1233269

Contribution à la modélisation mécanique et numérique
des problèmes de contact-impact.
Chokri Zammali
To cite this version:
Chokri Zammali. Contribution à la modélisation mécanique et numérique des problèmes de contactimpact.. Mécanique [physics.med-ph]. Ecole Centrale Paris, 2005. Français. �tel-00186569�
HAL Id: tel-00186569
https://tel.archives-ouvertes.fr/tel-00186569
Submitted on 9 Nov 2007
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Remerciements
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Table des matières
1 Formalismes et schémas d’intégration du contact en dynamique. Un bref état de l’art
" !$#&%')(+*,# -.*/-10*325476- #&%"*98;: (<"0 76- ="!3%>0$(<?0$-7# @A!3-7'')-BCCCCC
G!3# %')(+*$#H-.*/-7<I#JK6- 2K%>< (+LNM - 8 MO27<N03%"2P0QCCCCCCC
RE ST254/76- #&%*/8;: (<"0 76- ="!3%>0$(<&03-7# @A!3-7'')-UC CCCCCCC
REXWY76- 0$4 T8T-.*/<ZM #J76- !3(LNM -.*@["M !\')-]27<N03%"2P0\-7<O8T^Z<G%#H(+LNM -_CCCC
RET a<[email protected]!$"b 'd-7c # - 8 -C! 76- e.76- !3-7<[21-gf ()# @G%"2P0h8T-]8T-KMTijb[%>!3!$-.*hCCC
RETRE G!3# %')(+*$#H-C8TMk27<N03%"2P0\@G%>!/@/76- <G%')(+*$%>0$(<j-10h(<"0 76- ="!3%>0$(<l-K<I0$-7#
RET o G!3# %')(+*$#H-'+%>="!3%< =(-7<?8 MO27<N03%"2P0\-10h(<N0 K6- =!5%p03()"<l-7<I0$-K# @G*U
o r9<[21'MG*,(< CCCC CCCCCC CCCCCCC
11
CCCCCCC
CCCCCCC
CCCCCCC
CCCCCCC
CCCCCCC
@G*HCCCCCC
CCCCCCC
CCCCCCC
DFE
DFE
D.V
D.`
D.`
mEn
mENq
o"s
ETtu"*$()0$(<[email protected] !$"b 'd-7c # - -10\< >05%p03()"<G*vCCC CCCCCCC CCCCCCC
ETRE r9(<76- #&%>0$(+L"MG-8 -K*\="!3%<G8T-K*903!3%<G*wex"!$#&%>0$(<G*yC CCCCCCC CCCCCCC
ET o z 6 LNMG%>0$(<G*\8{:w.6- LNM ()'(b !$-?CC CCCCCC CCCCCCC CCCCCCC
ET o z 6 L"M[%p0$(<[*/')T27%')-.*/8;:wK6- LNM (')(b !3-mCC CCCCCCC CCCCCCC
ET o RE t!$(<G21(@A-8 -K*h}~!3%Fp%MTil€(!,03M -7'+* CC CCCCCCC CCCCCCC
ET | {"(8T-C21"# @["!,03-7# -7<N0CC CCCCCC CCCCCCC CCCCCCC
ETR‚„ƒ <N0$-K!3%"2P0$(<[*\8T-C21<N05%2P0jC CCCCCC CCCCCCC CCCCCCC
ETR‚T t†%!3%#J16- 0$!3(+*$%>0$(<&[email protected]{-7c !3-K*9')T2K%>MTiOC CCCCCCC CCCCCCC
ETR‚TRE ‡@ @G%>!3(-7# -7<N0ˆCC CCCCCC CCCCCCC CCCCCCC
ETR‚T o r9"<G8T()0$(<k8T-< <O(<[email protected]/K6- </76- 0$!5%p0$(<&-70hZ(‰03-K*3*,-.*/!$-K'%>0$(-K*ŠC CCCCCCC
ETR‚T | t!$(<G21(@A-8 -C'd: %210$(<I-108T-C'+%H! .6- %210$(<‹ CCCCCCC CCCCCCC
ETR‚TR‚ (+*\8T-]21"<"05%210/-70h8T-ex!$0,03-7# -7<N0ŒC CCCCCCC CCCCCCC
ET sD "!$# -.*/')T27%')-.*?K6- LNM (p%>'-7<N0$-.*/8T-K*/'(+*\8T-]21"<N03%210\-108T-ex!3>0$0$-7# -K<"0C CCCCCCC
ET s t!$-K#H(Ž-Kc !$-.*&.6- 21!3(‰03M !3-K*&K6- LNM (p%>'-7<N0$-.*\CC CCCCCCC CCCCCCC
ET s RE -7MTiT(d-7c # -.*?K6- 27!$()0$M !3-K*l.6- L"MG()p%>'-7<N03-K* C CCCCCCC CCCCCCC
ET s o G!3#M '+%p03()"<j'+%>="!3%< =(-7< <[email protected] !3b 'Ž-Kc #H-8T^Z<G%#H(+LNM -C8T-]21"<N03%210/ex!$0,03%<N0CCC
ET‘q„ "!$#M '+%p0$(<O*,03%>bG()'(* K6- -fZM < ()’[27%>0$(<k8T-.*ex!3# %')(+*$#H-.*/'%=!5%>< ="()-K<G*C CCCCCCC
ET V r9<[21'MG*,(< CCCC CCCCCC CCCCCCC CCCCCCC
o"V
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2 Contact entre solides : du local au global
37
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3 Formulation hybride faible-forte pour les problèmes d’impact et discrétisations
o " !$#M '+%p0$(<I4Z^Zb !$(+8T- e %>(b '-Žex!$0$-C21"<N0$(<NMG-DC CCCCCCC CCCCCCC
o ()# ()0$-K*8TMO#HT8;-7c '-8 -CST()="< !3()<G(;-7<k8T^Z<G%># (+LNM -?CCCC CCCCCCC
o RE (+*\8T-]21"<"05%210\8 -CST()="< !3()<G( Wk!3-K%M CCCCCCC CCCCCCC
o o z 6 21!3(‰03M !$--K< .6- LNMG%p03()"<G*/8T-K*\'(+*\8T-]SZ(=< "!$(< ( W !3-K%>M CCC CCCCCCC
o | G!3#M '+%p03()"<?e %()b '-Ž[email protected] !3bG'd-Kc # - 8T-C27<N03%"2P0h8T^Z<G%># (+LNM -CCCCCCC
o R‚ t!$(+*,--K<I27# @T0$-]8 Mjex!3>0$0$-K#H-K<N0yCC CCCCCCC CCCCCCC
o s IK6- *$M '‰05%p05*\8T-C21<[*,-K!$p%p03()"< CCCC CCCCCCC CCCCCCC
o REX‡[email protected]@ !$FiT(#&%p0$(<[*/-10h! K6- *$'MT0$(<?<NMG#J76- !3(LNM -.* C CCCCCCC CCCCCCC
o RET G!3#M '+%p03()"<I*$-7# ( 8T(+*321! 16- 03(*T76- [email protected] !$"b 'd-7c # -8T-]21"<"05%210\8 ^N<[%># (LNM - CCCCC
o RETRE (*321!G76- 0$(+*$%>0$(<G*-7<I-K*[email protected]%"21-]CCCC CCCCCCC CCCCCCC
o RET o (*321!G76- 0$(+*$%>0$(<j8 Mj03-7!3# -8 -]*w05%>b (')(+*3%p0$(<DCCCCCC CCCCCCC
o o SN03!3%>0 K6- =(- 8T-C!G.6- *,"')MT03()"< C CCCCCC CCCCCCC CCCCCCC
o | r9<[21'MG*,(< CCCC CCCCCC CCCCCCC CCCCCCC
4 Approches multi-échelles pour les problèmes de contact
|GWkT8;-Kc ')-C8;: (<"03-7!$e %27- #M ')0$(Ž< (-.%>M CCCC CCCCCCC
|G Wk0$(p%p0$(< CC CCCCCC CCCCCCC
|GRE WkT8;-7c '- =†76- "#J16- 03!$(+LNM -8T-.*\()<N03-7!$e %21-.* C CCCCCCC
|G o z 6 L"M[%p0$(<[*/')T27%')-.*/8TMO#HT8;-7c '-8{: (<N0$-7!$e %27-#M ')0$(Ž<G()"-K%>MCC
|G | G!3#M '+%p03()"<?e %()b '- Ž[email protected] !3bG'd-Kc # - 8T-C27<N03%"2P0\#M ')0$( < ()"-K%M
|GR‚ t!$-K#H(-7!5*! K6- *$M '‰05%p05*9<ZM #K6- !$(+LNM -K*9CC CCCCCCC
|GREX : %>@[email protected] !$T254 -C‡!$'-KLNM (<[email protected]!h')-.*/@ !3b 'd-7c # -.*8 -27<N03%"2P0 CCCCC
|GRET OK6- 27-K*3*,()0 6- 8T-.*\%>@ @ !3T254 -K*#M '‰03( ".6- 254 -7'')-.*QCCCCCCC
|GRETRE ‡@[-K! 27 MO*$M !\'+%H#J16- 0$4GZ8 -‡!$'-KLNM (<XC CCCCCCC
|GRET o : %@ @ !3Z254G- ‡!$'-KLNM (<O-7<k8T^Z<G%># (+LNM -OC CCCCCCC
|GRET | : %@ @ !3Z254G- ‡!$'-KLNM (<[email protected]["M !\')-.*\@ !3b 'Ž-Kc #H-.*98 -]21<N05%2P0\-K<O8 ^N<[%>#
|GRETR‚ (*321!G76- 0$(+*$%>0$(<G*/[email protected] !3b 'Ž-Kc #H- # T8 -7c '-„ CCCCCCC
|GRET s SN03!3%>0 K6- =(- 8 -C! .6- *,"')M 0$(< CCCCC CCCCCCC
|GRET‘q r9"<G*,-K!$p%>0$(<j8 -C'd:w76- <G-7!3=(- %F"-K2'd: %>@ @G!$T254 -‡h!3')-.LNM ()< CCC
|G o r9<[21'MG*,(< CCCC CCCCCC CCCCCCC
5 Exemples numériques
‚TmSN05%>b (')(+*3%p0$(<fGLNM -7';(<N0 K6- ! -70 CCCCCC
‚T r9"<"05%210\8 -h-7!$0J CCCCCC
‚TRE t†%>0$(<?ex!3>0$03%><N0\-7<k8T^Z<G%# (LNM - CC
‚T o t†%>03254 Ž0$-.*w0\8T-]}g%F^N'! CCCCC
‚TREX "!$#M '+%p0$(<I-7<OZ()0$-K*3*$- C CCCCCC
‚TRET ƒ # @G%"2P0h-7<N03!$-bG%!$!3-K*ŒCCCCC
‚TRETRE ƒ # @G%"2P08;: M <k21^Z'()<G8 !$-C*$M !\M <Ob 'Z2!3(=(+8T-
CCCCCCC
CCCCCCC
CCCCCCC
CCCCCCC
CCCCCCC
CCCCCCC
CCCCCC
61
sNE
sNE
sNE
sN‚
s"s
s"V
q>n
q>o
q>o
q‚
VG
VNE
VN‚
87
CCCCCCC
CCCCCCC
CCCCCCC
CCCCCCC
CCCCCC
CCCCCCC
CCCCCCC
CCCCCCC
CCCCCCC
CCCCCCC
(LNM -XCCCC
CCCCCCC
CCCCCCC
CCCCCCC
CCCCCCC
V"V
V"V
V"V
`"n
`G
`NE
`|
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`"s
UKn"n
UKnNE
UKnNE
UKn"o
UKnN‚
CCCCCCC
CCCCCCC
CCCCCCC
CCCCCCC
CCCCCCC
CCCCCCC
CCCCCCC
UKn"V
UKn"V
U.n
U"
U"
U.o
UF‚
107
T‚ RET o ƒ # @G%"2P08T-]}g%F^Z')"!k CCCCCC CCCCCCC CCCCCCC
‚T oDWkM ')0$(K6- 254 -K')'-vCCCC CCCCCC CCCCCCC CCCCCCC
‚T o WkT8;-7c '- #M ')0$(Ž< (-.%>M CCCCC CCCCCCC CCCCCCC
‚T o RE WkT876- '(*3%p03()"<[email protected]%!\'d: %>@ @G!$T254 -‡h!3')-.LNM ()< CCCCCCC CCCCCCC
‚T | z iT-K#[email protected]')-.*/()<G8 MG*w03!$(-7'+* CC CCCCCC CCCCCCC CCCCCCC
‚T |G
!5%>@ @A-C8T-]21"# # %<G8T- CCCCC CCCCCCC CCCCCCC
‚T |GRE r9!5%F^"<j8T-C21"#b M[*w03()b '-C-10h="!$('')-8T-C#&%>(<N0$(-7<Cf M <G- @G!$-K#H(Ž-Kc !$- @[-K!3*[email protected]()"- CCC
‚T |G o ƒ <N0 76- ! -10\8T-'d: %>@[email protected] !$T254 -C‡!$'-KLNM (<j-K<[email protected] ! K6- *$-7<G27- 8T-4G%MT0$-.*/ex! K6- LNM -K<G21-.*CCCCCC
‚TR‚ r9<[21'MG*,(< CCCC CCCCCC CCCCCCC CCCCCCC
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Introduction
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> '!& / S12 > A N 8 = & ! > / LTE5 !12 L 512>L ] 12
F41!K= += %@( ] 1 / y:512"H T M12U
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8 = & ! > F!ABBB
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& = "5 U
> ! ! <12Iz>+= / 0
= >T'($N4> I = !'& >T> / ' "X> X= 512>12 >l9
512 8 "@G12'J4> ! >>^ _= '9 ' BGi !12( (
= !&'!& '% = '
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> +' D' ++= ! 4 B0ky12F M512>124 >l9 [12$ ( I >l9 / 2
1 XG12' !12 4>5
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= ! >! > >l"9 " 512> 12 0 41J4>D !12" ! Iz ! / 6;Z $ (X>U[= 8 >1 G "XS = & 1 X6;W12 > _= '9 / 6; 5
/ 2
1 "g!12 ! >! > 41!&[F
= F! * 1J4>D B%F>5= J7124 E5 = "L
= !12 !0= 0 ] 1' > 12 /0 >l05 / 1C0T 12 +F0= 512> 12 F _= '9 M M 1 JK>m F !12E !
9" 12 9 Iz5 9 / [email protected]>+= JK += Byky12 !12 8 'X> 123& 8 / 12 % 8 1Cl12 w >D;W12 8 A ^= ! EP' ' } [email protected]> . 1JK>m ! 12" !c
l0 (
9 / 124< 8 1Cl12
G12S PJKJ4>1 4& -"4_= '9P= ! E5
(>U :5
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= 2 > " 0V5 C8 0 SaS1 / X> '^!12>> JG1 5 '
`{B B / } / / } / / / hLG12% S= !> A @ %>
= ! 94<!12E !%
Ul0 (
9B
5G ! '= >'% &+= ' E( > O (5 8A >q;ZT !'@!12!12"5'J412 ! > !4>q _= '9( 1J4>D N!12" ! Iz !
