for all 7) E Hg(w) = D(w). By the same inequalities,
Ilgll-~,~ =
sup
~(~)
l < 9, qp >1 _< clglo,l,~ "
II~ll~,~
Hence L I ( w ) ~-+ H - 2 ( w ) as announced. Let 0 " - ~ b - 00. Then the pair ( { - F, ~b)C H02(w) x H02(w) satisfies zx~ (,~ - F ) - [g, + 0o, ,'] ** ,' - F - B ( ~ + 0o, ,~),.
~V3 - -[~, ~] r ~ - - B ( ~ , ~), and thus - F -
B(-B(~,
~) + 0o,
~),
and the proof is complete.
I
We gather in the next theorem useful properties of the bracket [., .] and of the operators B, C, and A defined in Thm. 5.8-1. T h e o r e m 5.8-2. (a) The following implication holds" ~CeH~(w) and [sc,sc ] - 0
=> ~ - 0 .
(b) Let ({, r/)zx "- ~ A{Ar/dw. Then (B(~, r/), X)zx - (B({, X), r/)A for all ({, r/, X) E H2(w)xHg(w)xHg(w).
Sect. 5.8]
Existence and regularity of solutions
413
Consequently, for any ~ C H~(co),
(c~, ~)~ = (B(~, ~), B(~, ~))~ >__0, ( c ~ , ~ ) ~ = 0 ~. ~ = 0.
(c) The nonlinear operators B : H2(co) • H2(co) ---+H3(co) and C : H3(co) ---+H3(co) are sequentially compact, hence afortiori continuous, in the sense that (as usual, strong and weak convergences are noted and ---~): ({k, r / k ) ~ ({, r/)in H2(co) x H2(co)=> B({ k, r/k) -+ B({, ~7)in Hg(co), {k _ { in H~(co) ~ C({ k) --+ C({) in H~(co). (d) The linear operator A : H3(a;) -+ Hg(a~)is compact, and symmetric with respect to the inner product (., ")zx. Pro@ (i) The trilinear form
T ' ( ~ , r/, X)E H2(co) x H2(co) x H2(co)--+ o/[~, r/]x dco is continuous; moreover, T becomes a symmetric form if at least one of its three arguments is in Hg(a;), and in this case there also exists a constant C such that
~[{, r/]X dco < c1~12,~1~11,4,~1x11,4,~. The definition of [~, r/] and the imbedding H2(co) ~-+ C~ that there exists a constant c such that s [{, r/]X dco
show
I[~,~]1o,1,~1~1o,~,~ ~ c1~1~,~1~1~,~11~112,~.
Hence the trilinear form T is continuous. Let the functions ~, r/, and X be in C~176 we may then write
414
The von Kdrmdn equations
~
[Ch. 5
[{, r/]x dw - ~ (X011~0227~ -- X012~O~12T]) dw
--- fw 02(~011~02T] -- ~O~12~(~lT])da; -
-
s 027]02(~011~)dw + J2 01r/02(X012{)dco
+ ~ 01 (X022{01q - X/)12{02r/)dw -
~/)lq0X (X022{)dco + f~ 02rfi)l (X/)12{)rico.
If at least one of the three functions is in 7)(co), the integrals f~ 0~(... )dco vanish and we are left with ~ [{, r/]x dco
Since (7~176 - H2(w) and D(co) - H~(w), and both sides are continuous trilinear forms with respect to [[. 112,~ (recall that H2(co) r Wl'4(co)), this relation remains valid if the functions {, r/, and X belong to H2(w), one of them being in H02(c~); hence the announced inequality holds, and the trilinear form T becomes symmetric in this case: The left-hand side is unaltered if { and r/ are exchanged and likewise, the right-hand side is unaltered if r/and X are exchanged. (ii) Let ~ E Hg(w) be such that [~, ~] - 0 and let the function 1 2 Jr- X22). Hence [{, X] - A{ X e H2(co) be defined by )(~(Xl, x2) -- ~(X and, by the symmetry of T established in (i), 0 - f [sc,{]xdw - s
x]{dco - f~ sCAsCdw- ]sc]21,.,.
Therefore ~ - 0 and (a) is proved. (iii) Let (~, r], X) E H2(w) x Hg(w) x Hg(w). By definition of B
Existence and regularity of solutions
Sect. 5.8]
415
and by the symmetry of T, (B((, r/), X)~ - ~ AB((, ~)A X dw - ~ [ ( , ~]X dw
= ~[~, x]rldw - ~ AB(~, X)A~] dw - (B({, X), r/)zx. Let { C H~(w); by definition of C and by the relation just established, (C~, ()A -- (B(B((, ~), ~), ~)A -- (B(~, ~), B((, ())A k 0 so that, by (a), ( c ~ , ~),, - 0 ~
[{,~] - 0 ~ ~ - 0,
and all the assertmn of (b) are proved. (iv) We recall that (Thm. 1.5-1(a), part (i)of the proof) I~1~ - I ~ 1 o , ~
-1~12,~
for all ~ 6
H~)(w).
Hence [ - l a is a norm over the space H{(w), which precisely corresponds to the inner product (-, ")a. By definition of the operator B and by (i), (B(g, ~),
X)~ - ~ [~, r/]x dw - / ~ [X, 4]~]dw < C]~]Al~[1,4,wlT][1,4,w
for all ({, rl, X)E H2(w) x H2(w) x Hg(w). Hence IB(,~,,7)IA --
sup
(B(,~,,7), X)A
xcHo~(~) ,--/=o
Ixl~
for all (~, ~) C H 2(co) x H 2(w). Let (~k, r/k) ~ (~, r/) in H 2(w) x H z (w); using the bilinearity of B, we may write B(~,
~k) _ B ( { , ~) - B ( ~ ~ - ~, ~) + B(~, V~ - V) + B ( { ~ - ~, ~ - V),
and thus, by the last inequality,
416
[Ch. 5
The yon Kdrmdn equations
IB(~ ~, ~ ) - B(~, w)l= C(I~ ~ - ,~11,4,.., Ir/[ 1,4.~ + 1~1,.4,~o I~v~ - r/I 1,4,.., +
1~k -- ~ I 1,4,a~ IT]k -- T]I 1,4,c0) 9
The compact imbedding H2(co) e Wl'4((a2) then shows that B(~ k, r/k)--~ B(~, r/)in Hg(a~); hence the operator B is sequentially compact. The definitions of the mappings C and A then show that they are in turn sequentially compact. Thus (c) is established. (v) Let ({, r])E Ho2(Co) • H~(co). Then, by (ii), (Ag, r/)zx = (/3(00, g), r/)A = (/3(00, rl), g)Zx = (At/, g)A; hence A is symmetric with respect to the inner product (-, ")A.
m
Remarks. (1) The equation [~,~] = 2 det ( 0 ~ )
= 0
solved in (a) is called the Monge-Amp&re equation. (2) As there is no general agreement about various definitions of compactness for nonlinear mappings, the definition of "sequential compactness" used here may differ from others, m We are now in a position to establish an ezistence result. As in the case of a clamped plate (Thin. 4.6-1), it relies on the ezistence of a minimizer of an associated functional. When ~0 = 2/21 = 0, Lions [1969, Thm. 4.3, p. 54] has given a different proof, based on the Brouwer fized point theorem. T h e o r e m 5.8-3. Assume that f E H-2(co), d2o C Ha~2(7), and el ~ H1/~(~). C~t th~ ' ~ ~ " op~ato~ C " H3(~) --, Hg(~), th~ linear operator A" H3(a~) --, H2o(CO), and the function F ~_ Hg(w) be defined as in Thm. 5.8-1. (a) Define the functional j : H g ( c o ) ~ R by
j(~) -
1 ~(C(~), ~)~ + ~1 ( ( I - 1)~ , ~)~ - (F,~)~
Sect. 5.8]
Existence and regularity of solutions
417
where (~, rl)~x = f~ A~A~lda~. Then solving the reduced yon Kdrmdn equation, i.e., finding ~ such that
EHg(a~)
and
C(4)+(I-A)~-F=0,
is equivalent to finding all the stationary points of the functional j, i.e., those ~ that satisfy C H~)(w) and
j ' ( ~ ) = 0.
ib) There exists at least one ~ such that E Hg(a~)
and
j(~)-
inf
j(~).
Hence any such minimizer ~ is a solution of the reduced yon Kdrmdn equation, to which there corresponds (Thin. 5.8-1) a solution (~, O) E Hg(~) • H:(~) of th~ ~ a ~ o ~ l ~o~ I C ~ . ~ ~q~at~o~, o b t ~ g by l~tti~9 ~ = Oo - B(~, ~). Proof. (i) The functional j is diyerentiable over the space H~(w), and solving the reduced yon Kdrmdn equation is equivalent to finding the critical points of this functional.
Define the functional j4://02 (c~) ---, R by letting for all r/E H~(a~): 1 1 j4(r/) "-- ~(C(r/), rl)A - ~1 (B(B(TI, r]), ~7),r/)~ - ~ (B(r/ , r/), B(~7, r/))/,,; cf. Thin. 5.8-2(b). Note that j4(~]) > 0 and that j4 is "quartic" in the sense that j4(c~r/) = c~4j4(r/) for all c~ E R. As the bilinear operator B is continuous (Thin. 5.8-2(c)), it is (infinitely) differentiable, and for the same reason, the inner product (., ")zx is (infinitely) differentiable. Hence j4 is also differentiable by the chain rule. A simple computation, combined with another application of Thin. 5.8-2(b), then shows that j~(~)r/, i.e., the linear part (with respect to r/) of the difference (j4(~-t-r/)- j4(~))is given by
j;(~)~- (B(~, ~), B(~, ~))~ -(B(B(~, ~), ~), ~)~ -(C(~), ~)~.
418
The von Kdrmdn equations
[Ch. 5
As the linear operator A is continuous and symmetric with respect to the inner product (., ")zx (Thm. 5.8-2(d)), the quadratic functional j 2 ( r l ) ' H g ( c o ) - + R defined by 1
j2(r/) "- ~ ( ( I - A)r/, r/)A is likewise differentiable, and j;(sC)rl - ( ( I - A){, r/)zx. The continuous linear functional jl 9H~(co) --+ R defined by jl(
) -
(F,
is clearly differentiable, and j [ ( ~ ) r l - (F, r/)zx. To sum up, we have shown that the functional j is differentiable, and that j'(sC)r/- ( C ( ( ) + ( I - A){ - F, r/)zx for all ~c,r / e H~(w). As (-, ")A is an inner product over H02(co), finding the critical points of the functional j is thus equivalent to solving the reduced von Kgrmgn equation. (ii) The functional j is sequentially weakly lower semi-continuous Let r/k --~ r/ in H02(co). As B is a sequentially compact operator (Tam. 5.8-2(c)), B(r/k, r/k) ---, B(r/, r/)in Hg(co), and thus j4(r/k) -- ~1 (B(~Tk , rlk ), B(r/k , rlk ))A -+ j4(r/). As A is a compact operator (Thm. 5.8-2(d)), At/k --+ At/in H~(w) and thus (At/k, r/k)A --+ (Arl, rl)A, on the one hand; on the other, the square of the norm associated with the inner product (., .)~ is weakly lower semi-continuous. Hence j2(r/) < lim inf j2(r/k). k---+ o o
Finally, jl(r/k) ---+ jl(r/) by definition of weak convergence. We have thus shown that j (7-/) _< lim inf j (7/k). k---+ o o
Sect. 5.8]
419
Existence and regularity of solutions
(iii) The functional j is coercive on H2o(a;), i.e., r/E Hg(a~) and ]r/Izx "-IAr/Io,~ ~ +oc =~ j(r/) ~ +oc. Assume the contrary. Then there exists M _> 0 and a sequence (r/k)F=l such that r/k E H~ (w), It]k ]zx --' +oc, j(@) <_ M for all k. Without loss of generality, we may assume that r/k :/: 0 for all k. Let 1
Ok .= ~ r / k . Dividing the inequalities j(~k) < M by [r/kl~ and remembering that j4 is "quartic", we obtain 2 9 ~) < M + 1 -1 _ -l(A0k 0k)zx + [r/k1~3~(o 2 2 ' I~kl~ ~
(F, Ok
)~
As [0k[~ - 1 for all k, there exists a subsequence (0l)~=l of (Ok)kC~=l such that 0 ~ ~ 0 in Hg(cz), hence such that j4(0 t) ~ j4(O), by (ii). The last inequality then implies j4(0) -- 0; for, if otherwise j4(0) > 0, its left-hand side approaches +ec while the right-hand side approaches 0 as 1 -~ ec. But o - j~(o) -
1
~lB(o,
0)1~
B ( o , o) - o zx~B(o, o) - [o, o] - o ~
0 - o,
by Thin. 5.8-2(a). Passing to the limit in the same inequality, we thus reach a contradiction, since (the operator A is compact) 1=
lim{1
l(AolO1
} < liminf{ 1
9 0 z) ' I(A0/ 0l)~+[~121A34( )
while the right-hand side approaches 0. Hence j is coercive. (iv) The functional j has at least one minimizer over H~(aJ), and any such a minimizer is a solution of the reduced yon Kdrmdn equation.
420
The von Kdrrndn equations
[Ch. 5
These classical consequences of the properties proved in (ii) and (iii) are established as in part (iv) of the proof of Thm. 4.6-1. I
Remarks. (1) A more powerful existence theorem, also valid for multiply connected domains, can be directly established for the twodimensional "displacement" problem of a von Ks163 plate (Thm. 5.4-2), by minimizing the associated energyin the manner of a clamped plate (Thm. 4.6-1); see Ciarlet &: Rabier [1980, Thm. 2.1-1] and Ex. 5.3. (2) The existence and uniqueness of a strong, global in time, solution of the time-dependent yon Kdrmdn equations have been established by Puel & Tucsnak [1996]. In this direction, see also Lions [1969, p. 43 if.I, yon Wahl [1981], Koch & Stahel [1993], BShm [1996], Tataru & Tucsnak [1996], and Boutet de Monvel & Chueshov [1996, 1997a, 1997b]. Thermoelastic, time-dependent, yon Kdrrndn plates have been likewise analyzed by Bisognin, Bisognin, Perla Menzala Zuazua [1997] and Perla Menzala & Zuazua [1997]. (3) The controllability of yon Kdrmdn plates has been studied by Puel & Tucsnak [1992, 1995]. I
We next show that the solution of the canonical yon Ks163 equations possesses the expected regularity if the data are smooth enough. The proof given here is that of Lions [1969, Thm. 4.4, p.56] (naturally, the same result holds for the scaled and de-scaled von Ks163 equations given in Thms. 5.6-1 and 5.7-1). It is no surprise that analogous ideas were already used in establishing a similar regularity of the solution of the two-dimensional equations of a nonlinearly elastic, completely clamped, plate (Thm. 4.6-3).
T h e o r e m 5.8-4. Assume that f E L2(~),~Po E H7/2(~/), and ~Pl E H5/2(~), and that the boundary'7 is smooth enough. Let (~, ~) C
Sect. 5.8]
Existence and regularity of solutions
421
H3(co) x H2(cj) be a solution of the canonical von Kdrmdn equations" /%2~ _ [~b, ~] § f in co,
--/).~C--0on 7, --~o and 0 . ~ -
ff)l Oil ~ .
Then E H3(co) M H 4 (co) and
~ E g 4(w).
Proof. By Thin. 1.5-1(c) and the regularity assumptions on the functions ~0 and r the boundary value problem A20o = 0 in w, 0o = r
and 0,0o
=
~1 on 7,
has a unique solution 0o E H4(cJ). Given any solution (~,~p) E Ho2(CJ) • H2(cJ) of the yon K~rm~n equations, let r
-
0o.
Hence the pair ((, ~ ) E H~(a;) • H~(oa)satisfies A 2 ~ - [~, ~] + [0o, ~] + f in co, A2~ -- -[~, ~] in co. We then proceed to show that, if 0o E H 2 ( w ) a n d f E L2(~), any solution (~, ~b) E H~(~) x H~(co) of these equations possesses the following regularities: E H~(co)rq H4(co) and ~ E Hg(co)A wn'q(co) for any q > 1. Note that only the regularity "0o E H2(co) '' is needed here; it is only for showing that r (~+0o) E H4(co) that the "full" regularity "0o E Hn(w) '' is required. Since ~ E //o2(Oo)and A2~ E LI(~), we infer from an argument already used in the proof of Thin. 4.6-3 that ~c E H 3 (co) I'1 H 3-6 (w) for (5 > 0 small enough,
The von Kdrrndn equations
422
[Ch. 5
so that 0 ~ { e H1-5(CO). The continuous imbedding H1-5(CO) r L2/e(co) for 6 > 0 small enough then implies that
[~, ~] E Lq(co) for all q >_ 1; hence (cf. Thin. 1.5-1(c) for q - 2, and Agmon, Douglis & Nirenberg [1959] for q >_ 1)"
~b E H~(w) and A2~
E
Lq(co) ~ ~b E w4'q(w).
This regularity implies that
r c on the one hand, and the imbedding H~-a(co) ~ with the assumption 00 E H2(w), implies that [00,~]EL ~(w)
L2/a(co), together
for all l < r < 2 ,
on the other; besides, f E L2(co) by assumption. Hence E
H3(co) and A2~
But W2'~(co)~-~ C~
E
L~(co)~ ~
E
wn'r(co) for all 1 _< r < 2.
for all r > 1; thus [00,~] E L2(w),
which in turn implies that ~
E
H4(CO), as was to be proved,
m
Returning to the two-dimensional "displacement" problem from which the von Ks equations originated, we obtain as an immediate corollary of Thins. 5.6-1, 5.8-3, and 5.8-4" T h e o r e m 5.8-5. Assume that the domain co is simply connected, its boundary 7 is smooth enough, pa E L2(w), and h, E Ha/2('y). Then the scaled two-dimensional "displacement" problem 7)(co) of a yon Kdrmdn plate (Thm. 5.4-2(c)): 8#(A + #)A2~3 _ N ~ O ~ a 3(A + 2,)
- Pa in co
OzN~ =
0
in co,
I3 = 0u~3 = 0 on "y,
N~zu~ = h z on -y,
Sect. 5.9]
Uniqueness or nonuniqueness of solutions
423
where 4)~#
o
o
N~z= ~+2t, EO~(r - ~1 ( G ~ + c%G + o~C~0~), has at least one solution ~ - (~) with the following regularity"
~ E H3(w)
5.9.
and
~3 ~ H2(co) f"l H4(w).
m
THE VON K/kRM/kN EQUATIONS: UNIQUENESS OR NONUNIQUENESS
OF SOLUTIONS
The von Ks163 equations have been justified under the crucial assumption that specific applied surface forces act along the lateral face of the plate; they correspond in the "original" three-dimensionM problem to the boundary conditions:
1/~ (a~z + crkz ~ 0 ~ku ~~)v9 dx~ - h :
on 7"
It has been further assumed that the functions h ~ ' 7 --+ R satisfy h : ( y ) - e2h~(y) for all y C 7,
where the functions h~ E L2("/) are independent of c. In this section (and in the next ones), we further assume that the f u n c t i o n s ha are given by
where (v~(y)) denote the unit outer normal vector at y C 7, and p is a real parameter. We first note that such functions ha are in L2(7)
(they are even in L~(7)) and that they automatically satisfy the compatibility conditions
hi
-
h2
-
(Xlh2- X2hl)d ,
424
[Ch. 5
The von K d r m d n equations
required for the existence of solutions to the yon Ks163 (Thm. 5.6-1).
equations
Remark. The parameter p will turn out to be in effect a bifurcation parameter (Sect. 5.11), for which the notation ~ is usually preferred. The notation p is nevertheless chosen here, in order to avoid any confusion with a scaled Lam~ constant! II The boundary conditions
if
C
( .9 + kg0k
dx;-
h: on
correspond to an applied surface force that is a dead load (Vol. I, Sect. 2.7). A more realistic pressure load (Vol. I, ibid.) would mean that the scaled surface force density (recall that it is integrated across the thickness) remains normal to the deformed boundary, while keeping its magnitude - p (Fig. 5.9-1). As the corresponding limit two-dimensional problems nevertheless coincide (see Ciarlet & Blanchard [1983] or Ex. 5.2), the subsequent analysis applies verbatim to such pressure loads. Our objective consists in keeping the thickness 2e fixed and counting the number of solutions that the von K&rmgn equations have when p is considered as a parameter, and whenever possible, in "following" these solutions as functions of this parameter. The results obtained in this fashion have an important mechanical interpretation: Assume for instance that these are no "transverse" forces; this means that f~ - 0 and g~ - 0 in the original threedimensional problem, and that accordingly F = 0 in the reduced von Kgrm~n equation (this equation was the key to the existence theory of Sect. 5.8). Then it seems intuitively clear that ~ = 0 is the only solution when p is < 0 ("uniform traction"), while when p is > 0 ("uniform compression") and large enough, there might be several distinct solutions, corresponding to the phenomenon of b u c k l i n g (see Fig. 5.9-1, and also Vol. I, Fig. 5.8-5): This is exactly what we shall prove in Thm. 5.9-2(b). To begin with, we describe the effect of the particular choices h~ = - p u ~ on the reduced yon Kdrmdn equation.
Sect. 5.9]
Uniqueness or nonuniqueness of solutions
425
Fig. 5.9-1: A yon Kdrmdn plate subjected to a pressure load. The plate is drawn as seen "from above". The scaled surface force density remains normal to the lateral face of the deformed configuration (indicated by a dashed line) and keeps its magnitude - p . If there is no transverse force, there exists px > 0 such that the solution is unique if p _< pl. If p > pl, there are at least three distinct solutions: The plate "buckles". Theorem 5 . 9 - 1 . Assume that h~ - -pu~ along 7. Define the linear operator L " H3(w) ---+ H~(w) as follows: For each ~ E H 3 ( w ) , L~ is the unique solution of
L~ E H3(w) a n d D A 2 L ~ -
where D -
8 # ( A + t-t) + ,)
"
- A ~ in w,
Then L is compact and symmetric and posi-
tire definite with respect to the inner product (., .)A defined by (~, ~])a = When expressed in terms of L, the reduced yon Kdrmdn equation ( T h i n . 5.8-1) takes the form: Find ~ such that EHo2(w)
and
C(~)+~-pL~-F,
426
[Ch. 5
The von Kdrmdn equations
where the "cubic" operator C : H2o(~) ~ F e H~(a~) are defined as in Thm. 5.8-1.
H3(w) and the function
Proof. A simple verification shows that, when h~ = -pu~, the
functions ~0 and ~Pl appearing in the canonical von Ks163 are given by ~0(Y) -
P (yl2 + y~) and ~/21(Y) -2D
equations
P 0L,(y~ -~- y2)
2D
for all y = (Yl, Y2) E 7. Consequently, the linear operator A" Hg(w)---. Hg(co) defined by A(~) = B(00, ~) for all ~ C H02(w) (Thm. 5.8-1) is given by A = pL,
since A2B(00,~) - [00,~] - - p D - 1 A ~ . Hence the reduced yon Ks163 equation takes the announced form when h~ - -pL,~. The compactness and symmetry of the operator A established in Thm. 5.8-2(d) imply that the operator L shares the same properties; its positive definiteness is a consequence of the relations (L~, ~)/, - f~ (AL~)A~ da; - f~ (A2L~)~ dw -
Dl f ~ (A~)~ da~
-- D1 ~ O ~ O ~ d w > 0 for all ~ E H~(w), ~ 5r O.
