Classical and advanced theories of thin structures: mechanical and mathematical aspects

CISM COURSES AND LECTURES Series Editors: The Rectors Giulio Maier - Milan Jean Salençon - Palaiseau Wilhelm Schneider...

Author: Antonio Morassi | Roberto Paroni

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CISM COURSES AND LECTURES

Series Editors: The Rectors Giulio Maier - Milan Jean Salençon - Palaiseau Wilhelm Schneider - Wien

The Secretary General Bernhard Schreﬂer - Padua

Executive Editor Paolo Seraﬁni - Udine

The series presents lecture notes, monographs, edited works and proceedings in the ﬁeld of Mechanics, Engineering, Computer Science and Applied Mathematics. Purpose of the series is to make known in the international scientiﬁc and technical community results obtained in some of the activities organized by CISM, the International Centre for Mechanical Sciences.

INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES COURSES AND LECTURES - No. 503

CLASSICAL AND ADVANCED THEORIES OF THIN STRUCTURES MECHANICAL AND MATHEMATICAL ASPECTS

EDITED BY ANTONINO MORASSI UNIVERSITY OF UDINE, ITALY ROBERTO PARONI UNIVERSITY OF SASSARI, ITALY

This volume contains 32 illustrations

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned speciﬁcally those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. © 2008 by CISM, Udine Printed in Italy SPIN 12048489

All contributions have been typeset by the authors.

ISBN 978-3-211-85429-7 SpringerWienNewYork

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K )* 2 * 2EBB933 % 4 . D : D !D D 2 3 D 2% 3 D 2% 3

N% 3 N% 3 N% N% N%

2HH3

( > N : ! )* .**)5 !23 D 23 )*. K) 5 D 5 ! >D D ; * ! !(* .*) 3N= 0 *)) * : ! *K ( ))( )-4" : 2 * 2EBB933 2% 3 D 2% 3 % * % & 2HJ3 >7;

#A-)( -)

! * ( )*. K) * * * .*)) . = )5 * % )*: 23 *.) )*. . K) ! !23 ! 23 D /

* / B%

2HL3

* !* K . ( * !/ /= 5 / * * . *) **. * 2/ 3 * . / = # %*)! 2/ 3 .* * . ) ) !)= 0 *%

D 2/ !3 D / %

2H93

( D !5 *

D D ; 4 /2 4 3 4 2/ 3&

2H<3

#

/ & D ; 4 2 4 3 4 2/ 3 E * . )* . . D & ( *%

K

2JB3

/ D ; 4 2 3 4 2/ 3& E

2JC3

C .% 2 4 3D E

2JE3

Strain, Stress and Linearized Elasticity C &(& 2 3D/ E

23 2JF3

# ( 2/ 3 .5 ( *% & D ; 4 % (

D /%

D ; 4 / %

2JG3

/ D //%

2JH3

* *)) 8-)( ) * 8-)( ) 5 %)= # */ % !) ( * ( / !O) .*))5 -* )*

. D & )* / *

*) . .. * &( *> D 4 / = ( ) . ) ( :%*) / * * / * * 8-)( -) = K.*) .* 5 . * ) * K/5 * **5 * % )!. * * *) /( ( K/= $ *5 / ! * E=G5 ) *) ) ( * / %*)!* *

D

D 2; 4 E 4 2/ 33 D

D C 4 4 2/ 3% *

D %

2JJ3

( . 2/ 35 .* ( = 2EB3= ) ))5 / -* .!)* 2EC3 * ! *: !. D B5 D C5 D C% E5 ( *%

! D C 4 4 4 2/ 3%

2JL3

* 5 ( . 2/ 35

! D 4 &

2J93

.)*)5 / )** = 2EG35 ) )( ( *) K/ D 5 * D C% E5 % / 0 D E &

2J<3

24

A. Morassi

) # ( -5 ) / * %)

) ! * D - = )* . . ( *%

D 2 & 3 D & D & = *)) K

.* ( /*

&

2; 4 3 D 2; 4 3 2; 4 3 2; 4 3 C 4

D C 4 &

# 5 * % )!. . / * >

" D &

" >7<

2LB3

!$ !) A-)( -)

. )* 2HH3 K.*)

.* = # .) * 5 * ( . * * * ( ( * = *!. !)* !5 * % )) ( ( > 23 D 23

% * % % 2LC3 23/ D /23 % / &( * % & 2LE3 5 % * ! D 235 ( / 2 3 %*% = 2LC35 2LE3 ()) %*)!* * D 5 * 2 3 D

/ % * = 0 .*)) %) ! / #*) *.)

)*. / ! * / B= 0 *%> / 2 3 D 2; 4 / 3 D 2;3 4 /2;3 4 2;3 4 2/ 3% 2LF3 E ( 2/ 3 * . ( / 5 == ). 2/ 3/ D B= 0 *)) / 6 D 2;3 D 2;3 " 2LG3 )( 5 * ( / ( ! K!* = 23 D "2 3 *

6 ( *% 2;3 D "5 6 .& " 2LH3

Strain, Stress and Linearized Elasticity

25

(5 . .. %*%

3 3 3 3 D D D % 3 3 3 3 ! ! * 2;3 * .. !: =

2/ 3:. #*) %) . 2LF35 / ! ( *% 2LE3 ( D ; * / D /5 4 3 D 2;3 D 2;32/ D " 6 4 / " 6 4 2;3 D/ D 6 " 6 4 / " 6 4 2;3 D "

2LJ3

% 6 " 6 4 2;3 D " ( )* ) D 5 *)) 5 5 5 * / ! = # 5 ( *% 2 3

6 4 / " 6 4 D 2;3 4 /" E / " 6 3 4 2/ 3& 4 2 2;3 E

2LL3

0 * ( .! )* !% !* )*: , P .*)) /= = 2LL3 ( *%

D 2 3 D 2 3 D 2; 4

3

D

" 6 3 4 2/3 D 6 4/" 6 4 / 2 2;3 D" E / " 6 4 2/3 D 6 4/" 6 4 / 2;3 D"

4 2/3% 6 4 // " 6 4 / 2;3 D"

2L93

* / B&

# " ( ' ! ) ! 1 " . ' ) !! . ) ! " ' ! " " . $ % !! $ % " ! . ' . - / 0)

26

A. Morassi

*)) = 2GJ35 *! !*) "

4 2/33 2" 6 4 // " 6 4 / 2;3 4 2/332; 4 D 2C 4 / 4 2/332; 4 4 2/332" 6 4 // " 6 4 / 2;3 D 2C / /23 "32; 6 6 4 // " 6 4 / 2;3 4 D 2"

3

3 3

D

D

4 2/3 D

6 4 /" 6 4 // " 6 4 / 2;3 /23 " 6 4 2/3 D D" 6 C / 4 E/ " " 6 4 / 2;3 4 2/3& 6 D"

/ 2L<3 .* = 2L93 * 2L<3 % * : 6 D :5 )*:, P K!* *!*)5 * " P . *! = %5 / = 2L<3 6 D : *! *) )) ( * " &(:.. * )*. * = 6 D :5 =2L93 * 2L<3 ! %) $ " D 4 2/3%

" D 4 2/3

* / B%

29B3

* D " ( .= 5 !: .. 29C3 D 2;3 () .**) K!* = # )* *K :*)) - -- D D %

% % % D C% E% F&

29E3

0 )! ( * 2LC3 * 2LE3= # K )) ( / P* *.: P 23 D 23 % ( ) % * % 5 ( = ! ! ! 23 D = ! %* !) ( /*

3 2 233 3 23 3 23 D D 23 & I 238 D 3 3 3 $ % 2LE3 ( ! )) ( ()):& ( !) 2 -= F E=J3> * / &( * * + IB% E,85 2+3 D ; 4 +/ 4 2C +3 / &

29F3

Strain, Stress and Linearized Elasticity

27

2LC35 ( *% 23 23 D 23 23 % &

D 2+3 * = 29F35 ( + .*))= 2+3 D ; 4 +/ 4 + 5 * / &(5 ( /*

; 4 +/ 4 + 23 23 D ; 4 +/ 4 + 23 %

* 2*!. D 2% 3 !)* !3 +/ 23 D + 23 / 4 + & % / + * *& ). * + B5 )) (= >7=

" . !) $)(! .(- () ()

0 *)) )* !)/!. /). K )* K: .*) .* * . * K!* ( * *!*) * * :)* .**)= = 29B3 ( *% % D % 2 3 4 2/3

N% * / B&

29G3

0 *!. * *!*) / : .*))> . 2,3

D /.2,3

2N3&

29H3

# 5 / 2HL3 ( *% . 23

D . 23 4 2/3

N * / B&

29J3

= 29H35 29J3 = 2HH3 5 !)/!. !* / . 4 2/3 D B 4 . %

N * / B

* 5 ) .5 % 2 3 4 . D :

N%

29L3

( ()):& ( . **) P*) !* )*

)*= # / ! * 2HH3 * / ! * )* .)*)=

28

A. Morassi

$ )! 5 !/ / * 5 / ! * %*)! /: ). )* )* * / .!)* * )) (> K : )*. K) !5 ! > N 5 ! () 5 )! % 2!3 4 . D : N% 2!3 D 3 N% 23 2993 !D 3 N% ( * * !.* * ) 2!)*3 **5 %)5 * 3 N 3 N D 3N= $ - ( *)) . !! * - !) /). 23 K = 29935 *) 2EBBG35 * 2C7>

- , ) !! !( () ()5 .(-

$ ( *)) .*) ( 23 /). 2993 . ! ) * !.* / ! * > % 2!3 4 . D : N% 2!3 D : 3 N% 29<3 23 !D: 3 N% 6 ( 3 N D 5 3 N D 5 36 N 3 N D 5 3 N 3 N D 3N= 0 *)) ! )) ( *: *!. > 3 2 -) N3 N * 5 / ! * !/ 5 D F5 * * / ! * 3N )* ( * 5

A . %5 N 5 ( B= 3 2 -))( 3 # )* > N .. 2 D 5 . 5 D 5 % % % D C% E% F35 * . .. D D 5 % % % D C% &&&% A . %5 ) %- N> - ) ( B ! * 23 ) 5

! 2' 2 ,' , ' 2' ! . , $% 2 $,% . 3 $ %' . $ %' ' ' 3 $,% ' 3 $% . , 3 $ % ) 3 $ % . 3 $

%

& 3 $

% )

Strain, Stress and Linearized Elasticity

29

% . *== NA K*))5 2N35 * 2N35 % % % D C% &&&% = 3 2 ))3 # / .5 . > N 5 !* # " ' = */) N> . D

%5 ( *)) *!. * / ! * 3 N ( )*. % 2* !*) 3 () = )> - 3 N ! * 3N 2 3 3 N% ( 2 3 D ' = 0 *% )) ( ()):& ( !)= " - >7;7 1, ) -, 12 .(- ' - ( ) -) ))2 % ). # ) - 3 .(- 29<3 ) ) ( 2N35 # 27<7 2N3 = D 4

)/ * $ 2N35 ( .

5 0 *)) ! )) ( * .5 2C<9J3= " - >7=7 . ) (. ) ) ( !2% 3 > ) .()3 3 # - 5 4233 4 > 3 . ) () ) )( 5 !3 , ) ) ! 2% 3 D 4 23 # & 2

= $ ! ! * * 0 K D $ 2N3 % $ D : 3 N )/ * ( . = .!)) !)/!. !*

"' $2%) # 1' 2' $% . $% ! ) $2% 2) ! . 2' "

$2% 1 )) ) 1 )

30

A. Morassi

/ $ 5 * / * * *& / ! * 29<3 5 29<3 * !5 ( /* = 2 *

4 23 > K / ! $% 2<E3 ! 2!% $3 D

4 2$3 D

. $&

2

* = 2
* 4 23 ! != # . ! 2% 3 /% !) /)* * ! ! 5 / *!: (* !*)5 ( *% ! 2$ % $ 3

$ $ %

2
( D ! 23 = # % % ! 2% 3 ( ( * - * * * ( B ! * ! 2$% $3 *$

% $ &

%- ( *% 2$3 & ! 2$% $3 )

2<J3

$ .* . /) ( : . K.*) * ( *)) ( . * !*)5 5 -*.)5 2$3 , *1% * )& 2C<9<3= )9 !)( N / * / ! .* )* ( * 5 * ! * N 5 E= % $ 2N% 3 ! * $D:

3N 23%

( . 3N5 ( *% $ $ %

2
Strain, Stress and Linearized Elasticity

31

( ( B * * ) * =

+ 0 !)( N / * / ! .* )* ( * 5 5 * ! * N 5 E= % $ 2N% 3 ! * 3N 23%

$D:

( . 3N5 ( *% $ 2$3 %

2<93

( ( B * * ) * = (5 / *@1 * , 1 !*) ( *% $ 2C 4 3 2$3 %

2<<3

* ! 2$% $3 % > ! 2$% $3

) $ 2C 4 3

% $ &

2CBB3

* . ( * )! * - * !! )! 29<3= *))5 / $ D ! 2 !

2C 4 3 . & )

2CBC3

# -5 !! * */) )* )* /). * / % .)*)= $ !)5 ( *)) *: ( * ( 3 N D 3 N D =

32

A. Morassi

" - >7>7 1, ) ( .(- ' - ))2 % ). # ) - 3 .( # 23 4 D : N% 2CBE3 12 D: 3N%

) ) ( 2N3 D > N 2N3 % D : 3N )

2CBF3 % ' ( B ) ) ( % % % ) 5 5 # )) ( *. ) 23 = 5 ( *)) )) ( ()):& ( % *@ !*) 2N3= N / * / ! .* 5 E5 ( / ! * 3N ( * 5 * ! * N = # - ( B5 ) * 5 ! * $ $ % $ 2N3& 2CBG3

# , 1 !*) 2N3 * * )) (> ! */ % *!.: N5 ( *% $ E 2$3 % $ 2N3& 2CBH3

$ *5 * )* !)* ! 5 */ % !*) )) ( . C C 2$3 D $ 4 2 % $3 % $ 2N3& E E

" - >7?7 1, ) -) .(- ' - ))2 ) ( .)( (. - 6 D :3 D :5 !3 ). # ) - 3 .( # 23 4 D : N% 2CBJ3 12 23 D : 3N%

Strain, Stress and Linearized Elasticity

33

) ) ( 2N3 ) D :% 23 D :&

2CBL3

# %

2CB93

' ( B ( 3 3 3 ) 5 5 $ * % ( & !/* 2N3

K / $ D :% $ 2$3 D : & D $ 2N3

# % /)* . ! 2% 3 )) ( . ( !*) /) (= )9 0 !)( ' */ % *!. N5 ( *% $ C $ $ % $ 2N3 % N

2CB<3

( ) * = + 0 !)( ' */ % *!. N5 ( *% $ C $ 2$3 $ N

% $ 2N3 %

2CCB3 ( ) * = ! ( !*)5 # . )) ( *. ) * 23 * 23 =

-)% >7@7 $ *)) % ! /).5 /)* . ! 2% 3 .: .5 * ! 2!% $3 D ! 2$% !35 % $% ! = $ * )! ! * )* /). * / ! / )%: %** *) /). 2!3 D . 2$3 %

( 2$3 D

! 2$% $3

4 2$3 *) /).=

34

A. Morassi

% '(- $ ( *& = !* ))* * ** .*!=

= 5 )( )(5 ! 5 * )5 C<9J= == *)5 )-))( ()5 * ( - 76 ! - )( ()4 5 )%5 . *.5 EBBG= = ) 5 - )5 ' ) 1 C<9J5 *5 #!5 C<9J= = *5 , - ()= $ ) . $ 5 *?E5 5 ); )/; ( &5 C>5 C9L;C

= #! ))5 ! (- - )5 5 ); )/; ( &5 C<JJ= #= +)5 ) )( .(- ()5 5 ); )/; ( &5 C<99=

! %&&'&( #'&! % $&! %&&) 0I
"" """# & $ !" "# !! && # " && # "% " " #" " " " % ! ""% ! $ " !"" !" " " "! #! " # ! "

&$!'&!