"[email protected]> ++= ] 1' JK> M M5 ] 1 ' 12 B >G E5'. F> ! 6;Z .!12>T> JG1 12
"5>@> JG1 [email protected] bIda L @>D; Yq= !12> O E5 >
k 'ML>@+= 5 "e% >l05 Ha = ! T9"
k/ ]
/
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> 712 ' F5 8A [email protected]!12>> JG1 12 E= ' ' / += 8 >14+= 4F U! @9 Iz5 9 / B
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mH1 8A 12(4 55' >>[ H!H5 A8 >4 5q>H!
! 8 $0= !5 !> = Byky12!$5l05 / > 512>>T!
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> !4>6; 5 NJ4> A [!12J445J4> gr;Z 12 @= ' / 4 (! >1AI
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= T 5
5 ! h^!12 5 E w512!U6;W 0I
12w 5F> = X > ! "[email protected]> G
!12
T >> 12 !> = ' -M= 8 (> ! 0= [email protected]! T H!'& 4L = ! 49" / T! > D "H ] 1'! H !12E ! B
O P10! "U5P!12G125 !49
!'& K5 BHi?c4 (
<!12G15 J ' ] = SP>D; 'U
= &10 ? = 'T9" 6+= 4= 5 5 "? 6 '1J4>m rj!12" !y @l0 (9B O r = &10
!1 7124 E !125 "? ] 1' > Gj> _= ! 9q !12" !||
r!&[= |r;ZE= 12
<5 KH12"H12 F 47 >12 M9" T> 9" X= > = "BGir H4 L= "5 L = & 1 / [= 8 >1 M
= % "
= (9L6;Z( ! jH 0
!55L&6 / 512E >l= > > 4(mF6;Z ! ! [
J ' B
ir.5 !12 $!'& 45% 1G125L %12 8 >> = ! [ |= 9 12 [>12q>10! > +!12" !FG5I
5 " X+= ' 8 ] 1' > 12_> A (!12"" / 9 >QM= $5 J4T>= /? = += >T "
! >> .> A ^!> '9" N(`m9 Ivh> A ""= Byi 12 8 >T> ] 1' > 1231 G
' (! 'LG12q 4Q j> jT4> = " 12 [" = 9" q [_= &1 [> A j !5 B
, ^> (Dw ' X V!& 45 / V 8A EN 5 !<> 5 C8 0V<aU1 V(2 /
12 + '
1 71212 [4 ] 1 ' > 12w!12" &ElJ ' ]m J4>I ] 15 /0 "= 041J4>D F6;ZT !B
Yq>T>L+= ' 8 6;Z4<= !' `C= 9 8A > E5 h[ ?= "
9 12 ^ M>12[@b0 1'T IdaS1 / 1 l
"
>D;Z"510 !12< !'& (4 @dl"GN> 8 > Iz5 / [= Q4 H'> 12G15 " >> !12E !B
ir L>12M ] 155 "L O 2
1 >12NJU512"$= 2 > "<= !'5 H ?= 9" 12 L12EH""= = > w x ] 1' > 126B TE= x ( '! ><!55 ] 1'N4> 12_ 5(9" / ^4!0= 12 /
>>7L6; '>@ 5G !L %!12 12 H!+= 9" L!12E ! . UK> ! "L% 8 I
5 5 [ > ] / > ] 12B2, + H 0D H4!'& 45 / ^= > = "ML!12" ! Iz ! 512"%+= 8 = ( N> ] 1' > 12S!12"" B 12 %>12 F 8 ' E5 AIz'!'&+= G12
"!
> T!= 12-
5 4Bri !0= T 12- P 5 !( 5 ]m 5 >J4 .(> = & 1X
#%$'*&( ),+.-0/2143,-.35/ +7& 68+:981;%<8-.+ $
=$>@? 68+ +.& ) +.& AB+.</4CEDF<F3,CG/4HF+014A ? I A +0& -J;%<83LKM8+N6O (QP 9
O/]
= >= "XQK ^6;Z4U_= &1033!12>>10! 126Bji (D35(G12 41!&> X!'& (4
'T 0w w%> 0D NG12 '1!&> %!'& 4L N N`m0= 4>'F S5 4
$ 5 ! hB % > 1& %( EM +55 "= HG12E[Q0 qF = &10 Ldl"G LL512
= [= >= 5L >=.G12H> 0= 512>T12U = '[email protected] H 1J4>D L ! B
ij; 7 !g > IA= !& >> `{ 5 ! h| y 1J4>D ( y +!12" ! Iz !q 5 JG1'+F
= 4|>M!'& 45 B
12 y 1G12512 / 512* T! >D " / N(16 >F6;ZTE5 ]m !MN > Iz 8 NG'5 "* [
4
!12 5N H!12G15 "L>1! 0w >1J 0w FE5 ]{ ! @!12E !B O @106 >@4= 5 E5
""= x @" = 9" @ !4>HG12%> 0
= 512>T12c L41J4>D .!'&10! / 5 "= . . ] 1 '0I
> 124!> 9 B*ij; K>! 12V<> _
= &10UeL> 9"4 _ 1JK>m N4> I = !'& >T> !12" !F 5[4= 5 E= @ 4[ 5 !12 'B"i G12J4>H
= 96;W1 G%!55 = &10%G12q5 5
q4&[= 12< +>10! 0S`m& 5 ] 0= 9 ! hj q>
12 +!'9 g6;Z( !jq q4&+= 12< >1J `m10 @r;d= J4 > "h.>12TP(> 12!9 / ! @4= A Nr;W12 / 5N>D;Z
FG12"F '9 "[email protected]!N!'& 45B
ir%' j!'& 45H 5+[= = K>! 12 q" = 9" ! += (9 [ 0 ' (m 44> I
! 12 5' >> B ^ [
= 8 > 1 G "@ ] 1' T9" . ] 1 ' > 12441G12"=
= TE= 0= / F>@! @!55N&G 5 / 4
B
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/
Chapitre 1
Formalismes et schémas d’intégration
du contact en dynamique. Un bref
état de l’art
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B i J4J4>1 K& 'H>@ :5
h4
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= !12= 8 !
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8A ' = = !&[= ."4_= '9 H6;Z""= 123 U5 KD4!& 9" N!12 E%12 "
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5 / 12 L SJ > l12 .> L4> H!12 E5 B 12 %>
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!12 EqF!'&+= +6;Z""= 12 ^5 (4 8 !> ] 1' >' *512Ej>T5"= / 5 5= j >l0"=
D ! >T [email protected]>D;Z( [email protected] 0wJ ' X= > 59 B
> >T (D 6;Z w0 4>@4>?
1.1
Formalismes et schémas d’intégration temporelle
1.1.1 Formalismes en mécanique du contact
*!12 += 124| 512>Tq+= ] 1' 4
J >FE / 10!! "y>D; 4&[= '! J4>U 3 "5$ !12" ! 8 !P
1J45 !> PO !12" !B
[email protected]` 8 12HQ ' B hB
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J 12 / > P= 9 12 >10! > 6;v= 9 >J 12> ( _5 9 / > >12j X!12G15 " / >
- " %$&(' )+ ,-./' 0 123", H 3
w>T(5 M;d= !' 8 "!12((N -D
= 9" 12 L @!12 J4T>"= L> H!124 12
N
N D 4 8
N
1r
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` B h
LN ` h N
G
>
` B h
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5M>H5
+ [!12"5 "5 / +>5 5 + +[= ] 1 ' 12 +>+= '= /
5+>L5 [ '^= > 5T9" 8 = Q "+> [ 14'= "= +!> 9 / +>%!'& ($
++= 4> ! (
E / 5
> 4"
= 6;W G1 8 12>4(9 M X L> '"
= 6;W G1H ]{ !9 F124[= B
!5l05 / T > ]m . :[email protected]> .>12.!12" !LQ R [email protected]! > 0/ 12 .15124LM S> T 5 !
1' ""= @> {] ! G15 " >>[email protected]!12" !^ >D;W1JK5 !>B %J4 8 12 M9"r;W # [
k k `m&"l"G1&6 5 G5 FG'J 12 h / 124 8 12 SD
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5F>j: w > "5 >512>TLM>D;W1J4 !>^` 8 12
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] 1'N4> 12 0S[= 4> ! " ]m J4> S41J4>D _= ! 9[= !'.! Iz 4 / -4712 E
[email protected]>@!12" ! %6;W H ! ] `
N / Q!R'h / L> 8 "5HD
&('*),+.- 02/ 13034
f|12 8 65879:
1r
79
;q` h
N
(
<=?>[email protected]
;q`BA
h
L>D;W 5 !N H+= 4> ! ( E%!+= 9 " JK> qL16LD
;q`BCh*N D B` C,ECh !GF `BC0h
8 !
D ` E Ch*NIH2J ` hD `BC0h M7G
F `BC h N HKJ L C M7G HKM*N C7M>Q
` B h
!"#%$&(' *)+ ,-./' 0 123",4)-56$ ", 7$89 ,:);#,< =#5>" ? ,@ AB"' 1C)1? + "8
.( j9" H!L 1JK>m L
= " J4>(9" 7L9
;
1 (H>R;d= HG15 " >> U 5 ( x 5 0I
q4H512>12 9B O 2
5 P 5 !X # >JG(`m! ] B? B B h / 2
1 @ C8 12 ^9"X!55$512>T12
= [email protected];ZY[ > Bkq>T M 0= !5 ( E / > 1J4>D 5<= 9 8A > E S 8A ECD
]{ H>R;Z ""@
&('*),+.- 02/ 130 f|12 8 65879: A 5879
1r
YqU5
;
j; ` h A
D ` E A.h ! F B` A
N
h*N
` B 2h
+= > += ' 8 = S5
( x
;
5 S gB
E!12 5$ 4= 5 E U!12" !
&('*),+.- 02/ 130
f|12 8 65879
;q` N
h
1r
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8 "CD
:
<=?>[email protected]
T> "= 6
> ! / > '1J4>m ;q`BA.h
A : M B` A.h
` B h
QSR
` B h
BA
! 5N> ! 12"5 "5 4> "= >X` M ` h h / 12 G12 8 12 L4> L>H>D;Z ""N
= 6;ZY[0I
> B 12 q712 8 2
1 * >d x !&[> !12"5 "5r;Z > = >T"F
= 712 g (5 j! 12( 5 4g>D;d= B O !
12 H w <41J4>D ' 8A E9D
e
&('*),+.- 02/ 130
f|12 8 65879:
;q` h
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q
; `BA.h `BA.h
1r 5> ^] 124!12 >>.! !"= 59
`BA h
`BA h
N
N "
!
A 5
12
` B h
+= Q4 D
- " %$&(' )+ ,-./' 0 123", H 3
a P12 PVG12 8 12 -512A: 12- P>T5c>R;Z ""= 6;ZYq >3G12 ! != 'T5_> 512>T12
V '1J4>m X! ! 53U>D;Z0= 4> 'U
= B O 55 ] 12 !12 >T><6;W 5 ( x 5 0I
L= " J4>B Y[>>. 5L5124 Iz L= E J4>`m! ] B B B /
h+L124FG12 8 12 $= !''> 1JK>m 512 ] 1'(@6;Z+= 9 12 `{12U6;v= 9 12 h 8 ' 12 >> +!12N4 D
&('*),+.- 02/ 130
f|12 8 65879
: A 5 7L9
! F `BA ! h D ` E A ! h G
` B
h
O H5l05 L+4x 5L >? > 0= 512> 12"4_= '9B, ] 1 ' > 4
1!&[= gq6; 5 * !
12"w= "
= += 8 >14+= B 12 > >12 M G
>FJ m 8 E> F4>4+!> 9 H6;W "5 0GB
Formalisme par pénalisation
ij;v= !'`mkj1J4>D ( "hq X 1J4>D %!12" ! F :5125% ]m + AM= >"= ^Q4 N >D;v= 6 F96;Z $!'& 8 >
% 5G !5. M> !2
1 E5 TE5 .12w"5+= [=5 126B i? ] 1' >T
A
F
M= >T 12P`{12<> = &[email protected]= >T 12Khq!> T9".!12 T55 >5 `{12w0= > 5 h B
O ! r ]{ H4> EM UTE510 T EM> ^] 12 ! 12 > > M= > T 12LD
1r
B` A.h*N H M M
B` A h M7Q
` B > h
5q4= >455!5 "qG12 ]v/ G >H
= <5'FM= T> 12[163` h
G12 8 $` hB
+= > ir 1J4>D M= >=;v= !' >1'CD
&('*),+.- 02/ 130
f|12 8 5 7L9:
; ` hHN
; B` A.h
N
1r
8 !
(
<=?>[email protected]
; B` A.h
;q`BA h B` A.h
+= QK N` B > hB
` B "h
}
!"#%$&(' *)+ ,-./' 0 123",4)-56$ ", 7$89 ,:);#,< =#5>" ? ,@ AB"' 1C)1? + "8
>?; A .6;Z U 1J4>D 6;W1 12Ur;Z ] 12 !12 >> ( x 5 "Iz L= " J4>^% < 5 !
# >J7'B L124[G12 8 124[124!L>T5q>D;ZT E".
= 6;ZY[ >qG12+! !"= '5H> 512>T126BEY[12 5 /
G12 512 / > 1JK>m }$ [email protected] 3G12=(N> 512>12 4 10!'&+= 5 8 '
`m512> 12P 1JK>m h / 5 >12<> 1' '51! = <
79
/ >19" 5
8 L>D;Z Q4 *`{B B
hB
i > T(54 9X
<5'$X+= >
= V!'&12 V 5: <w> = &10<w+= > 12 5>= > ! >w
5X' > 8 " 4 += "= '1<>X!12 124 " 126B % w
1JK>m B %( <5'L > 8 E[G = G q"5M= +=5 12 [ 12 ! 8A J4> j "5%>
0 512>T B|b0 >12 . QK 9"r;Z>*0T55X6; 5 @dl"G (= > 'T 12 9( @+= > 12
`{B B } hB
Formalismes lagrangiens
>[email protected]"G H ] 1' > F> A B
>F!12 5_ "514 w
Formalisme purement lagrangien
G'5 "?j!12"5 12x >y> !12E5 E5JM 7` h BCi = &10qji A H>@!'& [email protected]+= 4> ! ( E U @!12"5 "5H>@N >4>! 5 `m!
ir 1J4>D w= 9"4 8 > E <41J4>D ' 8A E9D
>4>T! 5 ^Ui A j5 ] 1 'j> !12"5 "5
] B hB
&('*),+.- 02/ 130
f|12 8 N` E Kh
587L9 :
%` E7h
1r
N
=
<=?>?] @
`BA E |h
` B
N 5 : ` U5 [email protected]> >T"= h N ` Q!R'h
`BA E |hJN ;q`BA.h ! E M `BA h
%
E 0= ] = E P!10!'&N( >"= "5(>D;W !
` B }2h
8 !