I
We now begin our investigation of the uniqueness or nonuniqueequations, according to the values of the parameter p. We follow here Ciarlet & Rabier [1980, Thm. 2.3-1]. hess of solutions of the reduced von Ks163
T h e o r e m 5.9-2. (a) Let (~,~)~ - f~ A~Ar/dw, and define
Pl : z
Then Pl is > O.
r
inf ~r
(~' ~)/' (L~, ~c)/x
Sect. 5.9]
Uniqueness or nonuniqueness of solutions
427
(b) A s s u m e that F = O. If p <_ Pl, ~ = 0 is the only solution to the reduced von Kdrrndn equation. If p > pl, this equation has at least three solutions: {o = 0, ~1 # 0~ and ~2 = -~1. (c) A s s u m e that F r O. There ezists p~ = p l ( F ) < px such that the reduced yon Kdrrndn equation has only one solution if p < p~; moreover, P*I may be so chosen that p~ ~ pl if f ~ 0 in H2o(CJ). Pro@ (i) As L is a compact, symmetric, and positive definite operator with respect to the inner product (',')zx (Thm. 5.9-1), the spectral theory for such operators (see, e.g., Taylor [1958, Thms. 6.4-1 and 6.4-B] or Dautray & Lions [1985, p.51]) shows that L has an infinite number of distinct eigenvalues qk > 0, each of finite multiplicity, that can be arranged as ql > q2 > . . . > qk > . . . , with lim qk = 0 as k --+ oc; moreover, ql --
sup ~#o
This shows in particular that Pl := 1/ql is > 0, as stated in part (a). (ii) When F = 0, solving the reduced von Kgrmgn equation consists in finding ~ E Hg(co) such that C([) + ~ - p L [ = O.
Since C ( 0 ) = 0 (Thm. 5.8-1), ~0 = 0 is always a solution. A s s u m e first that p <_ Pl. Any solution { of the reduced yon Kgrmgn equation satisfies (C(~), ~)A + [~1~ - p(L~, ~)/x - O,
where I~I2A "-- (~, ~)a; hence it also satisfies
I~l 2 -- pl(L~, ~)/, + (Pl - p)(L~, ~)A - - ( C ( ~ ) , ~)A. The left-hand side of this equality is >_ 0, since Pl (L~, ~)A by definition of pl, and ( P l - p)(L~,~)A > 0. Hence ~ = 0 by Thm.
5.S-2(b).
428
[Oh. 5
The von K d r m d n equations
A s s u m e next that Pl < P. From T h m . 5.8-3, we infer t h a t there is at least one function ~1 E H3((.d ) such t h a t
j(~l) --
inf
j(~?) , where j(r]) 9- 1
+
1
- p~ (Lr], r]) A.
Let 01 be an eigenfunction of the operator L corresponding to the eigenvalue qx = 1/px (part (i)), i.e., plLO1 = 01, normalized so t h a t 10xl/x = 1. T h e n inf
r/EHg (~)
j(r/) < 0,
since
OL4 Ct2 ( j(o~01) -- -~-(C(01) ,01)A -~- y
Pl
< 0 for c~ > 0 small enough,
and thus [, -y= 0 as j(0) - 0. Since C ( - { l ) - - C ( ~ l ) ( T h m . 5.8-1), [2 " - - ~ is also a solution of the reduced von K~rmgm equation. (iii) The more delicate proofs of the assertions of part (c) (case F r 0) are left as a problem (Ex. 5.4). m 5.10.
THE VON KARMAN EQUATIONS" DEGENERACY INTO THE LINEAR MEMBRANE EQUATION
In this section and the next one, the functions ha are again assumed to be given by h~ (y) - - p . ~ (y) for all y e 7. The behavior of the solutions of the yon K d r m d n equations as p approaches - o c or +oc are in sharp contrast. The next result, due to Ciarlet & Rabier [1980, Thin. 2.3-2] covers the case where p --~ - e c (the case where p --~ + o c is discussed in the next section); it establishes in particular a somewhat surprising link between the von
Sect. 5.10]
Degeneracy into the linear membrane equation
429
Kgrm~n equations and an emblematic equation of linearized elasticity: T h e o r e m 5.10-1. ( a ) A s s u m e that A2F = f E L2(w)(Thin. 5.8-1), and let X denote the solution of: x E H ~ ( w ) and - A X - f
into.
For each p < pl(F), let ~, denote the unique solution of the reduced yon K d r m d n equation (Thin. 5.9-2(c)). Then
(p) -~p
---+ X in Hl(w)as p---+ - o c .
(b) Let c > 0 be fixed. For each p < Pl (F), dcJ~Tbe the Tb~Tfbbe7~T; and the functions ~ , . E H~(w) and p~ E L2(w) by T; "- - p c 3, ~,p "-- cD1/2E-1/2~p,
where D "- 8#(A + p) and E "-
pea "- c4Da/2E-1/2f,
~+#
-Tp A~3,p --~ P3 in H - i ( w ) as
9 Then
p --*
-oc.
Pro@ (i) There exists a constant c such that
I--P~pll,w ~ cDifIo,w and I~pizx < cD1/2{-P}-l/21flo,~ for all p < min{0,pl(F)},
where ]rill,w -
{f~0~0~r/d~} 1/2 and
For each p < min{0, pl(F)}, the function ~p satisfies
6
Hg( )
A 6- -[B(6, 6),
P A~v + f,
The von Kdrmdn equations
430
[Ch. 5
by definition of the operators B, C, L and of the function F (Thms. 5.8-1 and 5.9.1); hence for all r/E H2o(w), s A~pA~ dco - -
j / [B(~p, ~p), ~plr/dw + ~P f~ 0~p0~ r] d w + f ~ f~dw,
so that (again by Thm. 5.8-1) (~cp,r/)a -- -(B(~Cp, ~cp),B(~p, r/))a + ~
O~pO~rldw +
frldw.
Noting that there exists c such that 1,10,~ ~ c17711,~ for all r/ E H~ (co) and letting r / - ~p, we obtain
p
2
I~pl~ - ~ I G I I , ~ -
-IB(G,G)I~ + s f~p dw <_clflo,~lGIl,~o.
As p is < O, we first infer that t9 I~pll,~ < clflo,~, D then that IGI~ ~
clflo,~lGIl,~o ~ -c=Dp lfl~,~
p (ii) The function - - ~ p p---, -oc.
converges strongly to X in Hi(w) as
As the norms l-P~pll,,o are bounded independently of p by (i), there exist X E Hi(w) and a subsequence, still indexed by p for convenience, such that (as usual ~ denotes weak convergence) P D~p
X in H01 (w) as p ~
-oc.
As I~pla _< cD1/2{-p}-l/2[f[o,a, by (i), it further follows that [Gla
~
0 as p ~
-oc.
We already noted in (i) that
(G, ,)A - - ( B ( G , ~), B(~, ,7))A +
-~
frl dw
Degeneracy into the linear membrane equation
Sect. 5.10]
431
for all r/E H3(co). Hence the above convergences, combined with the continuity of the operator B (Thm. 5.8-2(c)), show that
c9~xO~rldco - s frldco for all~ E H~(co), as H02(co) is dense in Hi(co). Thus X E Hi(co) solves - A X - f in the sense of distributions. To show that the function - ~ converges strongly to X in Hi(co) as p --+ - o c , it suffices to show that --~p[1,w ---+ IX[1,co a s p--+ - o c (cf., e.g., Brezis [1983, Prop. 111.30]). The relation (established in (i))
I~1~ - ~1~1,,~ - - I B ( ~ , ~ ) l ~ +
f{p dw
shows that -~1~1~,~, ___L f ~ , henc~ that 2
- ~ {v
-<
f
- ~ {v dco for all p < min{0, Pl (F) };
1,co
consequently,
-~v
lim
p--+
l,w
_< limsup p---+-- cx)
f
-~scp
dco-
L
f x d c o - Ix121,~o
as - ~ p ~ X in H~(w). As a norm is a weakly lower semi-continuous function, we also have (Vol. I, Thm. 7.2-2) IXI21,w < liminf --
p
p--+ - c ~
1,CO
and thus p 1,w a s p
----+ - - 0 ( 3 .
1,w
Besides, the whole family converges as the limit X is unique, and (a) is proved. (iii) To prove (b), we note that P ~p + X in H~(co) => A ~ v D
-+ - A X -
f in H-l(co)
432
The von Kdrmdn equations
[Ch. 5
as p --+ - o c . Hence the relations between T~ and p, ~,~ and {p, and p~ and f defined in the statement of the theorem immediately imply that (recall that c > 0 is held fixed): - T ~ A ~ , p --+ p~ in H - l ( w ) as p--+ - o o . II A major conclusion is that we have justified in Thm. 5.10-1(b) the famed l i n e a r m e m b r a n e e q u a t i o n
- Tp A ~ , p -- p~ in co,
~,p - 0 on 7. In this equation, the number T~ - - p e a - Ipeal measures the tension of the membrane (exerted through a surface force density along 7 with components of the form h~ - -pc2u~, p < 0), the function ~,p is the transverse displacement of the membrane (the factor eD1/2E-1/2 multiplying {~ in the definition of ~,p is precisely that used for deriving the canonical von Ks163 equations in Sect. 5.8), and the function P3 fa dx~ + + ga-~ is the transverse force acting on the membrane (the factor c4Da/2E -1/2 multiplying f in the definition of p~ is again precisely that used for deriving the canonical yon K~rm~n equations). More precisely, we have shown that, for large enough values of the tension (i.e., for large enough values o f - p ; recall that c > 0 is fixed) and for a fixed transverse force p~, the transverse displacement of a yon Kdrmdn plate behaves like the solution of the linear membrane equation. This provides a mathematical justification of the definition found in Landau & Lifchitz [1967, p. 79]: "On appelle m e m b r a n e une plaque mince fortement tendue par des forces appliqu~es g ses bords". A w o r d of c a u t i o n . It may be surprising that the Lamd constants nc longer appear in the linear membrane equation. But, as already noted in Sect. 4.1, they can only describe the behavior of an elastic material near a reference configuration that is a natural state,
Sect. 5.11]
Bifurcation of solutions
433
i.e., "stress-free". This is certainly not the case when the tension is large! II For more details about this "degeneracy" of a plate that is subjected to a large "uniform traction" along its boundary into a membrane, see Fife [1961], Srubshchik [1964a, 1964b], Landau & Lifchitz [1967, p. 79], John [1975], Schuss [1976], Berger [1977, p. 206], and Sanchez-Palencia [1980, p. 194]. Remark. A similar, but only formal, link has already been noted between the linear membrane equation and the nonlinear membrane theory of Fox, Raoult & Simo [1993]; cf. Sect. 4.12. I 5.11 ~
T H E V O N K A R M / ~ N EQUATIONS" B I F U R C A T I O N OF S O L U T I O N S
When p ~ +oc, the picture changes drastically, as we enter the realm of b i f u r c a t i o n t h e o r y . We already got a glimpse at it when we proved (Thin. 5.9-2(b)) that the von Ks163 equation have at least three solutions when F = 0 and p > Pl. W e n o w briefly describe the considerably more precise results that can be gathered about the bifurcation, or "branching", of solutions of the yon Kdrmdn equations. For detailed and self-contained proofs of these results, we refer to Ciarlet & Rabier [1980, Sects. 2.4 and 2.5, and Chap. 3]. We begin by considering the case where F = 0, i.e., when there are no transverse forces. The resulting "unperturbed bifurcation diagram" is drawn and interpreted in Fig. 5.11-1. T h e o r e m 5.11-1. Assume that F = O. Let qk = 1/pk be a simple eigenvalue of the compact operator L (see proof of Thin. 5.9-2, part (i)), let Ok be a corresponding eigenfunction (pkLOk = Ok), and let
There exists a neighborhood blk of (Pk, O) in R • H3(cJ) in which, apart from the trivial solution (p, 0), the only solutions (p, ~) 6 blk of
The yon Kdrmdn equations
434
[Ch. 5
0
Unperturbed bi]urcation diagram for the yon Kdrmdn equation with right-hand side F = 0. Let 1/pk be a simple eigenvalue of the o p e r a t o r L. In an ad hoc n e i g h b o r h o o d L4k of (Pk, 0) in 1~ • H02 (w) (without loss of generality, L4k m a y be
Fig. 5.11-1:
chosen as a rectangle), the trivial solution (o : 0 is the only solution of t h e r e d u c e d von K&rm~n equation if p _< pk; if p > pk, there appear two additional solutions (1 and (2 = -~1 t h a t "bifurcate" from the trivial solution. These two solutions lie on a continuous, "parabola-like", curve, s y m m e t r i c with respect to the p-axis. Naturally, e x t r e m e care m u s t be exercised for interpreting such a "bifurcation d i a g r a m " , as the vertical axis is m e a n t to represent an infinite-dimensional space!
the reduced yon K d r m d n equation
C(~) + ( - p L (
lie on a p a r a m e t r i z e d curve
: 0
Sect. 5.11]
Bifurcation of solutions
435
~ h ~ to > o, ~ d w ( t ) e ~ a~d ~(t) e U3(~) ~ti~fv pk(t) -- pk + t 2pk# (t),
v~ (t) > o if t # o,
where
v~ (t) - o(1),
and
~ ( t ) - te~ + t 3(~# (t), (~(-t)--(~(t),
(~(t)e
{e~} •
where I~# (t)]A - 0(1),
the order symbols 0(1) being meant with respect to t. In particular then, this curve is continuous and symmetric with respect to the paxis (Fig. 5.11-1). m We continue by considering the "full" yon Kgrmgn equation, i.e., with a nonzero right-hand side. It is still possible to describe its solutions in a neighborhood of (Pk, 0) in R x H02(a~) when the righthand side is of the special form 6F, with F given in H~(a~) and 6 small enough. In other words, the right-hand side must be "small enough" in this restricted sense. The resulting "perturbed bifurcation diagram" is drawn and interpreted in Fig. 5.11-2. T h e o r e m 5.11-2. Let 1/pk be a simple eigenvalue of the compact operator L and let Ok be a corresponding eigenfunction. Let F E H~(a~) be given such that (iV, Ok)A # O. There exist 6" > 0 and a neighborhood bl~ of (pk, 0) in R x H2o (a~) such that, for any 6 E ] - 6", 6"[, all the solutions (p, ~) E Lt[~ of the reduced von Kdrmdn equation C(~) + ~ - pL( - 6F
lie on two continuous curves, which are disjoint if 6 7L O. If 6 # O, there exists (P*k(6), ~; (6)) E Lt~ such that P*k(6) > pk, P~(6) --* pk as 6 ~ O, and such that there is exactly one solution in
The yon Kdrmdn equations
436
[Ch. 5
''
2*
~
Ii$sJ9
Fig. 5.11-2: Perturbed bifurcation diagram for the yon Kdrmdn equation with a nonzero right-hand side i~F. Let 1/pk be a simple eigenvalue of L and assume that F is not orthogonal to the corresponding eigensubspace. In an ad hoc neighborhood L/~ of (pk, 0) in R x Hg(a~) and for 151 small enough, there exists p~(5) > Pk such that there is exactly one solution for p < p~(~), two distinct solutions for p = p~(~), and three distinct solutions for p > p~(~), in the neighborhood L/~. If /~ ~ 0, these solutions lie on two continuous, disjoint, curves, one of them having a "turning" point (p~(~),~(5)). When ~ ~ 0, this bifurcation diagram "converges" to the unperturbed bifurcation diagram of Fig. 5.11-1, represented here with a dashed line.
l/l[~ if p < p*k(~), two distinct solutions if p = p'k(5) (one of these is ~(i~)), and three distinct solutions if p > p*k(~). I R e m a r k . As e x p e c t e d , s o m e " s i n g u l a r i t y " o c c u r s at t h e "turning point" (p~(5), ~ ( 5 ) ) . M o r e specifically, it c a n be s h o w n t h a t t h i s is t h e o n l y p o i n t (p, ~) in t h e n e i g h b o r h o o d L/~ w h e r e t h e F r ~ c h e t d e r i v a t i v e {C'(~)+I-pL} is not a n i s o m o r p h i s m of H02 (a~): see C i a r l e t & R a b i e r
Sect. 5.11]
Bifurcation of solutions
[1980, Sect. 3.4].
437
m
A welcome complement to Thin. 5.11-2 is provided by QuintelaEst~vez [1994]: Using the method of "matched aysrnptotic ezpansions', she has shown how to "connect" the "local" bifurcation branches corresponding to two consecutive simple eigenvalues of the operator L (see Fig. 12 in ibid.). If the von Ks163 plate is circular, i.e., if w is a disk in R 2, it is known (see, e.g., Keller, Keller & Reiss [1962], Wolkowisky [1967], and Berger [1977]) that the largest eigenvalue 1/pl of the corresponding operator L is simple and that any corresponding eigenfunction 01 has a constant sign in cv. If the plate is "horizontal" in its reference configuration and subjected only to its own weight as a transverse force, the function F E Hg(w) solves A2F = -2pg, where p is the mass density of the constituting material. Hence
(F, Ok)zx - -2pg J2 Ok da)~ and the condition (F, 0k)zx 7~ 0 reduces to f~ Ok dw r 0 in this case. This condition is thus satisfied for k = 1 by a circular plate. The condition that the right-hand side be small enough (in the sense of Thm. 5.11-2) does not preclude interesting applications: Instead of "weightless plates" (F = 0), it affords to consider "horizontal plates with weight", since their weight is certainly "small" compared for instance to pressure loads producing buckling. In addition to Ciarlet & Rabier [1980], there exists a vast literature on the bifurcation of solutions of the yon Kdrrndn equations and more generally, on the buckling of plates. See in particular Taylor [1933], Friedrichs & Stoker [1942], Keller, Keller & Reiss [1962], Bauer & Reiss [1965], Wolkowisky [1967], Knightly & Sather [1970], Berger [1977], Antman [1978], Golubitsky & Shaeffer [1979b], Ciarlet & Rabier [1980], Matkowsky, Putnick & Reiss [1980], and Brewster [1986]. Using homogenization theory, Duvaut [1978] and Mignot, Puel & Suquet [1981] have studied the buckling of yon K~rm~n plates
438
The von Kdrrndn equations
[Ch. 5
with "many" periodically distributed holes. The buckling of a yon Ks163 plate lying on an obstacle has been analyzed by Do [1977] and Goeleven, Nguyen & Th~ra [1993a, 1993b]. References on bifurcation theory that are more general, but still relevant to the yon KgLrm~.n equations, are Crandall & Rabinowitz [1970], Rabinowitz [1971, 1975], Keener & Keller [1973], Chow, Hale & Mallet-Parer [1975], Matkowsky & Reiss [1977], Golubitsky & Schaeffer [1978, 1979a, 1985], Keener [1979], Rabier [1982a, 1982b], Golubitsky, Stewart & Schaeffer [1988], and Rabier & Oden [1989]. More recent tratements include the illuminating and in-depth account of bifurcation theory given in the books of Antman [1995, Chaps. 5, 6, and 14] and Chow & Hale [1996]. Numerical approximation of bifurcation problems are extensively treated in Crouzeix & Rappaz [1989] and Paumier [1997]. 5.12 ~.
THE M A R G U E R R E - V O N K/kRMAN E Q U A T I O N S OF A N O N L I N E A R L Y E L A S T I C S H A L L O W SHELL
As shown by Ciarlet & Paumier [1986], the method of formal asymptotic expansions, applied in the form of the displacement-stress approach, may be also used for justifying the Marguerre-von Kdrmdn equations. These two-dimensional equations classically model nonlinearly elastic shallow shells that are subjected to boundary conditions analogous to those of a yon Kdrmdn plate; we give here only a summary of results, refering to Ciarlet & Paumier [1986] for details, proofs and extensions. The "geometry" of the shell is defined as in Chap. 3 (see in particular Fig. 3.1-1), i.e., its reference configuration is of the form { ~ } - , where ~)~ "- O ~(f~), f~ - - a ~ x ] - c, c I, aJ is a domain in R e with boundary 7 and the mapping O~: {f~}- ~ R 3 is given by O : ( x : ) - - ( x i , x 2 , 0 : ( x , , x 2 ) ) + x a a a ( x , , x 2 ) for a l l Z e - - ( X l , X 2 , X~) E
,
where a~ is a unit vector normal to the middle surface O ~(~) of the shell and 0 ~ : ~ ~ R is a function of class Ca such that c9,0 ~ = 0 along 7.
The Marguerre-von Kdrmdn equations
Sect. 5.12]
439
odm,ss,ole dis placement 7-'
E.~- [ ~ - ~---....
F
~
Fig. 5.12-1: A Marguerre-von Kdrmdn shallow shell. T h e lateral face of the shell is a vertical t r a n s l a t i o n of the lateral face of the set 9F = w x] - e, r of c o n s t a n t m a g n i t u d e along the lateral face. T h e only possible displacements along the lateral face are horizontal ones, of equal direction and m a g n i t u d e along each vertical segment. T h e shell is "shallow" in t h a t the m a p p i n g 0 ~ : ~ ~ R is of the order of the thickness of the shell, up to an additive constant (for a b e t t e r representation, only a "cut" has been drawn).
Hence 0 ~ is a constant along-y and the lateral face O~(7 • I-s, s]) of the shell is "vertical"; cf. Fig. 5.12-1. We let c3~ - 0/02~, where 2~
(2~) denotes a generic point in the set { ~ } - . The shell is subjected to applied body forces of density (/~) (0, 0, f~) 9f)~ ~ R 3 in its interior, to applied surface forces of density (t~) - (0, 0, t~) " F~- O F~ ~ R 3 on its upper and lower faces F~+ "- O~(F~_) ~ where F ~+ 9- - ~ X { i s } and to applied surface forces on the entire lateral face O~(V • [-a, a]), whose only the resultant (h~, h~, 0) after integration across the thickness is given along -
._ o
The boundary conditions along the lateral face then take the following form, highly reminiscent of those corresponding to a v o n
440
[Ch. 5
The yon Kdrmdn equations
Ks
plate (Sect. 5.1), viz., 5~ independent of ~ and t2~ = 0 on O ~('7 x [-c, c]), 1
_
{(6;~ + ak~Oku~) o O ~ }u~dx; - h~ ^~ ^ ^~ ^~ o O
~
on 7,
^~
where u~ are the components of the displacement vector f i e l d / { and ^~ a~j are the components of the second Piola-Kirchhoff tensor field now ^~ As for a yon Ks163 expressed as functions of the coordinates x~. plate, the functions ]~ 9~ ~ R must satisfy the compatibility conditions (also given in "scaled" form in Thm. 5.12-2):
~ hl d'7 - ~ h~ d~/ - ~ (Xlh~ - x2h~) d'7 - O, where h~ "- h[~ o O ~. Assume for simplicity that the nonlinearly elastic material constituting the shell is a St Venant-Kirchhoff material, with Lam~ constants M and #~. Then the displacement vector field it ~ and the second Piola-Kirchhoff tensor field E~ solve the following problem Q ( ~ ) (which reduces to that of Sect. 5.1 when 0 ~ - 0 ) "
,s
(~.)E V(fi e) .__{,~e. (?)~) E wl,4(fie); 1); independent of 5:~ and 9j = 0 on O ~('7 x [-~, ~])},
~E
- ( 6 ~ j ) E L~(~ ~) - {('~i~)E L 2 ( ~ ) ; "~i~-"~j~},
(fTi~ + ~ j ~ t c i ) O j
+ ~1
~
(~;o
fa va dJc~ +
Vi dxr -O~
)dx;
~+uf'L
~d~ ~ for a l l ~ V ( f i
h~
--s
(u
.-
+
+
A~
1
g3v3 d ~ ~) ,
The Marguerre-von Kdrrndn equations
Sect. 5.12]
441
This problem is then transformed in the usual m a n n e r into an equivalent problem posed over the set 9 - ~ x [-1, 1]. To this end, we define the scaled displacements u~(e) 9 ft ~ R and the scaled stresses crij(e) : f~ --, R by the s c a l i n g s
<(~) - ~,(~)(~), ai(~) - ~(~)(~), {5.;13(~)_ c2cra/3(E)(:;c), o_a3(~e)^e_ E30.c~3(x), o.33(2~e)^e__ C40_38(x) for all s _ O~(7r~x) E f)~, where 71e(a;1,X2, X3) (Xl,X2, g*3). We also make the following a s s u m p t i o n s o n t h e d a t a : There exist constants A > 0 and # > 0 and functions fa E L2(Ft), ga E c~(r+ur -) where F + := co x {+1}, h~ E L2('7), and 0 E ca(~) t h a t are all independent of e, such t h a t -
l~=l ]~(:~) - eafa(x) g3(~e) __ c 4ga (X ) ^~
0~(zl, ~)
and
-
S=#,
for all 5:~ - O~(Tr~x) E ~ ,
for all ~ - O e (Tc~x)E F^e + U ~,e_,
= ~0(~1, ~ )
fo~ ~H
(z,, ~ ) e m.