'H NB?M? HIN?M ' ' >CM=OMM NB? ?RCMN?H=? I@ MIFONCIHM NI NB? HIHFCH?;L ?F;MNC= JLI NB? HIN?M I@ NB? F?=NOL?M >? FCP?L?> ;N !'1+ ' >C> HIN NLS ;N ;FF NI NI ACP? ; MCGJF? ;H> >CL?=N CHNLI>O=NCIH NI NB? MOC?H=? B;> P?LS >C?L?HN <;=EALIOH>M ' >?=C>?> NI CHNLI>O=? NB? =IH=?JNM IHFS ;@N?L B;PCHA MBIQH NB;N NB?S Q?L? CH>??> H?=?MM;LS @IF FIQCHA NB? F?;LH H?Q G;N?LC;F IH=? CM EHIQH QB;N CM H??>?> @IL M ; L?MOFN NB? G;HOM=LCJN FIIEM ; ; J;NB QBC=B F?;>M NI NB? >CM=OMMCIH I@ ?RCMN?H=? I@ MIFONCIHM NI NB? HIHFCH?;L ?F;MNC= JLI ?NIOLM ;L? N;E?H NI CHNLI>O=? NB? G;NB?G;NC=;F 7G;=BCH?LS H??>?> NI AI ;FIHA $IFFIQCHA NB? J;NB ' NLC?> NI ??> ;FF NB? N?=BHC=;FCNC?M B;P? ?L NB? =;LJ?N MI NI MJ?;E ;H> IHFS NB? G;DIL L?MOFNM ;H> C>?;M ;L? JL?M?HN?> .LII@M ;L? ACP?H IHFS C@ MCGJF? ;H> ;FIHA NB? Q;S G;HS L?MOFNM ;L? F?@N ;M ?R?L=CM? 2B? L?;>?L ;==OMNIG?> QCNB 1I Q?;E =IHP?LA?H=? =;H MECJ ;FF NB? >?NIOLM ,??>F?MM NI M;S NB? M?LCIOM L?;>?L MBIOF> =IHMOFN NB? L?@?L?H=?M KOIN?> MCH=? NB? JL?M?HN J;J?L BIJ?M NI O=NCIH NI NB? MO
36

R. Paroni

-1 1(-, 2B? HIN;NCIH OM?> NBLIOABION CM KOCN? MN;H>;L> 4CNB CH 1SG ;H> 1- Q? >?HIN? NB? M?N I@ JIMCNCP? FCH?;L NL;HM@ILG;NCIHM NB? M?N I@ JIMCNCP? >?HCN? MSGG?NLC= NL;HM@ILG;NCIHM ;H> NB? M?N I@ LIN;NCIHM L?MJ?=NCP?FS 2B? #O=FC>?;H $LIO=N CM >?HIN?> <S ;H> NB? =ILL?MJIH>CHA HILG <S ! ! 2B? AL;>C?HN CM >?HIN?> <S # ;H> ; <;L IP?L >?HIN?M NB? =FIMOL? I@ $CH;FFS ; M?N >?HIN?M NB? =FIMOL? @IL CHMN;H=? NB? =IHMN;HNM ;L? >?HIN?> <S NB? F?NN?L NB? P;FO? I@ G;S =B;HA? @LIG FCH? NI FCH?

%& )"$%& &$%

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(&2/$ ?N OM =IHMC>?L ; M?=IH> JF;=?G?HN I@ NB? S ;H> F?N JF;=?G?HN =IHMC>?L?> 4? MB;FF >?HIN? <S # NB? >?@ILG;NCIH AL;>C?HN $,(1(-, G;N?LC;F CM M;C> NI ?N?LGCH?> <S NB? >?@ILG;NCIH AL;>C?HN 2B;N CM NB?L? ?RCMNM ; G;JJCHA V MO=B NB;N 5 V

QB?L? 5 2B? G;JJCHA V CM =;FF?> NB? L?MJIHM? QCNB L?MJ?=N NI

Constitutive Equations and Variational Elasticity

37

1CH=? NB? L?F;NCIH NB? !;O=BS MNL?MM N?HMIL M?? +IL;MMC CM ACP?H <S >?N

Q? B;P? @IL ;H ?F;MNC= G;N?LC;F >?N V

$,(1(-, H ?F;MNC= G;N?LC;F CM '6.$/$ 01(" C@ NB? .CIF;)CL=BBI MNL?MM B;M ; JIN?HNC;F NB;N CM C@ NB?L? ?RCMNM ; M=;F;L @OH=NCIH CH MO=B NB;N

2B? M=;F;L @OH=NCIH CM =;FF?> NB? MNIL?>?H?LAS @OH=NCIH 2B? H?RN NB?IL?G QBIM? JLII@ =;H CH NB? GINCP;N?M NB? >?HCNCIH I@ BSJ?L?F;MNC=CNS '$-/$+ ( &-.$ )3 $- #3 , &-.$ $! ( )(&3 $! "$0 ( (3 ,. ( (3 .$' $(. ,0& 6+ + 8 .# 1),% 9 +

$- ()(( ".$0 $( &)- ,) -- -

! $ ! '$&!

?N OM =IHMC>?L NQI L?@?L?H=? =IHAOL;NCIHM ;H> I@ NB? S ;H> F?N

?N M?? $CAOL? ;H> L?MJ?=NCP?FS NI NB? M;G? JF;=?G?HN ;H> F?N #

#

1CH=? Q? F?N @OLNB?LGIL?

38

R. Paroni

(&2/$

?N ;H> L?MJ?=NCP?FS '@ >?HIN?M NB? !;O=BS MNL?MM N?HMIL Q? B;P? >?N

2BOM

>?N

>?N

;H> B?H=?

>?N >?N

>?N >?N

?N ?H?LAS >?HMCNS QCNB L?MJ?=N NI NB? L?@?L?H=? =IHAO L;NCIH ?N OM M?N >?N QCNB

3MCHA NB? =B;CH LOF? ;H> ?KO;NCIH Q? =;H ?;MCFS MIFP? NB? @IFFIQCHA 5$/"(0$ #)1 .#.

Constitutive Equations and Variational Elasticity

39

#KO;NCIH ;FFIQM NI =;F=OF;N? NB? MNIL?>?H?LAS >?HMCNS QCNB L?MJ?=N NI ; L?@?L?H=? =IHAOL;NCIH IH=? CM EHIQH NB? MNIL?>?H?LAS >?HMCNS QCNB L?MJ?=N NI ;HINB?L L?@?L?H=? =IHAOL;NCIH FMI ?KO;NCIH MBIQM NB;N NB? >?HC NCIH I@ ; BSJ?L?F;MNC= S >I?M HIN >?J?H> IH NB? L?@?L?H=? =IHAOL;NCIH

&$ )&$)

?N M?? $CAOL? ?N OM =IHMC>?L ; J;LNC=OF;L =;M? I@ NB? G;JJCHA =IHMC>?L?> CH NB? JL?PCIOM M?=NCIH QCNB 1- QB?L? 1- >?HIN?M NB? M?N I@ LIN;NCIHM 2B? G;JJCHA MCGJFS LIN;N?M NB? S ;LIOH> NB? JICHN ?N OM >?HIN? <S NB? JF;=?G?HN ?N ;H> ?H?LAS >?HMCNC?M QCNB L?MJ?=N NI NB? L?@?L?H=?M ;H> L?MJ?=NCP?FS

(&2/$ '@ Q? =;H H> ; 1- MO=B NB;N

BIF>M @IL ?P?LS CH Q? MB;FF B;P? MBIQH NB;N NB? L?MJIHM? I@ NB? JICHN CM DOMN NB? M;G? QCNB L?MJ?=N NI NB? NQI L?@?L?H=? =IHAOL;NCIHM =IHMC>?L?> ;H> 2B;N CM NB? MNIL?> ?H?LAS >?HMCNS =;H HIN >CMNCHAOCMB NB? NQI L?@?L?H=? =IHAOL;NCIHM 2B? LIN;NCIHM 1- @IL QBC=B BIF>M >?HIN? MSGG?NLC?M I@ NB? G;N?LC;F I==OJSCHA NB? JF;=?G?HN 4? >?HIN? <S 1- ?KO;NCIH BIF>M

40

R. Paroni

2B? M?N CM =;FF?> NB? 06++$1/6 &/-2. I@ NB? JICHN CH NB? JF;=?G?HN

5$/"(0$ #)1 .#. $- ",)/ $LIG QCNB ;H> Q? H> QBC=B CM

CH

2BOM @LIG ;H> Q? =;H ?;MCFS H> NB;N 1-

CH

$,(1(-, H ?F;MNC= G;N?LC;F CM M;C> NI

&$ $ $

$IL I@ NB? JLCH=CJF? I@ . ,$& ,' ($ , ( 2B?M? NQI JLCH=CJF?M ;L? ?KOCP;F?HN M?? JA I@ 2LO?M>?FF 'H NB? JL?PCIOM M?=NCIH Q? LIN;N?> NB? L?@?L?H=? =IHAOL;NCIH HIQ Q? E??J NB? L?@?L?H=? =IHAOL;NCIH R?> ;H> Q? MOJ?LCGJIM? ; LCAC> GINCIH NI NB? ;=NO;F JF;=?G?HN

(&2/$ ?N ;H> ?@ILG;NCIH M?? $CAOL? ?N OM =IHMC>?L ; LIN;NCIH 1- ;LIOH> NB? JICHN NB;N CM

Constitutive Equations and Variational Elasticity

41

QCNB 4? NB?H B;P? # #

# 6 8

2B? ,$($& )! . ,$& ,' ($ , ( GIL? JL?=CM?FS NB? JLCH=CJF? I@ /* ,$')- $"$ ).$)(- MN;N?M NB;N NB? MNIL?> ?H?LAS >?HMCNS CM CHP;LC;HN OH>?L MOJ?LCGJIM?> LCAC> GINCIHM NB;N CM

@IL ;HS 1- CH ;H> ?N OM 1SG 2BOM <S +;N?LC;F $L;G? 'H>C?L?H=? Q? B;P? QBC=B G?;HM NB;N NB? MNIL?> ?H?LAS >?HMCNS >I?M HIN >?J?H> IH NB? @OFF >?@ILG;NCIH AL;>C?HN

IL ?KOCP;F?HNFS

1-

'N NB?H @IFFIQM @LIG NB? MJ?=NL;F NB?IL?G NB;N >?J?H>M IHFS IH NLIOAB NB? JLCH=CJ;F MNL?N=B?M

#'$' $!

?N ?> MGIINB >IG;CH QCNB ;LS ?N

42

R. Paroni

(&2/$

4? ;MMOG? NB? S NI IH MI NB;N IH QBCF? IH Q? ;MMOG? NB;N NB? S CM MO?HIN? <S NB? S @IL=?M ;H> NBLIOABION NB? M?=NCIH Q? ;MMOG? NB;N ;FF ?F>M ;L? MGIINB '@ >?HIN?M NB? IONQ;L> HILG;F NI NB? ?KOCFC "CP CH IH CH CH 1SG CH # IH

?N GCMMC IH

?N OM =IHMC>?L NB? @OH=NCIH;F >?H?> <S

#

4? B;P? NB? @IFFIQCHA 5$/"(0$ #)1 .#. .# -..$)(,3 )$(.- )! .# !/(.$)(& , . & -. !),'&&3 .# -)&/.$)(- )! .# ,)& '

Constitutive Equations and Variational Elasticity

43

-/# .#. )( -) .#. $! % $- -..$)(,3 . )$(. )! 1 #0 .#. % + !), 0 ,3 + # ( -./3 .# $ (.$.3 FCG

2B? KO;HNCNS NIN;F ?H?LAS

% + %

+

CM =;FF?> NB? MNIL?> ?H?LAS QBCF? CM =;FF?> NB?

' $) ! '$&! %

'H NBCM M?=NCIH Q? MB;FF MNO>S NB? JLI

2BCM JLIO=? M?P?L;F N?=BHC=;F NIIFM H??>?> CH NB? =IOLM? 2B? ;CG I@ NBCM M?=NCIH CM NI >CM=OMM NB? ?RCMN?H=? I@ NB? GCHCGOG JLII MI Q? H??> NI MI NB;N

#

IH

4? =;H N;E? NB?H

FMI CN H;NOL;F NI ;MMOG? NB;N NB? MNIL?> ?H?LAS @OH=NCIH QCFF ?@ILG;NCIHM ;L? F;LA? ;H> CH J;LNC=OF;L

C@

!!

4? ;OAG?HN NBCM JLIJ?LNS <S ;MMOGCHA NB;N !!

@IL MIG? ;H> '

/01 #$1-2/

. / #(&+ 1(" 0" * / ./-!$+

?@IL? NLSCHA NI GCHCGCT? NB? ?H?LAS F?N OM LMN =IHMC>?L ; MCGJF?L M=;F;L JLI?MJCN? CNM MCGJFC=CNS QCFF BCABFCABN NB? G;CH MN?JM NI ?L NI >?>O=? NB? ?RCMN?H=? I@ MIFONCIHM

44

R. Paroni

?N ;H> 6
- !-!

-

?N OM M?? OH>?L QBC=B ;MMOGJNCIHM NB? JLI

QCNB

- - -

B;M ; GCHCGOG NB;N CM NB?L? ?RCMNM - MO=B NB;N CH@ - -

?N -

$

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% - !- ! -

2I =IHNCHO? Q? H??> NB? @IFFIQCHA 5$/"(0$ . ' ( (

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' ' ' ?N NB?H @LIG #R?L=CM? Q? B;P?

!! !- ! - - ' '

Constitutive Equations and Variational Elasticity

45

;H> B?H=? =BIIMCHA MG;FF @LIG Q? >?>O=? !- ! % !! NB;N CM - CM ; ?> M?KO?H=? S NB? IFT;HI4?C?LMNL;MM NB?IL?G Q? =;H >?>O=? NB;N NB?L? ?RCMNM ; MO<M?KO?H=? - - ;H> ; JICHN - MO=B NB;N - -

!F?;LFS - CM NB? =;H>C>;N? GCHCGCT?L 'H>??> ;MMOG? NB;N FCGCN ;H> NB;N FCG - -

- B;M ;

NB?H - FCG

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2B? ;MMOGJNCIH NB;N FCG

-

?RCMNM =;H <S =IHMC>?LCHA NB? FCG CH@ -

QBC=B G?;HM NB;N Q? =IHMC>?L NB? GCHCGOG FCGCN I@ NB? =IHP?LA?HN MO<M? KO?H=?M I@ - 4? =;H MOGG;LCT? NB? JLII@ >IH? CH M?P?L;F MN?JM MBIQ NB;N NB? GCHCGCTCHA M?KO?H=? CM ?> ?RNL;=N ; =IHP?LA?HN MO<M?KO?H=? @LIG NB? GCHCGCTCHA M?KO?H=? ! ;MMOG?> - NB?H

NI
FCG CH@ - -

46

R. Paroni ") 1- 1'$ +(,(+2+ $,$/&6 "-,&2/ 1(-,

4? HIQ NLS NI GCGC= ;M GO=B ;M JIMMCIH? CH 1?=NCIH ?N

$

2I MN;LN QCNB Q? H??> NI =B?=E C@ CM CH MIG? M?HM? ?> NB;N CM JICHN I@ 1?=NCIH 4? B;P? !# !

*?N !
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! !

4? MB;FF >?HIN? <S

!! !!

! !

$+ /) 2B? MJ;=? QCNB NB? HILG ;
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4? QLCN? CH C@ !! !! #KO;NCIH MN;N?M NB?L?@IL? NB;N C@ ;H> ! NB?H ! NBCM CM ;FMI ; =IHM?KO?H=? I@ &IF>?L M CH?KO;FCNS

5$/"(0$ &IF>?L M CH?KO;FCNS . ! .# ( ( !! !!! !! !! !!!!!

!

**&3 2 ,$- 1$.# ! ! !! !! ( !!! !!!!! ?N OM AI <;=E NI M CH NB? M=;F;L =;M? Q? B;P? !# ! ! ! !!

' ' 2B? N?LG ! ! =;H ? MG;FF <S =BIIMCHA ;H MG;FF F? CN Q? OM? NB? @IFFIQCHA

Constitutive Equations and Variational Elasticity

47

$++ . ($. 0)&/' ( )-$.$0 , # , 2$-.- )(-.(. -/# .#. ! ! ! ! !# !

/--% 4? ACP? NB? JLII@ @IL ;H>

(&2/$

?N NB?H 4? B;P?

*

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!# !

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! ! ! !

!# !

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48

R. Paroni

NB;N CM !!

! !

!# ! ! !

MMOG? HIQ NB;N CH@

NB?H MCH=?

!!

MI NB;N

! ! @LIG ?KO;NCIH Q? >?>O=? NB;N ! # ! ! !

0?=;FF NB;N ;H> B?H=?

!/($% #$1-2/ -!-$3 0. "$0

?N OM >?H?