>D;W 5 ! F5 ! H+= 4> ! E%1' 0w'
%J4 8 12
Q!R
B
h
` B
h
` B 2
h
>K>! 5 '%(i A @9> ] 1' >> A <
1J4>D X( !&'!'& X
(G12" Iz5 >>^ > 9 >.> < >4>T! 5 ' X512EU4 > = > ( E<! 12"5 " m` > <+= 4> ! ( EU >D;d= EUK> hB
- " %$&(' )+ ,-./' 0 123", H 3
E
ij;v= !'(> 5 124 '(
= -> A - P> 512>12V` Kh% ^< / ! ]m / = 9" 12 8A ' 12 > > / HT[= 9" 12 8A ' 12 >> M!12( H> '1J4>m B
ij; C8 " A L
E >4 ] 1' >TM> A E=
Formalisme lagrangien augmenté
!> '9" / } j G1 ! >T > A G
>H
= ! Iz / [96;Z>< >DW; 1 12
6;Z 3> A S0= >T. %!12"5 "5 %r;Z[= 2 > = .%> >4>! 5 / `m! ] B 4
/ T 712 H ] 1 >(%41!& hB
ir 1J4>D @ @!12E ! .;d= !'H U 7SD
&('*),+.- 02/ 130
f|12 8 N` E Kh
587L9 :
%` E7h
1rU>@> A E`BA E |h
E= ;v= !'
N
]
= <=?>[email protected]
` B h
] 1 >> "H!12(N 9D
E`BA E |hJN ;q`BA.h ! ` ! M `BA h5h 1rp` h += ' w> X+= 2 8 Sc` h^1r 5 _
!'& 12 $ S 9 6;W 5L F
<5<55'T!5 E^G12 ] 12E^>
[=B
,.w712TEM 8 % = '9 / > ] 1' >' qK "F> I
Formalisme lagrangien perturbé
= 12 E> 5! ^ "5 12V+= Q4 NG12 8 /
_(> A ""S
EF.! ]{ + M512> 8 ' 1 = Bi? ] 1' >TL> A $GJM
= G'M >>
!55w ! >"(
= = 'T9"B >j!12 '55c G [email protected]> 512>12P > A P -5'X 0I
= G1 UN4>4>T! 5 [email protected] A / } / } B
] 1! s E !1'! 8 "@
ir 1J4>D X 0= 12 @ L>@ 8 "9D
&('*),+.- 02/ 130
f|12 8 N` E h
5 7L9 :
8
%` E
1r
h
N
]
= <=?> @
B` A E |h N q; `BA.h !
B` A E |h
` B h
!"#%$&(' *)+ ,-./' 0 123",4)-56$ ", 7$89 ,:);#,< =#5>" ? ,@ AB"' 1C)1? + "8
H1r
5H U <555'!5 "HG12 ]d/0 J45 "M7B
nH '9" 12 r9 / ! 12E5 " ] 1' >Tg> A ""q
= [email protected]>r;Wl rj!12"5 "5 /
! ] 1' >TH!12 8 > !12E5 T E5%q>N >4>! 5 *!12>D;W %> ] 1 >(F4 "
> A 6BAi 512>12N124[=+ j
!55H_= &1 + 10!'&+= B2Y[>>q5 4 8 g> 12> 12 !5
>1''9"
5 8 ' B
Remarque 1.1
' # # * $ # $* J 2$# $ $#W
F ( V $# $* # # W % % $
% # *
O ] 1' > F< E.5 >12U9>R;W12U>T5 > 5 !'& 49" @M= >T 12<123! >>4UN4>4>I
! 5 %@i A 0
w 1J4>D ( FzlG H 8A "CD
\1 >( F+= > 12
'` h R ` h N !"#
\1 >(@> A ` B0 h
` h ? R ` E7h*N$ !%"#
1r
JR ` E7h
N
+= Ey 5G ! 8 ( E*>
H0"= ' ' / H1r S
R
8 !5 ' 510!= [email protected] 71'yTE= ' ' / +!12E !
0= 5 "5 E%>@ > 4>T! 5 M
!12" !LL> L>[email protected]!12" !B
249"r; 0= 5 " / 12 C8 2
1 *[= = ! 9. !12" !B, M> 1JK>m F !12E ! ] 10! >
h
` B T h
`m>[email protected]!12" ! h
& / ?R?' !"#
` B
> +
= >j5'+6;ZjG12?12
] 10! >?y> ] 1 ' > T r+> 5 !12w4 8 "5 / 12 q124q""= 512 @ >D; 5G ![
l0 (T9"
EFF> M L= "H!12 EM1M= 0= % M> >5"= "5
>
] 1' >T G >= @! Iz ' N> @!'&+= r;ZE= 12 P5 (4 / _= '[email protected] F41J4>D H @!12E !%
Ul0 (9B
8 X(0= 512> 12
1.1.2 Schémas d’intégration temporelle
YqVl0 (T9"X>+= / > 112 ^w!&[= (6;Z""= 12V 5 404>!5 K>!5
512"j5 [= `{B B hBEi? j ' (
'g512"q512 8 Eq >= *G12j> 0= 12>12X g 1JK>m 1 "' 2.)1 ,95 ' 2=.5> H15 $ ", 7$8 , );#,< =#5>
J '5 ] = 9" 4! B4ij; 4 10!'& 04>!5%+= !12K>
& 5 ] 0= 9 ! LH> F5 !12 H712 M! > * !124" y 1JK>m +4! / *+= ! 5H N!12"5 12x >F *+5 4y g>M *6;W !
] 12 !12 8 5 '5 [ j12 / G12= 8 5M>D;WK>1212( L> 12>12( 04>!5 +512"M712 [!55 12$4[!124 12 >> "q5 J4> Bi? M!'&+= !12 12 >> Ej J4> / +[= ! "F> 0= 512> 12X!1rx 5 5.5l05
512"L!124> = B
= 'T9"B"ir +!'&+= L4>T!5 q512"M0I
M1r$> + !12 Yq
l0 (9S 12 >+= / > $< x <112 512" = 2 > "w >= $1Cl 4 E<9 >9
0= ! 12 BeT / > *!'&+= j04>!5 yF512"g *J4 = 0N '1J4>m *l0 (9
8 ! [!12G15 ( EX= > 551K> 59 + M "= ' 0X!1245 " BYqX 7 / >D;ZT> 12
6;v= > = "<Q4 T6;W1' = > 8 = <G12$E= 3! $!12(71'5 E / !124 P $ 5'! w
. 12<!12 JK> Bij; 4
1!& 4>T!5H 5FK> "= $ ! . '1J4>m 8 !N
!'& 12 X% ML5 (4
+ > 8 "MGB"Y[w!%9 7!12 !' > [ 1J4>D ( + %!12" ! ! /
8 12.!&12|> ] 1' >T N!12" ! / >D;Z""= 12-N> l0 (9 G x 5 ]m 5Nr;Z D'@51 > "(4>!512U04>!5%12S( 05B
, L> 8 "5 / 12 F4!512 5 8 ' <0 4> ! += 9" / > FG ] 1' ! L
= &10 = 'T 9" 4 !12E !X l0 (9 / 1J 5 " ( (
!12T5 " ] 1' >' ^
$!'&+= X 5 4 B[ir X0= > $ = 'T9" 12E = "P
= 1J 5 "4 $4> = ( E 12 c!
] 1' >'
1.2
!&[= L 4
B
Méthodes numériques pour le contact en dynamique
1.2.1 Un problème de référence : impact de deux barres
> ; A +L>D;Z( !jL J ' q+= ] 1' J4> q q>L5 q>12 4 6
> `{Q B h Bi?L!12E !
L> = 12S12E >
l = <G12"F 8 @> ' 1 A2 12U H12 H>12 > B
V0
2g0
V0
L,S,ρ,E
L,S,ρ,E
x
O
B &
@. ! #V
# B $
! %
!"#%$&(' *)+ ,-./' 0 123",4)-56$ ", 7$89 ,:);#,< =#5>" ? ,@ AB"' 1C)1? + "8
i Q B = 5 "5> 512>12 >l"9. >D;Z !M . 0J ' F[= ] 1' J4> F "9 /
512 5 % 0 8 5 +T >
1 G12"= M+ 1'( (= 2 > B 12 +l(12"512 M> +!'& (4
/ [= 4> ! " /78 5 '5 L.!12"5 "5 > 34 8 S G2
1 "HG15 E >L!12E !B6i?
C#
0 12 L .45 E%^!12E !&R / = 12 / > !12"5 "5 0
>
@!12" ! R 512"L124[= H D
JN
1r
%16
N
# / _ .> (] 1'!
#
C$#
# C# R
NC # &R*N
@ky1212
` B "h
` B
h
` B }2h
+= ' " / 7 ! 8 E / > j: X > / > > 12 q+> 5 !12X + 0
J ' /
+= ' "%>N14 > j12 %> = 8 2
1 > (T9" / 5G ! 8 " > 10 >
= "U= 2 >+ = '1B
:
ky12H4>T [email protected]+= > / 12 F 8 1Cl124^ R> ; B|`m! ] B T h
0
v
u
v0
g0
tc
t
ts
tc
t
ts
−g0
−v0
σ
σ
tc
ts
t
−σ
B
*
( #
B $
! %
1.2.2 Formalisme du contact par pénalisation et intégration en temps
12
L4= 5 E5124 / G12> ! > '"= / >! >N4> $= > = ( E '^r;Z 10+= > 12U += > 126B
>7 ; A F%(1+= >T5F>1! > EF>.!12" !H F> = !12U6;Z X 1F>+= .% 0
B i R ;W0 '( ] [email protected]> += ] 1' 12S w '51N` M 0h !12(N 9D
] 1'[email protected]!12" !
1 "' 2.)1 ,95 ' 2=.5> H15 $ ", 7$8 , );#,< =#5>
0
1r
R N !
` M h
` B
5H>% [email protected]= >T 12P`{ <5712' ] hj+1r ` h
h
5H> %712' 8 .(` hB
e > _= &1 XM= >T 12 / >R;ZE= 12 5 G1 >>^ > C 8 12N10+= >=X>(!12E ! l0 ([email protected] c 0= ! 3!12" ! G
]m ^^ DK>!5N123(4>!5BK, 4%!9 / 12 j15124j > 5'!L 5Hq / N
> 8 !5 '[L+= 4> ! ( E /"8 5 '5
C
!!K= > = 12 / ' 5G ! 8 "B
D
Pénalisation et intégration explicite
O 55 10!'& 5%> 4
>T H(4>X (55N @ 8 L 3!1 N! >! M
> `{B B / } / hB
Yq>T>G125( h% 41 12 ] 1'! N (!12E !^ >(16 >3` B h hN '!= 12<04>T!5 @>D;d= 9" 12S6;v= 9 >J .!12(49D
N ! "# ! &
N C C
! R
D D . N C D
#
# C NC #
D
#
N
` B 2
h
` B
h
` B h
` B 0 h
#
% N>( N5 [email protected] 50= ] D> ;Z E !0= T 12 B|, X` B 0 h
X+= ' "(> ^12 += > By, 4>D;d= 9" 12 ` B 2h / > ] 1! <!12E ! / TE= ' ^
0"= 512E!12 " N >D;Z45 E B
1r
C$#
ir!'&+= L L= ! .! "5= (` B hH` B hL 5>N!&[= >4>4H>T"= 712 !zl7
'!= 12 / [
! / G12[ [ 12 qL = !12-`{ 0= ! $5 !12 X1'H $l0 (9H>T[= 0 hByYqPl0 (9>+= / 512 @5 4 .12 8 = Q (!12 12 (5 J4>T"S
= `{12
O \*iyh / 12 N G12512 9"6;W 0= 5
)+ >l!'&u1_ >H12"X12"50= <5'N M= >T 12U 12
1r
"
R &
&R FN
!w _!12E ! !55w!12 12_12P= 2 > Ewx 5 8 = 'Q+= B
9"3>U ( U5 4
X!'9U+= G = 2 > ( Ew " 74>4F 55'T!12<L>@!'&12 <
"
[email protected]> 4 > 12S41 ^ >^ 10G >("4_= '9N
!'T9" += G "L w
w5N +
= > 12 B
B
` B
&R &
[email protected]>N h
5 (4
!"#%$&(' *)+ ,-./' 0 123",4)-56$ ", 7$89 ,:);#,< =#5>" ? ,@ AB"' 1C)1? + "8
eQ4SNJ+= [=Q4!. S! 04>!5> !0= T 12 / 4 ] 1'!124 => 5'!
H12+x 5L>T"=B O F
!'&12 ^G'g ^! >! >0K> | 4MH> 512>T12 / 6;Z M ' / +5 4
!12 5%> 1 A2 12c >R;d= !& >>.L>R;d= > = ( E / 6; 5'H = "F12 +=%9"> q ] 1' 12
0<1 F512"L+= !124> = %> H4 H 5 B
:
Remarque 1.2
@ $ % J 2$# $ 4 # $* * @ !$ # * Q @ #W . 2# $* % T % & % . W# $* : $ @ . # W J # % &% * # % .2$# $* #
iq; > 1'&4jg0= 512> [email protected] %= 9 12 |j>D; 41!&* rM= >T 12
Algorithme de résolution
8 !N""= 12S04>T!5. 5L12 += H> Q ' B B
%J4 8 12
B
% #
@9> 5'! 5 5N
'T [email protected]` 8 !51' >> "hM !
D
R N !
. $
( 5'!!124
G = / 9"X>
] 1'! N (!12E !;WI
` M h
M9"%> q M qG12"q !12" !M12EM[= 5'(+= @ +6;Z L4& 5%L 0= 4!12 / 1 "' 2.)1 ,95 ' 2=.5> H15 $ ", 7$8 , );#,< =#5>
T
> 9 >>.123 "L H!12( 5N L12 8 0U5 !12" !B7i !1 !12 `{5
<!12" !hFG'([email protected]! >! >> 8 r4> ! ( EB
"!12( 5
12 6 0= 5 "512 |r> Q B > r!& 4?j[= 4> ! " /
Analyse des résultats numériques
8 5 5 /y !!4= > = 12V ' 12 X!12" !6;Z4 G12"@> J (G15 " >> " _!12E ! /
12 += + M>D; 41!&H +M= >T 12X+ ""= 12$K>!5 +>!&[= [4 L= 4!