Note t h a t the last relation is the "shallowness" assumption, already used in Sects. 3.8 and 4.14. As a consequence of these scalings and assumptions, the s c a l e d d i s p l a c e m e n t field u(c) = (u~(~.)) and the s c a l e d s t r e s s t e n s o r field E = ( ~ j ( c ) ) solve a problem Q(c; ft) of the form "U,(C) ~ V ( ~ ) ; =
{'U = (Vi) ~ wl'4(~'~); Va independent of x3
and va = 0 on ~/• [ - 1, 1]},
~(e) ~ U ( a ) -
{(~j)~ L~(a); ~ j - ~j~},
+ e2T2(e; E(e), u(e), v) - L(v) + eL 1(e; v) for all v E V ( a ) ,
E~
+
e2E2(e; u(c)) - (B ~ + e2B 2 +
c4B4)~-](~),
442
The von K d r m d n equations
[Ch. 5
where the linear form L, the bilinear form/3, the trilinear form T ~ the matrix-valued mapping E ~ and the fourth-order tensors 13o B 2 134 are all independent of ~, and there exists a constant C such that the linear form LI(e; .), the bilinear form B2(c;-,-), the trilinear form T2(c;.,., .), and the matrix-valued mapping E~(c; . ) - (E~j(c; .)) are all "of order 0 with respect to e", in that there exists a constant C such that sup
O<e<_eo
sup
0<~<_eo
IL~(r v)l ~ Cllvlll,n,
IB=(~; ~, v)l _ Cl~10,~llvll~,~,
sup IT~(~; ~, v, w)l ~ C[~lo,~[Ivlll,4,~llW]]l,4,~,
0<e_<eo
sup lEVy(C; V)lo,~ _< C{llVlll,~ + Ilvll ~1,4,f2},
0<e<_eo
for all ~ E L~(~), v E Wl'4(~'~), w E Wl'4(~'-~). These estimates, together with the strong resemblance between problem Q(~; ~t) and that found for a von K~rm~n plate (Thin. 5.2-1), suggest that we again use the basic Ansatz of the method of formal asymptotic expansions: Assuming that the scaled unknowns can be expanded as u(r
- u ~ + h.o.t,
and
E(e) - 5] 0 + h.o.t.,
we find that the leading terms u ~ and IEO satisfy
B(E ~ v)+ T~
~ u ~ v) - L ( v ) for all v E V ( ~ ) ,
E ~ (u ~ - BOEO. These equations then lead to the following extensions of both Thms. 5.3-1 and 5.4-2: T h e o r e m 5.12-1. Assume that u ~ - ( u ~ )0 E W l , 4 (f~), 03uO C~ u~o are independent of x3 and u ~ 0 on "7o, and :E ~ E L~(9,). -
-
The Marguerre-von Kdrmdn equations
Sect. 5.12]
443
(a) Define the space V ( w ) "-- H i ( w ) • H i ( w ) • H3(w ).
Then there exists ( = ((~) 6 V(w) such that o
u~ - (~ - x30~(3 and u ~ - (3.
(b) Let -0
1
/1
P3"-
f3dx3+g ++g;,
g~'-g3(',4-1).
1
Then ( =
(~i) 6 V(w) satisfies the s c a l e d t w o - d i m e n s i o n a l
t i o n s 7)(w) of a M a r g u e r r e - v o n
K f r m ~ i n s h a l l o w shell:
p3r/3dw + ~
-
equa-
h~q~ d'), for all rl 6 V(ft),
where
4A~ "~'
-
N~
-
-
3(~ + 2.)
4Art
-o
.- A + 2, G~(r
AGGz + +
0~z~3 5-
4#/9oz
'
(~;)
(c) Assume that the boundary 7, the functions P3 and h~, and the solution ~ of problem 7)(w) are smooth enough. Then r = (~) satisfies the boundary value problem:
444
[Ch. 5
The von Kdrmdn equations
- 0 ~ 9 m ~ 9 - N ~ O ~ ( ~ a + O) - P 3 in co, -OaN~a - 0
in co,
~3 - 0~3 - 0 on 7,
N ~ v ~ - h~ on 7. II Remarks. (1) We also have -O~om~ o - 8>(/~ + >)A2r
3(~ + 2~)
(2) The limit scaled stresses, i.e., the components of the tensor E0 can also be computed" cf. Ciarlet & Paumier [1986 Thm 4 1].m ~
~
9
.
The two-dimensional "displacement" problem found in Thin. 5.12-1 is itself equivalent to another two-dimensional problem, constituting the (scaled) Marguerre-von Kdrmdn equations. The proof of this equivalence, given in Ciarlet & Paumier [1986, Thin. 5.1], closely follows that of Ciarlet [1980, Thin. 5.1] for a yon Kgrms plate (cf. Thin. 5.6-1; the notations 7(Y) and v~(y) have the same meaning as in this theorem). T h e o r e m 5.12-2. Assume that co is simply connected and that its boundary "r i~ ~.~ooth r Lr thr162 br given a ~ol~t,on r = (~) of problem P(co) (Thin. 5.12-1) with the following regularity"
~o~ (~ H3 (w)
and
~3 E H 4(co) A Ho2 (co).
Then the functions h~ are in the space H3/2(7 ), they satisfy the compatibility conditions
~hld"~--~h2d~- ~(Xlh2-X2hl)d'Y-O, and there exists a scaled A i r y s t r e s s f u n c t i o n r E H4(co), uniquely determined by the requirements that q~(0) - 0 1 ~ ( 0 ) - 0 2 ~ ( 0 ) - 0, such that
The Marguerre-von Kdrmdn equations
Sect. 5.12]
445
Furthermore, the pair ( ~ 3 , r {Hg C~H4(a~)} x H4(a~) satisfies the s c a l e d M a r g u e r r e - v o n K 4 r m 4 n e q u a t i o n s :
8p(A + p) A2(8 _ [~ (3 + 01 +p3 in co s(a + 2.)
'
'
A2 5 _ _#(3A + 2#)[(3, ~3 + 20] in co, A+p ~3 - 0~G - 0 on 7, r
r
and 0 ~ r
r
on 7,
where
[)(~,~] "-- 011X(~22r -11-(~22)C(~11~)-- 2012)(~012~/), r
"----Ylj~7 h2d0/-~-Y2j/7 hld'~'-Tt-Js (Zlh2 -z2ht)d")/t (y)
(y)
J~ (y)
(y)
Jr (y)
"y.
II
Remark. Conversely, a solution of the "displacement" problem 7)(a~) can be constructed from any solution of the Marguerre-von KArmAn equations, as in the case of the von KArmAn equations (Thin. 5.6-1(b)); see Ciarlet & Paumier [1986, Thm. 5.1]. II It remains to "de-scale" the equations found in Thm. 5.12-2. To this end, we define the functions Ca " ~ + R and r " -co ~ R by the de-scalings
~'-c4a which immediately give:
and
r162
The von Kdrmdn equations
446
[Ch. 5
T h e o r e m 5.12-3. The de-scaled functions ~ and -r satisfy the M a r g u e r r e - v o n K~irmRn e q u a t i o n s : 8 # ~ ( X ~ . + / z ~) e a A 2
3iA-7 q2 2p~)
~
~
-~
4a - c[r ~, (a + 0~] + Pa in w,
/~2(~e
.~(aa~ + 2 . ~) A~ + # ~
[ ~ , ~ + 20 ~] in co,
(~ - O . ~ - 0 on 7, r
- q50 and 0~r ~ - r
on 7,
where
p~-~._
]~(o~(., ~))~ d~+g~(O~(~ ^~ ., ~))+g~(O~( ,^~ . _~)), (~)
r (y) "-- --l.I1(y) ~
(y) (y)
h~ d'y + u2(y) /i
(y) (y)
h~l aT, y C "y,
~
II The Marguerre-von Ks equations are due to Marguerre [1938] and von K~rm~n & Tsien [1939]. The function r is the A i r y s t r e s s f u n c t i o n ; flom its knowledge, one may again compute the "limit" stress resultants across the thickness of the shell, as in the case of a von K~rms plate (Thm. 5.7-2). As in Sect. 4.14, we conclude t h a t
both the vertical deflection 0~ and the "vertical" displacement ~ of the points of the middle surface should be of the order of the thickness of the shell in order t h a t the Marguerre-von Kgrmgn equations may be deemed asymptotically equivalent to the original three-dimensional equations. For questions of existence, regularity, bifurcation of solutions of the Marguerre-von K~rmgn and other related equations, or their degeneracy toward the linear membrane equation, we refer to Rupprecht
Exercises
447
[1981], Kesavan & Srikanth [1983], Rao [1989], Paumier & Rao [1989], Kavian & Rao [1993], and Rao [1995a, 1995b]. A Marguerre-von Ks shallow shell corresponding to a mapping 0~ 9 ~ -~ R can be imbedded in a one-parameter family of shells corresponding to mappings tO ~ 9 ~ ~ R, 0 <_ t <_ 1; these mappings are special cases of those introduced by Ciarlet [1992] for allowing general shells to "converge to a plate". Keeping the thickness 2c > 0 fixed, Alexandrescu-Iosifescu [1995] has then shown that the transverse displacements ~ ( t ) of the corresponding Marguerre-von K d r m d n shallow shells converge in H2(w) as t ~ 0 and that their
limit for t - 0 is the transverse displacement of the von Ks plate that is the natural "limit" of the Marguerre-von Ks163 shallow shells. EXERCISES
5.1. (1) Prove the assertion ( b ) i n Thm. 5.4-2 (argue as in the proof of Thm. 4.5-2; see also Thin. 4.1 in Ciarlet [1980]). (2) Prove assertion ( c ) i n Thm. 5.4-2 (argue as in the proof of Thm. 4.6-2). (3) Prove Thin. 5.5-1 (argue as in the proof of Thin. 4.8-1; see also the proof of Thin. 4.1 in Ciarlet [1980]). 5.2. The object of this problem, based on Ciarlet & Blanchard [1983], is to show that the boundary conditions defining a yon K d r m d n plate (Sect. 5.1) may be replaced by more realistic boundary conditions. More specifically, let the boundary conditions involving the disp l a c e m e n t u ~ - (u~) along the lateral face be the s a m e as for a yon Ks163 plate. Next, let ~ 9gt~ ~ R a be the deformation defined by (all notations are as in Sect. 5.1)" ~ ( x ~) - x ~ + u ~ ( x ~) for all x ~ e ~ ,
let T ~ ( ~ ( x ~ ) ) - (T~(~(x~))) denote the Cauchy stress tensor (Vol. I, Sect. 2.3) at each point ~ ( x ~) e ~ ( ~ ) , let (u~(~(y))) denote the unit vector normal to the deformed boundary ~ (~/) at each point
448
The von Kdrmdn
[Ch. 5
equations
~ ( y ) , y E 7, and let p be a real parameter. conditions along 3' take the form: m
iF
e
T ~z(cp~(y))u;3( r e ~
Then the boundary
r (y))dx; - -pe 2 u~(cp~(y)), y e 7.
In other words, the surface force per unit length (after integration across the thickness) remains normal to the deformed boundary and keeps its magnitude _pe2, whatever the deformation ~a~ may be: This is an instance of a boundary condition "of pressure", which is a live load (Vol. I, Sect. 2.7). Carry out an asymptotic analysis analogous to that made in Sects. 5.2-5.5, and show that the limit equations are the same von Ks163 equations that correspond to the boundary conditions (now a special case of those considered in the text):
1;
e Oe e
dx~_
pg2ua
,.,/
C
Hint: The first, and crucial, step consists in expressing these new boundary conditions on 7 in terms of the second Piola-Kirchhoff stress tensor, by means of the Piola transform (Vol. I, Sect. 2.5). 5.3. As shown in Ciarlet & Rabier [1980, Sect. 2.1], it is also possible to establish directly the existence of a solution to the twodimensional "displacement" problem P(a~) found in Thin. 5.4-2, in which case the set a~ may be multiply connected. (1) To this end, show that solving problem P(c~) is equivalent to finding the stationary points of the functional j'rldefined by -
((r/~), 77a) E {H'(co)/V~
1/ {1
+
- ( ~ parladco + f h~rl~d'),) ,
x Hg(co) ~ R,
~
}
449
Exercises
5.4-2, the space V ~
where E ~Z ~ (~/) is defined as in Thm. Thin. 5.6-1, and
as in
4)~# a ~ , - = )~ + 2# (2) Show that the functional j is weakly lower semi-continuous over the space { H I ( w ) / V ~ (w) } • (w) (because of the compatibility conditions satisfied by the functions h~, the functional j is well defined over this space) and that j is coercive on this space if the norms IIh~llL~(~) are small enough. It is then classical that these conditions insure that the functional j has at least one minimizer (a similar argument was used for a clamped plate; cf. Thm. 4.6-1). Remark. A refined argument shows that the same conclusions can be reached without any restriction on the norms Ilh~ll~(~); cf. Ciarlet & Rabier [1980, Thm. 2.1-1]. 5.4. (1) The operators B and C being defined as in Thm. 5.8-1, let
IIBII -
sup
IU(~5, g) [~x A
~#0
denote the norm of the bilinear operator B " Hg(w)x Hg(w) ~ Hg(w). Show that
IC(~)-c(w)l~ _< 311BII~ max{l~l~, Iwl~}l~-w[~ for all ~, ~ C No(W ), where I~cl/x := IA~10,~ = Ir for ~c C Hg(w). (2) Let pl be defined as in Thin. 5.9-2, and assume that p < Pl. Show that any solution ~ of the reduced yon KgrmAn equation (Thin. 5.9-1) satisfies
I~1~_<
P-----5--11FI~,.
Pl - - P
(3) Assume that p < Pl, and let ~1 and ~2 be two solutions of the reduced yon K~rm~n equation. Using (1) and (2), show that
(pl)
3/2 IYlzx[41 - 421/,.
450
The yon Kdrrndn equations
[Ch. 5
(4) Assume that F -r 0. Show that there exists p~ = p l ( F ) < Pl such that the reduced yon K & m g n equation has only one solution {p if p < p~ and that Pl may be so chosen that f~ --+ Pl if F -+ 0 in (5) Assume that F -J: 0. Using the implicit function theorem, show that the mapping p E ] - oc, pl (F) [--+ {v E H~(w) is of class C ~. H i n t : See Lemma 2.2-5 and Thm. 2.3-1 in Ciarlet & Rabier [1980].
REFERENCES
ABRAHAM, R.; MARSDEN, J. E.; RATIU, T. [1983]: Manifolds, Tensor Analysis, and Applications, Global Analysis Series, No. 2, Addison-Wesley, Reading. ACERBI, E.; BUTTAZZO, G. [1986]; Limit problems for plates surrounded by soft materials, Arch. Rational Mech. Anal. 92,335-370. ACERBI, E.; BUTTAZZO, G; PERCIVALE, D. [1988]: Thin inclusions in linear elasticity: A variational approach, J. Reine Angew. Math. 386, 99-115. ACERBI, E.; BUTTAZZO, G; PERCIVALE, D. [1991]: A variational definition of the strain energy for an elastic string, J. Elasticity 25, 137-148. ADAMS, R. A. [1975]: Sobolev @aces, Academic Press, New York. AGANOVId, I; JURAK, M.; MARUS;d-PALOKA, E.; TUTEK, Z. [1997]" Moderately wrinkled plate, Asymptotic Anal., to appear. ACANOVId, I; MARUgId-PALOKA, E.; TUTEK, Z. [1995]" Slightly wrinkled plate, Asymptotic Anal. 13, 1-29. AGANOVId, I; TUTEK, Z. [1986]" A justification of the one-dimensional model of an elastic beam, Math. Methods Applied Sci. 8, 1-14. AGMON, S.; DOUGLIS, A.; NIRENBERG, L. [1959]: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12, 623-727. AGMON, S.; DOUGLIS, A.; NIRENBERG, L. [1964]: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. Pure Appl. Math. 17, 35-92. ALEXANDRESCU-IOSIFESCU, O. [1995]: Comportement de la solution des modules non lin~aires bidimensionnels de coque "faiblement courb~e" de W. T. Koiter et de Marguerre-von Kgrmgn lorsque "la coque devient une plaque", C. R. A cad. Sci. Paris, Sdr. I, 321, 1389-1394. ALLAIRE, (3. [1992]: Homogenization and two-scale convergence, SIAM Math. Anal. 23, 1482-1518. ALVAREZ-DIOS, J. A.; VIA~O, J. M [1995]: Une th6orie asymptotique de flexionextension pour les poutres ~lastiques faiblement courb~es, C. R. A cad. Sci. Paris, Sdr. I, 321, 1395-1400. ALVAREZ-VJ~ZQUEZ, L. J.; QUINTELA-EST~VEZ, P. [1992]: The effect of different scalings in the modelling of nonlinearly elastic plates with rapidly varying thickness, Comput. Methods Appl. Mech. Engrg. 96, 1-24. ALVAREZ-VJ~ZQUEZ, L. J.; VIA~O, J. M. [1994]" Derivation of an evolution model for nonlinearly elastic beams by asymptotic expansion methods, Cornput. Methods Appl. Mech. Engrg. 115, 53-66.
451
452
References
ALVAREZ-VJtZQU'EZ, L. J.; VIAI~O, J. M. [1997]" Modelling and optimization of non-symmetric plates, Math. Modelling Numer. Anal., to appear. AMROUCHE, C.; GIRAULT, V. [1994]: Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension, Czech. Math. J. 44, 109-140. ANTMAN, S. S. [1972]: The theory of rods, in Handbuch der Physik, Vol. Vial2 (C. TRUESDELL, Editor), pp. 641-703, Springer-Verlag, Berlin. ANTMAN, S. S. [1976]: Ordinary differential equations of one-dimensional nonlinear elasticity. I. Foundations of the theories of nonlinearly elastic rods and shells, Arch. Rational Mech. Anal. 61,307-351. ANTMAN,S. S. [1978]: Buckled states of nonlinearly elastic plates, Arch. Rational Mech. Anal. 47, 111-149. ANTMAN, S. S. [1983]: The influence of elasticity on analysis: Modern developments, Bull. Amer. Math. Soc. 9, 267-291. ANTMAN, S. S. [1995]: Nonlinear Problems of Elasticity, Springer-Verlag, Berlin. ANTMAN, S. S.; LANZA DE CRISTOFORIS, M. [1996]: Peculiar instabilities due to the clamping of shearable rods, Intern. J. Non-Linear Mech. 31, to appear. ANTMAN, S. S.; MARLOW, R. S. [1991]: Material constraints, Lagrange multipliers, and incompatibility, Arch. Rational Mech. Anal. 116, 257-299. ANZELLOTTI, G.; BALDO, S.; PERCIVALE, D. [1994]: Dimension reduction in variational problems, asymptotic development in F-convergence and thin structures in elasticity, Asymptotic Anal. 9, 61-100. ARGATOV, I. I.; NAZAROV, S. A. [1993]: Junction problem of shashlik (skewer) type, C. R. Acad. Sci. Paris, Sgr. I, 316, 1329-1334. ARNOLD, D. N.; BREZZI, F. [1993]: Some new elements for the Reissner-Mindlin plate model, in Boundary Value Problems for Partial Differential Equations and Applications (J. L. LIONS and C. BAIOCCHI, Editors), pp. 287-292, Masson, Paris. ARNOLD, D. N.; FALK, R.S. [1989]: A uniformly accurate finite element method for the Mindlin-Reissner plate, SIAM J. Numer. Anal. 26, 1276-1290. ARNOLD, D. N.; FALK, R.S. [1990]: The boundary layer for the Reissner-Mindlin plate model, SIAM J. Math. Anal. 21,281-312. ARNOLD, D. N.; FALK, R.S. [1996]: Asymptotic analysis of the boundary layer for the Reissner-Mindlin plate model, S I A M J. Math. Anal. 27, 486-514. ATTOU, D.; MAUGIN, G. A. [1987]: Une th~orie asymptotique des plaques minces pi~zoeSlectriques, C. R. Acad. Sci. Paris, Sgr. II, 304, 865-868. AUFRANC, M. [1989]: Numerical study of a junction between a three-dimensional elastic structure and a plate, Comput. Methods A ppl. Mech. Engrg. 74, 207222. AUFRANC, M. [1990a]: Sur quelques Probl~mes de Jonctions dens les Multi-Structures Elastiques, Doctoral Dissertation, Universite! Pierre et Marie Curie, Paris. AUFRANC, M. [1990b]: Plaques raidies par des poutres, C. R. Acad. Sci., Sdr. I, 311,835-838. AUFRANC, M. [1991]: Junctions between three-dimensional and two-dimensional nonlinearly elastic structures, Asymptotic Anal. 4, 319-338.
References
453
BABUSKA, I. [1961]: Stability of the domain under perturbation of the boundary in fundamental problem in the theory of partial differential equations principally in connection with the theory of the elasticity, Czech. Math. 11, 75-105 and ibid. 165-203. BABUSKA, I. [1973]: The finite element method with Lagrangian multipliers, Numer. Math. 20, 179-192. BABUSKA, I.; LI, L. [1991]: Hierarchic modelling of plates, Computers and Structures 40,419-430. BABUSKA, I.; D'HARCOURT, J. M; SCHWAB, C. [1993]: Optimal shear correction factors in hierarchical plate modelling, Math. Modelling Scientific Computing 1, 1-30. BABUSKA, I.; LI, L. [1992]" The problem of plate modelling- Theoretical and computational results, Comput. Methods Appl. Mech: Engrg. 100, 249-273. BABUSKA, I.; PITKARANTA, J. [1990]: The plate paradox for hard and soft simple support, S I A M J. Math. Anal. 21,551-576. BABUSKA, I.; PRAGEa, M. [1960]; Reissnerian algorithms in the theory of elasticity, Bull. Acad. Polon. Sci. Sdr. Sci. Math. Astr. Phys. 8, 411-417. BABUSKA, I.; SCHWAB, C. [1996]" A posteriori error estimation for hierarchic models of elliptic boundary value problems on thin domains, S I A M Y. Numer. Anal. 33, 221-246. BABU~KA, I.; SZABO, B. A.; ACTIS, R. [1992]: Hierarchic models for laminated composites, Internat. J. Numer. Methods Engrg. 33, 503-536. BALL, J. M. 1977]: Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63,337-403. BANKS, H. T.; SMITH, R. C.; WANG, Y. [1996]: Smart Material StructuresModeling, Estimation and Control, Masson, Paris and John Wiley, New York. BATHE, K. J.; BREZZI, F. [1985]: On the convergence of a four-node plate bending element based on Mindlin-Reissner plate theory and a mixed interpolation, in The Mathematics of Finite Elements and Applications V (J. R. WHITEMAN, Editor), pp. 491-503, Academic Press, London. BATRA, R. C. [1972]: On non-classical boundary conditions, Arch. Rational Mech. Anal. 48, 163-191. BAUER, L.; REINS, E. L. [1965]: Nonlinear buckling of rectangular plates, S I A M Y. Appl. Math. 13, 603-626. BECEHa, H.; GILBERT, R. P.; Lo, C. Y. [1991]: The two-dimensional nonlinear orthotropic plate, J. Elasticity 26, 147-167. BEN BELGACEM, H. [1997]: Une m~thode de F-convergence pour un module de membrane non lin~aire, C. R. Acad. Sci. Paris, Sdr. I, to appear. BEN DHIA, H. [1995]" Analyse math~matique de modules de plaques non lin~aires de type Mindlin-Naghdi-Reissner. Existence de solutions et convergence sous des hypotheses optimales, C. R. Acad. Sci. Paris, Sdr. I, 320, 1545-1552. BERCER, M. S. [1967]: On the von Ks163 equations and the buckling of a thin elastic plate. I. The clamped plate, Comm. Pure Appl. Math. 20, 687-719. BERGER, M. S. [1977]: Nonlinearity and Functional Analysis, Academic Press, New York.