!! !!

!# ! ! !

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-

#)1 .#. ( - !-!

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!!

- & V NB? JLII Q? >?H? NB? >?LCP;NCP?M $,(1(-, @OH=NCIH CM M;C> NI

#&

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Constitutive Equations and Variational Elasticity

49

!F?;LFS C@ NB?H # &?L?;@N?L Q? MB;FF >?HIN? <S # NB? MNLIHA >?LCP;NCP? I@ 4? HIQ >?H? # NB;N CM C@ ;H> NB?L? ?RCMNM ; @OH=NCIH # MO=B NB;N ;H> ; M?KO?H=?

CH

C?

!!

!!

NB;N CM !!

!! !!#

# !!

?N C? ;H> IH ; H?CAB I@ NB?H QCNB ;H> ;M ;
#

5$/"(0$ #)1 .#.

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$,(1(-, 4? M;S NB;N CM NB? 4$ ) #$/(3 1(3$ I@ C@ #

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") & (, 1- 1'$ +(,(+2+ $,$/&6 "-,&2/ 1(-, M;NCM?M M?? ?KO;NCIH 2B? GCHCGCTCHA M?KO?H=? MOJ !! !!

50

R. Paroni

4? B;P? NBOM =IH=FO>?> MN?J G?HNCIH?> CH 1?=NCIH 4? HIQ MN;LN NI QILLS ; NI ?RNL;=N ; =IHP?LA?HN MO<M?KO?H=? @LIG NB? GCHCGCTCHA M?KO?H=? 4? EHIQ NB;N (% NB?L? ?RCMNM ; MO<M?KO?H=? =IHP?LACHA NI CH NB? HILG NB?H ON NB? JLI ; =IHP?LACHA MO<M?KO?H=? 2B? ;HMQ?L ;M NB? @IFFIQCHA ?R;GJF? MBIQM CM HI Q? =;H HIN 5 +.$ ?N J ;H> -

2B?H # - ;H>

!! !!

MCH $

$

-

=IM $-

-

$

2BOM MOJ !! !!

1OJJIM? NB?L? ?RCMNM ; MCH=?

=IHP?LACHA CH NI

CH

;H> B?H=? !! !! !! QBC=B CM ; =IHNL;>C=NCIH

2B?H

Q? GOMN B;P?

!!

,-1'$/ !/($% #$1-2/ 4$ ) "-,3$/&$,"$

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C@

!

CH !

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Constitutive Equations and Variational Elasticity

51

2B? ;?HCNCIH CM CGJILN;HN
.# ( .# ,

2$-. (

( -/- +/ (

#&

)!

-/# .#.

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-

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MOJ !!

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FCG

/

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#&

!), 0 ,3 ' - .# .# ), ' )0 2B? H?RN ?R?L=CM? >?;FM QCNB ?> M?KO?H=?M CH 5$/"(0$ . ( )* ( - . ' . -/# .#. MOJ !! !!

#)1 .#. .# , -/# .#.

2$-.-

( -/- +/ (

#&

52

R. Paroni #

#&

), )0 , -#)1 .#. #

4? MOGG;LCT? NB? JL?PCIOM ?R?L=CM? ;M @IFFIQM '@ MOJ !!

!!

NB?H NB?L? ?RCMN

MO=B NB;N

;H>

CH

NB;N CM

CH

#

#

CH

'H>??> CN =;H

CH ;H>

CH

#

#

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&?H=? NB?L? ?RCMN

;H> MO=B NB;N

CH

1I Q? B;P? H;FFS ;=BC?P?> MN?J G?HNCIH?> ;N NB? ?H> I@ 1?=NCIH ,IQ Q? N;=EF? MN?J ! NB;N CM OH>?L QBC=B ;MMOGJNCIHM IH Q? B;P? NB;N FCG CH@

0?=;FF NB;N

#

;H> NB;N NBOM FCG

Constitutive Equations and Variational Elasticity

53

'N NBOM MO=?M NI MBIQ OH>?L ;JJLIJLC;N? ;MMOGJNCIHM I@ NB;N C@ CH

NB?H

FCG CH@

#

#

NB;N CM NB? @OH=NCIH;F

#

CM M?KO?HNC;FFSQ?;EFS FIQ?L M?GC=IHNCHOIOM ?@IL? CHP?MNCA;NCHA OH>?L QBC=B ;MMOGJNCIH IH NB? @OH=NCIH;F CM M?KO?HNC;FFSQ?;EFS FM= F?N OM MBIQ NB;N (% CN CM NB?H NB?L? ?RCMNM ; GCHCGCT?L 'H>??> @LIG Q? B;P?

CH@ # $ NBOM N;ECHA NB? FCG CH@ ;H> OMCHA ;H> Q? I

$+ /) 'H>??> NI =IH=FO>? NB? JLII@ Q? H??> NI MBIQ NB;N 2BCM ;M ;FL?;>S EHIQH CM HIN NLO?
4? L?@?L NI NB? 0$%;LC?JS @IL NB? G?;HCHA I@ IH C@ CM MCGJFS ; @OH=NCIH -H=? NBCM CM =F?;L IH? =;H CGG?>C;N?FS QILE QCNB NB? ECH?G;NC=;FFS ;>GCMMC I@ ;H> ;=BC?P? NB? ?H> I@ NB? JLII@

-,3$5(16 16.$ -% 002+.1(-,0

2B? IHFS NBCHA QBC=B L?G;CHM NI CM OH>?L QBC=B ;MMOGJ NCIHM IH NB? @OH=NCIH;F CM M?KO?HNC;FFSQ?;EFS FIQ?L M?GC =IHNCHOIOM 2B? @IFFIQCHA NB?IL?G BIF>M

54

R. Paroni 6 8 )(0 2 ( &)1 ,

'$-/$+ . ' 6 - '$)(.$(/)/- .# (

#,

,

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# ;H> HIN? NB;N

NQI AL;>C?HNM I@ >?@ILG;NCIH

2BOM

>?N ;H> B?H=? CN CM H;NOL;F NI L?KOCL? NB;N

FCG + +

ON NBCM =;H HIN B;JJ?H C@ CM =IHP?R MCH=? + + + +

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+$' =IHP?RCNS

Constitutive Equations and Variational Elasticity NBOM

55

>?N

QBC=B G?;HM

2;ECHA

QCNB L?MJ?=N NI NB? JLCH=CJ;F ;R?M I@ Q? H> QB?L? ;L? NB? ?CA?HP;FO?M I@ 1CGCF;LFS IH? H>M

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- +/ (.$&&31 %&34

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( !), 0 ,3 )* ( - .

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56

R. Paroni

'$-/$+ ,$ ( /-) . ' ( #

! $- +/-$)(0 2 ( .# ,

2$-.- -/# .#.

!!

.# ( $- - +/ (.$&&31 %&3 &- 2BCM NB?IL?G KOCN? I@N?H CM OM?> CH HIHFCH?;L ?F;MNC=CNS ?P?H C@ NB? ALIQNB @LIG ;?N & ;M & MI CN CM H;NOL;F NI B;P? ;M & ON <S Q? B;P? @IL ?P?LS & 2B? ALIQNB @LIG ;?> C@ CM .-("-,3$5 NB;N CM CN ?RCMNM ; =IHP?R @OH=NCIH MO=B NB;N =I@ >?N QB?L?

=I@ >?N

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Constitutive Equations and Variational Elasticity

57

'$-/$+ (?HM?H M 'H?KO;FCNS . )(0 2 !/(.$)( ( , # ( , ,

1# ,

!!

( 1# , !! (). - .# ' -/, )! '$-/$+ JIFC=IHP?RCNS KO;MC=IHP?RCNS /--% ?N NB?L? ?RCMNM ; =IHP?R @OH=NCIH MO=B NB;N =I@ >?N ;H> F?N OM @IL ?HIN? <S =I@ >?N

?N NB?H <S (?HM?H M CH?KO;FCNS Q? B;P? # # !! #

4? HIN? NB;N ;FF =IGJIH?HNM I@ ;L? NB? >?N?LGCH;HN I@ ; G;NLCR 'H @;=N >?N CM NB? >?N?LGCH;HN I@ NB? =IGJIH?HNM I@ =I@ ;L? NB? =IGJIH?HNM I@ NB? MO<G;NLC=?M I@ ;H> H;FFS NB? =IGJIH?HNM I@ ;L? NB? >?N?LGCH;HNM I@ NB? MO<G;NLC=?M I@ CNM?F@ 1I Q? =;H =IH=FO>? NB? JLII@ OMCHA NB? @IFFIQCHA " (+ ?N NB?H >?N #

>?J?H>M IHFS IH NB? P;FO?M I@ ;H> # IH 1OJJIMCHA NB? =F;CG BIF>M Q? H> # !! # !! !! !!

58

R. Paroni

2BOM NI =IH=FO>? NB? JLII@ CN MO=?M NI JLIP? NB? =F;CG , , , , >?N # , , , , , , ,

, NBOM

>?N #

, , & , & ,

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=IHP?RCNS

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& %

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L;HEIH? =IHP

!! !! >?N

Constitutive Equations and Variational Elasticity

59

NB?H

CM

=IHP?R

JIFS=IHP?R

KO;MC=IHP?R

L;HEIH? =IHP?R

QB?L? 4B?NB?L IL HIN

"

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An introduction to -convergence methods for thin structures 4A
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1 Introduction to lower semicontinuity, relaxation and -convergence. #8G 58 4 @8GE<6 FC468 4A7 " .8 F4L G;4G

+2#

62

D. Percivale

1..& #8G + +

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@
F4<7 6B8E6
1..& *

G;8A + @
A G;8 4CC?<64G

+

+ <9 +

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An Introduction to Γ-convergence

63

! #8G * * 4A7 * - <9* BG;8EJ

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Y G;8A G;8E8 8K

+;8A
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D. Percivale

" #8G +;8 9HA6G

G;4G
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An Introduction to Γ-convergence

65

?84E?L 4A7 BA8 64A CEBI8 F88 1 2 G;4G 5BG; 4A7 4E8 ? F 6 @BE8BI8E G;8 9B??BJ
FH6; G;4G * ) +;8 9B??BJ
66

D. Percivale

! #8G - * - <9 * BG;8EJ

J;8E8 2 R
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+ -

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+

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An Introduction to Γ-convergence

2

67

-convergence and thin structures

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68

D. Percivale

4A7 G;
+

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! ) ! ! ! ! ! ! ! ! L<8?7F

+ ! + + !+

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An Introduction to Γ-convergence G;8A

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+

69

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+

2 Q+ - - & 2 3

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3

+

Q+

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R R

70

D. Percivale

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0

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4 , 4 1,2 3 0 , Z

R

4 1,2 3 , Z 0

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An Introduction to Γ-convergence

71

4+%1 %04!3).- %3 , ( $"% (%- , Z 1 2 !-$ ,.1%.5%1 "

, Z 4

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The structure of non regular solutions

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An Introduction to Γ-convergence

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/

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Z * + 3 +Z

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1

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78

P. Podio-Guidugli

" , +,8 ,; % 8 0 0 , : , ,++ U , 08 &1,0 U , + ,, 0 5 :" "+8 BVVT6; % ,+8 /+ +" + P , ",,& "8 , ,++ * 0" P "' 8 1" ,+ , ",+,& "8 , + , /0,+ P + & ; 8 &8 , +,: " 1 ",,& , ",+,& "X /,0+8 10,' ,+ 0 , ,+ , "+ ",,& " /+ +" ; $ /,0+ 0 1&, , 0" " * ' "" 0", ,+ &+ 08 , 0 "0,+ ,,+ , 0,X ,8 , *

+8 - ) )( -)8 , ," , ; % 0, , ,,+ 0 8 0", ,+ 0, 0+0 ,0 0+ , !, ,0, :+' ++: +A8 1 " +" 8 ,1, 8 , ,0 1 " , ,,; "+ , , G ; ; ; , , ++ 0 ,

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

68 + ,+ , +, + , , +, 0,,+; : 0", ,+ 0, , ,1+ , & "" , ,8 0 ,+ 0,,+ ;

Concepts in the Mechanics of Thin Structures

79

++ ,++8 0 0 , , ++ , &, 0,,+ , 0,0,,+ " , "" 8 : ,++ + 0, , 1" , , ,1 " 0 +" " '; (')

+

$ , 0, , , "" 8 ,

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""0X '0, , 1 8 1,+, ", , 1: , 0 P +8 , "& , ). ; $ * "& , , & 1,+, ", , "& ,"0 +, 0 , , : 0 ,+ ; % , ,8 " , +0 , , /0, 1 0 8 '8 0, ; $ , , , +, , :

0 ,+ 1 ,+ , , , ,1+8 : 0, , 18 0 8 >+,:+'4; % , , ,8 &+8 )- )( -

, - )( )? -)( )( - 8 1 / " +, 8 , " 0 ,+ ,; 0,,& " , , 1 ,0 ,+ , 1 " ,+ , " 5 ,+ 8 ; :" "+8 ,

2;; 2++,08 E@@B6; ! % ,++ 0+ 0 , "+ ," "& , , , % P ,++ ,,&X "& 1 &+ ,8 , &,1+ , ,? , 8 ,+ , , & , "& , ; $ ,

# 3 ) ) $ - ( )() ) - ) ) 3 ) 3 $ )(( )(( - -) ) ) 6 (3 - ?)( D ) ') ') . - ) ) 9 ' ()3 ) -( ' 0 ) 3 ;I>G1L ?" - ' )$ $) ) (. - () ) 6 6 6 )$ ' % 3

80

P. Podio-Guidugli

, , &,, ,8 0, +, ,+ , , 8 1 " ,:&,+" 1+0 0, "1+ 1 " , ,; $

,1,'8 &8 , 1 " , 1PX , +, , 0 "8 % ,++ , * : 0 ,+ ,& P+ , 1 "+0 , & * : 0 ,+ ++ +" ; ('*

#

$ + ++ * ,& 0+8 ,, , 0 * * '; " " ! ' $ & 0 : 0 ,+ +, ", "" , + :, ",+ 1+0 ""0 0,8 "1, 1 0,,+ 1,& , +, : " " 1 "^ ,+ ,+, 8 , " +& , "++:1+ : 0 ,+ ,+<1 " ,:&,+" 1+0; % 1: +04 + 8 * 0, + , 0; +8 , 1 , 1,' 5BTET6 , ," 5BTEV68 1, , #" " " ,0 /, " , ' ,,0; +8 , , ,0 " , 1 . ] 5BTL@68 "" , " " 1 , , "&+ ,,1+ + 0, , P ; % , 8 - )- )( "

1 ,+ , " 5BVQV6 , ' /, "8 0, " + 0 " ,+ ,,+? , - )( ) % P 5 :" "+8 BVTV68 , ++ , , , , ++ 1 1 0, , ,+ * 5BVVB68 1 + 0+0, , , '0,; $ ,0 ,+ 0 , 1 ,+ , "01 +, , +, 0 + +, , ++? , 0,# ,&0

,+4 >#"P, 4 & .A ,0A, +, ", 5 ,+8 BVT@8 BVTQ6? , 5 ,+8 BVVQ6 , ,+ / , , +"& + , +,; $ 0 ,+ , 1, ++ " ,

. ()) ) 3 . . ) 6 6 6 # ) - - ) ' % ( . ) ( 3 ' . )(( )(')

3 ) ' ) ) (- ) ) '((3 ( ) H,.( ) . )( ) ( 6 6 6 D 0( )() . 6 " ((3 6 ;C: 0" ((3 ;JJ;116

Concepts in the Mechanics of Thin Structures

81

0+ 0,,+ ,X , '0,,+ ) +" , :

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,,1+ +, 5 , , :" "+8 BVVK6; $ " ,: ,+ , , +, +, +, , ++ 1, 1 0, , 0 , "1# 5 :" "+8 BVVB6 , 5 :" "+8 BVVL6 5 ,+ 5 :" "+8 BVV@66? , &, & .A ,0A, ", , 0 & 5 :" "+8 E@@H6; "" , , : 0 ,+ +,: & 1 , , ,+ " 0 0 +, " , , /0, ": + 0, ; $ ",+ , ,0: ,+ 0 8 , +,,+ +, ", 8 ,8 1, , , +0 1+0 +" 8 . ]: & +,: 0 P+ 8 + , /0, : 0 ,+ +,0 ' ; $ , 0 , , ] , : X 1R8 , , #" . ]: &4 , , , )