! "5= B O c0= 4> 512"P!12&[= E 8 ! > 30= ' > -1J 5 - O G "5P D> B B
ij;v= 51 > H> 0wJ / 0= 5 ""= ^'L> Q B } / +
= !'1 x LG "H> K
& 5
@!12" ! '[email protected]>@!12" !L 5H5 "
= HM= > 12rB
1512
H9 G12M 5G !5H> 120Iz"5M= +=5 12S = ' >T> / >. <
5N.+= > 12<12
x 5 > 8 " 6B %35 >?!'& 12 S!12E5 E.>!&12 S U @5 4
%9"4r12x 5'N > 8 I
"HGMG12H ]m @> !12 12< @5 J4>"
= `m! ] B B B hB
−3
x 10
1
20
10
v(m/s)
u(m)
0.5
0
−0.5
−1
0
−10
0
1
2
3
4
t(s)
−20
5
1
2
3
4
t(s)
6
5
−4
x 10
8
x 10
6
0
−4
x 10
2
x 10
4
Sigma(N/m2)
0
a(m/s2)
2
0
−2
−2
−4
−4
−6
0
1
2
3
4
t(s)
# $
B&
5
−6
0
! J & J @ #W $ 2
3
t(s)
x 10
Remarque 1.3
1
−4
4
5
−4
x 10
E $ * #
$# $*
!"#%$&(' *)+ ,-./' 0 123",4)-56$ ", 7$89 ,:);#,< =#5>" ? ,@ AB"' 1C)1? + "8
350
Ep
Ec
Et
300
250
Energies (J)
200
150
100
50
0
* # 2 # 0.5
1
1.5
2
2.5
t(s)
3
3.5
4
4.5
5
−4
x 10
@ % @. $ G B}
0
@ $ #
$
#
Pénalisation et intégration implicite
ky12H> _= &10 .+= > 12 / R> ;ZE= 12<4>T!5H 5F""= "[email protected] ' ! 96;W >>G'F
5 5 A 1 ( E <>
1 >+= ' "= >12 (!12" !N ] 155 ( E / 1Cl
2
E>R;W ' I
!'&45 " > 5! "5<`m! ] B7B B W} hBGk >T> ' / T>6 5!124"S 3l0 (T9" >T[= 9> [!'&+= F4>T!5 g512"+4!12 124 > "g5 4
J > [["510 EL`m 4[>%! [ FI
u h 4X T 12P" = 9" X!12"5 12x > J4> B O 5 '!12(wi '5 / e'1 / 12"H>=.!55 = &10 / 512 F <14Q "+>D;ZTE= 12S4>T!5 Q4< !2
1 5I
8 L>D;v= @51 >@H> H9" E= 12 8 "L>T[= . l0 (9B
ir ^= 9 124j[>R; > 1 '& ( '51! =
Algorithme de résolution
N` B T hzI` B }2hL L> Q B / 7 ! 8 EB
C
1r
D
#
N
N C N # R N ! "#
`
! h D D C . ` ! h D D C # NC$#
> / w 0= 5
[email protected]!12E ! !55._= &10512"q124[=
` B T h
` B "h
12Eq> g <5 [ L u 8 = Q "q> !12 12H5 4
J T>"N
= `{ (>T[= h
0 / L
1r
512"L>@+= 4> ! ( EL> ^8 5 5$ R> ;Z 5 "H D> B
# #
C#
` B
` B }2h
h
1 "' 2.)1 ,95 ' 2=.5> H15 $ ", 7$8 , );#,< =#5>
B
% #
J $ #
ir w0= ' > <1J 5 < U!55 10!'& `{Q B 2h
Analyse des résultats numériques
12"5 "@9"(> K> ! "@ .05= ("= N 0
3J ^ 5G !5 " > ! 12 412-
120IzTE5M= +=5 126Brir .!12'J7 % 8 5 5 /6 !!4= >= 12 % '12c!12" ! / 2
1 "5 " 12!>T> 12 [" = 9" jG
"q> 4& 5%!12" ! B ij; K> [ ! +12!>> 12 j ] 15 "
+= G "5 !&12 . <5' N = '9 / 8 12! > y+= > [email protected]> @+= [=5 12 N
! 0( !&[= L u!12E5A12x > E+>D; 1T5 "j"4_= '9B2ij;v= L51 > q>
0^J ' jH5L!1245 8 L ! 5H> M= >T 126;Z4F / H> ' 12^ _= '9
"510 5% L>@'!'&+= L u / 6
; 5 ` 8 12HQ B hB
irU!&12 A1 > P V
<53<+= > 12 ]{ < G12" 09" >T> _= &10
55 JK>S
D| (5 8 > !X <5$G [email protected]>!124 12 [email protected] > 5'T! ' T ".
= E5 / ! H >> 5 >( E= @ F5' M= >T 126B
}
!"#%$&(' *)+ ,-./' 0 123",4)-56$ ", 7$89 ,:);#,< =#5>" ? ,@ AB"' 1C)1? + "8
Remarque 1.4
F " # *V ! % # ' % J
& * ! $ #W % #&# # *V$ V # # *V$ # 2$* % $# *
−3
x 10
1
20
10
v(m/s)
u(m)
0.5
0
−0.5
−1
0
−10
0
1
2
3
4
t(s)
−20
5
0
2
3
4
t(s)
6
5
−4
x 10
8
x 10
6
1
−4
x 10
x 10
2
4
Sigma(N/m2)
0
a(m/s2)
2
0
−2
−2
−4
−4
−6
0
1
2
3
4
t(s)
B
# 0
1
2
$
! 3
4
t(s)
x 10
% J
5
−4
x 10
#
350
Ep
Ec
Et
300
250
Energies (J)
−6
5
−4
200
150
100
50
0
B
0
0.5
1
1.5
2
2.5
t(s)
3
3.5
4.5
5
−4
x 10
@ % . 4
$ #
1 "' 2.)1 ,95 ' 2=.5> H15 $ ", 7$8 , );#,< =#5>
−Rc
−Rc
dn
dn
(a)
*V$ V # B &
(b)
$
! % # !" %
Remarque 1.5
* % # $ PV # $ $# % # # $ $# # 2* 4 V $# *B
# %
Q % % % $ 1.2.3 Formalisme lagrangien du contact et intégration en temps
ir
! ] 1' >' @> A @ 1JK>m N X!12" ! / G >= @ 0= !K= (( E N
5 9 / G
8 " 'jx 5>= F H U! Nl0 (T9"Bf|12L!12((G12H> ] 1' >T
MM= >T 12 / > M!'&+= H6;Z""= 12< $5 4MG 8 "%x [email protected]+K>!5 / 512MK>!5 /
8 12 5 S`{12S5 IRK>!5 hB,U= !' 8 12 %! L4 L= "5
4 10!'& B
Formalisme lagrangien et intégration semi-explicite
O +zlG+6; 4 10!'&M!12 5 %
1 5 g 1C0T 125 ( IR04>!5[+>D;d= 9" 126;d= 9" T>J 1 l
!'&+= N L= ! ! E50= / N (4>B|i?! !
" ""= 12 5 (IR4>T!5w 5X>T= 0 5 5 "$4>T!5 ] 1'! $S!12" !$4>T!5 ( ] 1!
""= ' B O !&12 5F""= "H! F> M!12 12 [ %!12" !L512"M 5G !"= F $5 4MM
5 ! 8 >D;Z A . % >K>! 5 ' +.i A <!'& ! S M 5 "F w! >! >DB6`m! ] BB B /
= mH !5 / 8 "q>D;v= 8 12>12l0 (9
/ hBi? ] '155 E+ 5= 2 > EM5 "%
-5l05 = ! T9"*
DGT>yr;Wl (+= 4& A E5'(> ] 1! @!12E ! ] 155 "@>
5 ] 1' 12 H L512> M S!12E !B
!"#%$&(' *)+ ,-./' 0 123",4)-56$ ", 7$89 ,:);#,< =#5>" ? ,@ AB"' 1C)1? + "8
i? = "
9 12 *j>D; > 1'& 510!= !55 '1!&F512Eg124[=
Algorithme de résolution
N` B hzI` B hL L> Q B / 7 ! 8 EB
R
D
C
1r
#
R
/
N
`
R
NC h
N
! ! #" N ` B
` B 2
h
`m>[email protected]!12" !h
D D . N C D
#
# C NC #
12E 5!
h
` B
` B
h
h
` B
8 !5 '!12E [email protected]
h
[email protected]> >K>! 5 .
!12" !B
%J4 8 12
9"6;W _ T9" / >w5l (P` B hzI` B h 5^= 512> ` !12E !Q q= h$ >D; r;Z = &10X @
J4>1! 4> 9 >>> 5'! 5 5N!12 "=X > A 12 >B
O ;W 5g q!55L!12 4 12(9H>D;W12^G g!12"" % = 8 19q4! !04>T!5F ^!'&+= B
ij; 4
4>! 12V<>D; > 1'& {` Q Analyse des résultats numériques
6;Z( !c 0VJ X[= ] 1' J4> $12 S> (0= > X>T> 550= X> 0= 5 "512 N> !'& ([email protected]+= 4> ! ( E /|8 5 '5 /r !!4= >= 12_ "(
= B U
h D> ;W0 4>
Q B
1 r
12
] 1'! !12E ! 8 [email protected] 0wG12EMG15 " >> ( EF U!12" !B
Yq >l0 EH! L= > / 12 F' '912 H9HD
h+>@!12" !% 5F 5G !"
= !5 " /
h X12( EM !&10! / > [ + 8 5 '5 qM !!4= > = 12 H1 > + jG12"j $!12E !
G ! ] 512"H L= "F = '1B4Yq$ 8A !'& .T>[12E$= 2 0 = 1 0 5 F 5 "M !12E !B
%J4 8 12
9" / G "N> 4
& 5$ 8 12>+>TJ 9 [ >X+= !12>> E / > !'& 4^ 8 5 /
3
= ] 1 '! S!12E !w12!T>> " 512wS> 8 > '(1 l B+k !!4= >= 12 w$ "
>> ' / >D;d= .51 > M> M 0XJ + 5H!12 5 "5 H"5 8A >>..5 [email protected]`{Q B > hB
e ! 5-c>D; 1C0T 12 04>!5S ] 1'! w""= ' ' / 12 [= 5 !5 ]m J4> 12!>> 12
(>D;v= ([= ] 1 12 [email protected](>D;d= (!T[=9" R 4
"(> = N12!>> 12 .
>D;v= <51 > j
B i? ^= > (9<12 ^12412 ^!F512"^ !!1' C8 U
! ! 0V1J 5 "4^ O G "5H >DB %b0& >DB B
1 "' 2.)1 ,95 ' 2=.5> H15 $ ", 7$8 , );#,< =#5>
B
% #
@.
%
E Q $# W % $ Formalisme lagrangien et intégration implicite
, !wzlGwr; 10!'& / 12V5 5<>w!12" !w D w !5<12 >5X> ^!'&+= X
L u ## f G12.>D;Z""= 12-5 G1 >>B O 55( = &10^ 5 5 T>=<`{B B W} / / /
/ h / '!9"6;W >T>FG' / ( ! > / 6;Z""= ' A 14 " @> q 120Iz>+= '"= q ] 1 I
5 "qq H>D;d= > 5514> !"= / 6;Z4M / 6;ZTE510 [>D; 1'5 "j" = 9" `m!12E5 12x > = h /
6; 5'q B q
Y >>q 5|T>=g _= >10= + ?4>T 5 'M`{B B W} / hBL12 |!12( *s!124
L>R; >l05 @> ^8A ' "5> 4> F!> 9(`mi A LL uhB L124L+= !' 8 124 / 5 / >
= >1 12 F '1712= H> >5"= B
!"#%$&(' *)+ ,-./' 0 123",4)-56$ ", 7$89 ,:);#,< =#5>" ? ,@ AB"' 1C)1? + "8
−3
1
x 10
20
10
v(m/s)
u(m)
0.5
0
−0.5
−1
0
−10
0
1
2
3
4
t(s)
−20
5
1
2
3
4
t(s)
6
6
0
−4
x 10
5
−4
x 10
8
x 10
2
x 10
4
Sigma(N/m2)
0
a(m/s2)
2
0
−2
−2
−4
−4
−6
0
1
2
3
4
t(s)
# B &
$
−6
5
0
1
2
x 10
! . R
C
#
NC N # N
`
R
h
N /
5
−4
x 10
@ Q $# WJ !55._= &10512"q124[=
N !% #" ` B
! h D D C ` ! h D D C # NC$#
h
` B h
`m>[email protected]!12" ! h
`
/ ( $512Ew>cN >4>!
!12" ! / 5G ! 8 " D> ;Z Eq %! >!4> 1r
R
4
% % $
ir ^= 9 124j[>R; > 1'& ( '51! =
Algorithme de résolution
N` B hzI` B E}2hL L> Q B / 7 ! 8 EB
D
3
t(s)
−4
` B "h
` B
h
` B E}2h
5 3!12" !<w w 5'! 8 !5 '<
512"+> [ <5' + L u7B
/ [1r
ij; 44>! 12V<>D; > 1'& `{Q B "hU >D;W0 4>
Analyse des résultats numériques
6;Z( ! 0cJ 124> 0= > 0= 5 ""= N' > Q B B 12 lc!12 5 512
9<> !124 12V<12V"5M= +=5 12 5^ 5G !"= Bqf|125 ] 12T / V!12 "(! 0= > 0(512> 12 >l9 gL>R;Z !jH 0J = > 5T9" / 12 j!1245 512 +>D; '12 / 12 "@ P!&1! / 6;W12!>> 12 ."4_= '9 G1 "5 @> ! & 4 8 5 5 / 4"(
= 1
H> 12LF!12" !B %JK5 8 12 g9H>D;Z A H6;Z
] 1'! jH!12" !qq> !!4= > = 2
( 8 1 "' 2.)1 ,95 ' 2=.5> H15 $ ", 7$8 , );#,< =#5>
0
350
Ep
Ec
Et
300
Energies (J)
250
200
150
100
50
0
B >
0
0.5
1
1.5
2
2.5
t(s)
@ % @.
3
3.5
4
4.5
5
−4
x 10
%
E Q $# W % $ !&[= LL u 1 EjG'(qFQ4>5'q F ' L j& 5 ] = 9" ! B <= 124 /
! 1T5 "14Q > ^10 U J 5 ] 0= 9 ! Bgky12^ = 3 !55U! ' ! / 12
G N >5N ^!'&+= ## f 12E>D; C8 " A < G1 _!'&+= (
L 'u 5
Q4>5'H> H& 5
] 0= 9
! H+= %> F(1 J 5
] 0= 9
! B
ij;v= 51 > H> H 0<J F H 0= 5 ""= L> Q B }B L124L!12 5 51249
>D;v= 51 > 5.!12 5 8 % 4[>%! F6;Z !0= 12X 1 EM _= '9 /" >1'
96;W >> += !1 x %G L>1F9"6;W12SE5'1 4+ @> ' 12w _= '9B
nH 8 12 [q> j12!>T> 12 g = '9 L`m 120IR4&"lT9" hB2Yq>> q512E
Origine des oscillations
= q [> q T!12ET" "= j (5 4q q!& 4[ 8 5 5 `m>T= % >R;Z A L6;Z4H_= &10
> A 712 [5 5+>.!12" !h++>R;Z> 12(6;Z $!&[= 6;W1''w= > 8 = $5 K / "=
= F[= 8 >1 G "
>D; '1 12w ] 12 !12 M0= >Tm M w5 [email protected]`m&"l"G1&6 5 L 8 >T "%
Ff l0>1 h B2i 0
= !T12M5 >g!&[= j 8 "*+= > "5F ^ 0= 5 4!H6;Z ! BAY[>> !! E > *12!T>> 12 B O F
!12 q5F!12Q '(+ !12 "j! g0= > ! 0N1J45 " j j> _= &10
M= >T 12<1rS> F12!>T> 12 F512"H124HG1 "5 MG12H> < x E= 12SK>!5B
k >> ' / >T "% !'&+= !> '9" %r;ZE= 12
!12"5 12x > "U> 120Iz"5+= [=5 12p U -_= &1 ci A
!& 4( 8 5 !!4= >= 12 0 712TE<!12" ( !12 X!<9 [$ 5w" = 9" E<`m! ] B "5U 0 G12"^<!12E ! / (4> 9 >M12
P5 K`{zlGL u [email protected]
/ 12p!12 5 9" c> w ! G ! ] U512" "4>Bgky12
h / !12 += 124>R;W K>X6;ZT !
4>T9" 12 > = &10U> A 8 !