454
References
BERGER, M. S.; FIFE, P. C. [1968]: On von KS~rmS~n'sequations and the buckling of a thin elastic plate. II. Plate with general edge conditions, Comm. Pure Appl. Math. 21,227-241. BERMUDEZ, n.; VIAlqO, J. M. [1984]: Une justification des 6quations de la thermo61asticit6 des poutres ~ section variable par des m(~thodes asymptotiques, RAIRO Analyse Numdrique 18,347-376. BERNADOU, M. [1989]: Variational formulation and approximation of junctions between thin shells, in Proceedings, Fifth International Symposium on Numerical Methods in Engineering, Vol. I (R. GRUBER, J. P~:RIAUX and R. P. SHAW, Editors), pp. 407-414, Computational Mechanics Publications, Springer-Verlag, Berlin. BERNADOU, M. [1994]: Mdthodes d'Eldments Finis pour les Coques Minces, Masson, Paris (English translation: Finite Element Methods for Thin Shell Problems, John Wiley, New York, 1996). BERNADOU, M.; FAYOLLE, S.; L~N~, F. [1989]" Numerical analysis of junctions between plates, Comput. Methods Appl. Mech. Engrg. 74, 307-326. BHARATHA, S.; LEVINSON, M. [1980]: A theory of elastic foundations, Arch. Rational Mech. Anal. 74, 249-266. BIELSKI, W. R.; TELEGA, J. J. [1988]: On existence of solutions for geometrically nonlinear shells and plates, Z. Angew. Math. Mech. 68, 155-157. BIELSKI, W. R.; TELEaA, J. J. [1996]: A non-linear elastic plate model of moderate thickness: Existence, uniqueness and duality, J. Elasticity 42, 243-273. BISOGNIN, E.; BISOGNIN, V.; PERLA MENZALA, G.; ZUAZUA, E. [1997]: On the exponential stability for yon K~rm~n equations in the presence of thermal effects, to appear. BJORSTAD, P. E.; WIDLUND, O. B. [1986]: Iterative methods for the solutions of elliptic problems on regions partitioned into substructures, SIAM J. Numer. Anal. 23, 1097-1120. BLANCHARD, D. [1981]: Justification de Moddes de Plaques Correspondant d Diffdrentes Conditions aux Limites, Doctoral Dissertation, Universit(~ Pierre et Marie Curie, Paris. BLANCHARD, D.; CIARLET, P. G. [1983]: A remark on the von K~irms equations, Comput. Methods Appl. Mech. Engrg. 37, 79-92. BLANCHARD, D.; FRANCFORT, G. A. [1987]: Asymptotic thermoelastic behavior of flat plates, Quart. A ppl. Math. 45, 645-667. BLANCHARD, D.; LEBELTEL, C. [1991]" l~tude de la convergence de plaques thermo6lastiques, Report No. 1~66, I.N.R.I.A., Rocquencourt. BLANCHARD, D.; PAUMIER, J. C. [1984]: Une justification de modules de plaques viscoplastiques, RAIRO Analyse Numdrique 18, 377-406. BLANCHARD, D.; XIANC, Yah [1992]: Clamped plates as limits of genuinely clamped plates in linear and nonlinear elasticity, Asymptotic Anal. 5,495-516. BLUM, H.; RANNACHER, R. [1980]: On the boundary value problem of the biharmonic operator on domains with angular corners, Math. Methods Appl. Sci. 2, 556-581.
References
455
BOHM, M. [1996]: Global well posedness of the dynamic von Kgrm~in equations for generalized solutions, Nonlinear Anal., Theory Methods Appl. 27, 339-351. BOLLEY, P.; CAMUS, J. [1976]" R~gularit~ pour une classe de probl~mes aux limites elliptiques ddg4n6r4s variationnels, C. R. Acad. Sci. Paris, Sdr. A, 282, 45-47. BoRcnERS, W.; SOHR, H. [1990]: On the equations rot v = g and div u = f with zero boundary conditions, Hokkaido Math. J. 19, 67-87. BOURQUIN, F. [1990]: Analysis and comparison of several component mode synthesis methods on one-dimensional domains, Numer. Math. 58, 11-34. BOURqUIN, F. [1992]: Component mode synthesis and eigenvalues of second order operators: Discretization and algorithm, Math. ModeUing Numer. Anal. 26, 385-423. BOURqUIN, F.; CIARLET, P. G. [1989]: Modeling and justification of eigenvalue problems for junctions between elastic structures, J. Funct. Anal. 87, 392-427. BOURQUIN, F.; CIARt,ET, P. G.; GEYMONAT, G.; RAOULT, A. [1992]: P-convergence et analyse asymptotique des plaques minces, C. R. Acad. Sci. Paris, Sdr I, 315, 1017-1024. BOURQUIN, F.; D'HENNEZEL, F. [1992]: Numerical study of an intrinsic component mode synthesis method, Comput. Methods Appl. Methods JEngr9. 97, 49-76. BOUTET DE MONVEL, A.; CnUESHOV, I. [1997a]: The problem of interaction of von Kgrmgn plate with subsonic flow of gas, Math. Methods Appl. Sci., to appear. BOUTET DE MONVEL, A.; CHUESHOV, I. [1997b]: Uniqueness theorem for weak solutions of von Kgrm&n evolution equations, to appear. BOUTET DE MONVEL, L.; CHUESHOV, I. [1996]: Nonlinear oscillations of a plate in a flow of gas, C. R. Acad. Sci. Paris, Sdr. I, 322, 1001-1006. BRAESS, D. [1996]: Stability of saddle point problems with penalty, Math. Modelling Numer. Anal. 30, 731-742. BRAMBLE, J. H.; PASCIAK, J. E.; SCHATZ, A. H. [1986]: An iterative method for elliptic problems on regions partitioned into substructures, Math. Comp. 46, 361-369. BRENNER, S. C.; SCOTT, L. R. [1994]: The Mathematical Theory of Finite Element Methods, Springer-Verlag, Berlin. BREWSTER, M. E. [1986]: Asymptotic Analysis of Thin Plates under Normal Load and Horizontal Edge Thrust, Doctoral Dissertation, California Institute of Technology, Pasadena. BREZIS, H. [1973]: Opdrateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espace de Hilbert, North-Holland, Amsterdam. BREZIS, H. [1983]: Analyse Fonctionnelle, Thdorie et Applications, Masson, Paris. BREZZI, F. [1974]: On the existence, uniqueness and approximation of saddlepoint problems arising from Lagrangian multipliers, Rev. Fran~aise Automat. Informat. Recherche Opdrationnelle, Sdr. Rouge Anal. Numdr. R-2, 129-151. BREZZI, F.; FORTIN, M. [1986]: Numerical approximation of Mindlin-Reissner plates, Math. Comp. 47, 151-158.
456
References
BREZZI, F.; FORTIN, M. [1991]: Mixed and Hybrid Finite Element Methods, Springer-Verlag, Berlin. BREZZI, F.; FORTIN, M.; STENBERG, R. [1991]: Error analysis of mixed-interpolated elements for Reissner-Mindlin plates, Math. Models Methods A ppl. Sci. 1, 125-151. BURQ, N.[1994]" Un th~or~me de contrSle pour une structure multidimensionnelle, Comm. Partial Differential Equations 19, 199-211. BUSSE, S. [1997]: Sur Quelques Questions Lides ~ la Thdorie Mathdmatique des Coques en Coordonn&s Curvilignes, Doctoral Dissertation, Universit~ Pierre et Marie Curie, Paris. BUSSE, S.; CIARLET, P. G.; MIARA, B. [1996]: Coques "faiblement courb~es" en coordonn~es curvilignes, C. R. Acad. Sci. Paris, Sdr. I, 322, 1093-1098. BUSSE, S.; CIARLET, P. (].; MIARA, B. [1997]: Justification d'un module lin~aire bi-dimensionnel de coques "faiblement courb~es" en coordonn~es curvilignes, Math. Modelling Numer. Anal. 30, to appear. C AILLERIE, D. [1980]: The effect of a thin inclusion of high rigidity in an elastic body, Math. Methods Appl. Sci. 2,251-270. CAILLERIE, D. [1984]: Thin elastic and periodic plates, Math. Meth. Appl. Sci. 6, 159-191. C AILLERIE, D. [1985]: Homogenization in Elasticity, in Lectures on Homogenization Techniques for Composite Media, CISM, Udine. CAILLERIE, D. [1987]: Non homogeneous plate theory and conduction in fibered composites, in Homogenization Techniques for Composite Media (E. SANCHEZPALENCIA and A. ZAOUI, Editors), pp. 1-62, Springer-Verlag, Berlin. CAILLERIE, D.; SANCHEZ-PALENCIA, E. [1995]: Elastic thin shells: Asymptotic theory in the anisotropic and heteregeneous cases, Math. Models Methods Appl. Sci. 5,473-496. CAMPBELL, A.; NAZAROV, S. [1997]: Une justification de la m~thode de raccordemeAts des d~veloppements asymptotiques appliqu~e g u n probl~me de plaque en flexion. Estimation de la matrice d'imp~dance. J. Math. Pures Appl. 76, 15-54. C~;A, J. [1971]" Optimisation, Thdorie et Algorithmes, Dunod, Paris. CHARPENTIER, I.; SAINT JEAN PAULIN, J. [1996]: Domain decomposition methods for the study of the particular behaviour of a very thin long structure, Numer. Methods Partial Differential Equations, to appear. CHENAIS, D.; PAUMIER, J. C. [1994]: On the locking phenomenon for a class of elliptic problems, Numer. Math. 67, 427-440. CHO, J. R.; ODEN, J. T. [1995]: A priori modelling error estimates of hierarchical models for elasticity problems for plate and shell-like structures, TICAM Report 95-01, The University of Texas at Austin. CHOW, S.; HALE, J. [1996]: Methods of Bifurcation Theory, Springer-Verlag, Heidelberg. CHOW, S.; HALE, J.; MALLET-PARET, J. [1975]: Application of generic bifurcation theory. I, Arch. Rational Mech. Anal. 59, 159-188.
References
457
CIARLET, P. G. [1978]: The Finite Element Method for Elliptic Problems, NorthHolland, Amsterdam. CIARLET, P. G. [1980]: A justification of the von Ks163 equations, Arch. Rational Mech. Anal. 73, 349-389. CIARLET, P. G. [1982]: Introduction ~ l'Analyse Numdrique Matricielle et l'Optimisation, Masson, Paris (English Translation: Introduction to Numerical Linear Algebra and Optimisation, Cambridge University Press, Cambridge, 1989). CIARLET, P. G. [1990]: Plates and Junctions in Elastic Multi-Structures: An Asymptotic Analysis, Masson, Paris. CIARLET, P. G. [1991]: Basic error estimates for elliptic problems, in Handbook of Numerical Analysis, Vol. II (P. G. CIARLET and J. L. LIONS, Editors), pp. 17-351, North-Holland, Amsterdam. CIARLET, P. G. [1992]: Echanges de limites en th~orie asymptotique de coques. I. En premier lieu, "la coque devient une plaque", C. R. Acad. Sci. Paris, Sdr. /, 315, 107-111 CIARLET, P. (].; DESTUYNDER, P. [1979a]: A justification of the two-dimensional plate model, J. Mdcanique 18, 315-344. CIARLET, P. G.; DESTUYNDER, P. [1979b]: A justification of a nonlinear model in plate theory, Comput. Methods Appl. Mech. Engrg. 17/18, 227-258. CIARLET, P. G.; KESAVAN, S. [1981]: Two-dimensional approximation of threedimensional eigenvalue problems in plate theory, Comput. Methods Appl. Mech. Engrg. 26, 149-172. CIARLET, P. G.; LE DRET, H. [1989]: Justification of the boundary conditions of a clamped plate by an asymptotic analysis, Asymptotic Anal. 2, 257-277. CIARLET, P. G.; LE DREW, H.; NZENGWA, R. [1989]: Junctions between threedimensional and two-dimensional linearly elastic structures, g. Math. Pures Appl. 68, 261-295. CIARLET, P. G.; LODS, V. [1996a]: Asymptotic analysis of linearly elastic shells. I. Justification of membrane shell equations, Arch. Rational Mech. Anal. 136, 119-161. CIARLET, P. G.; LODS, V. [1996b]: Asymptotic analysis of linearly elastic shells. III. Justification of Koiter's shell equations, Arch. Rational Mech. Anal. 136, 191-200. CIARLET, P. G.; LODS, V. [1996c]: Asymptotic analysis of linearly elastic shells: "Generalized membrane shells", J. Elasticity 43, 147-188. CIARLET, P. G.; LODS, V.; MIARA, B. [1996]: Asymptotic analysis of linearly elastic shells. II. Justification of flexural shell equations, Arch. Rational Mech. Anal. 136, 163-190. CIARLET, P. G.; MIARA, B. [1990]: Justification of the two-dimensional equations of a linearly elastic shallow shell, Comm. Pure Appl. Math. 45, 327-360. CIARLET, P. G.; MIARA, B. [1997]: Overlooked models in the asymptotic analysis of plates, to appear. CIARLET, P. G.; PAUMIER, J. C. [1986]: A justification of the Marguerre-von Ks163 equations, Computational Mech. 1, 177o202.
458
References
CIARLET, P. G.; RABIER, P. [1980]: Les Equations de yon Kdrmdn, Lecture Notes in Mathematics, Vol. 826, Springer-Verlag, Berlin. CIBULA, J. [1984]" Equations de von K~rms I. R~sultat d'existence pour les probl~mes aux limites non homog~nes, Aplikace Matematiky 29, 317-332. CIMETII~RE, A:.; GEYMONAT, G.; LE DREW, H.; RAOULT, A.; TUTEK, Z. [1988]" Asymptotic theory and analysis for displacements and stress distribution in nonlinear elastic straight slender rods, J. Elasticity 19, 111-161. CIORANESCU, D.: OLEINIK, O. A.: TRONEL, G. [1994]: Korn's inequalities for frame type structures and junctions with sharp estimates for the constants, Asymptotic Anal. 8, 1-14. CIORANESCU, D.; SAINT JEAN PAULIN, J. [1986]: Reinforced and honey-comb structures, J. Math. Pures A ppl. 65, 403-422. CIORANESCU, D.; SAINT JEAN PAULIN, J. [1987]: Tall structures- Problem of towers and cranes, in Applications of Multiple Scalings in Mechanics (P. G. CIARLET and E. SANCHEZ-PALENCIA,Editors), pp. 77-92, Masson, Paris. CIORANESCU, D.; SAINT JEAN PAULIN, J. [1988]: Elastic behavior of very thin cellular structures, in Material Instabilities in Continuum Mechanics (J. M. BALL, Editor), pp. 64-75, Clarendon Press, Oxford. CIORANESCU, D.; SAINT JEAN PAULIN, J. [1995]: Conductivity problems for thin tall structures depending on several small parameters, Advances Math. Sci. Appl. 5, 287-320. COHEN, G.; MIARA, B. [1990]: Optimization with an auxiliary constraint and decomposition, S I A M J. Control Optimization 28, 137-157. CONCA, C; ZUAZUA, E. [1994]: Asymptotic analysis of a multidimensional vibrating structure, SIAM J. Math. Anal. 25,836-858. CONCA, C.; ZUAZUA, E. [1995]: Asymptotic analysis of a multidimensional platebeam model, in Asymptotic Methods for Elastic Structures (P. G. CIARLET, L. TRABUCHO and J. M. VItrO, Editors), pp. 275-291, de Gruyter, Berlin. COURANT, R.; HILBERT. D. [1953]: Methods of Mathematical Physics, Vol. I, Interscience, New York. COUTAND, D. [1997a]" Existence d'un minimiseur pour le module "proprement invariant" de plaque "en flexion" non lin~airement ~lastique, C. R. Acad. Sci. Paris, Sdr. I, 324, 245-248. COUTAND, D. [1997b]: Th~or~mes d'existence pour un module membranaire "proprement invariant" de plaque non lin~airement ~lastique, C. R. Acad. Sci. Paris, Sdr. I, to appear. CRANDALL, M. G.; RABINOWITZ, P. H. [1970]: Nonlinear Sturm-Liouville eigenvalue problems and topological degree, J. Math. Mech. 19, 1083-1102. CROUZEIX, M.; RAPPAZ~J. [1989]: On Numerical Approximation in Bifurcation Theory, Masson, Paris and Springer-Verlag, Berlin. DACOROGNA, B. [1989]: Direct Methods in the Calculus of Variations, SpringerVerlag, Heidelberg. DAL MASO, G. [1993]: An Introduction to F-Convergence, Birkhs Boston. DAUGE, M. [1988]: Elliptic Boundary Value Problems on Corner Domains, Lecture Notes in Mathematics, Vol. 1341, Springer-Verlag, Berlin.
References
459
DAUGE, M.; GRUAIS, I. [1996]: Asymptotics of arbitrary order in thin elastic plates and optimal estimates for the Kirchhoff-Love model, Asymptotic Anal. 13, 167-197. DAUGE, M.; GRUAIS, I. [1997]: Asymptotics of arbitrary order for a thin elastic clamped plate. II. Analysis of the boundary layer terms, Asymptotic Anal., to appear. DAUTRAY, R.; LIONS, J. L. [1984]: Analyse Mathdmatique et Calcul Numdrique pour les Sciences et les Techniques, Vol. 1, Masson, Paris (English translation: Mathematical Analysis and Numerical Methods for Science and Technology, Vols. 1 and 2, Springer-Verlag, Berlin). DAUTRAY, R.; LIONS, J. L. [1985]: Analyse Mathdmatique et Calcul Numdrique pour les Sciences et les Techniques, Vol. 2, Masson, Paris (English translation: Mathematical Analysis and Numerical Methods for Science and Technology, Vols. 3 and d, Springer-Verlag, Berlin). DAVET, J. L. [1986a]: Justification de modules de plaques non lin~aires pour des lois de comportement g~n~rales, Moddlisation Math. Anal. Numdr. 20, 225-249. DAVET, J. L. [1986b]: Correction du second ordre pour le calcul des fr~quences propres d'une plaque en flexion, C. R. Acad. Sc. Paris, Sdr. II, 303, 521-524. DAVET, J. L.; DESTUYNDER, P. [1985]: Singularit~s logarithmiques dans les effets de bord d'une plaque en mat6riaux composites, J. M~canique Thdor. Appl. 4, 357-373. DAVET, J. L.; DESTUYNDER, P. [1986]: Free-edge stress concentration in composite laminates: A boundary layer approach, Comput. Methods Appl. Mech. Engrg. 59, 129-140. DE GIORGI, E. [1975]: Sulla convergenza di alcune successioni di integrali del tipo dell' area, Rend. Mat. Roma 8, 227-294. DE GIORGI, E. [1977]: F-convergenza e G-convergenza, Boll. Un. Mat. Ital. 5, 213-220. DE (]IORGI, E.; DAL MASO, G. [1983]: F-Convergence and Calculus of Variations, Lecture Notes in Mathematics, Vol. 979, Springer-Verlag, Berlin. DE GIORGI, E.; FRANZONI, T. [1975]: Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. C1. Sci. Mat. 58,842-850. DENY, J.; LIONS, J. L. [1954]: Les espaces du type Beppo-Levi, Ann. Institut Fourier (Grenoble) 5, 305-370. DESTUYNDER, P. [1980]: Sur une Justification des ModUles de Plaques et de Coques par les Mdthodes Asymptotiques, Doctoral Dissertation, Universit~ Pierre et Marie Curie, Paris. DESTUYNDER, P. [1981]: Comparaison entre les modules tridimensionnels et bidimensionnels de plaques en 61asticit6, R A I R O Analyse Numdrique 15,331-369. DESTUYNDER, P. [1982]: Sur les modules de plaques minces en elasto-plasticit~, J. Mdcanique Thdor. Appl. 1, 73-80. DESTUYNDER, P. [1985]: A classification of thin shell theories, Acta Applicanda~ Mathematicae 4, 15-63.
460
References
DESTUYNDER, P. [1986]: Une Tttdorie Asymptotique des Plaques Minces en Elasticitd Lindaire, Masson, Paris. DESTUYNDER, P. [1989]: Remarks on dynamic substructuring, European J. Mech. A/Solids 8, 201-218. DESTUYNDER, P. [1997]: Explicit error bounds for the Kirchhoff-Love and ReissnerMindlin plate theories, to appear. DESTUYNDER, P.; GRUAIS, I. [1995]: Error estimation for the linear three-dimensional elastic plate model, in Asymptotic Methods for Elastic Structures (P. G. CIARLET, L. TRABUCHO and J. M. VItrO, Editors), pp. 75-88, de Gruyter, Berlin. DESTUYNDER, P.; LEGRAIN, I.; CASTEL, L.; RICHARD, N. [1992]: Theoretical, numerical, and experimental use of piezoelectric devices for control-structure interaction, European J. Mech. A/Solids 11, 181-213. DESTUYNDER, P.; NEVERS, T. [1988]: Une modification du module de Mindlin pour les plaques minces en flexion pr~sentant un bord libre, Math. Modelling Numer. Anal. 22, 217-242. DESTUYNDER, P.; SALAUN, M. [1996a]: Mathematical Analysis of Thin Plate Models, Math~matiques et Applications, Vol. 24, Springer-Verlag, Paris. DIKMEN, M. [1982]: Theory of Thin Elastic Shells, Pitman, Boston. Do, C. [1977]: Bifurcation theory for elastic plates subjected to unilateral conditions, J. Math. Anal. Appl. 60, 435-448. DUVAUT, G. [1978]" Homog6n6isation des plaques k structure p(~riodique en th(~orie non-lin6aire de von Kdrmdn, in Journdes d'Analyse Non Lindaire, pp. 56-70, Lecture Notes in Mathematics, Vol. 665, Springer-Verlag, Berlin. DUVAUT, G. [1990]; Mdcanique des Milieux Continus, Masson, Paris. DUVAUT, G.; LIONS, J. L. [1972]: Les Indquations en Mdcanique et en Physique, Dunod, Paris (English translation: Springer-Verlag, Berlin, 1976). DUVAUT, G.; LIONS, J. L. [1974a]: Probl~mes unilat6raux dans la th6orie de la flexion forte des plaques, g. M~canique 13, 51-74. DUVAUT, G.; LIONS, J. L. [1974b]: Probl~mes unilat(~raux dans la th(~orie de la flexion forte des plaques. II. Le cas d'(~volution, J. Mdcanique 13, 245-266. ECKHAUS, W. [1972]: Boundary layers in linear elliptic singular perturbation problems, SIAM Rev. 14, 225-277. ECKHAUS, W. [1979]: Asymptotic Analysis of Singular Perturbations, NorthHolland, Amsterdam. ER:NGEN, A. C. [1962]: Nonlinear Theory of Continuous Media, McGraw-Hill, New York. FABRE, C. [1992]: R~sultats de contrSlabilit~ exacte interne pour l'(~quation de SchrSdinger et leurs limites asymptotiques - Application k certaines (~quations de plaques vibrantes, Asymptotic Anal. 5, 343-379. FENG, Kang [1979]: Elliptic equations on composite manifolds and composite elastic structures, Mathematicae Numericae Sinica 1, 199-208 (in Chinese). FENG, Kang; SHI, Zhong-ci [1981]: Mathematical Theory of Elastic Structures, Science Press, Beijing (in Chinese).