,-) 8 , 0 ' "0, . ]: &4 , ,) : 0 ,+ "+1"0 +" +8 , 0+ , & ' * : 0 ,+ "? , /0, , 0, +0

" ) ) )- )( ' % )( .(- ) 3 ' - )3 - (( .' () . N D - D3 ) .( %3

()3 3 0 ) )(3 ;JIJ9 )(3 6 ) 6 ')3 ;JIJ9 )(3 ;JJ:16 $).(3 - )( ) ' % ) '(( )- -)L 3 663 0)$7K3 ;JIJ1 ) )

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() $ ) )( ()6

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82

P. Podio-Guidugli

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1 ,+ "; % 5 :" "+8 BVVL68 0+0, + * 0: ++", , : 0 ,+ +, "+1"0 , , " 00, " ,+; ++: ' * &,, ,+ ,,, "+1"0 , ,&,+,1+? ,": +,8 0/ &,, ,+ + ,& 1 " & + : 0 ,+ 1 , ,+; , &,, ,+ ,0 0+ , "+, +" 8 , +, , "+, ,+ +, , & 1 X /,0+8 ,+ P+ 0, 1 ,0 00"0 + " 8 , ,+ ++ 0 ,+ +,0 , "& +,8 , , 0/ , , + " 8 , ,+ +,0 ,

, + ; "+8 " 0 ,+ " 1, ] &,, ,+ ,0 0, ]; 2, &,: , ,+ ,0 , 1 0 , 0, , 1" : 0,',1+8 % 1+& : 0+0, + ,+ , , 0

* "+ 0 +, ,++ ,; + 8 5 :" "+8 BVVL6 , , P : 0 : ,+ 1 ,& 0 , + ,1,8 1" ,8 8 , +, , , , ,/; $ ,

0+X 1 ,0 ,+ 0 , , 0 , , ++ , "& 8 & ++ , 0, 0,+ 0+, 8 0 , , 0 &,+ 5 ,+8 E@@@8 E@@L6; " [ " ' ++ +,,+ +, "" , 0 , &,, ,+ 0"+, 8 00, , "+, " ,+ , "+, " ,; ,,& 0, ," "" ,1+ " , >+ : 0 ,+4 " ,+

# F ') ) ) $ # ) ) ;JII ) &$ ) # $) 66 )(6 )3 ' ) -. (( - )(%3 ' % 3 ) 3 . #)( ) ). ) 9 -) 0 5 (3 ;JJC1 ') () . $ .( 3 , % 0 5 (3 ;JJG16

Concepts in the Mechanics of Thin Structures

83

" ,+ 0 , : 0 ,+ +,; ,,++8 , ""+ , 8 0,,+8 ,,+,+; $ ,+ , 0 #" & , "" , " 00 ," , +, 0,: ,+ 1 , , ,+ ,8 , ,1+8 , 01+ 0 8 , ; /+ ,+ , , 8 , ",1+ : 0 ,+ " ,+ " 8 " , +

+, " ,+ 1 0 ' , 8 ')# - ; $ 0,,+ , , , &,, ,,+,+ , , ,0 /, 0 , [ # - 8 1 ,& ,0 ,+ 1 ,' , ",1+ +0 , , , : 0 ,+ " ,+ 0 ; + '8 , , ,1 " + : " &,, ,+ & 1 / , ,& , , ,+ +

&+ 0? / , 0 0 , ,+ % ,0 + ",+P ; "+ ,'8 & , ,,0 ,0+ 00"0 1+0 0 568 0, , , , 00"0 1+0 0 56

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1 " ,+ & 00 , 00, 1+0 "+ 1 " ,+ ? ,8 ,0+ " ,+ , [ & , " ,+ ; $ ,1+ +, : : 0 ,+ +, , & "" 0 + " [ &8 , P , ,0+ : 0 ,+ ": ,+ ,8 , ' ,,0 + @78 [ & , :

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$ & $, ,++8 +, , , & ,,+; % ,++ , "P 0 " U P+ 5 :" "+8

84

P. Podio-Guidugli

BVVL6 , , & +" 0 ,+ ,8 , 1, , 0,+ ,+ 1 ,, , "' * 0 ': U , + , ++ " ,+ & , +,+ +, ""; $ ++ , 1 18 , 8 /,"&? ,"+,8 +" > : 0 ,+4 " ,+ "+:,, ", , !- $ 0 ) )).( , > : 0 ,+4 " ,+ , , 4 (0 ) )).( (); -,+ , ,0 +, ", 1 , &,, ,+ & 0 ,8 0 ' + 8 ,&,+,1+; % ,++ " , 8 , ++ , , +'+ 0 ; &+ 0 ++ * , ,' 0 * # , , ,, 5,, , :" "+8 E@@O8 E@@Q6; )'(

" % & % &" &

1 , & , : 0 ,+ "+ , , ; , +:,0+ ,& 08 , P/

+ S@% BS8 , + 5+6 D 5+6 5+68 : 5+6 ,

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, 5+6 , , ,+ + " & 8 ,,+++ 5+6; 1 " ,, , ,

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Concepts in the Mechanics of Thin Structures

85

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86

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Concepts in the Mechanics of Thin Structures

87

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88

P. Podio-Guidugli

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B 5+65 56 7 $5+65 % + D

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E B 5+6 7 7 5+6 & & & & ++ D E 5V6 0,,+ 8 5K68 , 5T6 &,+ , , 0 P&

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Concepts in the Mechanics of Thin Structures

89

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90

P. Podio-Guidugli

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Concepts in the Mechanics of Thin Structures

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A Brief Introduction to Mathematical Shell Theory

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" , ,+ 0 C 7 8 ,+ 0 /, ,1 " , 5 , 56 &+ , , +"0 & , , 0,/ 566 5 7 6 56 C 56 56 7 5 6 C 7 56 567 7 5 6 % * 8 ,+ , + 1: 5 7 6 , 56 7 56 567 ; $ 1&, " P , 0,/ 5$ 566 * 1 +

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+; $ 8 + + 56 C 56 56 +, 56 56 C 7 ; $ & 56$ 56 C 7 = 56 C $ 568 5$ 566 WC 5$ 566 * ! 56 C $ 56 56; $ +, " 0+ , 56 56 C $ 56$! 56 56 ! 56 C $ 56$! 56$! 56 C $ 567 C $ 56) , " & 56 , +,+ 0,/ 5$ 566 & P; 2 * "+ +' ,1+ , 56 C $ 56 56; $ & 56 0 '!'' ,' '

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118

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+, , &,, , ,&,, 0 P " ,: 0,+ W $ 56 C 56 56 , $ 56 C 56 56) 56 C $ 56 56 , 56 C $ 56 56 0, W C , ++ , 00 , , C8 ;;8 ],1+ C , * & B 56 , +,+ , , C; % W C , 00 8 & P+ W C ,

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A Brief Introduction to Mathematical Shell Theory

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D

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122

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A Brief Introduction to Mathematical Shell Theory

123

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124

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125

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A Brief Introduction to Mathematical Shell Theory

127

516 /+. 5' 5663 1 0 ) .3 0 > 0 ' 56 WC $ 56' 56) ) 5$ 566 C 5$ 566 5 A;D6 + /+. 5' 566 5 > + $ 0 A;J;H6 $ A A 7 C 5' 566) 56 56

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128

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A Brief Introduction to Mathematical Shell Theory

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A A C 8 & ! , ! , ""+ 56 56

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130

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A Brief Introduction to Mathematical Shell Theory

131

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1+; $ A* /0 ; + "+< , &++

132

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&+& + & > > + "+ , ,+ ,",++ :

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A Brief Introduction to Mathematical Shell Theory

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134

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8 , , " X , P $ 0 A;Q;A ]+ 01 + " & " = /+, , 1 ,0 01 +6; $ +, 8 , 0, + ,++ 1) 4) <) > A) D8 : &+ " '68 , '0'' / ' ; + 41?141 C 0 + "0 3 5C= 6 0 + // 3 + $ WC B B , ' WC B

B B

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A Brief Introduction to Mathematical Shell Theory

135

, ,1 & +, & B B ! C X B ! 7 ' B ! C X X! 7 ' '! *

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136

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$ " X! C ,

A 5B $! 7 B $! B! $ 6 C B ! C X! D X C $! X! C B C X

, C& +, & >8 , / " ; 2 ,++ , ]+ 01 + ' ,+ ,",++ ,, , ] / 5, &,, ], = ; A;N6; ,++8 " ! WC B X! B X! 7 X X! X X! , !'' + +' !' ' 5C6; $ P " X , X! 0+ , . +": "+ + +> + + +> 1 C A) 4 C D8 < C A) > C D , , +5++ "+ + +> +

* + +> 1 C A) 4 C D8 < C A , 1 C A) 4 C D) < C D 5 ,0 , +,+ 1+6; % 8 ,"E ", , ,:,, ", , &+ " , ) ", ; 1 C D) 4 C A) < C D) > C A ,"E ", & ,"+, C 5' 6*

"+8 +"+ "&+" , , B56 ", 5C6 , 1 , A 56 C ) C) 56 56

5$ 566

5' 56 A C 5$ 0 A;K;A6; 56 56

5$ 566 " 8 " , , +" ,8 + + "+3 + & & ;"&+"< "+3 &63 +"+ "&+"3 / 0 ' ) ;/< "+ + +/ 8 &;8 0 P " ,0: ,+ 0 , ,,+ &,& D , ,0 9 $ ,+ +" ,",++ & , 0W

+ 41?181 C 0 + "0 3 5C= 6 0 + // 3 $ C B B &++ / >

A Brief Introduction to Mathematical Shell Theory

137

" +/+ / "+ 5C63 + " X! + 0 > 0 A 5B $! 7 B $! B! $ 6) D A WC 5DB $ B

$ B $

6 7 $ 5X X X

X 6 D

X! WC

$3 + + 56 "+ 5C63 +* "&+" + +

56 A C ) C* 56 56

5$ 566 $ 0 A;Q;H " ,0 +' + ,"E MATDTS8 ,0 1 ,"E , 1 0+ , " 1 &; 41@

'+'& + '

) 8 , ,++ ", 0, 8 ,++ 00 0, * 8 , ,++ 008 & P 0, * ; ,8 ,& , , & , C , , 0 " 00 W C 8 " ,++ " P P+ 5$ 6 W C , 5' 6 W C 8 $ W C , ' W C , &,, 0 > , " +/+* / ", 5C6 ; +'" 41@141 $ // 0 #& , 0,/ P+ 1 ++ P ; 2 " +* " W & , "1 C , * 0 " 0,/ P+ 5$ 6 W C , 5' 6 W C 8 , P ,

" ,0,+ 0 , ", 5C6 8 ;;8 / , 00 W C " , B B

CI $ C B B , ' C B B B % " , 00 /8 , / ""I $ , " " " 1 0,',1+ 0+W % C / 5 3 + $ 0 A;Q;A8 ;;8 +": + +5++ "+ 3 + + "B . " + // 6 % C 3 // "" " / ;

138

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+'" 41@151 2 , 00 " , #& , ] "8 , + "+ 1 +& 1 ] 0,; $ / , "" "+ " '0 +'& + ' ; $ 0 0 : ,++ ,8 , * 0+* . " 5$ 0 A;T;H6 , , ""5 " 5$ 0 A;T;J68 +, 1 ,+ ,++ /; , * "+ , ,1+ " A +"/ C , 0 P+ 5$ 6 , 5' 6; 2 1 " .; " ,0,+ 0 ", & .+1 MAVQH8 $ 0 H;T;TS; ,+ ,+ 3 , " MD>>AS

,+ MD>>KS , +: , 8 , ,++ +0,8 ; G+ ,+L & 08 " 2 /8 " 8 ;;8 ,0 MAVQNS .O "+ MD>>DS; + 41@181 C 0 + + / 5 "0 + 5$ 6 5C= 6 + 5' 6 5C= 6 0 ) /+. >* + + +": + +5++ "+ 3 &63 B X! B X! 7 X X! X X! C ' '! ' '! C) B ' B ' 7 X ' X ' C > C) ) A 5B $! 7 B $! B! $ 6) D WC $! X! 5$! 6 WC 5$ 6

X! WC X

$ . + // 5C= 6 " + $ C B B , ' C B

B B

C* B B

$ "+ +"/ 0, $ 0 A;T;H 0,/ P+ 5$ 6 , 5' 6 , 1 P,+ +,/ &,+ ,; % 8 !,0, 3 2 MAVK>S P , /: 0 ++ + 5$ 6 5C= 6 , 5' 6 5C= 68 , "+ 0, , 5C= 6; $ ; , , MD>>H1S

,1+ , 5$ 6 ( 5C= 6 , 5' 6 5C= 6 , *

A Brief Introduction to Mathematical Shell Theory

139

0,/ P+ , , ,"E , ,:,, ", 1" 8 / , 0, ( 5C= 6 " , 5$ 6 , 5' 6 , " ,0,+ 0 ", 5C6; $ +, * 0 1+ ; , , MD>>KS8 , ,1+ " "

"+,8 , ( 5C= 6 P+ 5$ 6 ,

5C= 6 P+ 5' 6 , . D8 , "+ 0, 5C= 6; , ( $ 0 A;T;H ,1+ . , 00 W C

& , ", 5C6 1 P , " ,0,+ 08 & 0 , + "^ ; 2 * " " "" " 00 ; $ 1# / 08 ,++ 0 5C= 6 + '; $ "+ , ,8 00

, 5C= 6 , ,0 " ,0,+ 08 ", 5C6 1, 1 "1# ", 5C6 , + 5 1 , ,+ 0,/ C A68 1 "1# , ": , , ++ 5 1 , & 6; , , & +,+ + ; " , 0 , 0, ", 5C6 00 ,++

, G + /+ L8 0 , , , G L ; $ 1&, 0 &, 0 + G 0L; + 41@1:1 C 0 + "0 + 5C= 6 0 ) // " + + +

5C= 6 + > + " +/+ / + 5 +:/+,, , 6 $ C $ , ' C ' C $ . + & + + /+. " + 56 C 7 56 ,++ C* ,+ ,1 " &, " +,,+ ],+ 0 , A , " +, / " , ' MAVNVS8 .+1 MAVQHS8 ,0 MAVQNS8 3 ,"/ MAVTQS8 &,' MAVVVS8 .O "+ MD>>DS;

140

P.G. Ciarlet

% ,8 ,&+ + , A8 1 1 , ,+

+, , +, ", 1 2;$; . 0 + +, ++; $ ", , G* : 0 ,+L8 , , / 0 * "&+, , " P 0

+ ", ++; $ /8 ""8 , "+, +" +, . ", , ,1: + 8 ,' , " ,0,+ G. ",+ , ",L , , GP0,+ +,0 +00, , ",L; 514

-' ' % 0+ '& && ,&+9

/+ C , 8 1 " 8 "1 8 1 " , : " "8 C 1 + ,++ 1 " ,; $ 1 8 1R ,"+, 0 0 , " , " , A; , ,++ / ) " " ,++ P &8 " 8 ,1, "0 , A ,,+ /+ ; ' , / 5/ 3 , B 6 , A) D8 , , / , A) D) H 5,& , " / "68 , "00, & , , / 0,,++ " ; $ , , : 0 ,+ "+ , ,8 "+ , ,+, " , / " ) , ,

8 , "+ , 0 ; C 1 , 0, ; C 5 6 , C8 , + B WC B+B ; 1 & , // 5C= 68 ;;8 , 0, " , * & 56 WC B 56 , +,+ , ,++ C; $ * & " , , +, "+ WC 5C6 , 568 , " & 56 WC

56 56 56 56

0,+ , 56; $ & 56 " &++ 0+ , 568 + & 56 P 1

A Brief Introduction to Mathematical Shell Theory +,

141

56 56 C 7 )

7 . ' 01 +8 " +&++ 0+ , 56 5,++ , 56 C 56 , , & 56 , ,+ , +, , 566; $ &,, , ,&,, 0 $ , $ > " +/+* / 8 A* /0 X 8 , &,, ,

0/ 0 ' , ' " +/+ / ,

P 1 +W $ WC )

$ WC )

X WC B )