!"#%$&(' *)+ ,-./' 0 123",4)-56$ ", 7$89 ,:);#,< =#5>" ? ,@ AB"' 1C)1? + "8
% B
#
J
%
@
% #
.""= 12< w5 KM F> ]{ (T>> F!'&+= L LL uKB 12 F471212 [9 > M 0
G12"y12Eg512 0!12 4 12 y > |' 8A E5 CD {` 1r +[= "j> yG12124|4 >
H 0$G12"h
C#
D
#
# N #
NC$# C # N
N # N D
N
ij; > 1'& = !4= E(`{Q S>D;d= 2 >"
= ' 8A E5HD
N
` B
` B 2
h
h
` B
B "hG125X >D;Z 5 "L4 8 " ' > >4>! 5 H !12E !
h
` B h
1 "' 2.)1 ,95 ' 2=.5> H15 $ ", 7$8 , );#,< =#5>
T
Newmark(β=0.25, γ=0.5)
Newmark(β=0.4, γ=0.7)
Newmark(β=0.5, γ=0.5)
Exact
*
# B B
! # C # ! # C # 0 !
E!12 5 L!12 124F > /
D ! D N
i > 12w "5>
C ! C N
%
@
% $ #
] 12 !12U L9" "= L!124" ^ >D;Z EH4 >D
b12 / S>D;W4' "M
YqU5
J J
D # D N
D # D 12 F1J 5 124`{G12
C$# 8 5 '5 FL>
! C$# ` B0 h
h
D
` B
!!4= > = 12 .12 `{G12
h
D
h
` B T h
ir @ > 12 (` B [email protected]` B T [email protected]' E9(> @ !'& [email protected] 8 5 5 N6; ! !4= >= 12
1' > `m" = 9" h512E " >T P12( [email protected] -!&1!AB >.+= G " . <5 N
!"#%$&(' *)+ ,-./' 0 123",4)-56$ ", 7$89 ,:);#,< =#5>" ? ,@ AB"' 1C)1? + "8
320
300
energies(J)
280
260
240
220
beta=0.25, gamma=0.5
beta=0.5, gamma=0.5
beta=0.49, gamma=0.9
200
B }
0
1
@ %
2
3
t(s)
4 @.
4
5
6
−4
x 10
& %
@
% $ #
h [4^ qL5 4 B2k[> g 0= !T5 " / >H q6; !!4= >= 12 M q 8 '5 "
uU` / *
1G112 > ^X5 4 / !<9" qK>9X>D; 4 '12 6
;Z
12 "
^" = 9" V
-!'& 1!U`{Q B h% > 8 5 5 . [email protected] !5 ". 1G112 > P . 8 5 '5
> B
L
Améliorations proposées dans la littérature de l’approche lagrangienne/implicite
kq>T 5 'r5q12E?TE= = F > 10+= > 12 > A j K>!5`{B B W} /
BBB hB v>L>D;W12E >l0"=NH12"L41G12"=N H(1
G ] 1' ! H = 'T9" B 12 [= !' 8 12 H H> [email protected]> H10 QK!
!12" ! C 8 !q jE= 12
Q4! 12 FG12 = >T1
12 M!> T9" B
F
& M >DB W} 12"F 1G12"[email protected] !1' W
Modifications des schémas de Newmark # 12715 " >>(!12" !N N > 8 5 '5 /? !!K= > = 124^ ] 1!X(!12" ! > = 8 5 5 .N
>l05>10! >(^ 1 A2 12 6;W12 ]{ !s 12 ]m ^> !12 J4>
!!4= >= 12 B O ]{ T E / > g12!>T> 12 g [!'& (4g1J 5 j [>T 12^ ('!'&+= FI
u3 5 5 "BKi (1 Q4! 12c!12 !'!'& "
9 !12K> N
G12".! -!12" [email protected]
5 ]{ q = G
% >1296;Z>g12E[ 4& 5LL!12" ! M12"= 126B O 55%_= &1 %"510 K>!5 " V= !& >>$(5 4N> = c > !4= >= '5w @124 [email protected]> 12>
!$9"4jG125
>D; 4 12^ *M5 4yG12*1J 5 * *0= > * = ! Bk >> ' / > *10 QK! 12
:
1G12= M12"X= "N
= 5 5"= L M F! M16<> M1" 4 M !12E !L12EM
!12" !% 4 ] '155 ELH UG5 H+= ] 1' 12 B
8 I I 8 +H M>.! H
1 "' 2.)1 ,95 ' 2=.5> H15 $ ", 7$8 , );#,< =#5>
) & O & 4& l 12"L 1G12=>D;Z>T 12w6;Z U'!'&+= L u /KC8 !N>!'&12U 5I
! >T
N N B O !6G'Lr; 8 12 / H>@! 6;Z !H
0< HG12 ! >> / 8 5 5 @ 1' > >Brf|125 ] 12 / [email protected]>(! (>D;Z [email protected] ( 0-J / > @!'& (4
8 5 F 55 "L12!T>> "B
f l0>1q+k 1G12 > 12 12E[
>4>T! 5 ' ' "|6;Z4q 1' > y7 4 Ey> 4& 5MM!12E !^9" | 8 "^ >L>N'!'&+= 1G12"=LH!1 [>H'!'&+= L u +>D; : 12 jL 0
D'[ !5F> >>"+
= y y 8 5 |* ! !4= >= 12
!B O y!124 12 y512"* >d x !'&+= (12 EgM> "= 12 /
!> '9" N L L uKB O 55!1 !123G'(!12E5 12x >
> [ + 8 5 5 [[ !!K= > = 124F1' > . > ] 12Tj T % 4'j> q12!>> 12 B
Yj " 12 +=9"^!55_= &10E5'1 4HN 12 8 > > % !12 / > > >12' > 0= 512>T12- 1JK>m @"4_= '9 / 512 F <5' T 124 >DB
Y[ I
Algorithmes numériques conservant l’énergie et les quantités de mouvements
8A E(> ^5 8A 0V Ub1c >DB / i 5 >DB (eL1- >DB 12E41G12"=U
!&[= H6;Z""= 12w $5 4M + '1J4>m M.!12E ! Iz( !L!12 5 8 "F>D;d= / > M9 0I
= %@12 8 "L>T[= 4> <5l05 _= ! 9B
12
8 = 'Q412 / +>!& 45 / 9" >D;v= %51 >.r;Z Xl5 .512>T q+= ] 1' J4> / 120I
(% !12 +L!12" !+ Xl0 (9`{ J45 !% ] 1'! q0"= ' ' h / 5%!2
1 5 8 .9 > !12 12 G > = w$75 !$ 5 8 'QM= Bg, ^>w! w!12ET" / ! > $G125$ (41J4>D '! > / 4 {] N9"X> !12 12P!12" !NG'T5 [email protected]+= 8 ' >> "
> !12 12$ %!12K>= E "
= F!12 12 + !12E !B7Yq$ 8A !'& / L !0= 12
5 G1 >> / !55$!124 12_6;W 5K> '= Byky12>D; _ ! / i 5 >DB e'1 >RB
12"H 1G12=> M' w ' 8A E9D
Y[U"510 "F H > K>! 5 '[email protected] A "H> !12 412U G''5 ! 0
T 5 "[ ! >' / i 5 ( >DB ' 8 ". !12 5 8 H>R;d= % 4 '
> 12!T>> 12 / N> !12 412 X120IzTE5M= +=5 12V6;W 54> @ ]m 5- 512 >
T 5 "B7Y[S G / >^N > 4>! 5 L^!12E ! @6;W ! ] 9"6; 3 5 "
r;W 5H H!12"5 12x > = 0
U 5 "F! >!4> B
> 2 e''1 >DB 12"L>T"= > w x N+= !'& / M <M= >T "H> !12 12$ 7 I
4!BKi !12 5 8A 12UN>D;d= @ 0= L> !&12 <6;Z M= > 12w ] 124!12
4(!'& ($ 2 $l0 (T9" 0B O 7 4 E / > !12 12(L 120Iz"5+= [=5 12$6;W 5[ 2 3= > r;W 5H
]m 5! L>
H!12"5 12x > = B
} !"#%$&(' *)+ ,-./' 0 123",4)-56$ ", 7$89 ,:);#,< =#5>" ? ,@ AB"' 1C)1? + "8
, w $5 8A 0 4>"= ' ' / i '5 $5
< x +
= '!& .!12 5 8A 12 / +!55
> 8 5 5 %G12 '.> !12 5 8A 126BGi w!12>> J71 5 ' } / 12"w 1G12"=c 3 8 c> ] 125Iz!4 w!12"5 12x > "M> q4> ! E++ w!1' "
!1 !12P 8 5 5 % 5.J = % 41!&
'!5B ,% 0N>T( 124r5q4= 5 E5 "D2 G
h > = &10M6;W 5y 44>! J4>j5l"= T9" (
E 8 !
> . = &10 > A ^`m!> T9"12 ""= h% hF>R; 10!'&^ '!5^ > !1 ! 12
8 5 712 F @x5 4>T! J4> F> H! [email protected] >> A HT !12 JK> B
a3!'& >L X5 X!12>> JG1 5 '3`m! ] B[B B W} / } / / h12E
Formulations en vitesse
1G12=(0= 12 (>^ '1J4>m (6;Z( !. 8 5 5!12( 8A ' JK>^ ' >N PG'!412
!12( 8 ' J4>H > 0^>T 0 j[= 4> ! "[[ ] 1'! [L!'& 1!B O 55 41!&F 5jK>
$r;Z ! N '!<9"r;W >>G'(N$= 'X>D;W1' X 5l05 U L= " >
= 1JK>m .6;Z>T5H c!12 "= 9" ! S!&[= 6;W1''N124$= > 8 = / c>D;W10!! 4!N 8 E5 .> AIz = &10Bir j>12TjL!12" !+12E= !5 [r;Z H D'L ! 5L ] 12 !12 8 5 5 B O 55
!
JG12 [= > B0Yq$ '! > / > q12'!>> 124q EjG E+> K& 5
'1!&% 12 L
U!12" !B+f|125 ] 12 / ! 45S<>D;W1' U AIz!&[= `{1' h12 0
= !r'3U> 45G'12
_= '9B
!
1.3
Conclusion
, !X!'& 45 / 12 C 8 12 G >
= > N L= ' E ] 1' > @N!&[= ^6;Z""= 12 5 ^I
G1 >>^
>T"= % P_= ! 9 @ P!12" !X`{ P0= ((l (9 hB L12 C8 12 @5 "
= > l0"=
L= "5 = &10 ( = '9 ^1J 5 (!125 " ] 1' >T '!'&+= X
5 4g'j ^! ! += (T9"Hr;Z !gH NJ ' = > 59 B2ij;v= HJ4J4T>1 4& T9"j( [=
L!!'& [email protected] L!12 ]m 8 15L> L!'&12 U4 8 "9D
BL>
] 1 >( F> A 4
BL>R;ZE= 12U5 G1 >>.T4>!512U 5B
, >U!'& 45U 8A "^ 12 (12 E= '512 ] 1' > 12 > A 1JK>m [email protected]!12" ! F5 ] 1' 12 B
!12E X
Chapitre 2
Contact entre solides : du local au
global
<L
<L=<
<L.O
<L
L< <L.Q
<L
<L 3 /J+AJ7E !"/E3$4/9;;:;:;;:;:;;:;:;;:;
P:5/(' 4DBAE +"2 */E+E!"$*/ *$/E :;;:;:;;:;:;;:;
' DBAE4$53/E +-, ' DBA5 E :;:;;:;:;;:;:;;:;:;;:;:;;:;
O
O N
B<
3 +EG# 87-3&$/0 :;:;;:;:;;:;:;;:;:;;:;:;;:;
/0$!*#H/F+EG# /0$ #H ;:;;:;:;;:;:;;:;:;;:;:;;:;
;L#@*6' DBA**/0!"+E;535:+E #@3/0*#HF! +53 !)!/0 ;
<
A54$53/6$$ 51!' EA/E # 4/J+ 2 */E235/ :;
P:/#@AE$53/ :;":;;:;:;;:;:;;:;:;;:;:;;:;:;;:;
Q
ET o ET o R E
z 6 NL MG%>0$(<G*/'T27%')-.*8{:w.6- LNM ()'(b !$- CC C
CCCCC C
CCCC
t !3(<G21(@[-]8 -K*\}~!3%Fp%>M il€(!,03M -7'+* C
C C
CCCCC CC
CCC
ETR‚T
ETR‚TRE
ETR‚T o
ETR‚T |
ETR‚TR‚
t†%>!5%>#76- 0$!3(*3%p03()"<&[email protected]{-7c !3-K*9')T27%MTiOC CCCCCC
‡[email protected]@G%>!3()-K# -7<N0 CCCCCCC CCCCCC
r9<[8T(‰03()"<k8T-C< <I(<"[email protected]/K6- <16- 03!3%>0$(<l-70\N()0$-.*$*$-K*/!3-7'+%p03()"-K*Š
t !3(<G21(@[-]8 - 'Ž: %2P03()"<j-708T-C'+%!G.6- %2P03()"<CCCCCC
{"(*\8 -]21<N05%2P0\-708T- ex!$0,0$-K# -7<N0ŒC CCCCCC
CCCCC
CCCCC
CCCCC
CCCCC
CCCCC
ET s ET s R E
ET s o
t 3! -7# (d-7c !3-K*l.6- 21!3(‰03M !$-.*lK6- LNM (p%>'-7<N0$-.*/CC C
CCCCC CCCCC
-7MTiT(d-7c # -K*jK6- 21!3()0$M !3-K*&K6- LNM (p%>'-7<N0$-.* C C
CCCCC CCCCC
"!$#M '+%p0$(<I'+%>=!5%><G=(-7< < - [email protected] !3b 'd-7c # -8TZ^ <G%># (+LNM -]8T-C21"<"05%210ex!3>0$03%<"0‹
|ZE
|No
|Z‚
T| q
|NV
|N`
|N`
‚"E
‚o
‚s
C 1
",- 7$8J ,- L " )1 F)-5:3.$ !.5 /> [email protected]
O -!'& K
5c 5<! +3
= 0
B[i 4 (Dc 0= 5 "5P>c! = += >!> 9 /
! ] '15 E / l0 (T9" N5 ] 1 ' 124 / "5
} /
1J4>D (X!12" 512>T @+= ] 1' J4> B|i 0m 12"5w!12( "N124+= ' 8 124N> U] 1' > 2
1 > A 5 J4T>= = += >T E$> ] 1' > 12 !> 9 1 l
= 9 12 X
" = !' U
>12LN!12" ! . ] 155 ( E%%>R;ZE5'1 4
!12<!& 4%N' N`m @zlGNir 8 > Ivb > h
!124" B O 712TEL!1245 / 15!124 4! / >D;Z4 1' >"= N!5 8A >*&6 B
2.1
Position du problème et notations
H!12 T[= 12 H> 41J4>D @@!12" ! % E5N 0<512> / 1!! E F> H!12 QI
hM / H> L!12Q 12
12U > >D; &+= ! H12 G 6G ILKX` M4N
G Bij;W 5 ! I K 5j 6;Z r'F1&121' H
= 4 !
! >> / >R; &[= !H g 12 HG *
` h ` 8 12HQ B hB
12
E E E
K
/ G12`HN
1512
12
E
B#
#
E
h /
Q r471212 G9"[!
>
] 12ED' L H 12 G # BG /
5G ! 8 "B
"6= >D ??9" j!'& !