References
461
FICHERA, G. [1972a]: Existence theorems in elasticity, in Handbuch der Physik, Vol. Vial2 (S. FL~JGCE and C. TRUESDELL, Editors), pp. 347-389, SpringerVerlag, Berlin. FICHERA, (]. [1972b]: Boundary value problems of elasticity with unilateral constraints, in Handbuch der Physik, Vol. Vial2 (S. FLOGGE and C. TRUESDELL, Editors), pp. 391-424, Springer-Verlag, Berlin. FIFE, P. [1961]: Nonlinear deflection of thin elastic plates under tension, Comm. Pure Appl. Math. 14, 81-112. DE FIGUEIaEDO, I. N. [1989]: ModUles de Coques Elastiques Non Lindaires; Mdthode A symptotique et Existence de Solutions, Doctoral Dissertation, Universit~ Pierre et Marie Curie, Paris. DE FIGUEIREDO, I. N.; TRABUCHO, L. [1992]: A Galerkin approximation for linear elastic shallow shells, Computational Mech. 10, 107-119. DE FIGUEIREDO, I. N.; TRABUCHO, L. [1993]: A Galerkin approximation for curved beams, Comput. Methods Appl. Mech. Engrg. 102, 235-253. DE FIGUEIREDO, I. N.; ZUAZUA, E. [1996]: Exact controllability and asymptotic limit for thin plates, Asymptotic Anal. 12,213-252. FOSDICK, R. L.; SERRIN, J. [1979]: On the impossibility of linear Cauchy and Piola-Kirchhoff constitutive theories for stress in solids, J. Elasticity 9, 83-89. Fox, D. D.; RAOULT, A.; SIMO, J. C. [1993]: A justification of nonlinear properly invariant plates theories, Arch. Rational Mech. Anal. 124, 157-199. FRAEIJS DE VEUBEKE, B. M. [1979]; A Course in Elasticity, Springer-Verlag, Berlin. FRIEDRICHS, K. O.; DRESSLER, a . F. [1961]: A boundary-layer theory for elastic plates, Comm. Pure Appl. Math. 14, 1-33. FRIEDRICHS, K. O.; STOKER, J. J. [1942]: Buckling of the circular plate beyond the critical thrust, J. Appl. Mech. 9, A7-A14. GANCHOFFER, J. F.; SCHULTZ, J. [1996]: Geometrically nonlinear modelling of contact problems involving thin elastic layers, J. Mech. Phys. Solids 44, 11031127. GE, Z.; KRUSE, H. P.; MARSDEN, J. E. [1996]: The limits of Hamiltonian structures in three-dimensional elasticity, shells, and rods, J. Nonlinear Sci. 6, 19-57. GENEVEY, K. [1996]: Sur Quelques Questions Lides ~t la Thdorie Mathdmatique des Coques Minces, Doctoral Dissertation, Universit6 Pierre et Marie Curie, Paris. GEaMAIN, P. [1972]" Mdcanique des Milieux Continus, Tome 1, Masson, Paris. GERMAJN, P. [1986a]" Mdcanique, Tome I, Ecole Polytechnique, Ellipses, Paris. (]ERMAIN, P. [1986b]: Mdcanique, Tome II, Ecole Polytechnique, Ellipses, Paris. GEYMONAT, G.; KRASUCKI, F. [1996]: A limit model of a soft thin joint, in Partial Differential Equations and Applications (P. MARCELLINI, G. T. TALENTI and E. VESENTINI,Editors), pp. 165-173, Marcel Dekker, New York. GEYMONAT, G.; KRASUCKI, F.; LENCI, S. [1996]: Analyse asymptotique du comportement d'un assemblage co116, C. R. Acad. Sci. Paris, Sdr. /, 322, 1107-1112.
462
References
GEYMONAT, G.; KRASUCKI, F.; MARIGO, J. J. [1987a]: Stress distribution in anisotropic elastic composite beams, in Applications of Multiple Scalings in Mechanics (P. G. CIARLET and E. SANCHEZ-PALENCIA, Editors), pp. 118-133, Masson, Paris. GEYMONAT. G.: KRASUCKI, F.: MARIGO, J. J. [1987b]: Sur la commutativit~ des passages/~ la limite en th~orie asymptotique des poutres composites, C. R. Acad. Sci. Paris, Sdr. II, 305,225-228. GEYMONAT, (].; SUQUET, P. [1986]: Functional spaces for Norton-Hoff materials, Math. Methods Appl. Sci. 8, 206-222. GILBERT, R. P.; HSIAO, G. C.; SCHNEIDER, M. [1983]: The two-dimensional linear orthotropic plate, Applicable Anal. 15, 147-169. GIRAULT, V.; RAVIART, P.-A. [1986]: Finite Element Methods for Navier-Stokes Equations, Springer-Verlag, Berlin. GLOWINSKI, R.; LIONS, J. L. [1994]: Exact and approximate controllability for distributed parameter systems, Part I, in Acta Numerica 199~ (A. ISERLES, Editor), pp. 269-378, Cambridge University Press. GLOWINSKI, R.; LIONS, J. L. [1995]: Exact and approximate controllability for distributed parameter systems, Part II, in Acta Numerica 1995 (A. ISERLES, Editor), pp. 159-333, Cambridge University Press. GOELEVEN, D.; NGUYEN Van Hien; TH~RA, M. [1993a]" M(~thode du degr~ topologique et branche de bifurcation pour les in~quations variationnelles de von Ks163 C. R. Acad. Sci. Paris, Sdr. I, 317, 631-635. GOELEVEN, D.; NGUYEN Van Hien; THI~RA, M. [1993b]" Nonlinear eigenvalue problems governed by a variational inequality of von Ks163 type: A degree theoretic approach, Topological Methods in Nonlinear Analysis, 2, 253-276. GOLDENVEIZER, A. L. [1962]: Derivation of an approximate theory of bending of a plate by the method of asymptotic integration of the equations of the theory of elasticity, Prikl. Mat. Mech. 26, 668-686 (English translation: J. Appl. Math. Mech. [1964], 1000-1025). GOLDENVEIZER, A. L. [1963]: Derivation of an approximate theory of shells by means of asymptotic integration of the equations of the theory of elasticity, Prikl. Mat. Mech. 27, 593-608. GOLDENVEIZER, A. L. [1964]: The principles of reducing three-dimensional problems of elasticity to two-dimensional problems of the theory of plates and shells, in Proceedings, Eleventh International Congress of Theoretical and Applied Mechanics (H. GORTLER, Editor), pp. 306-311, Springer-Verlag, Berlin. GOLDENVEIZER, A. L. [1995]: Internal and boundary calculations of thin elastic bodies, J. Appl. Math. Mech. 59, 973-985. GOLDENVEIZER, A. L.; KAPLUNOV, J. D.; NOLDE, E. V. [1993]: On TimoshenkoReissner type theories of plates and shells, Internat. J. Solids Structures 30, 675-694. GOLUBITSKY, M.: SHAEFFER, D. [1978]: A theory for imperfect bifurcation via singularity theory, Comm. Pure Appl. Math. 32, 21-98. GOLUBITSKY, M.: SHAEFFER, D. [1979a]: Imperfect bifurcation in the presence of symmetry, Comm. Math. Phys. 67, 205-232.
References
463
GOLUBITSKY, M.; SHAEFFER, D. [1979b]: Boundary conditions and mode jumping in the buckling of a rectangular plate, Comm. Math. Phys. 69, 209-236.
GOLUBITSKY, M.; SHAEFFER, D. [1985]: Singularities and Groups in Bifurcation Theory, Vol. I, Springer-Verlag, Berlin.
GOLUBITSKY, M.; STEWART, I.; SHAEFFER, D. [1988]: Singularities and Croups in Bifurcation Theory, Vol. II, Springer-Verlag, Berlin. GOODIER, J. N. [1938]: On the problems of the beam and the plate in the theory of elasticity, Trans. Roy. Soc. Canada 32, 65-88. GORDEZIANI, D. G. [1974]: On the accuracy of one version of the theory of thin shells, Dokl. Akad. Nauk 216, 751-754 (in Russian). GREEN, A. E.; ADKINS, J. E. [1970]: Large Elastic Deformations, Second Edition, Clarendon Press, Oxford. GREEN, A. E.; ZERNA, W. [1968]; Theoretical Elasticity, Second Edition, University Press, Oxford. GREGORY, R. D.; WAN, F. Y. M. [1994]: Boundary conditions at the edge of a thin or thick plate bonded to an elastic support, J. Elasticity 36, 155-182. GRISVARD, P. [1992]: Singularities in Boundary Value Problems, Masson, Paris and Springer-Verlag, Berlin. GRUAIS, I. [1993a]: Mod~lisation de la jonction entre une plaque et une poutre en (~lasticit~ lin~aris(~e, Moddlisation Math. Anal. Numdr. 27, 77-105. ORUAIS, I. [1993b]: Modelling of the junction between a plate and a rod in nonlinear elasticity, Asymptotic Anal. 7, 179-194. (]URTIN, M. E. [1972]: The linear theory of elasticity, in Handbuch de Physik, Vol. Vial2 (S. FL/JGGE and C. TRUESDELL, Editors), pp. 1-295, Springer-Verlag, Berlin. GURTIN, M. E. [1981a]: Topics in Finite Elasticity, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia GURTIN, M. E. [1981b]: An Introduction to Continuum Mechanics, Academic Press, New York. HANYGA, A. [1985]: Mathematical Theory of Non-Linear Elasticity, PWN-Polish Scientific Publishers, Warsaw, and Ellis Horwood, Chichester. HELLINGER, E. [1914]: Die allgemeine Ansatze der Mechanik der Kontinua, in Encyclopiidie der Matematischen Wissenschaften, Vol IV~/ ~{, pp. 602-694, Leipzig. D'HENNEZEL, F. [1993]: Domain decomposition method and elastic multi-structures: The stiffened plate problem, Numer. Math. 66, 181-197. HERMANN, G. [1956]: Influence of large amplitudes on flexural motions of elastic plates, Technical Note 3578, NACA, Washington. HERMANN, G.; MIRSKY, I. [1956]: Three-dimensional and shell theory analysis of axially symmetric motions of cylinders, J. App. Mech. 20, 563-568. HLAVJ~(~EK, I.; NAUMANN, J. [1974]" Inhomogeneous boundary value problems for the von Ks163 equations. I, Aplikace Maternatiky 19, 253-269. HLAVJ~EK, I.; NAUMANN, J. [1975]" Inhomogeneous boundary value problems for the von Ks163 equations. II, Aplikace Matematiky 20, 280-297. HOMMAN, R. [1982]: Justification d'un module de plaque avec contact unilateral, C. R. Acad. Sci. Paris, Sdr. II, 294, 765-768.
464
References
HORGAN, C. O. [1995]: Korn's inequalities and their applications in continuum mechanics, SIAM Rev. 37, 491-511. HUGHES, T. J. R. [1987]: The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, Englewood Cliffs. JAFFARD, S.; TUCSNACK, M. [1997]: Regularity of plate equations with control concentrated in interior curves, Proc. Royal Soc. Edinburgh, to appear. JAMES, D.; PLEMMONS, R. J.[1990]: An iterative substructuring algorithm for equilibrium equations, Numer. Math. 57, 625-633. JOHN, F. [1965]: Estimates for the derivatives of the stresses in a thin shell and interior shell equations, Comm. Pure Appl. Math. 18, 235-267. JOHN, F. [1971]: Refined interior equations for thin elastic shells, Comm. Pure Appl. Math. 24, 583-615. JOHN, F. [1975]: A priori estimates, geometric effects and asymptotic behaviour, Bull. Amer. Math. Soc. 81, 1013-1023. JOHN, O.; KONDRATIEV, V. A.; LEKVEICHVILI, D. M.; NECAS, J.; OLEINIK, O. A. [1988]: On the solvability of the system of yon Kgrms equations for non-homogeneous boundary conditions and non-regular domains, G. Petrowsky Seminar 13, 197-205 (in Russian). JOHN, O.; NE~ZAS, J. [1975]: On the solvability of yon Kgrmgm equations, Aplikace Matematiky 20, 48-62. KAIL, R. [1994]: Moddlisation Asymptotique et Numdrique de Plaques et Coques Stratifides, Doctoral Dissertation, Universit~ Pierre et Marie Curie, Paris. YON K~RM~.N, T. [1910]" Festigkeitsprobleme im Maschinenbau, in Encyclop~idie der Mathematischen Wissenschaften, Vol. IV/4, pp. 311-385, Leipzig. YON Ks163 T.; TSlEN, H. S. [1939]" The buckling of spherical shells by external pressure, J. Aero. Sci. 7, 43-50. KARWOWSKI, A. J. [1990]: Asymptotic models for a long elastic cylinder, J. Elasticity 24, 229-287. KARWOWSKI, A. J. [1993]: Dynamical models for plates and membranes. An asymptotic approach, J. Elasticity 32, 93-153. KAVIAN, O.; RAO, Bo-peng [1993]: Une remarque sur l'existence de solutions non nulles pour leE (~quations de Marguerre-von KgLrms C. R. Acad. Sci. Paris, Sdr. I, 317, 1137-1142. KEENER, J. P. [1979]: Secondary bifurcation and multiple eigenvalues, SIAM J. Appl. Math. 37, 330-349. KEENER, J. P.; KELLER, H. B. [1973]: Perturbed bifurcation theory, Arch. Rational Mech. Anal. 50, 159-175. KELLER, H. B.; KELLER, J. B.; REISS, E: [1962]: Buckled states of circular plates, Quart. J. Appl. Math. 20, 55-65. KERDID, N. [1993]: Comportement asymptotique quand l'(~paisseur tend vers z~ro du probl~me de valeurs propres pour une poutre mince encastr~e en (~lasticit~ line~aire, C. R. Aead. Sci. Paris, Sdr. I, 316, 755-758. KERDID, N, [1995]: Mod(~lisation des vibrations d'une multi-structure fortune de deux poutres, C. R. Acad. Sci. Paris, Sdr. I, 321, 1641-1646.
References
465
KESAVAN, S.; SRIKANTH, P. N. [1983]: On the Dirichlet problem for the Marguerre equations, Nonlinear Anal., Theory Methods Applic. 7, 209-216. KIRCHHOFF, G. [1850]: /0bet das Gleichgewicht und die Bewegung einer elastischen Schribe, J. Reine Angew. Math. 40, 51-58. I~LARBRING, A. [1991]: Derivation of a model of adhesively bonded joints by the asymptotic expansion method, Internat. J. Engrg. Sci. 29, 493-512. KNIGHTLY, G. H. [1967]: An existence theorem for the von Kgrmgn equations, Arch. Rational Mech. Anal. 27, 233-242. t~NIGHTLY, G. H.; SATHER, D. [1970]: On non-uniqueness of solutions of the von Kg~rmgn equations, Arch. Rational Mech. Anal. 36, 65-78. KNOPS, R. J.; PAYNE, L. E. [1971]: Uniqueness Theorems in Linear Elasticity, Springer Tracts in Natural Philosophy, Vol. 19, Berlin. KOCH, H.; STAHEL, A. [1993]: Global existence of classical solutions to the dynamical von Kgrmgn equations, Math. Methods Appl. Sci. 16, 581-586. KOHN, R. V.; VOGELIUS, M. [1984]: A new model for thin plates with rapidly varying thickness. I, Internat. J. Engrg. Sci. 20,333-350. KOHN, R. V.; VOGELIUS, M. [1985]: A new model for thin plates with rapidly varying thickness. II: A convergence proof, Quart. Appl. Math. 43, 1-22. KOHN, R. V.; VOGELIUS, M. [1986]: A new model for thin plates with rapidly varying thickness. III: Comparison of different scalings, Quart. Appl. Math. 44, 35-48. KOITER, W. T. [1970a]: On the foundations of the linear theory of thin elastic shells. I, Proc. Kon. Ned. Akad. Wetensch. B73, 169-182. KOITER, W. T. [1970b]: On the foundations of the linear theory of thin elastic shells. II, Proc. Kon. Ned. Akad. Wetensch. B73, 182-195. KOMORNIK, V. [1988]: ContrSlabilit~ exacte en temps minimal de quelques modules de plaques, C. R. Acad. Sci. Paris, Sdr. I, 307, 471-474. KOMORNIK, V. [1991a]: On the exact controllability of Kirchhoff plates, Differential Integral Equations 4, 1121-1132. KOMORNIK, V. [1991b]: On the zeros of Bessel type functions and applications to exact controllability problems, Asymptotic Anal. 5, 115-128. KOMORNIK, V. [1992a]: On the exact internal controllability of a Petrowsky system, J. Math. Pures Appl. 71,331-342. KOMORNIK, V. [1992b]: Stabilisation non lin6aire de l'dquation des ondes et d'un module de plaque, C. R. Acad. Sci. Paris, Sdr. I, 315, 55-60. KOMORNIK, V. [1994a]: Exact Controllability and Stabilization. The Multiplier Method, Masson, Paris and J. Wiley, New York. KOMORNIK, V. [1994b]: On the nonlinear boundary stabilization of Kirchhoff plates, Nonlinear Differential Equations Appl. 1,323-337. KONDRATIEV, V. A. [1967]: Boundary value problems for elliptic equations in domains with conical or angular points, Trans. Moscow Math. Soc. 16, 227313. KONDRATIEV, V. A.; OLEINIK, O. A. [1983]: Estimates for solutions of the Dirichlet problem for the biharmonic equation in a neighborhood of an irregu-
466
References
lar boundary point and in a neighborhood of infinity. Saint-Venant's principle, Proc. Roy. Soc. Edinburgh 93A, 327-343. KONDRATIEV, V. A.; OLEINIK, O. A. [1989]: On Korn's inequalities, C. R. Acad. Sci. Paris, Sdr. I, 308,483-487. KONDRATIEV,'V. A.; OLEINIK,O. A. [1990]: Hardy's and Korn's type inequalities and their applications, Rendiconti di Matematica, Ser. VII, 10, 641-666. KOZLOV, V. A.; MAZ'YA, V. G.; MOVCHAN, A. B. [1994]: Asymptotic analysis of a mixed boundary value problem in a multi-structure, Asymptotic Anal. 8, 105-143. KOZLOV, V. A.; MAZ'YA, V. G.; MOVCHAN, A. B. [1995]: Asymptotic representation of elastic fields in a multi-structure, Asymptotic Anal. 11,343-415. KOZLOV, V. A.; MAZ'YA, V. G.; SCHWAB, C. [1992]: On singularities of solutions of the displacement problem of linear elasticity near the vertex of a cone, Arch. Rational Mech. Anal. 119, 197-227. LADEV~;ZE, P. [1975]" Comparaison de ModUles en Mdcanique des Milieux Coatinus, Doctoral Dissertation, Universi% Pierre et Marie Curie, Paris. LADEVEZE, P.; PI~CASTAIGNS, F. [1988]" The optimal version of Reissner's theory, J. Appl. Math. 55,413-418. LADYZHENSKAYA, O. A. [1969]: The Mathematical Theory of Viscous Incompressible Flows, Gordon & Breach, New York. LAGNESE, J. E. [1989]: Boundary Stabilization of Thin Plates, SIAM Publications, SIAM, Philadelphia. LAGNESE, J. E.; LEUGERING, G.; SCHMIDT,E. J. P. G. [1994]: Modeling, Analysis and Control of Dynamic Elastic Multi-Link Structures, Birkhs Boston. LAGNESE, J. E.; LIONS, J. L. [1988]: Modelling, Analysis and Control of Thin Plates, Masson, Paris. LANDAU, L.; LwCHnz, E. [1967]: Thdorie de l'Elasticitd, Editions Mir, Moscow (English translation: Theory of Elasticity, Pergamon Press, 1970). LEBELTEL, C.[1992]: Sur quelques modules bidimensionnels de plaques en thermo61asticit6 lin6aris6e, C. R. Acad. Sci. Paris, Sdr. I, 314, 1069-1072. LE DREW, It. [1989a]: Folded plates revisited, Computational Mech. 5, 345-365. LE DREW, H. [1989b]: Modeling of the junction between two rods, J. Math. Pures Appl. 68, 365-397. LE DREW, H. [1990a]: Modeling of a folded plate, Computational Mech. 5, 401416. LE DREW, H. [1990b]: Vibrations of a folded plate, Math. Modelling Numer. Anal. 24, 501-521. LE DREW, H. [1991]: Probl~mes Variationnels dans les Multi-Domaines. Moddlisation des Jonctions et Applications, Masson, Paris. LE DREW, H. [1994]: Elastodynamics for multiplate structures, in Nonlinear Partial Differential Equations and their Applications, Coll~ge de France Seminar, Vol. XI (H. BREZIS and J. L. LIONS, Editors), pp. 151-180, Pitman Research Notes in Mathematics Series, John Wiley, New York. LE DREW, H. [1995]: Convergence of displacements and stresses in linearly elastic slender rods as the thickness goes to zero, Asymptotic Anal. 10, 367-402.
References
467
LE DRET, H.; RAOULT, A. [1995]: The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity, 3". Math. Pures Appl. 7"4, 549578. LE DRET, H.; RAOULT, A. [1996]: The membrane shell model in nonlinear elasticity: A variational asymptotic derivation, J. Nonlinear SeN. 6, 59-84. LEGUILLON, D.; SANCHEZ-PALENCIA, E. [1990]: Approximation of a two-dimensional problem of junctions, Computational Mech. 6, 435-455. LEMBO, M. [1989]: The membranal and flexural equations of thin elastic plates deduced exactly from the three-dimensional theory, Meccanica 24, 93-97. LEMBO, M.; PODIO-GUIDUGLI, P. [1991]: Plate theory as an exact consequence of three-dimensional linear elasticity, European J. Mech., A/Solids 10, 1-32. LE TALLEC, P. [1994]: Numerical methods for nonlinear three-dimensional elasticity, in Handbook of Numerical Analysis, Vol. III (P. G. CIARLET and J. L. LIONS, Editors), pp. 465-622, North-Holland, Amsterdam. LEWII~SKI, T. [1991a]" Effective models of composite periodic plates. I. Asymptotic solution, Internat. J. Solids Structures 27, 1155-1172. LEWII~ISKI, T. [1991b]" Effective models of composite periodic plates. !I. Simplifications due to symmetries, Internat. J. Solids Structures 27, 1173-1184. LEWIIs T. [1991c]" Effective models of composite periodic plates. III. Twodimensional approaches, Internat. J. Solids Structures 27, 1185-1203. LICHT, C.; MICHAILLE, G. [1996]: Une mod61isation du comportement d'un joint co116 61astique, C. R. Acad. Sci. Paris, Sdr. I, 322, 295-300. LIONS, J. L. [1957]: Lectures on Elliptic Partial Differential Equations, Tata Institute, Bombay. LIONS, J. L. [1961]: Equations Diffdrentielles Opdrationnelles et Probl~mes aux Limites, Springer-Verlag, Berlin. LIONS, J. L. [1969]: Quelques Mdthodes de Rdsolution des Probl~mes aux Limites Non Lindaires, Dunod, Paris. LIONS, J. L. [1973]: Perturbations Singuli~res dans les Probl~mes auz Limites en Contrgle Optimal, Lectures Notes in Mathematics, Vol. 323, Springer-Verlag, Berlin. LIONS, J. L. [1981]: Some Methods in the Mathematical Analysis of Systems and their Control, Science Press, Beijing, and Gordon & Breach, New York. LIONS, J. L. [1985]: Remarques sur les probl6mes d'homog6n6isation dans les milieux ~ structure p6riodique et sur quelques probl6mes raides, in Ecole d'Etd d'Analyse Numdrique C.E.A.-E.D.F.-I.N.R.I.A., Etd 1983, Collection Direction des Etudes et Recherches, Electricit6 de France, Vol. 57, Eyrolles, Paris. LIONS, J. L. [1988a]: Exact controllability, stabilization and perturbations for distributed systems, John von Neumann Lecture, S I A M Rev. 30, 1-68. LIONS, J. L. [1988b]: Contrglabilitd Ezacte, Perturbations et Stabilisation de Syst~mes Distribuds, Tome 1: Contrdlabilitd Exacte, Masson, Paris. LIONS, J. L. [1988c]" Contrglabilitd Exacte, Perturbations et Stabilisation de Syst~mes Distribuds, Tome 2: Perturbations, Masson, Paris. LIONS, J. L. [1990]" ContrSle de structures souples multidimensionnelles" Remarques introductives, Matapli 24, 27-42.