' WC B ) ' WC $ ' * $ ++ +0 ,+ $ 8 $ WC 5$ 6

, ,+ , $ C ; Z WC C S 9) 9M8 + % C 5% 6 , Z 5 % C 68 , + B WC B+B% ; , *+ ** /

"+ C 5C6 , ' D9 . >8 ;;8 , +, 1 >"+ B5C M 9) 9S68 5; " D;A;A6 B5) % 6 WC 56 7 % 56 ,++ 5) % 6 C M 9) 9S * ,",++8 P 0,' ,++ + 0, B + 1,++ #& Z; ",+8 , 00 + + 1,++ #& C , 9 0,++ "8 , ++ "+ 5 " ,+ MD>>>,8 $ 0 H;A:AS6; + 514141 C 0 + /+ + 5C= 6 0 + #& // 6 $ . 9 . > " + /+

B W Z > 0 B5) % 6 WC 56 7 % 56 ,++ 5) % 6 Z) ) Z WC C S 9) 9M3 + 5 A / / / Z B5Z6 +

5 ) ) 6 . > Z3 ) WC B B6 6 $ ,"0 "+, 0+ , B 5C M 9) 9S = 6 , 9 . >; $ +,

C B B C 7 % B , C B B C

142

P.G. Ciarlet -

514141 $ >"+ + *+ **6 C 1 ,

0, 8 + Z C C S 9) 9M . >8 + 5C= 6 1 , 00 8 , + 0, B W Z 1 P 1 B5) % 6 C 56 7 % 56 ,++ 5) % 6 Z; $ 0, B + 1,++ #& Z 00 + 1,++ #& C , 9 . > 0,++ " 5$ 0 D;A;A6; % ,8 B5Z6 0, 1 & , P: ", , +, ++ ' D9 , 0

+ ", C 85C6; $ , 5 ) ) % 6 , ,1, % Z , & , "&+, , % C B5%6 P", ++; 0+ ,

5 ) ) 6 C 5 ) ) 6 . > C ! 5 ) ) 6 . > C M 9) 9S 9 . > 0,++ "; $ 8 / " / , 1 ,+ 9 0,++ "W % ,8 * +**8 0, B , : ] 0 0W & , C8 / , 1 ! 56 C , 956 . > " , B , : ] 0 0 0 ! 56 M 956) 956S B5! 56

M 956) 956S6; 8 ;;8 , MAVVD8 , HS 5 0+ " 08 ,+0 &,,1+ & " P

& 8 , 1 ,+ / " P & * "

/+8 " , C M 9) 9S= 8 ;;8 MAVQ>S6;

A Brief Introduction to Mathematical Shell Theory

143

$ ,1+ , 0, B W C M 9) 9S #& &

9 . > 0,++ "8 1 , W % ,+8 / 9 . >) 5 ) % 68 , 5 ) % 6) >8 " , 9 > , )

C)

C)

% 9 )

% 9 )

) % 6 , B5 ) % 6 C B5 ) % 6 5 ) % 6 C 5 C 0,8 / C , C8 , / , "1"8 ++ / 1 &8 " , )

)

% >)

% > , *

! ) % 6 C 5 6) 56 C +0 B5 ) % 6 C +0 B5

1 " 0, B , " C 0, #& 1 ,"0 ; " , + ,+ #& 5 ,1 &6 0, B; ! / 9 . > " , B #& Z C C M 9) 9S; % , ++ 8 ,"0 , 9 . > 0,++ " , : +" $ 0 D;A;A + ; % ,"+, 8 , ) ) % % Z " , 0 G: 0 ,+L "&+ + 1 P", B5Z6 ++; $ , % Z8 +,+ & 5%6 WC B B5%6 " &++ 0+ , B5%68 , 5+' +,+ 6 & 5%6 P 1 +, 5%6

5%6 C 7 " +&++ 0+ , ,0 ; = 1 , 0,",1+ "1 1 " , = WC BC , ,P = . >; % ,"0 , ++ "1# , / " 0 " + + ,+ B5= M 9) 9S6 +,,+ , B5= M 9) 9S6; $ 0, , +,0 P+ ++ &, B5= M 9) 9S6; $ ++ "1# +

0 B5Z6 , +

"+ G"L , G+ L , B5X 6 , B5X 68 X WC C 9; " ,+ , & 1 G ,&,,L 0 5Z6 , 5X X 6 " & +"0 , " ,,8 &+ 50, , 5%6 5%6 5%6 % 1 ,+ & +"0 5%6 % , , % Z8 5%6 WC 5 5%6 5%66= , 0+, , + ", 6;

144

P.G. Ciarlet

"

"

514151 *+ ** / + + 5 / + 0/6 Z C C S 9) 9M; $ B5Z68 B5) % 6 C 56 7 % 56 ,++ % C 5) % 6 Z8 P", , ++8 ': D9 , 0

+ ", C 5C6 5" D;A;A68 "1# , 1 " , +, ,+ B5X 6 +,,+ , 5;;8 +,0 &, B5X 668 X C = M 9) 9S ,

= = C BC; $ ++ "1# ,+ 1 B5Z6 , ,+ ", " , + , B5X 6 ,

B5X 6 X C C 9; ( R" 8 , B5%6 " , +,0 5%6 5%68 & 5%6 0 ,&,, 1, , B5%6; $ "' 1: +0 , &,, 0 W Z +,0 P+ W Z B5Z68 " , 1 " , C > X ; $ 1#& P + : ,] G+,0L : 0 ,+ 1+0 1 , G* : 0 ,+ 1+0 & 0

+ ", L 9 G0,++ "L= " D;A;H; ,8 ,' &",+ +,8 ' &+ /,: , ;

A Brief Introduction to Mathematical Shell Theory

145

,++8 +, 0,,+ " ++ ,"0 1 5 / " , , P", B5Z6 ++ ,"0 1 , +"+* +; ! 0,,+ ,, 1 * +/9 + 0 , - , H0 7 D- . > , - . > 5 , ,+

+, 0,,+8 8 ;;8 ,+ MAVTT8 , HS6; " , ++8 "&+, ,8 5 ) ) % 6 Z8 , " 1 0 + 1 0, "+ 5 / +3 + +3 + "&+ + 5 , ,+ ", 8 ,+ MD>>KS6; % " ", 8 G,",+L "' ) , ; &++< / W Z +5 / >* W Z >"+ B5Z68 & P+ ,&,, 1,= ; " D;A;D; % , G) 5 / + +

+L8 ,1 & : 0 ,+ 1+0 G+, L 1 ,8 "0,1+ 0" 0+8 ;) 5 / +< 0/8 0 G & /

* "+ ** L; $ 0, , * "' "+ 1 &++ / 5 W C +/ > 5 W C /

"+ C 5C6= ; " D;A;H; " , , 8 ,1+ , 1 0, , , " #"P, " , G+,0L; $ ,+ , , +/ ++@ % ,8 ,, , + 0," 8 5 / +* +/ & > 5 + G,+ L6 & + +

+ " + + 9 > + ;/< & > + + 0 * / " 0 & + ) 5

/ + 0/6 % 8 ,+ MD>>>,8 , S , " && + , , ' 1" 3 , "+ MAVVNS ,

'8 ),08 , 3 O "++ MD>>HS + ,; 515

&' ) && '

2 * 1 &' ) && ' 8 ,0

, . MAVNNS8 , , * : 0 ,+ +, 0 + 0", ,+ 0, 5 +, , ,0 ,,+ , 9 > 1R " , 6; $ 1 8 P * 0 , P+ ; & , ,1, +,0 P+ ? W C ", 0 " 0 ? W C 8 P & P+ ! WC 5? 6 W C , + $ 5!6 WC 5!6 5!6) 5!6 WC B 5 7 ? 6)

&,, 0 P " ,0,+ 0 5

146

P.G. Ciarlet

- +

*

)

)

514181 *+ ** / + + ) 5 / + 0/; 9 . > G0,++ "L , ,, + 0," 8 : 0 ,+ ++ 1+0 5" D;A;D6 G+, L 1 , G :

0 ,+ ++ 1+0L; $ 0, , * "' , &,, 0 5 W C +,0 P+

5 W C 0

+ ", C 5C6; % 8 G: 0 ,+L 1 " , X 1 +, 1 + G : 0 ,+L 1 " , = ; ,8 G1 " , +,0L 5 C B 5 C > = 5" . 4 +, ", = ; D;H6 0, , 8 , , , 8 0 , " 0 0

+ ", ,+ 5= 6; / "+ 5 7 ? 65C6; $ " A 5$ 5!6 $ 6 D

!'' + ' + , , +,0 P+ ? ; % & $ 5!6 , +,+ , ,++ C8 + 5!6 WC

A B 5 7 ? 6 5!6 5!6) ' 5!6 WC $5!6 $5!6 WC 5$ 5!66)

A Brief Introduction to Mathematical Shell Theory

147

&,, 0 " ,0,+ 0

/ "+ 5 7 ? 65C6; $ " 5!6 WC ' 5!6 '

!'' + ' !' >& , , +,0 P+ ? ; , $5!6 C 5!6 5!6; , 0 ", 5C6 , 5 7 ? 65C6 , " +/ "&+, , ) ; , ,"8 , +, ++ 0 + , , :

0 ,+ 1+0 5 D;A6; $ +, : 0 ,+ ",: 1 . MAVNNS 0 + , +, ++ , : & 0 +, : 0 ,+ +, 1, * + ,"0 W ,"0 8 , /+* ,"8 - A5 & +"/ ; % , , , ", , 0,+ 0

+ ", 0, 0,+ 0 0

+ ", , 0, , ,' +, , ,8 ,

8 , 1 " , , 0

+ ", 0, ,; $ ,"0 8 , /++ ,"8 , , , ++ +,, , ,,+++ 0

+ ", 5 ,"0 + 1, +, + 0, " ) MAVNK8 AVQAS6; $,' + ,"0 , "8 2;$; . ,

+" , "' * & P+ C 55 6 "+ 1 , + +

8 ,"+, , //8 & , 0 " & P+ ! C 5? 6 W C , + 1 " , = 8 " ,+ P 1 5; . MAVNN8 ; 5J;D68 5T;A68 , 5T;H6S6W 9 A 9$! ! 5!6 5!6 7 $! ! 5!6 5!6 $ 5!6 C D " H ? $ ) "

" $! WC

J0- ! $ $ 7 D-5$ $! 7 $! $ 6 0 7 D-

'!'' + && &' , " 5C6 , P 1 WC % 7 5) 796 7 5) 96

148

P.G. Ciarlet

$ ,1 & " ,+ ,++ ) . ' &' &' &&; $ " # " . 4 " P

1 9 9 # 5!6 C $! ! 5!6 5!6 7 $! ! 5!6 5!6 D N + & P+ !; $ / "0 ;//0+<

+ 9 # 5!6 C $! ! 5!6 5!6 D , ;?."+< + 9 ! $ ! 5!6 5!6 N $ + :, " * "+ , #" +, * : 0 ,+ ", +, ++ 0 : 0 ,+ +, , P,++ + ' 1" 8 1 3 , "+ MAVVNS , 1 '8 ),08 , 3 O "++ MD>>HS8 &+ #"P ", , +* *+ //0+ ** , , + + ?."+ ** 1 0, X5 & 5, +,+ +, ++ , 001, ++ , , 01+ +,0 0

+ ", , & 0 = , R/",+ ++ 6; $ $ " # , +,+ +, 001, ++ , + ", &/ &+ 8 " " 1 + , " &,, 0 $ 5!6 P " ,0,+ 0 ": ' * 0 0

+ ", 5 ", &/8 +, , ,+ + ,+"+" &,, 8 " MAVKDS= , /: ++ " & , , MAVTV8 , $ " ,1 & G001,L , # . : KS6; $ " # 4 " # + , +, +,0 P+ ? 0

+ ",; ,8 " , +,+ +, R/",+ ++ ,+, ",+ ,1 & GR/",+L , # . 4 " # ; # 5!6 C

+'" 515141 %+8 , /+* ,0 ,,+ :

0 ,+ ", + ,,1+ +& ,1 & G L / # 5!68 1" ,+ & ,+ / 8 ;;8 &,+ ,++ +,08 " 1 3 , "+ MAVVNS; ,8 ,0 0,+ , , + / # 5!6; ,+8 ,, MAVVTS8 3 ,, MAVVTS8 , ,+ MD>>>,8 , S;

A Brief Introduction to Mathematical Shell Theory

149

+ + +, +, ++ ", G . 4 L , 1 1 ,+ MD>>>1S; % ", 8 0,: $5!6 , ,, " 5!6 C ' 5!6 ' 0+ +, 1 $8 1 ,& 1+ , &, 0,: / # 5!6; $ ,+ 3 " MD>>AS ,& , +, 0 , 0,+ ,0 /, , +" * : 0 ,+ 0 +8 ' D9 , G0,++L ,,08 , " 1 , 0,+ ,0 ,,+ :

0 ,+ ", ; $ "+ " , , , " #"P: , 8 ,, 1 0, X: & 8 +, . 4 0 + 0 1 1+; 518

' ) && '

. , +,+ +, ++8 P 1 5; D;A6 A 9 5!6 C 9$! ! 5!6 5!6 7 $! ! 5!6 5!6 $ D " H ? $ ) "

0 " & P+ ! C 5? 6 W C ; &" , , P * ,+ , "+ + / 0 &,, 0 5!6 , 5!6 , 08 , , "&,"8 , , , +,0 P+ ? 0

+ ", C 5C6 ++; % 1, + ", . MAVQ>S8 , ,1 " 18 "^8 G0 > L8 +, &,, 0 5!6 C

A 5$ 5!6 $ 6 , 5!6 C ' 5!6 ' ) D

1 + + ) ! C 5? 68 &+

= 5!6 , : 5!6 1+ ; +8 " P ,' P /+ / " +, ; $ 1 8 0" 0 = 5!6; % ' 1 $ & >* WC ? ) ! C 5? 6 , ! ) + 0 > C3 /" 0 +"** " 4 2+ +, , , ++8 0 , = + & , &+ &,, 0 ? ; 0, & P+ !

150

P.G. Ciarlet

+ 518141 C 0 + /+ + 5C= 6 0 + WC ? "+ C 5C6 // 6 & + +/ >* ! ) / " &++ / ? W C 3 " = 5!6 W C 0 > 0 = 5!6 WC

A M$ 5!6 $ S ) D

) $ + $ 5!6 + &++ / > " +/+ / "+ 5C6 + 5 7 ? 65C63 + M S +

+ ) ! C 5? 6 . M S6 $ A 7 B ! 6 C = 5!6 5B ! D A C 5? 7 ? 6 ' ? D A C 5B ? 7 B ? 6 X ? ' ? ) D

= 5!6 C

) &++ &+& ? + > 0 ? C B ? X ? 5$ 0 A;N;A66 % +"+ 3 ? 5C6 , ? 5C6 = 5!6 5C6* 6 $ &,, 0 $ 5!6 0 ", 5 7 ? 65C6 , 1 P & 1 6 B 5 7 ! 6* $ 5!6 C B 5 7 ! $ +, 6 C 7 B ! B 5 7 ! * , 6 5 7 B ! 6 $ 5!6 C 5 7 B ! 7 B ! 7 B ! B ! ) C $ 7 B ! , = 5!6 C

A A 7 B ! 6* M$ 5!6 $ S C 5B ! D D

$ / = 5!6 00 ,+ ++ * 0 / C B 5? 6 & $ 0 A;N;A; B !

A Brief Introduction to Mathematical Shell Theory

151

$ " = 5!6 , ,++ !'' + &' ' + , , , +,0 ? ", ; 2 / 0" 0 : 5!6; + 518151 C 0 + /+ + 5C= 6 0 + WC ? "+ C 5C6 // 6 & + +/ > ! ) / " + ;/+** "< &++ / ? W C 3 " : 5!6 W C 0 > 0 : 5!6 WC M' 5!6 ' S ) ) ' + ' 5!6 + &++ / " +5 /+ / "+ 5C6 + 5 7 ? 65C63 + M S + + ) ! C 5? 6 . M S6 $ X B ! 6 C : 5!6 : 5!6 C 5B ! C ? ' ' ? 7 ' ? 7 '! ?! 7 '! ?! C B ? X B ? ' ' ? 7' 5B ? X! ?! 6 7 '! 5B ?! X! ? 6 75B '! 7 X! ' X '! 6?! ) ) &++ &+& ? + > 0 ? C B ? X ? 5$ 0 A;N;A6 +

? WC B ? X B ? , '! WC B '! 7 X! ' X '! % +"+ 3 ? 5C6 , ? 5C6 : 5!6 5C6 $ " '! + // + '! C '! 6 &8 & P& ,; % , 56 ,

568 ,1+ +0, +, ,P 1 & , &,, , ,&,, 1, ,+ ; 56 $ ) & C B + C $3 ) $ C 5$ 66 0,/ ) ) , +"0 & ; "+8

C 5 6 C 5 6

C *

152

P.G. Ciarlet

8 5 6 C 5 6 C 5$ C $) C $ , C 7 ; ! C $; 56 $ & + + + 0 C $ + C $ ; $ & , & , 8 "^ & , C A) D) H; % ,8 5 6 C > , 5 6 C >) 5 6 C 5 6 C $)

$ C 1 568 , = , 8 $ C $ C > , $ C $)

C 7 ; ! C $ ; $ +, 0+,+ ,1+ ; 56 $ &++ / ' 5!6 + X B ! 6 7 ; ;;) ' 5!6 C ' 7 5B ! ) G; ;;L + ;5 /<8 ;;8 0 , +, ! C 5? 6; "+8

X B ! C : 5!6 : 5!6 WC M' 5!6 ' S C B ! & C B , +,+ C , P+ ! C 5? 6 , 0 " 1 ,"0 8 & B 5 7? 6 + + +* C & >* ! + ;/+** "<8 ;;8 0 , 5C= 6; $ ++ 0", , + , ,+ , +,, , " ! C 3; , 5!6 WC 6 C 7 B ! 5!6 WC B 5 7 !

