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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 _
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 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, 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 _ 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 _
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, /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 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~ 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 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
0o 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<e0, 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"
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 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, 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
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 _ 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 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 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,~
0 such that C5]]T]3]]2,w 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, 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 1 and 0z~a E L2(w),
OB{@aBcrTOcr 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 _ -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) 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 0, c~ > 0, fl 6 R, and 1 < p < ec such that ^
II~(F)I _% c~lF[ p + fl for all F 6 M 3. It can be verified that the stored energy function of a St VenantKirchhoff material satisfies such inequalities with p = 4.
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) _ 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,_ 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(@) 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, 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 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
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(/~ + >)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