468
References
LIONS, J. L.; NIAGENES, E. [1968]: Probl~mes aux Limites Non Homog~nes et Applications, Vol. 1, Dunod, Paris (English translation: Non-Homogeneous Boundary Value Problems and Applications, Vol. 1, Springer-Verlag, Berlin, 1972). LOPS, V. [1992]: Formulation mixte d'un probl~me de jonctions de poutres adapt~e g la r~solution d'un probl~me d'optimisation, Moddlisation Math. Anal. Numdr. 26, 523-553. LODS, V. [1996]: Modeling and justification of an eigenvalue problem for a plate inserted in a three-dimensional support, Math. Modelling Numer. Anal. 30, 413-444. LODS, V.; MIARA, B. [1995]: Analyse asymptotique des coques "en flexion" non lin~airement ~lastiques, C. R. Acad. Sci. Paris, Sdr. I, 321, 1097-1102. LODS, V.; MIARA, B. [1997]: Nonlinearly elastic shell models: A formal asymptotic approach. II. The flexural model, Arch. Rational Mech. Anal., to appear. LOVE, A. E. H. [1934]: A Treatise on the Mathematical Theory of Elasticity, Fourth Edition, Cambridge University Press, Cambridge (reprinted by Dover Publications, New York, 1944). MAGENES, E.; STAMPACCHIA, G. [1958]: I problemi al contorno per le equazioni differenziali di tipo ellitico, Ann. Scuola Norm. Sup. Pisa 12,247-358. MAMPASSI, B. [1992]: Un type de jonction bidimensionnelle d'une tige et d'un massif ~lastiques, C. R. Acad. Paris, Sdr. II, 315, 261-266. MAMPASSI, B.; SANCHEZ-PALENCIA, E. [1992]: Etat local de contraintes g une jonction de plaque avec un corps tridimensionnel, C. R. Acad. Paris, Sdr. II, 315, 129-135. MARGUERRE, K. [1938]: Zur Theorie der gekrfimmten Platte grosser Formgnderung, in Proceedings, Fifth International Congress for Applied Mechanics, pp. 93-101. MARSDEN, J. E.; HU(~HES, T. J. R. [1983]: Mathematical Foundations of Elasticity, Prentice-Hall, Englewood Cliffs. MASCARENHAS, M. L.; TRABUCHO, L. [1992]: Asymptotic, homogenisation and Galerkin methods in three-dimensional beam theory, in Computational and Applied Mathematics II (W. F. AMES and P. J. VAN DER HOUWEN, Editors), pp. 85-91, North-Holland, Amsterdam. MATHtJNA, D. 6. [1989]' Mechanics, Boundary Layers, and Function @aces, Birkhguser, Boston. MATKOWSKY, B. J.; PUTNICK, L. J.; REISS, E. L. [1980]: Secondary states of rectangular plates, SIAM J. A ppl. Math. 38, 38-51. MATKOWSKY, B. J.; REISS, E. L. [1977]: Singular perturbations of bifurcations, SIAM J. Appl. Math. 33, 230-255. MAUGIN, G. n.; ATTOU, D. [1990]: An asymptotic theory of thin piezoelectric plates, Quart J. Mech. Appl. Math. 43, 347-362. MAZ'YA, V. G.; NAZAROV, S. A. [1987]: Paradoxes of limit passage in solutions of boundary value problems involving the approximation of smooth domains by polygonal domains, Math. USSR Izvestiya 29, 511-533.
References
469
MAZ'YA, V. G.; NAZAROV, S. A.; PLAMENEWSKI, D. A. [1991]: Asymptotische Theorie eUiptischer Randwertaufgaben in singulr gestrten Gebieten, Band 2, Akademie-Verlag, Berlin. MIARA, B. [1987]: Numerical assessment of the validity of two-dimensional plate models, Research Report NA-87-06, Computer Science Department, Stanford University, Stanford. MIARA, B. [1989]: Optimal spectral approximation in linearized plate theory, Applicable Anal. 31,291-307. MIARA, B [1994a]: Justification of the asymptotic analysis of elastic plates. I. The linear case, Asymptotic Anal. 8, 259-276. MIARA, B [1994b]: Justification of the asymptotic analysis of elastic plates. II. The nonlinear case, Asymptotic Anal. 9, 119-134. MIARA, B [1994c]: Analyse asymptotique des coques membranaires non lin~airement ~lastiques, C. R. Acad. Sci. Paris, Sdr. I, 318, 689-694. MIARA, B [1997]: Nonlinearly elastic shell models: A formal asymptotic approach. II. The membrane model, Arch. Rational Mech. Anal., to appear. MIARA, B ; SANCHEZ-PALENCIA, E. [1996]: Asymptotic analysis of linearly elastic shells, Asymptotic Anal. 12, 41-54. MIARA, B ; TRABUCHO, L. [1992]: A Galerkin spectral approximation in linearized beam theory, Math. Modelling Numer. Anal. 26, 425-446. MIELKE, A. [1988]: Saint Venant's problem and semi-inverse solutions in nonlinear elasticity, Arch. Rational Mech. Anal. 102, 205-229; Corrigendum, ibid. 110 [1990], 351-352. MIELKE, A. [1990]: Normal hyperbolicity of center manifolds and Saint Venant's principle, Arch. Rational Mech. Anal. 110, 353-372. MIELKE, A. [1995]: On the justification of plate theories in linear elasticity theory using exponential decay estimates, J. Elasticity 38, 165-208. M~CNOT, F.; PUEL, J. P.; SUQUET, P. M. [1981]: Flambage des plaques ~lastiques multiperfor~es, Ann. Fac. Sci. Toulouse Math. 3, 1-57. MINDLIN, R. D. [1951]: Influence of rotatory inertia and shear on flexural motions of isotropic elastic plate, J. Appl. Mech. 18, 31-38. MORAND, H. J. P.; OHAYON, R. [1992]: Interactions Fluides-Structures, Masson, Paris. MORCENSTERN, D. [1959]: Herleitung der Plattentheorie aus der dreidimensionalen Elastizit/itstheorie, Arch. Rational Mech. Anal. 4, 145-152. MORGENSTERN, D.; SZABO, I. [1961]" Vorlesungen iiber Theoretische Mechanik, Springer-Verlag, Berlin. Moaozov, N. F. [1967]: Nonlinear vibration of thin plates with allowance for rotational inertia, Dokl. Akad. Nauk 176 (English translation: Soviet Math. 8 [1967], 1136-1141). MORREY, JR., C. B. [1952]: Quasi-convexity and the lower semicontinuity of multiple integrals, Pacific J. Math. 2, 25-53. MORREY, JR., C. B. [1966]: Multiple Integrals in the Calculus of Variations, Springer-Verlag, Berlin.
470
References
MOSER, J. [1961]: A new technique for the construction of solutions of nonlinear partial differential equations, Proc. Nat. Acad. Sci. USA 47, 1824-1831. MURNAGHAN, F. D. [1951]: Finite Deformations of an Elastic Solid, John Wiley, New York. NAGHDI, P. M. [1972]: The theory of shells and plates, in Handbuch der Physik, Vol. Vial2 (S. FL/JCCE and C. TRUESDELL, Editors), pp. 425-640, SpringerVerlag, Berlin. NARDINOCCHI, P.; PODIO-GUIDUCLI, P. [1994]: The equations of ReissnerMindlin plates obtained by the method of internal constraints, Meccanica 29, 143-157. NAZAROV, S. A. [1996]: On the accuracy of asymptotic approximations for longitudinal deformation of a thin plate, Math. Modelling Numer. Anal. 30, 185-213. NAZAROV, S. A.; PLAMENEVSKI, B. [1994]: Elliptic Problems in Domains with Piecewise Smooth Boundaries, de Gruyter, Berlin. NAZAROV, S. A.; ZORIN, A. S. [1991]: Two-term asymptotics of solutions of the problem of longitudinal deformation of a plate with clamped edge, Computer Mechanics of Solids 2, 10-21 (in Russian). NECAS, J. [1967]: Les M~thodes Directes en Th~orie des Equations Elliptiques, Masson, Paris. NE(~AS, J." HLAVX(~EK, I. [1981]" Mathematical Theory of Elastic and ElastoPlastic Bodies: An Introduction, Elsevier, Amsterdam. NECAS, J.; NAUMANN, J. [1974]: On a boundary value problem in nonlinear theory of thin elastic plates, Aplikace Matematiky 19, 7-16. NGUETSENG, G. [1989]: A general convergence result for a functionnal related to the theory of homogenization, SIAM J. Math. Anal. 20, 608-625. NIANE, M. T. [1988]: Contr61abilit~ exacte de l'~quation des plaques vibrantes dans un polygSne, C. R. Acad. Sci. Paris, Sdr. I, 307, 517-521. NICAISE, S. [1990]: ContrSlabilit~ exacte d'un probl~me coupl~ pluri-dimensionnel, C. R. Acad. Sci. Paris, Sdr I, 311, 19-22. NICAISE, S. [1992]: About the Lam~ system in a polygonal or a polyhedral domain and a coupled problem between the Lam~ system and the plate equation. I. Regularity of the solutions, Ann. Scuola Norm. Sup. Pisa, Ser. IV, 19, 327-361. NICAISE, S. [1993]: About the Lam~ system in a polygonal or a polyhedral domain and a coupled problem between the Lam~ system and the plate equation. II. Exact controllability, Ann. Scuola Norm. Sup. Pisa, Ser. IV, 20, 163-191. NIRENBERG, L. [1955]: Remarks on strongly elliptic partial differential equations, Comm. Pure Appl. Math. 8, 648-674. NOEL, W. [1959]: The foundations of classical mechanics in the light of recent advances in continuum mechanics, in The Axiomatic Method, with Special Reference to Geometry and Physics, pp. 266-281, North-Holland, Amsterdam. NOEL, W. [1966]: The foundations of mechanics, in C.I.M.E., Non Linear Continuum Theories (G. GRIOLI and C. TRUESDELL, Editors), pp. 159-200, Cremonese, Rome.
References
471
NOLL, W. [1972]: A new mathematical theory of simple materials, Arch. Rational Mech. Anal. 48, 1-50. NOLL, W. [1973]: Lectures on the foundations of continuum mechanics and thermodynamics, Arch. Rational Mech. Anal. 52, 62-92. NOLL, W. [1978]: A general framework for problems in the statics of finite elasticity, in Contemporary Developments in Continuum Mechanics and Partial Differential Equations (G. M. DE LA PENHA and L. A. J. MEDEIROS,Editors), pp. 367-387, North-Holland, Amsterdam. NORDGREN, R. P. [1971]: A bound on the error in plate theory, Quart. Appl. Math. 28,587-595. NORDGREN, R. P. [1972]: A bound on the error in Reissner's theory of plates, Quart. A ppl. Math. 29 551-556. NOVOZHILOV, V. V. [1953]: Foundations of the Nonlinear Theory of Elasticity, Graylock Press, Rochester. NOVOZHILOV, V. V. [1961]: Theory of Elasticity, Pergamon Press, Oxford. ODEN, J. T. [1986]: Qualitative Methods in Nonlinear Mechanics, Prentice-Hall, Englewood Cliffs. ODEN, J. T.; CHO, J. R. [1995]: Local a posteriori error estimation for hierarchical models for plate and shell-like structures, TICAM Report 95-02, The University of Texas at Austin. ODEN, J. T.; REDDY, J. N. [1983]: Variational Methods in Theoretical Mechanics, Second Edition, Universitext Series, Springer-Verlag, Berlin. OCDEN, R. W [1984]: Non-Linear Elastic Deformations, Ellis Horwood, Chichester and John Wiley, New York. OHAYON, R. [1992]: Structures intelligentes, in Science et Ddfense 92 - Nouvelles Avancdes Scientifiques et Techniques: Acoustique et Vibrations, Robotique Mobile, pp. 120-127, Dunod, Paris. DE OLIVEmn, M. P. [1981]: Alguns Problemas em Optimiza~o de Estruturas, Doctoral Dissertation, Universidade de Coimbra. OVTCHINNIKOV, E. E.; XANTHIS, L. S. [1995]: Effective dimensional reduction for elliptic problems, C. R. Acad. Sci. Paris, Sdr. I, 320, 879-884. OVTCHINNIKOV, E. E.; XANTHIS, L. S. [1996]: A new Korn's type inequality for thin domains and its application to iterative methods, Comput. Methods Appl. Mech. Engrg. 138,299-315. PANASENKO, G. P. [1980]: Asymptotic behavior of solutions and eigenvalues of elliptic equations with strongly varying coefficients, Dokl. Akad. Nauk SSSR 252, 1320-1325. PANASENKO, G. P. [1993]: Asymptotic solutions of the system of elasticity theory for rod and frame structures, Russian Acad. Sci. Sb. Math. 75, 85-110. PAUMIER, J. C. [1985]: Analyse de Certains Probl~mes Non Lindaires: ModUles de Plaques et de Coques, Doctoral Dissertation, Universit~ Pierre et Marie Curie, Paris. PAUMIER, J. C. [1990]: Existence theorems for nonlinear elastic plates with periodic boundary conditions, J. Elasticity 23, 233-252.
472
References
PAUMIER, J. C. [1991]: Existence and convergence of the expansion in the asymptotic theory of elastic thin plates, Math. Modelling Numer. Anal. 25, 371-391. PAUMIER, J. C. [1997]: Bifurcation et Mdthodes Numdriques. Applications aux Probl~mes Elliptiques Semi-Lindaires, Masson, Paris. PAUMIER, J. C.; RAO, Bo-peng [1989]: Qualitative and quantitative analysis of buckling of shallow shells, European J. Mech. A/Solids 8, 461-489. PAUMIER, J. C.; RAOULT, A. [1997]: Asymptotic consistency of the polynomial approximation in the linearized plate t h e o r y - Application to the ReissnerMindlin model, in ESAIM Proceedings: Elasticitd, Viscodlasticitd et ContrSle Optimal, to appear. PEETRE, J. [1966]: Espaces d'interpolation et th@or~me de Sobolev, Ann. Inst. Fourier 16, 279-317. PEISKER, P.; BRAESS, D. [1992]: Uniform convergence of mixed interpolated elements for Reissner-Mindlin plates, Math. Modelling Numer. Anal. 26, 557574. PERLA MENZALA, G. ; ZUAZUA, E. [1997]: Explicit exponential decay rates for solutions of von K~rms system of thermoelastic plates, C. R. Acad. Sci. Paris, Sdr. I, 324, 49-54. PITKs J.; SURI, M. [1996]: Design principles and error analysis for reducedshear plate-bending finite elements, Numer. Math. 75, 223-266. PLEMMONS, R.; WHITE, R.[1990]: Substructuring methods for computing the nullspace of equilibrium matrices, SIAM J. Matrix Anal. Appl. 11, 1-22. PODIO-GUIDUGLI, P. [1989]: An exact derivation of the thin plate equation, J. Elasticity 22, 121-133. PODIO-GUIDUGLI, P.; VERGARA-CAFFARELLI,G.; VIRGA, E. G. [1987]: The role of ellipticity and normality assumptions in formulating live-boundary conditions in elasticity, Quart. Applied Math. 44, 659-664. POINCARI~, H. [1890]" Sur les @quations aux d@riv~es partielIes de la physique math~matique, Amer. J. Math. 12, 211-294. PUEL, J. P.; TUCSNAK, M. [1992]: Stabilisation fronti~re pour les @quations de yon Ks163 C. R. Acad. Sci. Paris, Sdr. I, 314, 609-612. PUEL, J. P.; TUCSNAK, M. [1995]" Boundary stabilization for the von Ks equations, SIAM J. Control. 33, 255-273. PUEL, J. P.; TUCSNAK, N~. [1996]" Global existence for the full von Ks163 system, Appl. Math. Optimization 34, 139-160. PUEL, J. P.; ZUAZUA, E. [1993]: Exact controllability for a model of a multidimensional flexible structure, Proc. Royal Soc. Edinburgh 123A, 323-344. QUINTELA-ESTt~VEZ, P. [1989]" A new model for nonlinear elastic plates with rapidly varying thickness, Applicable Anal. 32, 107-127. QUINTELA-ESTt~VEZ, P. [1994]: Perturbed bifurcation in the yon Ks163 equations, Asymptotic Anal. 8, 161-184. RABIER, P. [1979]" R@sultats d'existence dans des modules non lin~aires de plaques, C. R. Acad. Sci. Paris, Sdr. A, 289,515-518. RABIER, P. [1982a]" t~tude locale des probl~mes non lin~aires perturb~s, Y. Math. Pures Appl. 4, 311-343.
References
473
RABIER, P. [1982b]: Une m4thode directe dans l'4tude des bifurcations perturbdes par un nombre quelconque de param~tres, Rend. Sere. Mat. Univers. Politecn. Torino 40, 135-171. RABIER, P.; ODEN, J.T. [1989]: Bifurcation in Rotating Bodies, Masson, Paris and Springer-Verlag, Berlin. RABINOWITZ, P. H. [1971]: Some global results for nonlinear eigenvalueproblems, J. Funct. Anal. 7, 487-513. RABINOWITZ, P. H. [1975]: Thdorie du Degrd Topologique et Application ~ des Probl&mes aux Limites Non Lindaires (Lectures Notes by H. BERESTICKY), Report 75010, Laboratoire d'Analyse Numdrique, Universitd Pierre et Marie Curie, Paris. RAHMOUNE, M.; OSMONT, D.; BENJEDDOU, A.; OHAYON, R. [1996]: Finite element modeling of a smart structure plate system, in Proceedings, Seventh International Conference on Adaptive Structures and Technologies, Rome, Sept. 23-25, 1996), to appear. RAO, Bo-peng [1989]" Sur Quelques Questions d'Elasticitd Non Lindaire Relatives aux Coques Minces, Doctoral Dissertation, Universitd Pierre et Marie Curie, Paris. RAO, Bo-peng [1993]: Stabilization of Kirchhoff plate equation in star-shaped domain by nonlinear boundary feedback, Nonlinear Anal., Theory Methods Appl. 2O, 6O5-626. RAO, Bo-peng [1995a]" Marguerre-von Kgrm&n equations and membrane model, Nonlinear Anal., Theory Methods Appl. 24, 1131-1140. RAO, Bo-peng [1995b]: Existence of non-trivial solutions of the Marguerre-von Kgrmgn equations and membrane model, in Asymptotic Methods for Elastic Structures (P. O. CIARLET, L. TRABUCHO and J. M. VIANO, Editors), pp. 161-170, de Gruyter, Berlin. RAOULT, A. [1985]: Construction d'un module d'dvolution de plaques avec terme d'inertie de rotation, Annali di Matematica Pura ~dApplicata 139, 361-400. RAOULT, A. [1988]: Analyse Mathdmatique de Quelques ModUles de Plaques et de Poutres Elastiques ou Elasto-Plastiques, Doctoral Dissertation, Universitd Pierre et Marie Curie, Paris. RAOULT, A. [1992]: Asymptotic modeling of the elastodynamics of a multistructure, Asymptotic Anal. 6, 73-108. RAVIART, P.-A.; THOMAS, J. M. [1983]: Introduction ~ l'Analyse Numdrique des Equations aux Ddrivdes Partielles, Masson, Paris. REISS, E. L.; LOCKE, S. [1961]: On the theory of plane stress, Quart. Appl. Math. 19, 195-203. REISSNER, E. [1944]: On the theory of bending of elastic plates, J. Math. Phys. 23, 184-191. REISSNER, E. [1945]: The effect of transverse shear deformations on the bending of elastic plates, J. Appl. Mech. 12, A69-A77. REISSNER, E. [1950]: On a variational theorem in elasticity, J. Math. Phys. 29, 90-95.
474
References
REISSNER, E. [1963]: On the derivation of boundary conditions for plate theory, Proc. Roy. Soc. Set. A 276, 178-186. REISSNEa, E. [1985]: Reflections on the theory of elastic plates, Appl. Mech. Rev. as, 1453-1464. REISSNER, E. [1986]: On small finite deflections of shear deformable elastic plates, Comput. Methods Appl. Mech. Engrg. 59, 227-233. RIGOLOT, A. [1972]: Sur une th~orie asymptotique des poutres, J. Mdcanique 11, 673-703. RIGOLOT, n. [1976]: Sur une Thdorie Asymptotique des Poutres Droites, Doctoral Dissertation, Universit~ Pierre et Marie Curie, Paris. RIGOLOT, n. [1977a]: Approximation asymptotique des vibrations de flexion des poutres droites ~lastiques, J. Mdcanique 16, 493-529. RIGOLOT, n. [1977b]: D6placements et petites d~formations des poutres droites: Analyse asymptotique de la solution/~ grande distance des bases, J. Mdcanique Appl., 1, 175-206. RIGOLOT, A. [1978]: Sur la d~formation due s l'effort tranchant dans les poutres droites ~lastiques, A nnales de l'Institut Technique du Batiment et des Travaux Publics 363, 35-52. RIGOLOT, A. [1982]" Equilibre ~lastique d'un composite plan coll,, J. Mdcanique Thdor. Appl. 1, 799-839. ROBERT, J. E.; THOMAS, J. M. [1991]: Mixed and hybrid methods, in Handbook of Numerical Analysis, Vol. H (P. G. CIARLET and J. L. LIONS, Editors), pp. 523-633, North-Holland, Amsterdam. RODRfGUEZ, J. M. [1997]" Analyse Asymptotique des Pales de Turbines et des Poutres it Profil Mince, Doctoral Dissertation, Universit~ Pierre et Marie Curie, Paris. F[ODRfGUEZ, J. M." VIAI~O, J. M. [1997]" Asymptotic derivation of a general linear model for thin-walled elastic rods, Comput. Methods Appl. Mech. Engrg, to appear. ROSEAU, M. [1984]" Vibrations des Syst~mes Mdcaniques: Mdthodes Analytiques et Applications, Masson, Paris (English translation: Vibrations in Mechanical Systems: Analytical Methods and Applications, Springer-Verlag, Berlin, 1987). I:[UPPRECHT, G. [1981]: A singular perturbation approach to nonlinear shell theory, Rocky Mountain J. Math. 11, 75-98. SAINT JEAN PAULIN, J.; VANNINATHAN,M. [1996]: ContrSlabilit& exacte de vibrations de corps 61astiques minces, C. R. A cad. Sci. Paris, Sdr. I, 322, 889-894. SANCHEZ-HUBERT, J.; SANCHEZ-PALENCIA,E. [1989]: Vibration and Coupling of Continuous Systems; Asymptotic Methods, Springer-Verlag, Berlin. SANCHEZ-HUBERT, J.: SANCHEZ-PALENCIA, E. [1997]: Coques Elastiques Minces : Propridtds A symptotiques, Masson, Paris. SANCHEZ-PALENCIA, E. [1980]: Non-Homogeneous Media and Vibration Theory, Lecture Notes in Physics, Vol. 127, Springer-Verlag, Heidelberg. SANCHEZ-PALENCIA, E. [1988]: Forces appliqu~es ~ une petite r~gion de surface d'un corps ~lastique. Application aux jonctions, C. R. Acad. Sci. Paris, Sdr.
References
475
H, 307, 689-694. SANCHEZ-PALENCIA, E. [1989a]: Statique et dynamique des coques minces. I. Cas de flexion pure non inhib6e, C. R. Acad. Sci. Paris, Sdr. I, 309, 411-417. SANCHEZ-PALENCIA, E. [1989b]: Statique et dynamique des coques minces. II. Cas de flexion pure inhib6e- Approximation membranaire, C. R. Acad. Sci. Paris, Sdr. I, 309, 531-537. SANCHEZ-PALENCIA, E. [1990]" Passage/~ la limite de l'~lasticit6 tri-dimensionnelle /~ la th~orie asymptotique des coques minces, C. R. Acad. Sci. Paris, Sdr. II, 311,909-916. SANCHEZ-PALENCIA, E. [1992]: Asymptotic and spectral properties of a class of singular-stiff problems, J. Math. Pures Appl. 71,379-406. SANCHEZ-PALENCIA, E. [1994]: Singularities and junctions in elasticity, in Nonlinear Partial Differential Equations and their Applications, Coll~ge de France Seminar, Vol. XII (H. BREZIS and J. L. LIONS, Editors), pp. 172-189, Pitman Research Notes in Mathematics Series, John Wiley, New York. SAPONDZHYAN, O. M. [1952]: Bending of freely supported polygonal plate, Izv. Akad. Nauk. Arrnyan SSR Set. Fiz.-Mat. Estestv. Tekhn. Nauk. 5, 29-46. SCHUSS, Z. [1976]: Singular perturbations and the transition from plate to membrane, Proc. Arner. Math. Soc. 58, 139-147. SCHWAB, C. [1994]: Boundary layer resolution in hierarchical models of laminated composites, Moddlisation Math. Nurndr. 28, 517-537. SCHWAB, C [1995]: Habilitation Thesis, University of Stuttgart. SCHWAB, C. [1996]: A-posteriori modelling error estimation for hierarchic plate models, Nurner. Math. 74, 221-259. SCHWAB, C.; SuRI, M. [1994]: Locking and boundary layer effects in the finite element approximation of the Reissner-Mindlin plate model, AMS Proc. Syrnp. Appl. Math. 48, 367-371. SCHWAB, C.; WRIGHT, S. [1995]: Boundary layers of hierarchical beam and plate models, J. Elasticity 38, 1-40. SCHWARTZ, L. [1966]: Thdorie des Distributions, Hermann, Paris. SHOIKHET, B. n. [1976]: An energy identity in physically nonlinear elasticity and error estimates of the plate equations, Prikl. Mat. Mech. 40,317-326. SIMMONDS, J. G. [1971a]: An improved estimate for the error in the classical, linear theory of plate bending, Quart. Appl. Math. 29, 439-447. SIMMONDS, J. G. [1971b]: Extension of Koiter's L2-error estimate to approximate shell solutions with no strain energy functional, Z. Angew. Math. Phys. 22, 339-345. SOKOLNIKOFF,I. S. [1956]: Mathematical Theory of Elasticity, McGraw-Hill, New York. SRUBSHCHIK,L. S. [1964a]: On the asymptotic integration of a system of nonlinear equations of plate theory, Prikl. Mat. Mech. 28,335-349. SRUBSHCHIK, L. S. [1964b]: Circular plates under the action of discontinuous loadings, Prikl. Mat. Mech. 28, 1024-1032. STOKER, J. J. [1968]: Nonlinear Elasticity, Gordon & Breach, New York. STRUWE, M. [1990]: Variational Methods, Springer-Verlag, Berlin.