5!6 5!6 ) $5!6

$5!6 WC 5$ 5!66 , $ 5!6 WC $ 5!6 5?6

A Brief Introduction to Mathematical Shell Theory

153

$ ' 5!6 C B 5!6 5!6 A 6 5 7 B ! 7 B ! 7 ; ;;6 5B 7 B ! C $5!6 A 6 $5' 7 B ! C $5!6 A 7 B ! 6 7 ; ;; ) 5X 7 ' 6 5 B ! 7 $5!6 ' C B , B C X 7 ' 1 0"+, ,"E 5$ 0 A;N;A 5,66; /8 6 5X 7 ' 6 5 B ! 6 ' B ! 5 6 C X 5 B !

7 ' B ! ) C $ X B ! 6 C B ! 5 6 C $B ! ,

8 1 568 5 B ! C $ = +'8 6 C $5 X B ! 7 ' B ! 6 5X 7 ' 6 5B !

"+8 $ F 6 7 5B ! X B ! 6 7 ; ;; * ' 5!6 C ' 5A 7 B ! $5!6 $ 0, P +, 0 ! C 5? 6 A A /, C 5A 7 6; $ 8 , $ $5!6

5 7 6 C 5 65A 7 7 566) WC 5$ 6 , 7 WC 5$ 5!66; ! 7 B ! 7 ; ;;6 ) C 5B ! 7 B ! 5$ 0 D;H;A6; $ 8 M$ 5!6 $ S C B ! 7 ; ;;6) $5!6 C 5$ 5!66 C 5$ 65A 7 DB !

154

P.G. Ciarlet

C 5$ 6= "+8 A A 7 ; ;;6 C 5A B ! $ $5!6 , , +, 0 ! C 5? 6 65A 7 B ! 68 P , " /, 8 &;8 " 5A B ! X B ! 6 7 ; ;; ' 5!6 C ' 7 5B ! 5&6 $ / : 5!6 + 0 + ) + : 5!6 C ? ' ' ? 7 ' ? 7 '! ?! 7 '! ?! ) ) " ? + '! + > + +/ /6 $ 0 A;N;A 5168 C 5B ? X! ?! ' ? 6 7 5B ? 7 '! ?! 6 B ! ! C X 5B ? 7 '! ?! 6) X B ! C 7 ; , 1 $ 0 A;N;A 5168 C B 5B ? X! ?! ' ? 6 7 5B ? 7 '! ?! 6 B ! C 5B ? X! ?! ' ? 6B 75B ? 7 5B '! 6?! 7 '! B ?! 6 7 5B ? 7 '! ?! 6B C ' 5B ? X! ?! 6 ' ' ? 7 B ? 7 5B '! 6?! 7 '! B ?! )

B C 5 X! ! 7 ' 6 C ' ) B C ' C >)

1 0"+, ,"E , 2, 5$ 0 A;N;A 5,66; 2 " 1, X B ! 6 : 5!6 C 5B ! C ' 5B ? X! ?! 6 ' ' ? 7 B ? 7 5B '! 6?! 7 '! B ?! X 5B ? 7 '! ?! 6*

A Brief Introduction to Mathematical Shell Theory

155

2+ +, 0+ & +& + &,, &,& ? , ? 8 0, 1 ,+ , & +& ,

" ?! , '! ; $ ,,0 0+ " +, X! ' ?! X! '! ? C >9 $ & : 5!6 C 5B ? X B! ? 6 ' ' ? 7' 5B ? X! ?! 6 7 '! 5B ?! X! ?! 6 75B '! 7 X! ' ' X '! 6?!

46

5&6 $ " '! + // ) 1 +

,8 1," 0"+, ,"E , 2,8 ,

3 C B ! B ! C B X! 7 '! B X! 7 '! C 5B X! 6 7 X! X X! ' 7 5B '! 6 '! ' 75B X! 6 X! X 7 X! ' 5B '! 6 7 '! ' *

"+8 > C 5B ! B ! 6 C B '! B '! 7 X! ' X! ' ) , ; , 8 00 ,+ 0 P: " '! , ,+ ,& '! '! C B '! B '! 7 X! ' X! ' ) , " 0+; $ " : 5!6 , ,++ !'' + &' ' !' , , , +,0 ? ", ; $ " ? C B ? X B ? , '! C B '! 7 X! ' X '! &+ , 0 !'' !'! !0 >& ? , , >0 !'' !'!

'+'& + 8 P 1 0, /. 0 '! ; +'" 518181 5A6 &,, &,& ' , 1 +' P ; P,++8 , " ' WC B ' X! '! X! '!

156

P.G. Ciarlet

, >5 &++ &+& " +/+* /8

P 1 0, &++ 0 ' ; , ,,+ : " , & $ 0 D;H;D ,1+ 00 +, '! C '! 8 , * , &,, &,& +' , 00 +, ' C ' ) , 0+& "&,+ +, B ' B ' 7 X! '! X! '! C >) ;;8 ,0+, ,:,, ", 5$ 0 A;Q;A69 5D6 $ " WC ' ' C ,, / : 5!6 , &,, 0 " +/+* / ; ,+8 8 ;;8 ' MAVNV8 ; VTS .+1 MAVQH8 ; JTS; 5H6 $ " ' 5!6 , ,+, ++ P 5 , 18 & $ 5!6 0" 1 +,+ C68 1" " : 5!6 , +)+ ++ P ; 5J6 $ 00 : 5!6 C : 5!6 ++ 00 ,+ 1 X B ! 6 " ; ,8 / : 5!6 C 5B !

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

A Brief Introduction to Mathematical Shell Theory

157

" 0,+ &,& , ,+ BC 5 C , 0,8 , " " 0,+ & 53 6 / =:,+0 & ,+ =8 ,

" B C 3 B 68 , + , 5C6 1 P 1 5C6 WC ! C 5? 6 5C6 5C6 5C6= ? C B ? C > = * $ "' * & P+ C 55 6 W C 8 " 5 , &,, 0 +,0 P+ 5 0

+ ", C 5C6 ++8 "+ 1 , + + & + 5C6 '& P 1 A 9 5!6 C 9$! =! 5!6= 5!6 7 $! :! 5!6: 5!6 $ D " H ? $ "

,++ ! 5C6; $ " ,+ ,++ ) . ' &'& &' &&; "&,++8 & P+ 5C6 "+ , &++ + "+ 9 9$! =! 56= 5!6 7 $! :! 56: 5!6 $ H " ? $ ,++ ! C 5? 6 5C6* C "

2 ,++ , " J0- ! $ $ 7 D-5$ $! 7 $! $ 6 $! WC 0 7 D +&++ / ** + 50 , , ,0@ , +, 0,,+ " ++68 = 5!6 , : 5!6 &,, 0 + + /8 , + "&+"3 , , , +,0 P+ ? 8 , & " 5C6 , " ,+

; ,++8 1 " , ? C B ? C > = / , ++ +/ ,+ 5= 6 0

+ ", 5 " D;A;H6; $ " , 5C6 , 5C6 ,,+ 0 ? , 0,+ 0 ? +,0 P+ ? " 1 ,",+ "0 , " = 5!6 , : 5!6 1 1 5C68 , " ++ P ! C 5? 6 5C6; , 1 ,' , 00 , ++ 0

+ ", ,& ++ "+, 5 + ", 3 MAVVVS6;

158

P.G. Ciarlet

+'" 5181:1 . 4 +, ", , 1 "++ #"P 1 0, , ,0 ,,+ G: 0 ,+L ", +,

+, , 9 >= ,+ MD>>>,8 , QS , ; , 1 " +, ++ ", 8 ,+ MD>>K8 J;KS= , : " , , /& + 8

,+ MD>>>,8 , S; 51:

'+'& ++' $11

2 P &* 0 ,+ P , , 8 , " ,0,+ //+ (66 5$ 0 D;J;A6; $ +00, ++ +, , ' B + . ",+ , ", 5 D;K6; /+ Z , 3 0 " 3 "1 , 5 " " 0 " + X8 Z 1 * +** X; X : " "8 , /+" X , 1 P ,+ X , , " " /+* & C 53 6 5G"L 0, , "+ , 0 6 / X:,+0 & ,+ X; Z 1 , 0, ; , & A) 5Z6 , 5Z6

"",+ 0 & +; % ,"+,8 5Z6 WC 5Z6= B 5Z6) A ) 5Z6 WC 5Z6= B 5Z6) A ) ) B , B ,,+ &,& 1" 8 ,

5Z6 WC 5Z6= C > X) +, C > X 1 " ,; $ 0 5Z6 , 0 5Z68 & A8

; % ,"+, 8 % 5Z6) WC

WC

7

B

5Z6*

2 ,+ 1 +& , 5Z6 WC ",+ , 5Z6* 1+ P , 5Z6 1 5Z6 C + " 5Z6 )

A Brief Introduction to Mathematical Shell Theory

159

5Z6 , P+ ],1+ ,+:&,+" ": P & Z " , 0, "1 Z8 +, , 5Z6 C 5Z6 , B 5Z6) A ) 5 ",+ 1 , 5Z6 , 5Z6 1 ) 6W ) A C A % A ) B ) A C ) B A C B A % A

,++ A 5Z6; % 0,',1+ 51" ,+ 0,',1+ ^"+ &96 , & 0+, + W + 51:141 Z 0 + /+ + 0 + 0" Z6 $ 5Z6 , B 5Z6) A C 5Z6 $ 0+, , P & 1 );; 8 , , , 3 ,0,, MAVKT8 ; HD>8 5 6S= , 8 ++ 1 , &++' $11 ; % P "1+

0, 0 1 " , ,, "&," 3 MAVQD8 ; AAAS= , , ,+ & 1 $,, MAVQTS; -, " / G"L 0,8 ;;8 : " " 1 " ,8 , & ++ 3 ,0" MAVQNS8 0 , 3 "" MAVTNS8 , 3 MAVV>S= 0 " 3 ,"+ MAVVJ8 D;A>S & & , 0 ,+ 0+, 5Z6 , B 5Z6) A C 5Z6 + +0+ & ; " 1#& / " / , "": +" ,1 & &,, ,+ ", ; $ 8 ,++ 5$ 0 D;N;A6 ,8 " ,"0 H0 7 D- . > , - . >8 / , , . > " , $! 56!

160

P.G. Ciarlet

,++ C , ,++ 00 0, 5 6; 2 = . >8 / , "" , +" &,, ,+ 1+0 1 0, +.5+/ //+ ++ 1 , " / , , " , ? " 7 ? "

= 5!6 " 7

: 5!6 "

,++ ! 5C6*

$ 1#& / + , " , " ,0,+ ;- 2 "+ + "+< * 5$ 0 D;K;H6; 51;

) . '& ' '

$ 1 8 ,1+ , ) . '& ' '/ 7%0 , ' <8 , , " //+ (66 5 5; $ 0 D;J;A6; 2 ++ ,+ 3 ,, MAVVDS 5 ,+ , "8 ,+ 3 ,, MAVVJS6; + 51;141 C 0 + /+ + 5C= 6 0 + #& // 6 & ! C 5? 6 5C6 5C6 5C63 7 B ! 6 5C6) 5B ! D X B ! 6 5C6 : 5!6 WC 5B ! = 5!6 WC

A

&++ / + + /3 +

+ + "&+"3 + + ) +/ >

WC ? "+ C 5C66 $ . + + C ! 5C) 6 " +

? " 7 ? "

? " 7 ? " 7

= 5!6 " 7

: 5!6 "

+** ! C 5? 6 5C6 5C6 5C66 6 $ G"++ /+L / " = 5!6 , : 5!68 , " $ 0 D;H;A , D;H;D8 , " 8 0+ 1," , 0 & " ;

A Brief Introduction to Mathematical Shell Theory

161

56 > + 5C6 WC ! C 5? 6 5C6 5C6 5C6= = 5!6 5C6) : 5!6 5C6 * $3 "

) / " > 0 ! " WC

? " 7? " 7

= 5!6 " 7

: 5!6 "

)

+ 5C6 + !0 +6 $ +, G= 5!6 5C6L , G: 5!6 5C6L ,,

P , 5C6 , 1 " 1": ; $ 0, , , & P+ ! 5C6 5C6 5C6 1+ 5C6 / " 5C68 = 5!6 , : 5!68 " , ,++ A 5C68 A 5? B A 7 ? B A6 7 X ? A 7 ' ? A ) = 5!6A C " " D B ? B A 7 X B ? A 7 ' ' ? A : 5!6A C "

"

7 ? B 5' A6 7 ' X! ?! A 7 ?! B 5'! A6 7 '! X! ? A

B '! 7 X! ' X '! ?! A 1 & , ," " 5! 6 +0 ! C 5? 6 5C6; $ P 0 " , / ? 5C68 ? 5C6) = 5C68 , : 5C6 " ,

? ? 5C6) ? ? 5C6) = 5! 6 = 5C6) : 5! 6 : 5C6

,

; & , " + +, A 5C68 = 5! 6A

C C * * * ,

: 5! 6A

C C * * * , = C " " = 5!6 , : C : 5!6;

56 $ + 5C6 + 5C6 5C6 5C6 6

+,+8 5C6 5C6 5C6 5C6; $ & +" 8 + ! C 5? 6 5C6; $ +, 5!6 WC

A 5B ? 7 B ? 6 C = 5!6 7 X ? 7 ' ? D

162

P.G. Ciarlet

0+ , 5!6 5C6 " X , ' , : " " C; $ 8 B ? 5C6) B 5B ? 6 C B 5!6 7 B 5!6 B 5!6 5C6) 6 5C6 0+ B 6 5C6; ! B ? 5C6 1 //+ (66 5$ 0 D;J;A6 , " ? 5C6; $ P " : 5!68 " & C ": X ) ' ) ' 8 , B '! 8 , +, : 5!6 5C6 0+ , B ? 5C68 , ? 5C6; 56 - 2 "+ ) " 0 " + ; $ 0, @ 0 , 5C6 5C6 5C6 "

" 0 ! C 5? 6 ? " 7 ? " , 5C6 " " #&8 " "8 , "#& 1 56; 1 , , 0+ 5; 5668 + / , & 0, @ ,+ " " "&,++8 , ",+ . 4 " 1 " , + ; % ,1+ , . 4 ",+ G 1 " , L8 ,& +, 1 " , 1 0 P+ ! C 5? 6 5C6 5C6 5C6 , , G L 0 ? " , ? " :, ,1 & ",+8 ;;8 ", /5 / ! C 5? 6

= 5!6 " 7

: 5!6 "

1 0 , /8 "+ 1 ,

"&+ " 0; $ 8 , >/+ +/ //+ ; + "+< 5 , #& GP0,+L 0 , + + , = 5!6 , : 5!6 G"++L , 0 , "&,"

5$ 5!6 $ 6 , 5' 5!6 ' 6 , " &, C68 " , " 3 ,+ MAVQN8 $ 0 K;A:A , K;D:AS= ,+ , "8 ,+ 3 ,, MAVVJ8 00, D;K , D;NS , + ", 3 MAVVV8 $ 0 NS; , 5,6 / 0 , >+'& &'+ &++' ' '/ 7% , ' <8 + , 516 , >+'& &'+ &++' ' '/ 7% , ' <;