] '12ED |512"? (
<512TF L {:512"5 / 1"= Q / Q Q R 5 >> H9HD
q6;W >> G ?[= !12G12"=
Q N Q Q Q R
` B h
O F5'12F H512"L[= Q4 F > ^]m !s 12S 8 "5D
Q 5> wUQ r
1 V+= 4> ! E
5((712=B*, > 4
> N ! / 12
' G12512 9"^!55^ 5% ! 550= S`
N h >^! .> !12 123 N,.'!& >%
12
& 121 712 EF L 1J4>D (@ !4> B
:
Q 5H> ' H> 9 >>. F 4"= H ] 1'! H ]{ !T9"
44>9[= B
/ Q R
5L> G15 E >> !12" !B
R
512E> F5 ] 1' = #
Q / Q = 2 > " > 1' > 12 T G G / ' 5G ! 8 "B
1512
#
Q R
% `m G12"= H!12 h+512"
H> !12 Q 12 ! >>B
] 12"D
4Q / 0"= 0
1 1 H
,(' 12 =#5> )1 F/0 ,9)1 ( ,A 123",#
.
UT
B
Q ! #V #
k >> E
/ Q4$+= !.> ! += 9" % M 512> q+r;d= ! = 9 124+r;d= 9 >TJ /
12 LE5'1 4 512 +> H1 12 H 8 "5 CD
D4>D;Z4 ! 8 > "
12
D?!&
B
[email protected];Z PG12" = ' >[ 4> !12 Q 12_ > ! >T> /
' 5G ! 8 "B
/ @kq12> #
D
N
D
D5
5 y r!12"5 "5 ? O !'&"l / ' (6? 0D q5
'!'&4&1 / ' 5G ! 8 "B
5
= ] 1'!
#*E 8 12>T (9 F <512>T
G B
8 12> (T9" F 5 ? r!12E5 E5
H> L 0<!12Q 12 B
>D;Z"5 8 >T><U5 4X "(> 9 >H124 = 124(> l0 (T9"< 5l (
= ! 49" B
2.2
Cinématique des grandes transformations
, %! A 4& / 124H 0= 5 "512 %> L4 L= "5 'H!+= T9" LG'5 "> I
!'412M512>T |+= ] 1' JK> / 5 ] 1' 124?Q44 B k >> ' / 14 Ej> !' 12
> A `m "= ' >> hM w12 8 " / 12 M G >12 +> M4 124+ M5 5 'F!> 9
@+= ] 1' [email protected]!12"5 "5 B
irN!'& [email protected][= 4> ! "4<512>
` E hHN
` E h
!
E
G12H`
51"=
E h
5
G #
L;v= !'D
` B h
C 1
",- 7$8J ,- L " )1 F)-5:3.$ !.5 /> [email protected]
i 8 5 5LM>D; ! !K= > = 12 m` > A hg%> '! >
;W1J E% q+= 8 12
>> M H G1 <
5 4H4U!'& (S
@+= 4> ! ( EB7e4 D
D
`
` E 5h N ` E h
E 5hHN ` E h N ` E h
C ` E h
N
` B h
` B "h
i 5 ] 1' 12X X
512> % !12 Q 12X >N
L>R; 4> ! 12
= Q4 D
+
!12Q 12 ! >> 5++= !5
G !
E h ! ` h
DG #
`
` B h
Yqw G12 E+9" .! M5 ] 1' 12 M512"M= >D / > 5 4 ] 1 12w M ]m ! F.!12E !
G .x 5P= !'5!12(' CD
` Q R E hHN
R
` B }2h
12
G125124U= 2
= 4>m / 1"= > ( E>D;W0T55 ! (! 5 0
!12" !B
Q R BKYq>> F12EL+= QK F> [email protected] 8 "5`{B B hD
1r
` E !^
=
N
E E
B
I ! `Eh
!! /
5H4
N
K
' ]m ! @
` B h
D !
` B 2h
= >= E. 0
= ] = ir "[email protected]+= ] 1' 12 / 1"=
N
/ [= !' 8A [email protected]>
h
/ H0 4> / >@! N
= 0
= ] = , I
` E h!
D !
/
!^@n / L K> /
B
5L+= QK G D
` B h
1 1 H
,(' 12 =#5> )1 F/0 ,9)1 ( ,A 123",#
1r 5>D;Z E<
= UI
K
> ?B
K
c
5(> "^ G1U > 38 J4>w<> !12Q 12
ij;ZTE510 !12UN!5
M N M ` B h
5 LG'%6;d= !'> 5 4 ] 1 12Sr;Z 8 !5 c= > = " :M
> !12Q 12UT >% U 8 !5 B
M
@> !12Q 12 ! T> >X`m! ] B B B h9D
ir5
5[= Q4 G D
5 HN[= ] 1' 12S .
0Izi A
N
1r ` h
h ! K (N
`
`
5H>D;W1M= 5 . 5 45G12126B
h `
h
` B h
5 H +
= ] 1' 12 5 [= !12G125 < % .>+= H4 5 12
>=
>+= G1X
B O >>(>+= += [email protected]>5 5 @+= ] 1' 124>+= "=
L>@! L H75 FGJ 12 -D
ij;W0 '12w X5
12
N
H1512 L
;
; ` E 5h*N
M `
h
>+: !1JK [email protected]> 5 [`
E h E
] 1' 12
G12L`
E 5h
G #
5
` B
D
` B > h
ij;Z&"l71&6 6;ZM= +=5 4
J >
= @> [email protected] "5> H 0<512> FG125 9" ; ` E 5h
E
G12L`
E h
5 G # h
D
#
` B "h
= > = "U 8 12> M7G
-> "U>D;Z"510 !12 P: !1J4 / > 5 ] 1' 12 r;Z
!12Q 12U 0= ] = ! U
= > = " 8 2
1 > M G > !12 Q 12<!12 E5;d= !'D
aS1Cl
M G N
; M7G #
#
` B
#
h
" 5 [email protected];Z4 = >= E^' m] !M > !12Q 12 T > / 1 ' EX
= >12
/
U
= > = " @ ]m !(M @> !12Q 12 ! >> / 1' ""
= 5 >12 / T >7 ;d= !L!12(N D
M N
; ` h # M # ` B }2h
C 1
2.3
",- 7$8J ,- L " )1 F)-5:3.$ !.5 /> [email protected]
Équations d’équilibre
ij;v= 9 >J F 5= ![6; JG1'X512 [ @
] 1'L>1! >^`{12 ] 15 h / 512 L ] 1' >1J >`{12 ]m J4> hB
56;Z L m |= 9 8A > "5
2.3.1 Équations locales d’équilibre
ir c= 9 124 >1! > (`m!> 9 h6;v= 9 >J N ;d= ! 8 E$`m D
` E h ` E h
N ` h
12
^ G12512 >> B
D ` E 5h E
N
4 8
G12L`
1r
5H> 471212
irP!&12 ` B
/ R
R
h
` B 2
h
] 1!
8 12>T (9
512"
;v= !' 8 ".!12(49D
` B h
` B h
` B0 h
Y = " 2
j
1 [[email protected]" 12 H 12 F""= ' 512
12 L > +12EL> F A8 E5 D
` Jh NC #
= !12 "[email protected] M G1H ! 12" !LH ] 155 "F [email protected]> F 0$12> B 12
F> ' 5 / 9
N `m!12 12<6;W ! 55' EhF N B
/ ` Jh N
5
G > 45 / G12 >> = > 4= 5 E 12 / 9"w>
ir L!12 124 0w>T(5 F H> H ]{ !
N E
N E
N E
E 5h
> !12Q 12 ! >> h%!12
#
U 1JK>m @l0 (9 / 12 H' G12512 [email protected]> H!124 I
` B
` B T h
h
P> ! 12
> A S4= !12 5P>D;v= !' P = 9" 12 w>1! > <6;d= 9" T>J > !12 Q 12PT >BGaU1 l
! 12E5 E5 N
"N>D;ZTE510 !123 -4 (.5 5 @
kq12> I '!& &1 / +
= Q4 G D
N ; ` h ` B "h
1 > ' =.5>123",#6)+&8=' #5 @
>> F ;d= ! 8 E.!12(N @`{B B
` E hHN # ` E h E
D
` h N ` h
,. 8
8 !N> H!12 412
N
#
#
E
N
0<>5
G12H`
E 5h
N I E
` B
` E hJN # ` h E
C ` E h*NC # ` h E
h
` B }2h
9D
Q '*Q Q R
L> L!12 124F >
` B
h
` B 2
h
` B
h
D
4
G #
` B h
*G #
` B 0 h
, @` B h / I M> .
= M 71'F12(
"r "5q> r 0
.12> / G1"= % > !12Q L> !12Q 12 ! >> / L>D;v= 9 12S
I M # N M eQ4
5 G E
h-D
0
$"5 !124+ .!12" !HF ] ' 155I
512 0= 5 " "
[email protected] >BY[>>* 5?> = .
/
5 5G1F 8 "5HD
` B
3 4 > X5 45 '$5l0_= 5'9 / 2
1
"510 / !12"5 "5 Lk[12> I '!'& & 1 / 1= L+= QK 6
C
D
= += > / >c5 !124
N ; ` h ` h N ` h 5
h
<
` B T h
>[5 8 ! 12(w5 !12"5 "5 > 10!'& P= !
512^ T! >D " /
! >>@ X= 9" 12 >1J > `{12 ]m J4> h[6;v= 9 >J / 5 EF>@kq' !G% Hf| 8A 0wt%' >
`mkqfHt hB
:
2.3.2 Principe des Travaux Virtuels
ir
= 9 124>1! > 6;d= 9" T >J $512" = !'5 (< 4m ?= 9 8A > "5 512 ] 1' >1J >w12
kj !G[ yf| C8 0^
t. > F`mk[fFt hB2ky12j! ]m / 124yTE510 T 512 y>D;W 5 ! F g+= 4> ! E
L+= Q44G D
!+= T9" ( E 4(J4> / 1"=
79
N A E
7 9
( "F= >'L*G
# A N
:
Q ` B "h
C 1
",- 7$8J ,- L " )1 F)-5:3.$ !.5 /> [email protected]
irkqfFt
;W1J " . hM >4>! 12S <= 9 12 %>10! > ^` B h c+= 4> ! ( E 8 >
T h*""= 12$j512+>L12 / T h 4>T! 12L> @] 1' > % X[ 8 h*4
A (!124 12 >T(5 <` B hzI` B hBLi?SkqfFt ;v= !' /+ / > !12Q 12 T > /
!12(N 9
D
A 5879 /
HG12F512H5 (4FL 4
A 8
5
7
9
` E A h
` EA h N R ` I EA h
K ` B
1rLD
h
` E A
K ` EA
hHN HKJ
h*N HKJ
# D A
R ` I EA h N H M I A M>Q
1r
M G
` h ` B }2h
%J4 8 12
G
` B
h
` B 2
h
8 124T>= > > 12 ` B T hB
c9"V> c!'& (4c !'&!'&+= P+= G
!NG12 >T>= >D;v= !' B
0= ] = '
M
^ 5>D;W1+= 5 5 [email protected]
BA h `
"c4
5 K3-9
12 S12(5512 P6;Wl
]m i X] 1'N4> 12 m] J4>@ 0= !4= "55 8 !12( ] 1 ' > 12SJ 5B L124> !124> = 5124% > 5$ > >2
1 +<!12G15 " ^ "= ' 0 !12 5 E> ^ _512> >w10G >
6;Z"5 !12U E5 07B
2.4
Loi de comportement
)M P96;W12P >^
= N @[= ] 1 12 (`m""= h / 12 F 12 H>512 M!q > 0= E 12c6;Z .>126!12G15 "L L "
= ' 0
&"l"G0= > 5T9" N` = > 59 L12<>T[= h+9 6;v= !'N`{B B h D
N
1r
#
5L @ !12G15 "c= > 551K> 59 .
` h
@
= '9" @6;d= TE5' @>1! >@[= Q4 @ H>@512> ` B
B
h
1 > &,- 7$823",#()1 $ ", 7$8
ϕ
i
Γc
i
γc
i
φ
i
i i
φt= ϕ(φ)
i
ω
2.5
T
B
#
i
% &
Interactions de contact
2.5.1 Paramétrisation et repères locaux
1 4[ G12"=F = !4= ( "[9L> g' ]m ! jG15 E >> gH!12" !Q
8 2
L H! 5
`m ( "H= >D +
h 5 >> H9HD
12
` h N Q R
1r
5L4
R
N
! (ILK
E Ch
1r `
`
` hHN
5
5
E
E
` `Eh E h
` hHN
1r
E B
` ` E h E h
E h N
h
B
12E ' / L ! ]{ / [= !'5 F F! 5
!9 GG'H6;v= !'@G12F512HG12"
512"q+= !'5
` B
= > = ( E. 0
= ] = '
R
ir L m] !
B 9
h D
R *Q R
R
HG12F512H5 K
+= Q44 M ` 8 12HQ ` B
h
` B
h
D
B
` B h
C E}
1
",- 7$8J ,- L " )1 F)-5:3.$ !.5 /> [email protected]
nit
ni0
τ i2
i
2
T
Γci
γ ic
τ i1
Ti1
ϕi
ez
ey
ex
LV J B ir
= !4= ( "B ir
8 !5 'F4(r > 10! >412EM[= Q4 + [! '5 F[= ! '5 + 0
"L L 0$!12 Q 124F512EH 1"
= ` N
h , {` Q B h+ +
,B >H512"L+= Q4 TF!12(49D
N
N
N
*QSR
N ` h N
E
E
QSR
N ` h E
E
E
R
# N
N
# N
N 1r
*Q!R
R
'
Q!R
R
` B
h
` B E}2h
R
i += Q4412 8 !5 '^ "G'^<!12 55 w >T> "N>
L> F!12Q 12 H4 >% ! >>B
8 !5 ` B "h
' = 5 "5N> '1 4 8 !51 >?!> '9B
` B
8 !5 1' 0
h
` B 2
h
` B
` B h
h
` B0 h
1 > &,- 7$823",#()1 $ ", 7$8
2.5.2 Appariement
ky12 = !'w> >12 $!12" ! / 12 8 12 ^JG 512T / 5 126; JG1' / ] 1 ' / $_= &10
6; ' " / > +!12 4> M%712TEM9 K12EF '! J4> [. "5M <!12E !B 2
1 M+= !' 8 124
J 'D 8 ( EL> ' [email protected]> = & 1L12 F' 8 1 l12 ^
G12F4> F @[= >B
O 12 '[= 12 F <!12 4>@.G12"F! U!12E !B O @!124> 5H[= Q4 6 H> !12Q 12
>j L`
hrg *> !12Q 12 ! >>+ F`
h?12^g> !12Q 12^M= ] = !
@`
hB0i? MG12E
E 0$' ]m ! Q R / R Bi? M712TE
/
4 /
512".> ' 8 5I I 8 %6
Q R / R % K>! 12 6; "BKi?!&12 39N 12
E E
E
8 12 F m] m] F%> ' ]m !6Q R ' ]m ! A !> 8 N= ] = !H%> ' ]m !6Q
Q R H>D; 4 >T! 12<6; " / > m] ! x5' ( 0= ] = ' !B
ir
4>! 12 F6; "F7 8 [email protected] 5N!12 55' 5 H 8A "H 0 4 10!'&
R /
>R;Z A CD
!
> 4 (D. 5 ] 2
1 += 'F> 112< 41 ((
= `{Q ' B 4` h5hB7i? 1J4>D &+= 9" T $!55 10!'&X!12 55P (T (.4 ! X5124!12"5 "5 B O 55 I
41!& G F 0= 5 "54< 1J4>D (@@120Iz 4!"