476
References
SURI, M.; BABUSKA, I.; SCHWAB, C. [1995]: Locking effects in the finite element approximation of plate models, Math. Comp. 64, 461-482. SZABO, B. A.; SAHRMANN, (3. J. [1988]: Hierarchic plate and shell models based on p-extension, Internat. J. Numer. Methods Engrg. 26, 1855-1881. TANG, Qi [1990]: Justification of elastoplastic models, Asymptotic Anal. 3, 57-76. TARTAR, L. [1978]: Topics in Nonlinear Analysis, Publications Mathdmatiques d'Orsay No. 78.13, Universit~ de Paris-Sud, Orsay. TATARU, D.; TUCSNAK, M. [1996]: On the Cauchy problem for the full von Kgrmgn system, Nonlinear Differential Equations, to appear. TAYLOR, A. E. [1958]: Introduction to Functional Analysis, Wiley, New York. TAYLOR, G. I. [1933]: The buckling load for a rectangular plate with four clamped edges, Z. Angew. Math. Mech. 13, 147-152. TEMAM, R. [1977]: Navier-Stokes Equations, North-Holland, Amsterdam. TIIHONEN, T.; PIETIKAINEN, R. [1995]: Thermal deformations of inhomogeneous elastic plates, Math. Methods Appl. Sci. 18, 423-436. TIMOSHENKO, S. [19511: Theory of Elasticity, McGraw-Hill, New York. TIMOSHENKO, S.; WOINOWSKY-KRIEGER, W. [1959]: Theory of Plates and Shells, McGraw-Hill, New York. TRABUCHO, L.; VIA~O, J. M. [1987]: Derivation of generalized models for linear elastic beams by asymptotic expansion methods, in Applications of Multiple Scalings in Mechanics (P. G. CIARLET and E. SANCHEZ-PALENCIA, Editors), pp. 302-315, Masson, Paris. TRABUCH0, L.; VIA~O, J. M. [1988]: A derivation of generalized Saint Venant's torsion theory from three-dimensional elasticity by asymptotic expansion methods, Applicable Anal. 31, 129-148. TRABUCHO, L.; VIAI~O, J. M. [1989]: Existence and characterization of higher order terms in an asymptotic expansion method for linearized elastic beams, Asymptotic Anal. 2,223-255. TRABUCHO, L.; VIA~O, J. M. [1996]: Mathematical modelling of rods, in Handbook of Numerical Analysis, Vol. IV (P. G. CIARLET and J. L. LIONS, Editors), pp. 487-974, North Holland, Amsterdam. TRUESDELL, C. [1978]: Comments on rational continuum mechanics, in Contemporary Developments in Continuum Mechanics and Partial Differential Equations (G. M. DE LA PENHA and L. A. MEDEIROS, Editors), pp. 495-603, North-Holland, Amsterdam. TRUESDELL, C. [1983]: The influence of elasticity on analysis: The classic heritage, Bull. Amer. Math. Soc. 9, 293-310. TRUESDELL, C. [1991]: A first Course in Rational Continuum Mechanics, Vol. I, Second Edition, Academic Press, New York. TRUESDELL, C.; NOLL, W. [1965]: The nonlinear field theories of mechanics, in Handbuch der Physik, Vol. III/3 (S. FLOCGE, Editor), pp. 1-602, SpringerVerlag, Berlin. TRUESDELL, C.; TOUPIN, R. n. [1960]: The Classical Field Theories, in Handbuch der Physik, Vol. III/1 (S. FLOGaE, Editor), pp. 226-793, Springer-Verlag, Berlin.
References
477
TUCSNAK, M. [1996]: Control of plate vibrations by means of piezoelectric actuators, Discrete and Continuous Dynamical Systems 2, 281-293. VALENT, T. [1979]: Teoremi di esistenza e unicits in elastostatica finita, Rend. Sem. Mat. Univ. Padova 60, 165-181. VALENT, W. [1988]: Boundary Value Problems of Finite Elasticity, Springer Tracts in Natural Philosophy, Vol. 31, Springer-Verlag, Berlin. VALID, R. [1977]: La Mdcanique des Milieux Continus et le Calcul des Structures, Eyrolles, Paris (English translation: Mechanics of Continuous Media and Analysis of Structures, North-Holland, Amsterdam, 1981). VASHAKMADZE, T. Z. [1981a]: On the accuracy of approximations of a problem in elasticity theory, Soviet Math. Dokl. 24, 568-570. VASHAKMADZE, W. Z. [1981b]: On the error estimation of transition from some three-dimensional problems of elasticity to two-dimensional models, Sem. Inst. Prikl. Mat. Dokl. 86, 57-66 (in Russian). VASHAKMADZE, T. Z. [1983]: For the construction of the refined theories of anisotropic plates deduced exactly from the three-dimensional theory, Meccanica 17. VASHAKMADZE, W. Z. [1986]: Some Problems in the Mathematical Theory of Anisotropic Elastic Plates, Tbilisi University Press (in Russian). VASHAKMADZE,T. Z. [1991]: On the mathematical theory of piezoelectrically and electric conductive elastic plates, in Trends in Applications of Mathematics to Mechanics (W. SCHNEIDER, H. TROGE and F. ZIEGLER,Editors), pp. 236-241, Longman, Harlow. VEIGA, M. F. [1995]: Asymptotic method applied to a beam with a variable cross section, in Asymptotic Methods for Elastic Structures, (P. G. CIARLET, L. TRABUCHO and J. M. VIA~O, Editors), pp. 237-254, de Gruyter, Berlin. VEKUA, I. N. [1970]: Variational Principles of Constructing Shell Theory, Tbilisi University Press (in Russian). VEKUA, I. N. [1982]: Some General Methods of Constructing Different Versions of Shell Theory, MIR, Moscow (in Russian). VIDRASCU, M. [1984]: Comparaison num6rique entre les solutions bidimensionelles et tridimensionnelles d'un probl6me de plaque encastr~e, Rapport No. 309, I.N.R.I.A., Le Chesnay. VILLAGGIO, P. [1977]: Qualitative Methods in Elasticity, Noordhoff, Leyden. VOGELIUS, ZVI.; BABUSKA, I. [1981a]: On a dimensional reduction method. I. The optimal selection of basis functions, Math. Comp. 37, 31-46. VOGELIUS, M.; BABUSKA, I. [1981b]: On a dimensional reduction method. II. Some approximation-theoretic results, Math. Comp. 37, 47-68. VOGELIUS, M.; BABUSKA, I. [1981c]: On a dimensional reduction method. III. A posteriori error estimation and an adaptive approach, Math. Comp. 37, 361-384. VON WAHL, W. [1981]: On a nonlinear evolution equation in a Banach space and on nonlinear vibrations of the clamped plate, Bayreuther Math. Schr. 7, 1-93. WANG, C. C.; TRUESDELL,C. [1973]: Introduction to Rational Elasticity, Noordhoff, Groningen.
478
References
WANG, Lie-heng [1992]: A mathematical model of coupled plates and its finite element method, Comput. Methods Appl. Mech. Engrg. 99, 43-59.
WANG, Lie-heng [1993]: A mathematical model of coupled plates meeting at an angle 0 < 0 < 7~ and its finite element method, Numer. Methods Partial Differential Equations 9, 13-22. WANG, Lie-heng [1995]: An analysis of the TRUNC element for coupled plates with rigid junction, Comput. Methods Appl. Mech. Engrg. 121, 1-14. WASHIZU, K. [1975]: Variational Methods in Elasticity and Plasticity, Second Edition, Pergamon, Oxford. WEINBERGER, H. [1974]: Variational Methods for Eigenvalue Approximations, SIAM, Philadelphia. WEMPNER, G. [1973]: Mechanics of Solids, McGraw-Hill, New York. WOLKOWISKY,J. H. [1967]: Existence of buckled states of circular plates, Comm. Pure Appl. Math. 20, 549-560. YAN, Jin-hai [1992]: ContrSlabilite~ exacte pour domaines minces, Asymptotic Anal. 5, 461-471.
INDEX
a priori assumption (of a geometrical or mechanical nature) : 76, 81, 86, 87, 88,
120, 321 adhesive ( e l a s t i c - - ) : 136, 177 admissible deformations (set o f - - ) : 336 AIRY stress function : 403, 406, 446 b o u n d a r y conditions for the : 403, 408, 446 scaled : 390, 444 anisotropic elastic material (plate made with a n - - ) : 85, 327 Ansatz of the m e t h o d of formal asymptotic expansions : 90, 269, 297, 338, 358, 381,442 applied body force: 16 assumptions on s : 27, 94, 113, 126, 139, 184, 216,265,329, 350, 358,379, 441 applied force : 16 compatibility condition on s : 377, 390, 404, 440 applied surface force: 16 along the lateral face : 373, 377, 439 assumptions o n - - s : 27, 94, 126, 216, 265, 329, 350, 358, 379, 441 assumptions on the d a t a : 27, 77, 83, 94, 108, 113, 139, 184, 192, 198, 216,244, 265, 325, 350, 358, 379, 441 justification o f - - : 94, 329 asymptotic analysis : 24, 33, 83, 137, 264 - - and F-convergence : 97 asymptotic expansions (method of f o r m a l - - ) : 82, 90, 268, 297, 356, 358, 381, 442 asymptotic m e t h o d ( f o r m a l - - ) : 82 BABUSKA'S paradox : 53 beam inserted in an elastic foundation : 176 bending m o m e n t : 321 l i n e a r i z e d - - : 70 bifurcation of solutions of the MARGUERaE-VON KXaM~S equations" 446 bifurcation of solutions of the YON K~aM~S equations 9433, 437 perturbed : 435, 436 unperturbed : 433, 434 bifurcation p a r a m e t e r : 424 bifurcation theory: 433, 438
479
480
Indez
b i h a r m o n i c o p e r a t o r : 49 b o d y force ( a p p l i e d - - ) : 16 boundary condition : s for a linearly elastic c l a m p e d p l a t e : 18, 181, 188 s for a MARGUERRE-VON KXRM~,N shallow shell 9439 s for a n o n l i n e a r l y elastic c l a m p e d p l a t e : 315, 327 s for a YON K~RM~N p l a t e 9 374, 376 s for the AIRY stress function : 403, 408, 446 s for the •IARGUERRE-VON KARMAN e q u a t i o n s 9 446 s for t h e VON K~,RMAN e q u a t i o n s 9 403, 407 of place : 16, 258 of pressure : 409 of t r a c t i o n : 259 linearized of t r a c t i o n : 18 periodic s: 84,323 two-dimensional s of c l a m p i n g : 317 b o u n d a r y layer : 102, 103 b r a n c h i n g of solutions : 433 BROUWER fixed p o i n t t h e o r e m : 416 b u c k l i n g of a VON KARMAN p l a t e " 424, 437 c a n o n i c a l VON K/~RMAN e q u a t i o n s " 409 CARTESIAN c o o r d i n a t e s : 218, 242, 361 CAUCHY-GREEN s t r a i n tensor ( r i g h t - - ) : 326, 337 c h a n g e of c u r v a t u r e tensor of the middle surface : 6 8 , 3 1 9 c h a n g e of m e t r i c t e n s o r of the middle surface : 319 c l a m p e d p l a t e : 15, 16, 181 completely : 16, partially : 17 clamping: b o u n d a r y c o n d i t i o n o f - - : 16, 1 8 8 , 3 1 7 , 327 "in average" : 101 classical t w o - d i m e n s i o n a l p l a t e t h e o r y : 8 1 , 3 2 1 coerciveness of a f u n c t i o n a l : 285, 347, 419, 449 c o m p o s i t e m a t e r i a l : 85 c o n s t i t u t i v e e q u a t i o n : 76, 260 of a ST VENANT-KIRCHHOFF m a t e r i a l : 260 general :325 inverted : 294, 375 scaled inverted : 295, 300, 380 three-dimensional :260 two-dimensional : 317, 322 x~-dependent 9 321 c o n s t r a i n t m e t h o d : 88
Index
481
controllability: of elastic m u l t i - s t r u c t u r e s : 203 of plates : 118, 420 convergence: of a formal expansion : 84 of scaled displacements :34, 83, 100, 109, 115, 121. 142. 199. 229, 352 of scaled eigenfunctions : 109, 194 of scaled eigenvalues : 109, 194 of scaled stresses : 58, 83, 84, 121 F: 83, 95, 101,348 t w o - s c a l e - - : 85 corrector : 102 COSSERAT plate t h e o r y : 86, 328 coupled multi-dimensional problem : 170, 175 e i g e n v a l u e - - : 197 time-dependent : 199 c u r v a t u r e (change o f - - tensor) : 319 curvilinear coordinates : 219 dead l o a d : 258,408, 424 d e f o r m a t i o n : 258, 260, 335 admissible :336 inextensional :343 l a r g e - nonlinear flexural theory : 343 large - - nonlinear m e m b r a n e theory : 340, 353 scaled : 337, 350 d e f o r m a t i o n gradient : 336 deformed configuration : 258 de-scalings of the displacements : 64, 116, 167, 240, 360, 386, 403, 404, 405, 445 de-scalings of the stresses: 69, 320, 405 displacement : 16 convergence of the s : 34, 75, 83, 100, 109, 115, 121 ,142, 229, 352 error estimates for t h e - - - s : 83, 102, 103, 104 in-plane : 64, 313, 404 limit s inside the plate : 64, 76, 3 1 3 , 4 0 4 limit - - s of the middle surface : 64, 313, 386 scalings of the - - s : 25, 83 ,89, 125, 137, 183, 215, 264, 295, 329, 357, 378, 441 s m a l l - t h e o r y : 73, 324 tranverse : 64, 313, 403 displacement approach : 27, 33, 34, 268, 300, 383 displacement gradient : 260 displacement of the middle surface : i n - p l a n e - - : 64
482
Index
transverse :64 d i s p l a c e m e n t - s t r e s s a p p r o a c h : 33, 83, 120, 268, 294, 295, 300, 327, 380, 381 j u s t i f i c a t i o n of t h e : 313 d i s p l a c e m e n t - t r a c t i o n p r o b l e m : 262 linearized : 17 d o m a i n d e c o m p o s i t i o n m e t h o d : 180 d o m a i n in R n : 7 as a NIKODYM set : 388 multiply connected : simply connected : 389 e i g e n f u n c t i o n : 105, 190 scaled : 108, 193 e i g e n s o l u t i o n : 105 scaled : 108, 193 e i g e n v a l u e : 105, 190 scaled : 108, 193 eigenvalue problem : c o u p l e d , p l u r i d i m e n s i o n a l , - - : 197 for an elastic m u l t i - s t r u c t u r e : 189, 199 for folded p l a t e s : 199 for r o d s : 112 for t h e s c a l e d flexural e q u a t i o n s : 111 in t h r e e - d i m e n s i o n a l e l a s t i c i t y : 105 e x i s t e n c e t h e o r y for s : 106 stiff--: 198 three-dimensional for a p l a t e : 105 t w o - d i m e n s i o n a l - - for a p l a t e : 111 elastic a d h e s i v e : 136, 177 e l a s t i c m a t e r i a l : 17, 260 a n i s o t r o p i c - - : 85, 327 general : 325, 407 homogeneous: 17, 260, 325 hyper:262 i s o t r o p i c m : 17, 260, 325 l i n e a r l y - - : 19 nonhomogeneous--: 85 n o n l i n e a r l y m : 260 ST VENANT-KIRCHHOFF : 260 thermo---: 85, 420 e l a s t i c m u l t i - s t r u c t u r e ( l i n e a r l y - - ) : 133 controllability of--s: 203 e i g e n v a l u e p r o b l e m for a n : 194, 199 e x a m p l e of a n - - : 134, 176, 177, 178, 182 l i m i t e n e r g y of a n : 171
Index
483
l i m i t e q u a t i o n s of an - - : 163, 168 stabilization of--s: 203 s u b s t r u c t u r e of a n : 133 t h r e e - d i m e n s i o n a l e q u a t i o n s of an - - : 135 t i m e - d e p e n d e n t p r o b l e m for a n : 199 e l a s t i c m u l t i - s t r u c t u r e ( n o n l i n e a r l y - - ) : 175 elasticity tensor : three-dimensional: 20, 24, 2 6 1 , 3 1 8 two-dimensional : 68, 319 elastodynamics : linear : 104, 112, 189, 199 nonlinear :327 e l a s t o p l a s t i c p l a t e : 85, 101 e l l i p t i c i t y of a b i l i n e a r f o r m : 21, 22, 23, 50, 55, 165, 240 energy: c o e r c i v e n e s s : 285, 347, 419, 449 e x i s t e n c e of a m i n i m u m : 69, 287 flexural--: 69, 317 i n f i m i z i n g s e q u e n c e of an - - : 287 infinite - - : 176 limit of a n elastic m u l t i s t r u c t u r e : 171 m e m b r a n e - - : 69, 317 m i n i m i z e r of a n - - : see " m i n i m i z a t i o n of a f u n c t i o n a l " scaled : 30, 43, 97, 283, 351 stored f u n c t i o n : 336, 349 three-dimensional : 23, 336 two-dimensional--: 69, 283, 316, 361 w e a k lower s e m i - c o n t i n u i t y : 2 8 4 , 3 4 7 , 418, 449 e q u a t i o n s of e q u i l i b r i u m : three-dimensional :259 two-dimensional--: 317, 322, 364 e r r o r e s t i m a t e s : 88 for t h e d i s p l a c e m e n t s : 83, 102, 103, 104 for t h e s t r e s s e s : 102 e x i s t e n c e of s o l u t i o n s : 21, 48, 53, 164, 238, 247, 263, 282, 287, 316, 323, 3 2 8 , 3 4 0 , 345, 3 6 3 , 4 0 4 , 4 1 6 , 4 2 0 , 446 face of a p l a t e : l a t e r a l - - : 15 lower - - : 15 upper : 15 face of a s h a l l o w shell : lateral--: 439 lower : 214
484
Index
upper :214 fibers (materials w i t h - - ) : 85 finite element : - - - o f class C O : 87 - - of class" C 1 : 87 finite element m e t h o d : c o n f o r m i n g - - : 24 hybrid :24 mixed :24 nonconforming :24 flexural energy : 69, 317 flexural equations (of the linear KIRCHHOFF-LOVE theory) : 67, 75 eigenvalue problem for the : 111 existence of s o l u t i o n s : 48, 67 regularity of s o l u t i o n s : 48, 67 scaled : 35, 48 time-dependent : 115, 117 uniqueness of s o l u t i o n s : 48, 67 flexural rigidity of a plate : 67, 319, 386, 404 flexural t h e o r y (inextensional, large deformation, nonlinear - - ) : 3 4 3 , 3 4 5 folded p l a t e s : 175, 180 eigenvalue problems for : 199 t i m e - d e p e n d e n t problems for - - : 203 formal a s y m p t o t i c expansion : convergence of a : 84 leading t e r m in a : 269 m e t h o d o f - - s : 82, 90, 268, 3 5 6 , 3 5 8 , 381 t e r m of order p in a : 269 formal a s y m p t o t i c m e t h o d : 82 force: applied b o d y : 16 applied : 16 applied surface : 16 frame-indifference : 19, 323, 3 2 8 , 3 3 7 , 364, 365, 407 frame-indifferent flexural theory : 343 frame-indifferent m e m b r a n e theory : 340, 353, 355, 365 functional : see "energy" and "minimization of a functional" F - c o n v e r g e n c e : 83~ 95, 101,348 F-limit : 95, 97 GREEN-ST VENANT strain tensor : 260, 267, 326
HELLINGER-REISSNER variational principle : t h r e e - d i m e n s i o n a l - : 23, 82, 120
485
Index
two-dimensional : 71, 76 hierarchic plate t h e o r y : 87 HILBERT uniqueness m e t h o d (HUM) : 118, 203 holes (plates w i t h - - ) : 85, 438 h o m o g e n e o u s elastic material : 17, 260, 325 homogenizat!on t h e o r y : 85, 437 HOOKE's l a w : 19, 76, 88 hyperelastic m a t e r i a l : 262 implicit function t h e o r e m : 263, 323, 364, 450 N A S H - M O S E R - - : 324 inextensional d e f o r m a t i o n : 343 inextensional flexural t h e o r y : 343 infinite energy : 176 in-plane displacement : 64, 313 integration across the thickness : 81 inverted constitutive e q u a t i o n : 294 s c a l e d - - : 295, 300, 380 isotropic elastic m a t e r i a l : 17, 260, 325 junction: between a shallow shell and a plate : 245 between a three-dimensional s t r u c t u r e and between a three-dimensional s t r u c t u r e and between a three-dimensional s t r u c t u r e and between nonlinearly elastic s u b s t r u c t u r e s : - - between plates : 175 between plates and rods : 175 between r o d s : 175, 199 - - between shells : 180 j u n c t i o n conditions : for a coupled, multi-dimensional problem : for a three-dimensional problem : 139 -
-
a plate : 163, 168 a rod : 176 a shallow shell : 175, 178 175
148, 170, 172, 175, 197
KIRCHHOFF-LOVE hypothesis : 76, 81, 120 KIRCHHOFF-LOVE displacement field: 65, 67, 120, 173, 188, 243, 314, 322, 406 scaled : 47, 110, 115, 142, 195,276, 278, 333, 344, 384 KIRCHHOFF-LOVE t h e o r y of a plate : linear : 67, 72, 75, 79, 86, 94, 342 n o n l i n e a r - : 317, 324, 326, 327, 334, 344, 364 KORN's inequality : 13 with b o u n d a r y conditions : 10, 22, 36, 54, 119, 135, 144, 214, 224 w i t h o u t b o u n d a r y c o n d i t i o n s : 10, 54, 119 generalized : 126, 144, 224, 231,236, 239 KUHN-TUCKER relations : 125
486
Index
LAGRANGE m u l t i p l i e r : 77, 125, 172 LAMI~ constants 9 17, 19, 261 a s s u m p t i o n s on the : 27, 78, 94, 108, 113, 139, 173, 181, 192, 198, 216, 265, 329, 379, 441 l a m i n a t e d p l a t e : 85, 88 large deformation nonlinear flexural theory : 343 large deformation nonlinear m e m b r a n e theory : 340, 353 lateral face of a plate : 15 applied surface forces along the : 373 lateral face of a shallow shell : 439 applied surface forces along the : 439 LAX-MILGRAM l e m m a : 21, 23, 39, 50, 55 leading t e r m in a formal a s y m p t o t i c expansion : 269 limit displacement : inside the plate : 64, 76, 313, 404 -of the middle surface : 64, 313, 403, 404 limit e n e r g y : 69, 171 limit scaled energy : 44 limit scaled stress : 58, 62, 302, 387, 444 limit scaled three-dimensional equations : for a linearly elastic c l a m p e d plate : 34 for a nonlinearly elastic c l a m p e d plate : 277 for a VON KARMAN p l a t e " 382 limit stress : 69, 76, 174, 319, 322, 328, 361, 404 linear KIRCHHOFF-LOVE plate t h e o r y : 67, 72, 342 eigenvalue problem : 111 existence of solutions : 67 justification of the : 72 as a small displacement t h e o r y : 73, 335 regularity of s o l u t i o n s : 67 time-dependent : 115 uniqueness of solutions : 67 linear m e m b r a n e equation : 341,432, 446 linearized bending m o m e n t : 70 linearized b o u n d a r y condition of t r a c t i o n : 18 linearized elasticity ( t h r e e - d i m e n s i o n a l - - ) : 13, 24 eigenvalue p r o b l e m : 105 equations of - - : 1 8 o p e r a t o r o f - - : 18, 105 t i m e - d e p e n d e n t problem : 104, 112 linearized e l a s t o d y n a m i c s : 104, 112, 189, 199 linearized strain : 19 tensor : 19 scaled :32 linearized stress couple : 70, 76