A Brief Introduction to Mathematical Shell Theory

163

0 8 , +, 0 /8 +,; $ & 8 1" , ,",+ W % + +,,+8 , 0" , &8 G: 0 ,+ P0,+

+,0 +00, ,, ,L " , 56 / 5, G L 8 " , ,++ " 1 , " 3

,+ MAVQNS8 "+ G,+L6; % / 8 " 5 6 , 56 , ++ ,

&++ / ;5 / +< + + ,

&+ , , , +,0 P+

W Z

, +,0 P+ W Z = " X , A* = /0 5 ' 6 , , 0, B W Z Z P,++8 " , &++ &+& & P+

W Z ; + 51;151 0 & + /+ C + + #& // 5C= 66 5,6 ! C 5? 6 5C6 5C6 5C6 0 " + = 5!6 C : 5!6 C > C* $ . ) & ) " + ? 56 56 C 7 56 ,++ C* 516 = 0 + =5/+"+0 "0 = C BC + +> + = . > + + & >* ! C 5? 6 5C6 5C6 5C6 0 " + = 5!6 C : 5!6 C > C , ? C B ? C > = * $ ! C 3 C6 6 $ & P& ,8 "01 56 5&6; 56 % + 56 568 Z + /+ + B W Z

+ 5 A / / / Z /+ B5Z66 "+8 & 5%6 WC B B5%68 B C B+B% 8 , +,+ : , ,++ % C 5% 6 Z , & 5%6 P 1

5%6 5%6 C 7 , +' +,+ , ,++ % Z; ,+ , 5Z= 6 , 5Z= 6;

1, 8 + % ,, , WC B5Z6; 1 + & >*

, % 8 + B WC B+B % 8 , + Z

164

P.G. Ciarlet

W Z ) WC 5 6 5Z= 63 ) + + + & >*

0

W Z %6

C 5%6 5%6 ,++ % C B5%6) % Z 5 6; WC 5 % +, , 6 5Z= 2 * * ,8 +** % Z8

B 5 C B5%6) %6 C # M S M # S 5%6) % )

WC B X , X WC B )

+

M 5%6S WC 5%6

5 / 5%6 & 0+

)

)

; % , ++ 8 0"+, " ,,, % , % , ",+ 0, , , +, 1 % C B5%6 , , ",+ " + ,++ % Z; %6 C \ 5 WZ B5%6 C \ 5%6

, B5 %6 8 B

C % ,++ % Z8 & 0, B W Z ; B5B5%66 , "+ , 0, B5%6 WC 5B \ 5%66 5 * / B5 %6 WC 5B \ 5 6 , %66 5 * / 6 , B5 %6B5%6 C ) "&,++8 B \ 5%6 5 5 5 5 %6B \ 5%6 C B \ %6 B \ % 6 B \ %6 B \ 5%6 C 7 * B \ B \ 5%6

$ 0 ,1 & +"0 & 1 + & 5%68 0 ,1 & * & 0" 1 & 5%6 5%6 ""+ P , / 1 +, 5%6 5%6 C 7 ) C A) D) H; ! : 0

)

)

, 1 ,+ / 0 &

5%6 & 1, 0, B8 ,W 5 %6* M 5%6S C B \ 2 / 0" &,& B# 5%6 5 ,+ , P+

C , +, Z B ,"0 1 +, 6; $

A Brief Introduction to Mathematical Shell Theory

165

&,& ++ 1 1+ / &,& B 5 %6 , " % 5,++ , 5 %6 C 5%6M 5%6S 6; ,++ , &

5%6 0 , 1,8 0, + B# 5%6 C X# 5%6 5%6) 1 ",01" "+ P " X# W Z ; $ P 1& , / 0 0, B , B8 X# 5%6 C X# 5%67 C X# 5%6 5%6 5%6 C B# 5%6 5%6 5 %68 1, !8 , B# 5 5%6 5%66 C > , M 5%6S C B \ 5 %6B# \ 5%6 C X# 5%6 X# 5%6 C 5%6 B# 5%6 C B \ 5Z= 6 1 ,"0 8 +, +,: B 5Z= 6 , B * , X# 5Z6; 2 , * , 0" ,,+ &,& B 5 %6 , " %8 1 0, +, 5 %6 C 5%6M 5%6S ; $ 8 P , , ],1+ " # W Z ,P %66 C B# #5%6B \ # 5 B #5B5 %6 C B# #5%6M # 5%6S ) 1 , "+; % ,"+, 8 %66M 5%6S 7 5%6B M 5B5 %66S B 5 %6 C B 5B5

C B# 5%6M # 5%6S M 5%6S 7 5%6 B# M 5%6S M # 5%6S C 5B# 5%6 X# 5%6 5%66 M 5%6S M # 5%6S ) B# 5%6 C X# 5%6 5%6; 2 ,& , %6 C # 5%6M 5%6S M # 5%6S ) B 5

# 5%6 WC B# 5%6 X# 5%6 5%6)

, M 5%6S , X# 5%6 , P , ,1 &; 56 2 , & P+ W Z WC 5 6 5Z= 68 / , , " 56 5Z6 P 1

56 WC

A A 5 7 6 C 5B 7 B 6 X ) D D

166

P.G. Ciarlet

WC 5

W Z , 8 , & P+ 6 5Z= 6 , , "

5 6 5Z6 P 1

5 6 WC

A 5B 7 B 6* D

6 + + + 568 5Z= % > 5Z= 6 + // +* ** ) / 56 +

5 C B5%6) % Z* 65 %6 C # 56M S M # S 5%6 ,++ % 56 + & >* C 5 6 5Z6 0 " +

56 C > Z* $ . ) & ) " + + + & >*

/ 5%6 5%6 C 7 B5%6 ,++ % Z* .3 X 0 + X5/+"+0 "0 0 " + BZ + +> ,, X . >3 + + & > C 5 6 5Z6 0 " +

56 C > Z + C 3 X * $ C 3 Z6 % ++ 0 , 56 ,

56 C > Z 0+ 5 6 C > Z $ B 5B 6 C B 5 6 7 B

5 6 B

5 6 5Z6 " , 0+ B 5B 6 C > 5Z6

5 6 C > Z , +,,+ "+ 0 1" 5 , MAVNN8 ; N>S68 , " , + 0,+ A &,,1+ % 8 Z ; $ " / , $ , ' " , %6 C $ 7 ' % ,++ % C 5 % 6 Z 5

A Brief Introduction to Mathematical Shell Theory

167

" 5 6 C > ,+ 0+ , ' C ' ; ! / * & ) " , 5 +"0 & 0 % 6 5 %6

C 7 ,++ % Z) "&,++8 " , 5%6 5%6 C 7 B5%6 ,++ % Z

&, ,+, " , & P+ ,, "+ C C 3 5, ,+ & = 8 ;;8 ,+ MAVTT8 $ 0 C 3; N;H:JS68 ,"0 ++ X . > 0+ , 5&6 1 ) Z WC C S 9) 9M ) + ) /+

B W Z 0 > 0 B5) % 6 WC 56 7 % 56 ,++ % C 5% 6 WC 5) % 6 Z) 9 . > , 1 " , , , 0, B , :

] 0 0 0 Z 0, B5Z6 5$ 0 D;A;A6; 1 + & >* ? ) &++ / ? 5C6 + ? 5C63 0 + + & > > Z 0 5) % 6 5) % 6 C ? 56 56 % 5B ? 7 ' ? 656 56 +** 5) % 6 Z3 ) & + > 0 C 7 6 $ &++ / & >* + 5Z6 + " 56 5Z6 > + + 56 + & 0

56 C = 5!6 % : 5!6 7

% ' : 5!6 7 '! :! 5!6 D' '! =! 5!6 ) D

56 C >* ,8 , ,1 & / " 568 : % /+8 1" C 0 8 " " ; $ /+ / " = 5!6 , : 5!6 0 " ? 5$ 0 D;H;A , D;H;D6 , " , ;

168

P.G. Ciarlet

$ & ,1 & , 8 * ,; 8 & " ? ) 5C6 + ? 5C63 & > 0

> Z 0 C ? 7 % * $ " + 5Z66 3 " 56 >

+ + 56 + & 0 A 5? 7 ? 6 ' ?

56 C D % 7 ' 5? ' ? 6 '! 5?! '! ? 6 7 D % 7 ' '! ! ) D A

56 C 5 7 B ? 7 ' ? 6) D

56 C >) ? C B ? X ? , C B X , &,, &,& P+ ? , C > 5 A;N6; $ 8 , B C ' 1 0"+, 2, 5$ 0 A;N;A6; !8 & &,, 1, , , 0, B C 7 % , & 1

C % ' , C * $ ,"0 "+, " ? , 0+ , C 5 6 C 5? 7 % 6 5Z6 5Z6; $ , " / " 56 , 1, 1 0+ 0", 8 1, +, C B 5 6 5, 566 , 56 C 5 7 6; 8 ,8 ) C 5B ? 7 ' ? 6) " 56 +0 & +' . + " +/ + 5&6; 2 P , 5C6 5 ' 5C66 , , 56 C > C 5B ? 7 ' ? 6; % " 0, P /+ 0

A Brief Introduction to Mathematical Shell Theory

169

" 56 ,; +, " 1 / , " 00 +, ' C ' 5$ 0 D;H;D68 P , A 7 ' 5? ' ? 6 '! 5?! '! ? 6 D C ? ' ? '! ?! '! ?! 7 ' ' ? ) ;;8 , % 56 ",+ : 5!6; ,++8 ' '! !

C ' ? 7 '! ?! 7 '! ?! 7 '! ? ! 7 '! ? 7 '! ?

C ' : 5!6 '! ?! 7 '! '! ? 7 '! :! 5!6 ' ?! 7 ' '! ? C ' : 5!6 7 '! :! 5!6 D' '! =! 5!6) ;;8 ,

56 , , " ;

5&6 Z C C S 9) 9M , + & P+ C 5 6 5Z6 1 P , , 5&6; , 5&68 ,"0 , = 5!6 C : 5!6 C > C 0+ ,

56 C > Z* $ 8 1 , 568 / * & ) " , 5) % 6 5) % 6 C 7 56 7 % 56 ,++ 5) % 6 Z* ! ? 56 56 C 5) % 6 5) % 6 C 7 56 ,++ C) , , 5,6 0 ,1+ ; / = = 1 " , = . >; % ,

? C B ? C > = 8 " 6 C 5B ? 7 ' ? 6 &, = 8 ? C B ? C > = 0+ B ? C > = ; , 5&6 , C 5 6 C 5? 7 % 6 C > X WC = M 9) 9S * ++ X . >8 , 56 0+ , C 3 Z8 , ! C 3 C , , 516 0 ,1+ ; 2 , * , & , " ) . '& ' '/ 7% , ' <;

170

P.G. Ciarlet

$ ",+ , P & 1 , " 3 ,+ MAVQNS; % , +, & 1 ,+ 3 ,, MAVVDS , , "8 ,+ 3 ,, MAVVJS= 1 ', MD>>HS , ,+ 3 ; , , MD>>AS8 , , 1 + & 0 : 0 ,+ . ",+ "&+, , 5 , 1,' " MAVTKS6= 1 + ", 3 MAVVVS8 , ++ + " , + 0 ,"0 0, ; 2 ++ * , "8 ,+ 3 ,, MAVVJS; + 51;181 C 0 + /+ 3 5C= 6 0 + #5 & // 3 = 0 + =5/+"+0 "0 = C BC + +> = . >3 + + 5C6 0 > +@ 5C6 WC ! C 5? 6 5C6 5C6 5C6= ? C B ? C > = & ! C 5? 6 5C6 5C6 5C63 A 7 B ! 5C6) 5B ! = 5!6 WC D X B ! 6 5C6 : 5!6 WC 5B !

&++ / + + / + 5 + + "&+" + + ) +/ >*

WC ? "+ C 5C66 $ . + + C ! 5C) = ) 6 " +

? " 7 ? "

= 5!6 " 7

: 5!6 "

+** ! C 5? 6 5C66 6 ! " " " WC

? " 7 ? "

*

% , " ",+ ,+8 / , " 5! 6 & P+ ! 5C6 " , ! " " " C A ,++ ) = 5! 6 " 7 : 5! 6 " C >* +0

A Brief Introduction to Mathematical Shell Theory

171

" 5! 6 1 " 5C6 5C6 5C68 , #

"1" 5! 6# & 5C6 5C6 5C6 1 **5 # - +8 & /; "0 8 , " 5= 5! # 66 # , 5: 5! 66#

,+ & 5C6 5 >8 1" 0, " , ,6 +0 = 5! # 6 " 7 : 5! # 6 " C >* #

$ "1" 5! # 6 # " , ," " 0 !

? " 7 ? " 7

= 5!6 " 7

: 5!6 "

)

0 " " " 1 - 2 "+ ) " 0 " + 5$ 0 D;K;A6; $ , 5C6 1 0+ , , + "1, , 5C6

5C6 5C68 / ! 5C6 " , ! # ! 5C6 5C6 5C6) , +0 ! ,P = 5!6 " C +0 = 5! # 6 " C >) #

: 5!6 " C +0 : 5! # 6 " C > #

! ! C 3 1 $ 0 D;K;D; " , +, !# " " " C A ,++ D A8 , 0+; % 0, 0 5 ) 6 C 5 ) ) >6 ,++ 5 ) 6 C8 ",+ $ 0 D;K;H " * ",+ 5 1,

1 + P ? C >8 ? C >6W ? "

B ? "

,++ ? 5C6 , ? C B ? C > = 8 ,

? "

A 5B ? 7 B ? 6 D "

172

P.G. Ciarlet

,++ ? 5C6 , ? C > = ; $ P ",+ , ++:' 1 +& ,; $ ",+ ) 5

/ +* - "+ ++ +; +, , ,+ B + . + ) 5 / +* + "+ 58 ;;8 ,+ MAVVQ8 $ 0 A;K:A , A;K:DS6; 1 + ", 3 MAVVVS8 MD>>JS8 , 8 3 , "+ MD>>KS8 "+, ,"0 0, 0, , P+ ! C 5? 6 1 P0,+ +,0 +00, , . ",+ , ", $ 0 D;K;D , D;K;H , 1 "1,,++ ,' ; % 0,',1+ ,8 > / + 0 " + 3 + - "+ + 0 +0 + & & + + / 6 P,++8 ,+ 3 MAVVNS , ,+ 3 ,:,+, MAVVNS ,& ,1+ ++ ) '& 7 ' && '<W C 1 , 0, , + 5C= 6 1 , #& 0: 0 , "+ C 5C6 && 8 , ,++ , ++ 5 0, , ,", "&,: " . > &* = ; A;K6; $ . + + C 5C) 6 . > " +

? " 7 ? "

= 5!6 "

+** ! C 5? 6 5C6 5C6 5C6; +'" 51;1:1 5A6 $ 0 ? " ,, +:, . ",+ , G,+L ", 5$ 0 D;K;H6 * +,

1 0 ? " ; $ +,0 R , " , ! C 5? 6 5C6 5C6 5C6 , = 5!6 5C6; , "+8 1 " , , 1 0 ? ; 5D6 $ . ",+ , ++ ", , P ,1+ 1 " MAVTK8 $ 0 N;A , N;KS8 " ,

,+ ,"0 , 5C6: 0 ]+ 01 + , 0,++ "; + 0, ", P 1 , #& 00 5C6 ,& 1 ,; ,8 , / + "+ ;) " 0 " +< 5" , , ++ , "6 P 1 0, , > "/0 D #& // 5C 6)A 8 C , 0, 8 " , , , C 5C 6; * 1 ; , , MD>>H,S8 . ",+ G " 1 " , L 5$ 0 D;K;A6 , 1 1 / " ", " 1 " ,;

A Brief Introduction to Mathematical Shell Theory 51=

173

' # + ' ) && '

, 5C6 1 P 1 5C6 WC ! C 5? 6 5C6 5C6 5C6= ? C B ? C > = ) = , =:0,",1+ "1 = WC BC , ,P = . >; " 0, 1#& , 1+, 0 W 5C6

5C6 P 1 9 5) !6 WC 9$! =! 56= 5!6 7 $! :! 56: 5!6 $ H " ,++ 5) !6 5C6 5C6 5C6:++; , +0,8 ,1+ " / &5 > + **8 & 1 0, +&++ /5

$! 5 , ,"0 ,0@ ,8 &;8 H0 7 D- . > , - . >8 , "",++ 0, +, 0,,+6; + 51=141 C 0 + /+ 3 5C= 6 0 + #5 & // 3 $ +&++ / / "+ 5C63 +&++ / ) 5

/ + *+ 0 & 0 $! C

J0- ! $ $ 7 D-5$ $! 7 $! $ 6) 0 7 D-

+ +"/ + H0 7 D- . > + - . >6 $ . + + C 5C) ) 0) -6 . > " + $! 56!