= `{B B hB
!
i 0m H 5 ] 2
1 += %q> 112( !12 H !'&
!1'' M E5' 5 g>M 1JK>m FF120Iz T!"+
= 4 Iz
> M712TEM! T <!12E !% 8A "M4%4 !12
= [= > "%! >T>@ U!& S 8 5 '!'& ('J4> {` Q ' B 4`{J7h5hBY[>>
"12 +=B"b12 'T !Gj
5jF!&'!'& (TJ4>`m+= Q4 w4= > J4> h+9 B
γ 1c
1
x
1
γ 1c
x
v
x
γ 2c
γ 2c
x
n
(b)
(a)
B
$# *
C 1
",- 7$8J ,- L " )1 F)-5:3.$ !.5 /> [email protected]
2.5.3 Condition de non interpénétration et vitesses relatives
O j4 A 4&[12 q> !12 [email protected][ 120Iz"5+= [=5 12 "= ' > >q "5[ 12> / '5 E >>
G12H>D;v= !' @ H>12T+ @!12E ! / L+= Q4 M> ^8 5 '5 > 8 [email protected] 0w12> M 4 5G 4
J > /
G12H>D;v= !' @ H>12TF ] 155 "B
i ! 124 12 j 2
1 0Iz"5+= [=5 12N 5? *!124 12 ! += 9" q"5' "G5125q"5M= +=5 12
= ' >T> ( 0
-512> BrY[>>^;d= !'@ ] 12 ! 12P (> 712'123 > 8 "5( 0
G12" ' = / 12 [email protected] !D
M4N
YqS[= !
` h
N
` 5h
"H >
1'
E h ! ` ` h E 5h
>X Q R / ""= `
` B
U12>
<G12"
/ DBWB /
! ` E h
N
h
` B T h
H U+= !12(712 EBMX H> J 5>1! >HD
M4N M
M
` B "h
> !12 412<@12IzE5'+= [=5 12c5 5 >1'F CD
M
12
N M
N
` 5h
H+= Q4 T512 F TE5
C ` E h N C
`
h
!
` h
` B
"L> ^8 5 @ > 8 "5 0$G12" = / >>. 512 [email protected] !D
! C`
` h5h
M N M ij;v= 9 12V 0= !K= "5S;W1J4 ES ^6;Z
8 "5HD
M
N
`
`
h
E h ! `
` B }2h
X+= ' 8A 12 ^ G1 5 4^w>D;d= 9" 12
E
` h 5h5h
9 7 55 8 .G12F H!'& 2
1 @: ! 0 F ]{ ! FG15 " >> [email protected]!12" !HHG12F
"H H41 (
= `{Q B 4` h5hB
` B
h
'I
1 > &,- 7$823",#()1 $ ", 7$8
i 8 5 5 > T5 " /
!12" ! G ! ] ` M <N hB
C ` E h
BC
N
C
1"= E h5h (N C
/ 5F> % ` `
`
! C`
h
ky12H4>T [email protected]+= > / 12 F 8 1Cl124^
` 5h5h
C L9
" >> w> M 0$G12E+512"F
N
` B 2
h
B
2.5.4 Principe de l’action et de la réaction
ir ' !7%@>D; [email protected]= !123;d= !'% 12 'r I
`
Yq J45
E M E h M `
`
N
` 5h h
E
F> !12Q 12 ! >>@!12(N 712 L`
E h 5
R
D
` B
h
F> !12 Q 12S >SD
E h M # I
E 5h M # `
N
E
G12L`
E h 5
Q R
` B}
h
! @!12E ! / >D; !12<> = !12c512E "9" EF >> B
2.5.5 Lois de contact et de frottement
, ! A 4& / 12 4= 5 E5124N> >[email protected]>10! > @ !12E ! ] 155 ( E9" *0= T 5 "
>D;Z"5 !12c "5> % 03512>T N.> ' % ]m ! %G15 E >T> !12" ! Q R L
Q R B
ky12L! ]m /
QSR*N Q R
12 H"510 5124 / 5 12 L6; JG1' / > H 1 12 H4 8 "5
E
R*N
R
E
E
N
Yq4 5 / 124L+= !12G12512 %> E >>HD
I(` E 5h N?`
c 5
E h I
`
E 5h
E
UN
I(`
E 5h*N
I
`
QSR
CD
E 5h
= L0= !124!12E !LI
'
` B}
3 @ 1 >@% h
` B}
> 4"
= 12( >1' > ] 1'!!12E ! (I ( 5 > ] 155 "B
'"^
= 12( >
] 1!
h
C 1
",- 7$8J ,- L " )1 F)-5:3.$ !.5 /> [email protected]
Lois de Signorini
ir L>12Fb0 1' 512"L> F>12TF !12E ! .
f !u BKYq>> F ;d= ! 8 E.!12(N 9
y
D
G` E 5h |` E h |` E h M ` E h
M
}
Q R *QSR
> "= > / G >= / !124 12 M
` B} h
N
'
& 0I
` B } "h
Q!R
` B}
h
i mU= 9 12w +>12+!12" !F5 4[> 120Iz"5M= +=5 12< "= ' >> / > 0
m ^ > ] 1'! 1' > % ^!12" !9 y512E !124 8 . > m 5
c= 9 [email protected];W!> 12$12U @!124> = " '= BGir H>12TFNb0 1 412EH0= ' _= 'H>
4& @ 8A ECD
λ
dn
B
' % Lois de Coulomb
!w5 8A >+512E^! >> O 12 >12NJ
ir ^>12T ] 155 (
E!'& 12 ;v= !' 8 " @> D N 8A E5HD
I
'
I
'
I
N
Q!R
>1'
>1'
C
N
: C LN ! !I
Q
R
C
`m! ] B*B B W }
/
hBgY[>>
` B }}2h
QSR
1r ( 5*>q!1! "y ] ' 155 E* O 2
1 >12NJ L 5|>+!'& ($`{1J0:5 ! ] h?
" >> H E5N 0w712TE '= / += Q44 N` B 2hB
` B}
` B} 2
h
h
8 5 r > 8 1 > &,- 7$823",#()1 $ ", 7$8
0
e%0^>12H` B }}2hzI` B } 2
h / >!12 8 "q6; 5
: 125q4H9" 5'm / (712 Eg> >>"F
= ] 1'! [ ] 1 I
5 "? ?>j! ?j [email protected]!12E !M{` B B hB Yq>>* 5|>D;d= 9" 8 > "rq>R;d= 9 12j!124> = " '=
G12H>@!12" !%4 > = > D
M I (N
*Q!R
` B}
h
e 8 ! ! 55!12 12F M>.! H.12w!12" ! M
= !12U .!12" !L9 / !A;W [email protected]> 0
5L > >%12<5 > ( EL @1 >`[email protected] M
N hB
ir L>12H ] 155 "L O 12 >12NJU512"L>>455= FH> 4&4 8 "9D
Rτ
µ|λ|
v
− τ
−µ|λ|
Remarque 2.1
' $ ' %J 2#* >V J @ ' B}
' %J 2$# *B # !
# # &% * B $ % [email protected]
# ! T 0V % % *
@
$
ir |>12r>1! > ?q!12" !|y ] 155 "|512"|124[= 0: 96; 0= 5 "y12
Yq>T> [email protected]@4x 5 E = "^ ] 1N > 12 ]m J4> B
] 1'q6;ZT[= 9" 12 B
12
?124?""= 512 F 0= EH >D;v= > JG1 12 ] 1' > 124r> A r5 JK>= B Yq>> rI
G125 "qq 0^T = "j E >TBEirL (j!12455 = !'L> q>12g L!12E ! ] 155 "
512 ] 1'(L6;d= 9" 12 L`{+12(6;Z+= 9 124h / 0 / q>H5 !124c "510 '[ [ !12 dl"G(!'& [email protected](' > 2
1 ?
i 8 > Ivb / !12(!& [email protected] !12 . - 1J4>D (B O @ 0
G12"F512E[= >> = L H!9 6
B
C nit
1
",- 7$8J ,- L " )1 F)-5:3.$ !.5 /> [email protected]
τ i2
Λ
Γ
i
c
τ i1
2.6
B
.2$# $*
# W # Formes locales équivalentes des lois de contact et de frottement
eQ49" H> j>12g H!12E ! [q ] 155 "q512 "= !'5 [512 ] 1' A8 ' 124 >>M!12(L! >T>
c= 9 12 N6;d= 9"4>J ' / X12 8 >> 3= !' 12 ] 1 6;v= 9 12 @12 6;ZT[= 9" 12 /
12"w= "
= += 8 >14+= B
2.6.1 Premières écritures équivalentes
ir X>12( S!12E ! wcb 1'4+ d; = ! 8 Ew6;Z S 8 "5` > J @>D; > 1 & Hr; hCD
! ` ! M 0h*N
1r
R
Q
1r
` B h
5L> 1 :5 !12UH>@ ( I [email protected] F0= >[= 2 ] L
I
N
N
!QSR # ` C h
= 9 8A > "535124> -] 1'c!> 9
ir ] 155 ( EL ;d= !<= 2 > "r;Z @ D
5 w= >?55!5 "HG12 ] B
D'c= 9" 8 > "5!12(( D
` B h
Q
R
` B
#
@ -5 ( Iz >K>! 5 ` 8 !51' > h% ] 155 "X`{Q B h /
1&1 12 >H> J7124> = <` +
h 5L w0= >|55'!5 "HG12 ] B
E
@> 1 :5 !12
h
1 - 1$ 3 8=' .5 3 ,- 6)1 3"B)1 $ ",- 7$8" )16A3 ,-
T
2.6.2 Deuxièmes écritures équivalentes
ky12<G12 8 12w> $K>125<"4_= '9 " / > = !' w 0= !K= "5 S512"
= 9"4 8 > E512 [email protected] 8 >> ] 1' $= 9 8A > "5 0
/ / B
12
^!12( *s!12 ^ > ^>12Tw!12E !X
G1 "H6;W1J4 8 [email protected] 0w <5
Proposition 2.1
' % * *
> = [
> 12 124 _= > 4> 9" >[>[ 5
512"L>T"= 9D
* ! ` ! M h N
CQ!R N ` ! M h
CQ!R V N 8 $LB * $# N J*:
F
` B Z h
F #
= !'5 U c 4m
` B "h
J
' $# *
$#V
ky12 ' 8 0 = 9" 12 ( 0= !K= "5 < (>12(U!12" !$4 > = >` B Z hzI
` B "h / 12 F '512 H ' !5 EH H>12F^b0 1'T y` B } hzI` B } hB
$#
! ,U= !12G12512 %> {] [email protected] " > >@ !12E !6Q!RM 3 @ ]{ !N!12" ! G ! ] Q R
5%>J 'B2e4 / > [>12q !12" ! M d; = ! 8 EM+!'& 9 12^`{+G12j512
hq!12(' 9D
!R 'Q
'Q!R
` h
FQ !R E
` 5h
E510 T512 M "5
N
1r
E
M
N
M
"L <!'& ` ! M h
QSR
` B
N
/ += QK 6
`
h
` B }2h
+= ' > ^] 12 !12S ! 5!% (I
5
QSR
CD
h
` B
h
B
O -!'& ( 47 > = !'& pc' 0B v> 5wczl7 ir 8 >Ivb
! > B
= 12_= 5'9:Q R
e% / 1 l !& XT !12 "+>R;ZE5'1 4!12 (
` B }2hL;d= !' 8 " "5 "L512 ] 1'(@6;d= 9" 12
D
/ > q!12 124j>10! >
X' m> 8 >D;Z !12 >5' 8 %` B
hzI
C 1
M N
` ! %J4 8 12
h
N
QSR
",- 7$8J ,- L " )1 F)-5:3.$ !.5 /> [email protected]
` B 2
h
Q!R
(9_` B 2
h(3` B
` B
h( 0= 5 "5 Ew 0
!12 12
0 >5 Bgi 4 (D< 5$
UG > L '12GL c N > /
zlG,.!'& > 4
> 5 !12 5.dl"GL 6B
/ >1'%9" N
1 l
= ! /
" &12(1 = += T 12c L w w5N 12U"4> / 1"N
M 12<
! h N
*QSR
B O !?12 HD
` B
h
D
!1'
`
h
N C8 !
! M *QSR N ` ! M h
QSR
O ! !&G 8 X
> 8 ^ ^> 41G1212S4 9N712 .
>R;W1J 5 E12P
H` B Z hzI ` B "h.>G H :N B
>12.b0 ` B
h
` B
h
1' +` B (
h Yq3 8A "H += !'& >1 < ! > >N9">D;W12 8 "+= 8 >1 GLG12L> >12H !12E !
T> "= >Gb0 1'T / 124[G12 8 2
1 4= K
J >F $= > H'(> HG12F> M>12M ] 155 "L
O 12 >12J D
Proposition 2.2
' J 2$#
$
#
!
* *
9QSR ` ! h ! ` C 2h ` ! 2h
N ` ! M h
CQ!R N # ` C 2h
9Q!R F V *B 8 $L $# $* F %
I
V
N
C
C
$#
512"$= 9"4 8 > E5
(N
-QSR ` B "h
` B
h
` B }2h
1 N[= 12E5'9> N>[email protected] ] '155 E^ O 12 >12NJ
8 2
= 9 124H 8A "5 `m! ] B h
[email protected]
0
` B h
` B }}2hzI` B }
h
1 - 1$ 3 8=' .5 3 ,- 6)1 3"B)1 $ ",- 7$8" )16A3 ,-
!QSR N # ` C h
M I (N
'Q!R
I
N
` B
Q
R
k L '12
M
M
1r
N
N
M
Q !R K
` B 2
h
` B
= 8 12> 8 @> ]{ ! G15 " >>@ @!12E ! / 12
(Q
C
N
(N
N
!R E
C
C8 12
E
h
!D
Q R Q !R h !
K
Q R
!
Q R
q`
E
C
h
` B h
` B0 h
` B
512" > % % G ! 8 !12" ! &+= "5
h
>' E5 / ' 5G ! 8 I
" / >D;ZT 5 "+B
i ' Dc= 9 12 ` B
55!5 "HG12 ] D
hH124 / C8 12"510 M < w5Nr;Z&121 = [= 12
N
C
C
N
YqSE5'1 4 "+>
N
1r
L
C
C
O & SNb # ` C 2h E
!R *Q
QSR
` B T h
/ += Q446 D
` B "h
w <5'N55'T!5 EL712' ]v/
> $= 9 12 L 0= !4= "5 ^` B hzI` B
! h N E ' Q!R *C (N E QSR ` ! 2h ! CC
`
h% ;d= ! 8 E`{ U5N5 8A EL4<!'& -h D
` B
h
` B }2h
N
E
Q!R
` B
h
i 00D (
N> 5 12D = 9 12 ` B 2
} hzI` B hNG 8 "^x 5V= !'5 5 > 6
/ > 4>T! 12^L> 00D (H q ( < 5.6;Z&121 = += 12 / 0= >712( > / [
512 5 ! I
12U H 0 = 9" 12 B O !?12 HD
C 1
C
! 2 h C
}
12<
! ` !1' / C 2h `
:512 "L> ' (m3= 9" 12_` B
! h ! ` C8 ! N
` ! M h
`
C 2h ",- 7$8J ,- L " )1 F)-5:3.$ !.5 /> [email protected]
N
E
QSR
hCD
C
! 2 h C (N E
`
C
#
h E Q!R `
N
` B 2
h
Q!R
` B
` B h
h
O ! !'&6 8 (> [= 12 55 123> '1712'12<4496;Z>7 5!> 96;W1237 H 12"5 S>12
O 12 >12NJc` B zh I` B h{` M 12 ! 0
>12q O 12 >12NJX!> 9 hH [^` B hzI` B 2
} hB
2.6.3 Formulation lagrangienne du problème dynamique de contact frottant
%
] 12> >12TNw!12" ! ( ] 155 ( E = !'5 512 ] 1$r;d= 2 >T"= (12V6;Z+= 2 >"= /
12 LG12 8 12 T"= E > 55^512 ] 1'( ]m J4>B O !|124L12 @4 (D = !' 1JK>m @l0 (9 <!12" ! ] 15 " / 12 += < 1G1212 B
Proposition 2.3
' @.