Index
487
linearized stress r e s u l t a n t : 70, 76 linearly elastic c l a m p e d p l a t e : eigenvalue p r o b l e m : 105 limit scaled t h r e e - d i m e n s i o n a l equations : 34 scaled t h r e e -d i m e n s i o n a l e q u a t i o n s : 28 scaled two-dimensional energy : 43 scaled two-dimensional equations : 42, 57 t h r e e - d i m e n s i o n a l b o u n d a r y conditions : 18 t h r e e - d i m e n s i o n a l energy : 23 t h r e e - d i m e n s i o n a l equations : 20 three-dimensional HELLINGER-REISSNER variational principle : 23, 82, 120 t h r e e - d i m e n s i o n a l principle of virtual work : 23 t i m e - d e p e n d e n t p r o b l e m : 112 two-dimensional b o u n d a r y conditions : 66, 181 two-dimensional energy : 69 two-dimensional equations : 67 two-dimensional flexural equations : 67, 111 two-dimensional HELLINGER-REISSNER variational principle : 71, 76 two-dimensional m e m b r a n e equations : 67, 169 two-dimensional principle of virtual w o r k : 71 linearly e l ~ t i c m a t e r i a l : 19 linearly elastic m u l t i - s t r u c t u r e s : see "elastic multi-structure" linearly elastic rod : 83, 85, 112 linearly elastic shallow shell in CARTESIAN coordinates : 242 scaled t h r e e - d i m e n s i o n a l equations : 218 scaled two-dimensional equations : 229 t h r e e - d i m e n s i o n a l equations : 213 two-dimensional equations : 242 linearly elastic shallow shell in curvilinear coordinates : 219 linearly elastic shell : 213 LIONS ( l e m m a of J . L . - - ) : 9, 11, 226 live l o a d : 327, 408, 424, 448 load: dead :258 l i v e - - : 327, 408, 424, 448 locking p h e n o m e n o n : 87 lower face of a plate : 15 lower face of a shallow shell : 214 MARGUERRE-VON KJ~RMJ~N e q u a t i o n s " 446 447 AIRY stress function for the : 446 bifurcation of s o l u t i o n s : 446 b o u n d a r y conditions for the : 446 d e g e n e r a c y into the linear m e m b r a n e equation : 446 existence of solutions : 446
488
Index
regularity of s o l u t i o n s : 446 scaled :445 MARGUERRE-VON KARMJ~N shallow s h e l l ' 439 scaled two-dimensional e q u a t i o n s : 443 three-dimensional equations : 440 two-dimensional equations : 446 mass density : I04 a s s u m p t i o n s on t h e - - : 108, 192, 198 m e m b r a n e (tension in a - - ) : 342 m e m b r a n e energy : 69, 317 m e m b r a n e equation ( l i n e a r - - ) : 341,432 degeneracy of the VON KXRM~,N equations into t h e - - " 432 degeneracy of the }vIARGUERRE-VON KARMAN equations into t h e - - " 446 m e m b r a n e equations (of the KIRCHHOFF-LOVE theory) : 67, 76, 112, 402 existence of s o l u t i o n s : 53, 67 regularity of s o l u t i o n s : 54, 67 s c a l e d - : 35, 53, 162 t i m e - d e p e n d e n t - - : 116 uniqueness of s o l u t i o n s : 53, 67 m e m b r a n e theory (large deformation, n o n l i n e a r - - ) : 340, 353, 355, 365 existence of solutions : 340 m e t h o d of formal a s y m p t o t i c expansions : 82, 90, 268, 356, 358, 3 8 1 , 4 4 2 Ansatz of t h e : 269, 338 metric (change o f - - tensor) : 319 m i c r o - v i b r a t i o n s : 199 middle surface of a plate : 16 change of curvature tensor of the : 68, 319 change of metric tensor of t h e : 319 displacement of the m : 64, 313, 403, 404 strain tensor of the : 68, 319 middle surface of a shell : 213 rain-max principle : 107, 109 minimization of a functional : 21, 22, 42, 43, 69, 135, 171,243, 262, 263, 284, 336, 349, 352, 416 m i n i m u m p r i n c i p l e : 107, 109 m o d a l synthesis by s u b s t r u c t u r i n g m e t h o d : 198 m o d e r a t e l y slanted plate : 245, 247 m o d e r a t e l y thin p l a t e : 86 MONGE-AMPI~RE equation 9 416 MONGE-AMP~;RE form 9 391 multiplicity of solutions : see "nonuniqueness of solutions" multiply connected domain : 402, 420, 448 multi-dimensional problem : 163, 170, 175, I97 m u l t i - s t r u c t u r e : see "elastic multi-structure" model: 204
Index NAGHDI plate t h e o r y : 87 NASH-MOSER implicit function t h e o r e m : 324 n a t u r a l s t a t e : 17, 19, 260 NIKODYM set : 388, 395 n o n h o m o g e n e o u s elastic m a t e r i a l : 85 nonlinear elasticity ( t h r e e - d i m e n s i o n a l - - ) : 263 nonlinear e l a s t o d y n a m i c s : 327 nonlinear flexural t h e o r y (large deformation - - ) : 343 existence of solutions : 345 nonlinear KIRCHHOFF-LOVE plate t h e o r y : 317, 3 2 1 , 3 2 6 , 3 4 4 , 356, 364 existence of solutions : 316, 363 generic character of the - - : 327 justification of t h e - - : 321,334 as a small displacement t h e o r y : 324, 335 regularity of solutions : 316 time-dependent :327 nonlinear m e m b r a n e t h e o r y (large d e f o r m a t i o n - - ) : 327, 340, 353, 355, 365 n o n l i n e a r l y elastic c l a m p e d p l a t e : limit scaled t h r e e - d i m e n s i o n a l equations : 277 scaled t h r e e - d i m e n s i o n a l equations : 265, 295 scaled two-dimensional e n e r g y : 283 scaled two-dimensional equations : 279 t h r e e - d i m e n s i o n a l b o u n d a r y conditons : 262 t h r e e - d i m e n s i o n a l constitutive equation : 260 t h r e e - d i m e n s i o n a l energy : 263 t h r e e - d i m e n s i o n a l equations : 262, 294 t h r e e - d i m e n s i o n a l equations of equilibrium : 259 t h r e e - d i m e n s i o n a l principle of virtual work : 259 t i m e - d e p e n d e n t p r o b l e m : 327 two-dimensional b o u n d a r y c o n d i t i o n s : 315, 317 two-dimensional constitutive equations : 317 t w o - d i m e n s i o n a l energy : 283, 316 two-dimensional equations : 279, 315 two-dimensional equations of equilibrium : 317 two-dimensional principle of virtual work : 317 nonlinearly elastic m a t e r i a l : 260 e x a m p l e of a : 260 g e n e r a l - - : 325 nonlinearly elastic m u l t i - s t r u c t u r e : 175 n o n l i n e a r l y elastic rod : 325, 328, 335 nonlinearly elastic c l a m p e d shallow shell : 356 scaled two-dimensional equations : 358 t h r e e - d i m e n s i o n a l equations : 357 two-dimensional equations : 360 nonlinearly elastic shallow shell in CARTESIAN coordinates : 361
489
490
Index
nonlinearly elastic VON KARMAN p l a t e " see "VON KARMAN plate" nonuniqueness of solutions : 288, 313, 426 numerical a p p r o x i m a t i o n : 24, 87, 179, 180, 198, 438 obstacle (plate. lying on a n - - ) : 53, 85, 124, 438 orientation-preserving condition : 217, 336 periodic b o u n d a r y conditions : 84, 323 piezoelectric plate : 86, 118 PIOLA-KIRCHHOFF stress tensor : f i r s t - - : 327 second : 18, 259, 260, 266, 448 place ( b o u n d a r y condition o f - - ) : 16, 258 plate: a n i s o t r o p i c - - : 85 clamped : 15, 16 completely c l a m p e d : 16 controllability o f - - s : 118 elastoplastic : 85, 101 folded s : 175, 180, 199 j u n c t i o n between a and a rod : 175 j u n c t i o n between a - - and a shallow shell : 245 j u n c t i o n between a and a three-dimensional s t r u c t u r e : 163, 168 junction between--s: 175 laminated: 85, 88 linearly elastic c l a m p e d p r o b l e m : 20 linearly elastic VON KARM~,N 9 122 m o d e r a t e l y slanted : 245, 247 m o d e r a t e l y thin : 86 nonlinearly elastic c l a m p e d problem : 262 nonlinearly elastic YON K J ~ R M ~ N - p r o b l e m " 374 lying on an o b s t a c l e : 53, 85, 124, 438 with holes : 85, 438 with rapidly varying thickness : 85, 327 with stiffeners : 175 with varying t h i c k n e s s : 85, 123 partially c l a m p e d - - : 17 piezoelectric : 86, 118 simply s u p p o r t e d - - : 102, 103, 122 thermoelastic :85 v i s c o p l a s t i c - - : 85 VON KARMAN 9375 wrinkled :85 plate theory : classical- : 81,321
Index
491
C O S S E R A T - : 86, 328 f r a m e - i n d i f f e r e n t - : 335, 340 hierarchic :87 linear K I R C H H O F F - L O V E - : 67, 72, 75, 79, 86, 94, 342 N A G H D I - - : 87 m o d e r a t e l y slanted : 245, 247 moderately thin: 86 non-frame-indifferent : 19, 323, 364, 407 nonlinear K I R C H H O F F - L O V E - : 317, 321, 326, 334, 356 derived by a constraint m e t h o d : 88 derived by integration across the thickness : 8 1 , 3 2 8 with rapidly varying thickness : 85, 327 with varying thickness : 85, 123 p r o p e r l y invariant : 335 REISSNER-MINDLIN : 84, 86 small d i s p l a c e m e n t linear : 73 small d i s p l a c e m e n t n o n l i n e a r - : 324 VEKUA : 87 POISSON ratio : 19 pressure: b o u n d a r y condition o f - - : 49 - - load : 424 principle of v i r t u a l work : scaled : 295, 380 t h r e e - d i m e n s i o n a l - : 23, 259, 294, 375 two-dimensional : 71,317 properly invariant plate t h e o r y : 335 quasi-convex envelope : 3 5 1 , 3 5 2 , 3 5 4 quasi-convex function : 351 quasilinear equations : 322, 340 quasi-static p r o b l e m : 116
RAYLEIGH q u o t i e n t : 107, 110 reduced VON K/~RM/f,N e q u a t i o n " 411,426, 429, 434, 435 reference configuration : 16 regularity of solutions of partial differential equations : 48, 54, 290, 302, 316, 404, 420, 446 REISSNER-MINDLIN plate t h e o r y : 84, 86, 88 n o n l i n e a r - - : 328 response function : t h r e e - d i m e n s i o n a l - - : 261 two-dimensional : 318
492
Index
rigidity: flexural of a p l a t e : 67, 319, 3 8 6 , 4 0 4 of an elastic m a t e r i a l : 17 rod : eigenvalue p r o b l e m f o r - - s : 112 junction between a plate and a : 175 j u n c t i o n b e t w e e n a t h r e e - d i m e n s i o n a l s t r u c t u r e and a j u n c t i o n b e t w e e n - - s : 175, 199 linearly elastic : 83, 85, 112 n o n l i n e a r l y elastic : 325, 328, 335 o n e - d i m e n s i o n a l - t h e o r y : 83, 85, 112 shallow : 85, 246 thin-walled : 175 r o t a t i o n a l i n e r t i a t e r m : 118 r o t o r b l a d e : 177, 246
: 176
satellite : 177 scaled d e f o r m a t i o n : 337, 350 scaled d i s p l a c e m e n t : 25, 83, 89, 125, 3 5 1 , 4 4 1 convergence o f - - s : 34, 83, 100, 109, 142, 199 scaled e i g e n f u n c t i o n : 108 c o n v e r g e n c e o f - - s : 109 scaled e i g e n s o l u t i o n : 1 0 8 scaled eigenvalue : 108 convergence of - - s : 109 scaled e n e r g y : 30, 43, 97, 283, 351 scaled i n v e r t e d c o n s t i t u t i v e e q u a t i o n : 295, 300, 380 scaled KIRCHHOFF-LOVE d i s p l a c e m e n t field : 47, 110, 115, 142, 185, 195, 276, 278, 333, 344, 384 scaled linearized s t r a i n s : 32 scaled MARGUERRE-VON K/~RMAN e q u a t i o n s " 445 scaled principle of v i r t u a l work : 295, 380 scaled stress : 57, 58, 83, 295 c o n v e r g e n c e o f - - es : 58, 84 limites : 58, 62, 302, 387, 444 scaled t h r e e - d i m e n s i o n a l e q u a t i o n s : - - for a linearly elastic c l a m p e d p l a t e : 28 - - for a linearly elastic m u l t i - s t r u c t u r e : 140 for a linearly elastic shallow shell : 237 for a n o n l i n e a r l y elastic c l a m p e d p l a t e : 265, 295 for a VON KJ~RMAN p l a t e " 380 scaled t w o - d i m e n s i o n a l e q u a t i o n s : for a linearly elastic c l a m p e d p l a t e : 42, 57 for a linearly elastic shallow shell : 229 for a MARGUERRE-VON K/~RM/~N shallow shell" 443
Index
493
for a nonlinearly elastic c l a m p e d plate : 279, 289 for a nonlinearly elastic shallow shell : 358 for a VON K~RM~N p l a t e " 384 scaled vector field : 27, 295 scaled VOU K~RM~N e q u a t i o n s " 390 scalings of the d e f o r m a t i o n s : 337, 350 scalings of the displacements : 25, 83, 89, 125, 137, 1 8 3 , 2 1 5 , 2 6 4 , 295, 329, 357, 378 justification of the : 93, 329 scalings of the s t r e s s e s : 57, 83, 295, 378 justification of the : 58,120 semilinear equations : 3 2 2 , 4 0 7 shallow rod : 246 shallow shell : definition of a - - : 216, 245, 246, 247, 362, 441 elastic m u l t i - s t r u c t u r e with a : 175, 178, 246 j u n c t i o n between a and a plate : 245 linearly elastic : 242 MARGUERRE-VON K/~RMAN " 439 nonlinearly elastic : 356 shell : 212 j u n c t i o n between s : 180 middle surface of a : 213 shallow :245 thickness of a : 213 simply c o n n e c t e d d o m a i n : 389 simply s u p p o r t e d plate : 102, 103, 122 singular p e r t u r b a t i o n s : 46 singularities of solutions of partial differential equations : 24, 53, 56, 174 sliding e d g e s : 323 small d i s p l a c e m e n t t h e o r y : l i n e a r - - : 73, 335 nonlinear : 324, 3 3 5 , 3 6 2 SOBOLEV space : 8, 58, 261, 291 solar panel of a satellite : 177 spectral t h e o r y : 1 0 6 , 4 2 7 ST VENANT-KIRCHHOFF m a t e r i a l : 260, 262, 263, 354, 357, 374 stored energy function of a : 336 stabilization of elastic m u l t i - s t r u c t u r e s : 203 s t a t i o n a r y point of a functional : 262, 283, 361 s t a t i o n a r y s o l u t i o n : 105, 189 stiff eigenvalue p r o b l e m : 198 stiff elliptic p r o b l e m : 171 stiff t i m e - d e p e n d e n t p r o b l e m : 202
494
Index
stiffener ( p l a t e w i t h s) : 175 s t o r e d e n e r g y f u n c t i o n : 349 of a ST VENANT-KIRCHHOFF m a t e r i a l : 336, 349 s t r a i n : 260 l i n e a r i z e d "---s : 19 scaled s : 32 s t r a i n t e n s o r : 319 CAUCHY-GREEN (right - - ) : 326, 337 GREEN-ST VENANT : 260, 267, 326 linearized : 19 t w o - d i m e n s i o n a l - - : 318, 322 s t r e s s : 18, 259 AIRY f u n c t i o n : 403, 406, 446 c o n v e r g e n c e o f - - es : 83, 121, 189 e r r o r e s t i m a t e s for es : 102 limit : 69, 76, 174, 319, 322, 328, 361, 404 limit scaled : 58, 6 2 , 3 0 2 , 3 8 7 scaling ofes : 57, 83, 295, 378 stress c o u p l e : 321 linearized : 70, 76 stress f u n c t i o n ( A I R Y - - ) : see "AIRY stress f u n c t i o n " stress r e s u l t a n t : 321, 406 linearized : 70, 76 stress t e n s o r : first P I O L A - K I R C H H O F F - - - : 327 linearized : 18 s e c o n d PIOLA-KIRCHHOFF : 18, 259, 260, 2 6 6 , 4 4 8 s t r i n g : 328, 348 s u b s t r u c t u r e : 133 s u b s t r u c t u r i n g m e t h o d ( m o d a l s y n t h e s i s b y - - ) : 198 s u r f a c e force ( a p p l i e d - - ) : 16 t e n b i o n in a m e m b r a n e : 342 tensor: c h a n g e of c u r v a t u r e : 68, 319 c h a n g e of m e t r i c : 319 CAUCHY-GREEN s t r a i n : 326, 337 first PIOLA-KIRCHHOFF s t r e s s : 327 GREEN-ST VENANT s t r a i n : 260, 267, 326 linearized strain : 19 l i n e a r i z e d stress - - : 18 s e c o n d PIOLA-KIRCHHOFF stress : 18, 259, 260, 266, 448 three-dimensional elasticity : 20, 2 6 1 , 3 1 8 two-dimensional elasticity : 68, 319
Index
495
two-dimensional strain : 318 t e r m of o r d e r p in a formal a s y m p t o t i c e x p a n s i o n : 269 t h e r m o e l a s t i c p l a t e : 85, 420 t h i c k n e s s of a p l a t e : 16 i n t e g r a t i o n across t h e : 81 plate with varying : 85, 123 rapidly varying : 85,327 t h i c k n e s s of a shell : 213 t h i n - w a l l e d r o d : 175 t h r e e - d i m e n s i o n a l elasticity t e n s o r : 20, 24, 2 6 1 , 3 1 8 t h r e e - d i m e n s i o n a l e n e r g y : 23, 263 three-dimensional equations : for a linearly elastic c l a m p e d p l a t e : 20 for a linearly elastic m u l t i - s t r u c t u r e : 135 for a linearly elastic shallow shell : 213 for a MARGUERRE-VON KJ~RMJ~N shallow shell" 440 for a n o n l i n e a r l y elastic c l a m p e d p l a t e : 262 for a n o n l i n e a r l y elastic shallow shell : 357 for a VON K/~RM/~N p l a t e " 376 of e q u i l i b r i u m : 259, 364, 375 t h r e e - d i m e n s i o n a l HELLINGER-REISSNER v a r i a t i o n a l principle : 23, 82, 120 t h r e e - d i m e n s i o n a l principle of v i r t u a l work : 23, 259, 294 t h r e e - d i m e n s i o n a l set w i t h a t w o - d i m e n t i o n a l slit : 163 time-dependent problem : c o n t r o l l a b i l i t y of a : 118, 203 coupled, p l u r i - d i m e n s i o n a l , : 202 existence t h e o r y for - - s : 113 s t i f f - - : 202 - - for a VON KJ~RMfi~N p l a t e 9 420 for an elastic m u l t i - s t r u c t u r e : 199 for folded p l a t e s : 203 in t h r e e - d i m e n s i o n a l linearized elasticity : 112 t w o - d i m e n s i o n a l - - for a linearly elastic p l a t e : 115, 117 two-dimensional for a n o n l i n e a r l y elastic p l a t e : 327 t r a c t i o n ( b o u n d a r y c o n d i t i o n o f - - ) : 259 t r a n s m i s s i o n c o n d i t i o n s : 136 t r a n s m i s s i o n p r o b l e m : 136 t r a n s v e r s e d i s p l a c e m e n t : 64, 313, 403 t w o - d i m e n s i o n a l b o u n d a r y c o n d i t i o n s of a linearly elastic c l a m p e d p l a t e : 66 t w o - d i m e n s i o n a l b o u n d a r y c o n d i t i o n s of a n o n l i n e a r l y elastic c l a m p e d p l a t e : 315, 317 t w o - d i m e n s i o n a l b o u n d a r y c o n d i t i o n s of a YON KJ~RMJ~N p l a t e 9403, 407 t w o - d i m e n s i o n a l c o n s t i t u t i v e e q u a t i o n s of a p l a t e : 317, 322 t w o - d i m e n s i o n a l elasticity t e n s o r : 6 8 , 3 1 9
Index
496
two-dimensional energy : 43, 69, 2 8 3 , 3 1 6 two-dimensional equations : - - f o r a linearly elastic c l a m p e d plate : 67 - - f o r a linearly elastic shallow shell : 242 - - - f o r a M~ARGUERRE-VON NJ~RM~N shallow shell" 446 for a nonlinearly elastic c l a m p e d plate : 315 - - - f o r a nonlinearly elastic shallow shell : 360 - - - f o r a VON KJ~RM~N plate 9386 two-dimensional equations of equilibrium : 317, 322 two-dimensional flexural equations : 67, 75, 172 two-dimensional HELLINGER-REISSNER variational p r i n c i p l e : 71, 76 two-dimensional m e m b r a n e equations : 67, 76, 112 two-dimensional plate theory : 79, 81 two-dimensional principle of virtual work : 7 1 , 3 1 7 two-dimensional response function : 318 two-dimensional slit in a three-dimensional open set : 163 two-dimensional strain tensor : 3 1 8 , 3 2 2 two-scale convergence : 85 -
-
uniqueness of solutions : 21, 48, 53, 164, 238, 247, 288,328, 426, 450 u p p e r face of a plate : 15 u p p e r face of a shallow shell : 214 VEKUA plate theory : 87 viscoplastic plate : 85 VON K/~RMAN equation ( r e d u c e d - - ) ' 411,426, 429, 434, 435 VON KXRMXN e q u a t i o n s " 403, 404 AIRY stress function for the : 403 bifurcation of s o l u t i o n s : 433, 434, 435, 436, 437 b o u n d a r y conditions for the : 403, 407 canonical :409 degeneracy into the linear m e m b r a n e equation : 432 equivalence with a two-dimensional displacement probl em : 389 existence of s o l u t i o n s : 404, 4 1 6 , 4 2 0 , 448 generic character of the : 407 MARGUERRE9 see "MARGUERRE-VON K~,RM/~N equations" nonuniqueness of solutions : 426 regularity of s o l u t i o n s : 404, 420 s c a l e d - - : 390 time-dependent :420 uniqueness of s o l u t i o n s : 426, 450 VON KARMA~N p l a t e " 375, 376, 425, 447 buckling of a : 424, 437 c i r c u l a r - - : 437 controllability o f - - s : 420
Index
linearly elastic : 122 scaled t h r e e -d i m e n s i o n a l e q u a t i o n s : 380 scaled two-dimensional equations : 384 t h e r m o e l a s t i c - - : 420 t h r e e - d i m e n s i o n a l b o u n d a r y conditions : 374, 376 t h r e e - d i m e n s i o n a l constitutive equations : 374 t h r e e - d i m e n s i o n a l equations : 374 two-dimensional b o u n d a r y conditions : 386 two-dimensional e q u a t i o n s : 3 8 6 , 4 0 3 lying on an obstacle : 438 with holes : 438 weak lower semi-continuity of a functional : 284, 347, 418, 449 wrinkled p l a t e : 85 YOUNG m o d u l u s : 19, 404
497
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