+ C + +** // /+ 5 66 6 % , ++ 8 , &+ , ,++ ,+ 0, * , ,++ ,+ 00 0, * ; $ +, , 56 & 1+ * " + , ,; 56 $ 1 8 ,1+ , ",+ ",+; 6 + - 0 ) + + 6 7 - . > + - . >6 $ . + + = C =56) -6 . > " + = 5 6 65 6 7 D- 5 6 ,++ *

174

P.G. Ciarlet

% 6 > , - . >8 ",+ + = C D-; % " 0, , - , 6 , > , - . >; & , 0,/ 8

P 0,/ 1 C WC 65 6 7 D-* $ +, 0, W P , , 1 ,+ & 6 7 - C > , - C >8 , C C

A 6 5 6 7 * J-56 7 -6 D-

, 1+, 0, 5) 6 W WC

P , " & , 8 " 1, 65 6 7 D- 5 6 C 56 W C W 6 A A 5 6 7 5 6 W C J-56 7 -6 DD , C - , 6 , > , - . >; / , , 4 C 456) -6 . > " ,

+,+

W 4 W ,++ C ) , " ",+ ,+ + - , 6 , > , - . >8 = C 5D-46 ,; 56 2 / * ,8 + C + + // /+. 5 68 $! 56! =$ 56$! 56! . >) ) = C =56) -6 . > + " 56; & , C , , 00 0,/ 5 68 + 56 C 5$ 566 , C 5 6) + 56 1 "" ", 56 5;;8 "" &:

P 00 0,/ , ,P 5566 C 5668 , + 56 WC 56 56 * $ A ! D0$ * 56! C 6 56 7 D- 56 56 6 WC D 0 7 D-

A Brief Introduction to Mathematical Shell Theory

175

",+ ,1+ , 568 " / , , 150) -6 . > " , A ! $ 56! 1 56 56 D 6 7 - . > , - . >8 "&,++8 H0 7 D- . > , - . >; 56 " @

0, C A $ 56$! 56! ) 5) 5 66 WC C 5 6 =

" " , 0, P 0,8 , 7 C 75C= 6 WC ! 7

$ 56$! 56! . >

$ 56$! 56!

, "

$! 56!

,++ C , ,++ 00 0, 5 68 WC 5=76 ;

01 . 4 ",+ G 1 " , L 5$ 0 D;K;H68 & P +, +, / , )+' +" 8 ;;8 , +" &,, ,+ ", + - ** "+ ; + 51=151 C 0 + /+ 3 = 0 + "0 = C BC ) + = . >3 + 5C= 6 0 + #& // 6 +**3 0 & + 0 + - + + H0 7 D- . > + - . >3 +

" 5C6 / . A + 5C66 $ + " C 55 6 &++ +

0/@ 5C6 C ! C 5? 6 5C6 5C6 5C6= ? C B ? C > = ) 9 9$! =! 56= 5!6 7 $! :! 56: 5!6 $ H " ? $ ,++ ! C 5? 6 5C6) C "

176

P.G. Ciarlet

) J0- ! $ $ 7 D-5$ $! 7 $! $ 6) 0 7 DA 7 B ! 6 , : 5!6 C 5B ! X B ! 6 ) = 5!6 C 5B ! D WC ? 6 ) ! $ > 5C6 + "" " //+ 05 /@ 56 C 5!6) $! C

"

) A 5!6 WC D

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9$! =! 5!6= 5!6 7

9 ! $ $ :! 5!6: 5!6 H

? $

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6 , + "1, 5C6 5C6 5C68 , 5C6 , !+1 ,; $ ,"0 0, 0, " ,"+, , & P+ , 1+ 5C= 6 , , " $! ) X 8 , $ , " " 0, C; ! 1+, 0 P 1 +:, &,, ,+ ", " " & + 5C6 5C6 5C6; $ " " 01

, 5C6 , 5C6 , A , , 5C6 , 5C6 * , +, 0 P 1 :, " " & ,0 ,; 00 0,/ 5$ 566 &: P ,++ C8 / $ " , $56 $ . > ,++ C; ,++8 . ",+ G 1 " , L 5$ 0 D;K;H6 , " 0 & P +, ++ 5$ 0 D;N;A6 0+ , 9 0 9)

$ ? " 7 ? " H 9 9$! =! 5!6= 5!6 7 $! :! 5!6: 5!6 $ H " ,++ ! C 5? 6 5C6; ! 0+ / 5C65** ; $ +.5+/ +00, , &,, ,+ ", ,& , + +" ; 1+, 0 008 +": ,+ "" +" 00, 1+0 , 0;

A Brief Introduction to Mathematical Shell Theory

177

$ ,1 & / , "" "+ ,+ +, " 5

+/ , +/5+ 1+08 ;;8 ,

= . >; 00 , " $ 0 D;N;D / , "": , +/ >* 5 /

"+ 8 0 5 5C6 , 5 5C6 , " 1, 1 P +" C 55 6 &,, ,+ ", $ 0 D;N;D; & P+ 0 1 &,, 1, 1+ , 5C= 6 1 ,: "0 8 & P+ 5 , 5 1+ &+ , 5C= 6 , 5C= 6; 2 / & 0 " + &+" 0/ , 8 , +, 0,++8 "&,+ &,, ,+ ", $ 0 D;N;D; % , ++ 8 = WC = = 8 53 6 " " 0,+ & ,+ =8 > WC 3 8 > WC 3 8 , B! 6 WC > B 6 ,,+ &,& 6

& 5> 6; + 51=181 C 0 + /+ + 5C= 6 0 + #& // 6 "/ + 0 " + = C + " + / "6 % " C 55 6 &++ +* "+ " $ 0 D;N;D / "3 + + " ** ) 0 " + &+" 0/@ & ' ' & ' C C) 5 7 ' & 6 ' 5& 6 C C)

5 C B 5 C > = ) & 3 3 C > = ) 5& 63 7 B! 5& 3 > 6 C > = ) 5 7 D' & 63 C > = )

)

9 ! :! 56) $ H + 3 + +0+ >* ) / " &++ / W C 3 WC 9$! =! 56 , & WC

7 X ) WC B 7 X

WC B 5 6 7 X 5 6* 6 0+8 & + , = C =8 ;;8 , 5C6 $ 0 D;N;D " 5C6 C 5C6 5C6 5C6

178

P.G. Ciarlet

$ / , = . > , , ; % , ++ 8 ,"0 , +" G0 "L , 5C6 , & 5C6; 56 2 P ,1+ +, B $ C $X * 0,/ ) ) , +"0 & 8 , $ C 5 , 56 $ 0 D;N;D6; : "+8 B $ C 5B ) ) 6 7 5 ) B ) 6 7 5 ) ) B 6 C 5X

7 X 7 X

6 5 ) ) 6 C $X B C X 7 ' 5$ 0 A;N;A6; 56 ( /"+ 0 & + 58 ;;8 ?, MAVNQS6 , ,"0 , " C , 5C68 P , 0 P ,+ ,, +:, &,, ,+ ", ; $ &8 ,++ ! C 5? 6 5C6 5C6 5C68 + ,++ ! C 5? 6 5C6 5C6 5C68

$! =! 56= 5!6 $ C

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$

C "

A D

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A Brief Introduction to Mathematical Shell Theory

179

+:, &,, ,+ ", 8 &;8 A $! :! 56: 5!6 $ C & : 5!6 $ H " " $ & B ? C "

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7 "

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,++ ! C 5? 6 5C6 5C6 5C6; ( 00 & C & 8 +, B $ C $ X 5; , 5668 , ,0 0"+, , , 568 1, & : 5!6 $ C $5B & 7 X & 7 X & 6B ? "

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"

"+8 ? & : 5!6 $ C $ D5' & 6 7 5' 6& " " 7 $ & ' ' & ? * "

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180

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, 00 +, ' C ' 5$ 0 D;H;D68 P,++ 1,

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: 5!6 $ C

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$ 5' & 6 7 ' 5& 6 ? $ ' ' & & ?

5&6 , 56 , 568 &,, ,+ ", A $! =! 56= 5!6 7 $! :! 56: 5!6 ? $ C > H " 0+ , " 7 "

$ 5 7 ' ? & 6 7 ' 5& 6 7 $ ' 7 ' ' & & 7 ? C >

,++ 5? 6 5C6 5C6 5C6; $ , " ,,+ ],+ ", , " ,P C; $ " C 9$! =! 56 , '!'' + &' &' >& ++8 , " & C

9 ! $ :! 56 H

, '!'' + &' &8 &' , + +/ >& ++; $ " C B 7 X 7 X )

C B 5 6 7 X 5 6) ,& ,",++ ,, " $ 0 D;N;H8 " /,0+ >0 8 , / !'' !'0 ! ' >& > ' '8 1 0, : ,&,, 0 W C ;

A Brief Introduction to Mathematical Shell Theory

181

,++8 , , "+ " , & , , )+' " 8 &;8 +" &,, ,+ ", 8 ,+ , +5 + " 8 &;8 , +" , , 1 " , &,+" 1+0; $ "+8 " +/, " MAVVJS8 + , +, , , + 1R ' ; + 51=1:1 C 0 + /+ ) C 0 + #& // 6 "/ +3 + / +* "/0 . A3 = *+ 3 ( 5C63 + ( 5C66 +**3 +"/ )+' " " $ 0 D;N;D +>

0 " + = + W / & > 5C= 6) + = C =6 $

C 55 6 ( 5C6 ( 5C6 ( 5C6* ' 6 $ 1 8 ,"0 , 1 " , = +, , 0, 1+ , 5C= 6; P &P , +, 0 ,,+ ],+ ", " $ 0 D;N;H 5 "' 5 , " "' 5 6 " /* ** , ,P "

/ ,

/ / 0 " + 8 0 8 "+ 3 1 MAVNJS; &P , ,0 0 ,+ * ** ?, MAVNQ8 ; ATKS; "+, "+ ?, MAVNQ8 00, H;D8 ; DN>S , ,' +" C 55 6 " $ 0 D;N;D8 1+ , 5C6 5C6 5C6 = C = 1 ,"0 8 ,P C 55 6 5C6 5C6 5C6 5C6 , 5C6; "+ 0 , MAVNK8 $ 0 H;KS ,1 " . , : , +, , 0+ , C 55 6 ( 5C6 ( 5C6 ( 5C6 ( 5C6 , 5C6 0 . A; "0 P,++ ,8 0 & A , 0 ,+ "01 . A8 = +, , 5C= 6; $ , "+, "+ 0 8 "+ 3 1 MAVNJS 0+ , C 55 6 ( 5C6 ( 5C6 ( 5C6 ( 5C6 , ( 5C6;

182

P.G. Ciarlet

, 0 23 !'0 23 -0 2 MAVNJSW 0, , 1 " , +" ++ ,,+ ],+ ", , ,+ 1 " , ; %%8 //6 "

6 +; 4?8 HKUVD; $(0 %22 MD>>HSW 0+ ++ . 4 0 +8 + +

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" 2;$; . 8 / "

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6 +; ;?8 HAQUHHQ; ''0 23 (,!0 %2 MAVQNSW @"+,@ " " +, 1+<0 ,"/ +0 ++" @@@@ &,, +8 66 + 6 6 +3 96 8 5@58 JKUJQ; 0 .23 0 2 MAVV>SW ", C , & C 1 " , 8 ! ''+ +6 (; 4A8 NQUTQ;

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('0 22 MAVTDSW % " 7 + 2+ "/9" +** 7 + 2 /+ 8 , 8 , 5+ ,+, W % " "/+ + 0+ + /+ 8 ,01 (& 8 ,01 8 AVTV6;

A Brief Introduction to Mathematical Shell Theory

183

('0 22 MAVTTSW +/++ +3 , "/ %@ $5 / +* +8 :! ++, 8 0 ,0;

('0 22 MAVVQSW +/++ +3 , "/ %%@ $ +8 :! ++, 8 0 ,0;

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+ /8 8 ;

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; ?;8 A>QUADJ;

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6 ; 448 AHQVUAHVA;

('0 223 ((0 2 MAVVDSW ++ +, ++ 0 +8 6 +)6 +6 ; :88 DJHUDKH;

('0 223 ! 0 2 MD>>ASW )"P, , * : 0 ,+ +, ++ 0 + . 4 8 6 +; 558 ADVU DJJ;

('0 223 (1('(0 2 MAVVNSW / , "": 0 * : 0 ,+ +, 001, ++ ", 8 (6 +6 "

; ?;8 KAUNQ; ( (0 2 MAVTVSW +"" ,++ 8 8 +; ! 0 2 MAVTKSW +,P, ++ 8 + 5

+ + +/++ :8 AKUNH; !#(!0 23 0 %22 MAVQDSW %9"+ 9+" "8 " 8 , 5+ ,+, W %"+ + + 8 :-+,8 +8 AVQN6; 5''0 2 MD>>HSW &,: $0 23 %(,0 223 (0 223 ! +, 1 ++ 0 0 ,+ : +, +, 1 ,00,: &8 66 + 6 6 +3 96 %8 88=8 NVQUQ>D; (!0 22 MATDTSW " ,+ , "P "&,8 //+ + + +"/ =8 ; , (0 2 MAVNKSW " 1+0 , +0 0 +, ++8 6 +6 "+

; =A8 D>QUDTJ;

184

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, (0 23 '( 0 2 MAVVTSW :/0+ <, +4@,+@ . ," +00 , 0, @"+8 "+ +". 9&9 +**

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6 ; @8 D>NUDDD; (,(0 23 .0 2 MAVK>SW 01

1+0 ]: ,+ 08 /6 (6 +; ?58 KKHUKNJ; % 0 2 MAVNKSW 0, &,& , ++ , ++ ", 8 //6 "

6 +; 4@8 DHKUDNQ; % 0 2 MAVQASW P ", +, ++8 //6 "

6 +; 4@8 DHKUDNQ; '-0 .2 MAVQHSW , " " =0 A+ /3 :-+,8 + 5+ ,+, W " A+* /8 :-+,8 +8 AVQT6; * 0 .2"2 MAVNNSW +, +, ++8 6 - 6 6 '+ 6 1; =A8 AUKJ; * 0 .2"2 MAVQ>SW " , +, +, ++8 6 - 6 6 '+ 6 1; ?88 ANVUAVK; 5'0 .2 MD>>DSW A+ /3 ; - 3 8 2: ! 1, 5+ ,+, W A+ /@ "&5 "+5 + * 3 0, ,0,,+ 8 & 8 D>>D6; 0 2 MD>>JSW 2++: . , , ++ , :0 ",8 + +

+ 58 HNKUHTT; 0 23 ( !'0 2 MAVVNSW $ 001, ++ 0 + +, +,W &,, ,+ ,0 &, 8 (6 + ; =8 KVU TJ; 0 )2 / ((0 2 MAVVTSW +,+ +, ++ 0 +; %%; $ R/",+ 0 +8 6 + +* 6 +; 4:58 HKKUHQJ; (0 23 (,((0 2 MAVKTSW % 1+0 ,+ + ", ],+ ++ 8 6 " + /6 " 6 + 458 DJQUHKT; ( (0 2 MD>>H,SW %",+ . 4 0, ", : " 1 " ,8 + +6 5:8 AVAUD>J ( (0 2 MD>>H1SW $ " ,0,+ 0 ", ", ++ "+,8 (6 + ?88 DKAUDV>; ( (0 2 MD>>KSW ,] 0 ^ , ,+, ],+ 08 (6 +6 "

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A Brief Introduction to Mathematical Shell Theory

185

4(0 %2 MAVNQSW 9 $9 "+ ** 5 "8 , 8 ,; &(0 2 MAVNNSW $9 0" 8 !0,8 ,; &(0 2 MAVVDSW + %%@ +"* A9* "+ A95 **8 !0,8 ,; #($0 2 MAVVVSW / & % " A+* /5 3 , "/ % ,3 $ 3 "1+ 8 '+; 0 2 MAVQ>SW "+ %+ + A+0 " 8 (& ; $0 %2%2 MAVNVSW A+* /8 ) 2+8 * '; "((0 2 MAVQTSW $ + +8 "1+, ,@: 0," 4, ; QT;AH8 (&@ ,: " 8 ,;

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Non-Brittle and Brittle Thin Films

189

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