! #V
* & * .2 * #
EE E 2h
`BA E E h
( # QJ j` :
` EA h K :
& ` EA h N H M A
* #&#
$ $# $ M>Q
` B h
C
H M ` ! 2 h C A M7Q
` B h
! HKM ! ! M M>Q N ! KH M ! ` C 2h ` ! 2h CC M>Q ` B h
H M ` ! h %M>Q N N ` ! M 0h
CQ R ` B "h
N # ` C 2h
9Q!R ` B h
V * #* F V * J* F
.
1 - 1$ 3 8=' .5 3 ,- 6)1 3"B)1 $ ",- 7$8" )16A3 ,-
n G >124F9"@>`[email protected]+;d= !'%!12([email protected]`m! ] B|` B $#
` EA h
K ` E A h N R ` I EA h E ?N
& R 512E12 += H
K /
h5h
E
` B }2h
` B }2h / ` B hHN` B 2h / 5G ! 8 "B
Yq ]m " TE5 "^> 512$' P` B }2h / V> [email protected]>$ 'T !G$>R; !12 ^w> = !123H U124H5 8A E H1 124` B } h / 12 F1J 5 124CD
0
1rU> H ' 12
1r
` EA h K A
& ` EA h N H M I A 7M Q
5L>@!'& 3 / 124[[email protected] D
A NA
! A ` Gh E
` Gh
` B 'Q!R
` B 2
h
h
L9HD
1512
I N
I N M
M:
N
` B ` B h
h
12 !
I N ! M N
` B h
` B h
e% "D
` EA h K & ` EA h N H M ! & E%9" HD
ky12
ky12
12<
E (N
(
N
E N N
C
! N
C
5 <` E
C
A
!1' /K = ` B "hHN` B hD
8 12 12MX
`
N C
! 2h C
N `
M>Q
H M A
h
C
M7Q
` B > h
`
` B "h
h
C
! Eh C
` B h
` B }2h
C 1
",- 7$8J ,- L " )1 F)-5:3.$ !.5 /> [email protected]
A
M>Q
M7Q
>@k[fFt~ ;d= ! >1'L!12(4CD
` EA h K & ` EA h N H M HM % *B J 2$#
`
Remarque 2.2
#
12
= !' 8 12 X E5
!'& 4 8 >T
$
$
! 2h
C
C
N N
A
` B h
( # "
% % $* *
# Ew> >12(c!12E !ww ] 155 "w12
12 D
! H M ! ! M K M>Q N KH M ! ! ` C 2h ] 1' ]{ JK> Bjky12$512
` B 2
h
`
C
! 2 h C H M ` ! h M>Q
12
2 '12 > U= 9 12 @ > "%> %!& 4 ^' . ! 12 = ! 9 !2
1 F w
'14
J >m @512 L> ' ] 1' ] 1 5 / 512CD
` ! M h E QSR # ` C 2h E QSR N
N
M>Q
N
` B h
0 5
!'& (4
` B
h
` B 0 h
i 8 @@> 1G1212$ 5 !& 8 = B
2.7
Formulation stabilisée: unification des formalismes lagrangiens
, .> 1J4>D w` B hzI` B h / . >! .6;Z c!12" ! 7 ! ] ` N
h 8 ! ] 155 "
h / > <= 9 124(` B hzI` B h ' " / ^ D' ]m J4> / > ]m .9" M N &+= "(` <N
N B -7 @12 !^ :5125 _kqfFt ` B h / .512 > @! Q / [email protected] Q 512 = [email protected]!12" / > F5 > 4 8 "CD
(C
1 -50123", [email protected]!' 50,#C$ 123", )1 A 3/0 ,1/>3 ,#
H M M A M>Q
H M C A
1r
M>Q
1r
` B
h
` B T h
Remarque 2.3
# V "J " @ @ % #W
@ Q * W * T $ # * ( $ $ * . * J" # # * W
J ir 1J4>D @ l 9" %N!12E ! ] 15 " ;d= ! >1'L!12(4B
E E :
` EA h
K E 2 h : `BA E*E h
& ` EA h N H M ! M A M>Q H2M *C ` ! 2h CC A M>Q
HKM ! ! M M>Q N ! ! ` C 2h ` ! 2h CC M>Q H M ` ! h %M>Q N ` ! M h E QSR # ` C h E Q R f|12 8 M`
` B
"h
` B
h
` B
}2h
` B
h
` B
2h
! KH M
N
N
12
#& % @
% $
!$ * % # W
< J4>= / H9 >Q 124+! ] 1' > N> A Proposition 2.4
% $ #
/
$#
B
#& Yq>>H 5[T(_= 5 D [>! +1r
N > A @!> '9B , L>@! L16
N N
' >> " > = &10$`m9 Ivhv> A 4 ""= B
(N
N
%
@
% % * / 125'12 8 %> @] 1 'N4> 12
N N 12U5'12 8 }
C 1
",- 7$8J ,- L " )1 F)-5:3.$ !.5 /> [email protected]
i 12 8 >> ] 1' > 12p` B "hzI` B 2hX Q X __= &10 ^G12>$!12" ! /j]m !> " T > %!1 A ] 1' 9" B ij;Z>T 5 N`mr;Z S!10 = > = ( E.Q44h[G H '5%r;Z = &10U
5 / %4>%!&12 [email protected] w
5 / >12U> FJG 512T [email protected]+= > 12 F12U @= 512>12
_= '9 Bk H0 4> / F>.! 6;Z w!12" !`m ] 155 " / G12F> !> '"= hM <5 9
j ^ 0= !LH12 8 E[F!14g' / > 5j""= "[6; :5125q> J4> 12<`
G12X= 8 5> !12 'N L512> 8 'H712 H 5' ! H' >D' @`m! ] B hB
Yq~125' / > ] 1' >'_> A M= >T 126B
nH ' 9" 2
1 L Q4S9N / } !&12T "
G'J 12< 8 "9D
HM 1r
2.8
h
5 J4>T"= G'-V!12 8 ' "= " > ] 1' >T_ L>w x ^ 5 ' / 12SG 1J 5 %> ] 1' >T> A S7'JM=
N `m! +L!12" !+ ] 155 "hg[ 5: 12 ". ` B h[> H5
M7Q
` B
h
5L < w5G12 ]? J45 "MGH9" @>D;W12EL7 M!&12$= 2 +
> B
Conclusion
e
N -= !' P= 9 8A > "5 @
= 9 12 N >[email protected]!12" ! / 1Cl 4 E>D;Z"510 !12
!'& K% N / 12 8 124. 1G
12"=4 ] 1' >'> A 3 J4>=NG12> 1JK>m .!12E ! = += > "H! 0X> A $!> 9M> A ""= / !> 'Q EM>D;Z""= x H
<5' r; " 12 4Q "> T4> = " 12 "4_= '9 B|, ^>$!'& K5
8 " / 124L12 ""= ' 512 0 5G !%l0 (9 H 1JK>m N!12"
512 / ( '! > / ] 1' N4> 12(&"l"J4' ]m J4>I ] 15 "= 0 '1J4>m ] 1'N4> 12< M F5 F> F> [= '!&N B IdB4aS1 U%a B42
12 1G1AI
!B qr;Z !B O 55
`{B B
/ }
hB
}
Chapitre 3
Formulation hybride faible-forte pour
les problèmes d’impact et
discrétisations
O1
O1=<
O1.O
O1
A54$53/6%0KE&5+ * 3&$G# /0$5/BA :;;:;:;;:;:;;:;
Q0<
77EL*$/E;!;E!' A$/J/[email protected]' 5DBA :;;:;:;;:;:;;:;
3O
41'235 + ' 35A1/ ;:;;:;:;;:;:;;:;:;;:;:;;:;
P:/#@AE$53/ :
;":;;:;:;;:;:;;:;:;;:;:;;:;:;;:;
0<
o o RE
o o
o |
o R‚
o s
o RET
o RETRE
o RET o
L
{(# (‰03-K*h8 MI# T8;-Kc ')-C8T-]SZ(=<G!3()< (;-K<k8T^Z<G%># (+L"MG-?CCC CCCCC sNE
{"(*\8 -]21<N05%2P0h8T-ST()="< !3()<G( Wk!3-K%M CCCCCC CCCCC sNE
z 6 27!$()0$MG!$--7< .6- L"M[%p0$(<[*8 -K*\'(+*\8T-SZ(=< "!$(< ( W !3-K%>M C CCCCC sN‚
"!$#M '+%p0$(<je %>(b '- Ž[email protected] !3b 'Ž-Kc #H-8T-]27<N03%"2P0\8T^Z<G%># (+L"MG-YCCC s"s
t !3(+*,--7<k27# @T0$-C8TMIex!3>0$0$-7# -K<"0 CC CCCCCC CCCCC s"V
I.6- *,MG'‰05%p03*\8 -]21<[*,-K!$p%p03()"<
CCC CCCCCC CCCCCBq>n
" !$#M '+%p0$(<O*$-7# ( 8T(+*$27! 16- 0$(+*T76- -8 [email protected]!$"b 'd-7c # - 8 -]21<N05%2P0h8T^Z<G%# (LNM - CCBq>o
(*321! 16- 03(*3%p03()"<G*-7<O-K*[email protected]%"21-CCCCC CCCCCC CCCCCBq‚
(*321! 16- 03(*3%p03()"<j8TMI03-7!3#H-]8 -*,03%b (')(+*$%>0$(<DCCCCC CCCCC VG
C % > 5123", 0;[email protected] )1 A @3 A28 FH15 3 *C [email protected] 6)H+ 7$8*"F) %$S'"212 ",#
}
Ub 1'4 IdaU1' / } / / } / / !12
= 9 12 / 1 l
"$
"$>D;ZTE510 !12 ] 1' > 12(> A F 8 5 ]m J4>I ] 15 /
G'5 "(w[= ' 8 _= > = " $!12" ! Iz !X 5G ! "(> ^!12 124N!+= 9" U!12" !c`{4> ! E( 8 5 5 h B L12 8 = 'Q 124 / T! > / 9" U!55 ] 1'N4> 12
!12"" !12 5 8 N>D;d= @51 >@> F9 ""= @12 8 "L>T[= 4> B
, !U!'& 45 / V1 "G12>$(16 >U
10G >3 3!12" !$
l0 (
9UX >R;d= ! 8
0(!& 4[L > / 12 j '171212 j 3.1
Formulation hybride faible-forte continue
3.1.1 Limites du modèle de Signorini en dynamique
ir_16 >_ b 1'4 / ! 12 (
"3T>=P
!12" !P5 9 / r;W 5U S5
1JK>m [email protected]!12" ! Iz ! `m! ] B B B } / } [email protected] 312 F X 12 CD
"= 0
>q!12"5 12x >q9j> r4> ! "? r712TErr;ZE5 ]{ !j[!12E !BAYq K>
12EL= >M <5 4B
>G 5 1! <
= w= 9 12 6;d= 9" T>J `m>10! > F12 >1J > hMr;W1' 0w <5 KB
/ > r!& 4|!12E5 12x > =
Y = "?124[=q>D;Z0= 4> 'g
j
= 5 Kr r!& 4r j[= 4> ! "| r712TE6j!12" ! / >D;Z""= 12
5 G1 >> F F'!'&+= !> 9 + > ] 1N > 12w[= ' 8A EF w10G > b0 1 = 6
g12'!>> 124* 5 0^(12 E[ j!&1! B O g12!T>> 12 g ; !! E "[r; Eq4> g9
>D;W1' U!&[= 5w= > 8 <
= `{B B W }
hB
ky12. = 03! ' ! 5Iz E12[= / 12 1G12512 hHr;Z>L>N(16 >^aU1' 9 "[email protected] !12E5 12
x > / > ] 12 / > y4> ! "y*> 8 5 y |G12"| !12" !jG15 " >
L{hq @[= ' 8 4 ] 1N > 12w 8 5 [email protected];[email protected] w L G1 <5 K B
3.1.2 Lois de contact de Signorini-Moreau
n 47 >12 [email protected]> H>[email protected]!12E !.Nb0 1' 7;v= !' 8 ".!12(49D
M
QSR
QSR
M <N
Q
R
` B h
` B h
` B h
E =
n 47 > 12 w= 2 > "@9 M 5%> ! ' [=N "5^ 3712TEL +` Q!R h .>G12"9 > 7 5 510!
= L>R; 4>! 12$r; ' ( E`m! ] B45 ! 12 B B U!& 4 5' h B
> -50123", 0;@ )1 A @3 A28 $ ",-2,-57
B B0aS1 9 6 9
D
12 +=( ! F>12[4U= !'. L= ' E5. wl0 (9B O 55c= !'% 5F! >>
ky12H512L5 4
|` E h
N
+H512HG12" 3(QSR / 12
5
` E h E |` E h C 6` E h C ` E h M
} C8 12 L>D; >5' 8 ' 8A E5HD
?` E 5h C 6` E 5h
N
G` E 5h
M
` B "h
` B h
1r
C
N
E h ! C ` E h ` E 5h
`
` B }2h
[= "[> [!'& (4[ 8 5 5 [> A q [512>T / 5G ! 8 "q
>HG12"jBQ R /" ' = X712TEw Q!R / >D;Z45 E B .J45 8 12 [9"r; 0( 5 " 6;Z0= 4> '=
!A;W 5 > [= 5'( 12_ '[email protected]
`5` h h+9"6;ZT>G 5 ]m F= ] = 4! 4F> H>12F!Iz B
C %C
2
i G
! H4(10G >L.aU1 X 5= !> = [>H0= ' > [9 ' 9DK`{1r$12 j G12512 j9
>D; ' "M 5L!12 5 " S!12'H U5 4h
Proposition 3.1
V Q
* ' % # % % % % 4 # W W $* #
b0 G125124X96; ` E # h 4M G` E # h |` E # h:N
|
#B
# *V
#
E h
N
`
E C# h H C
>G U 5L @w x @G12H> G12"
` E h
N
Q!RF;v= !' 8
ir L[= 4> ! "'H>D;Z"5 ]m !
`
E#
>D;Z45 E<4 > c<G12w512
Q!R$12 M ` Ch /
10G > / G
12512
BgaS12"512 >D;v= 9 8A > !w E5'w> 0_
$#
`
E
h
M
*V$
#
E
G12
5
% ] 124!12U
`
E h
8 5 / !12(N 9D
4CQ!R
= SL124 8 12
/4 X
` E C# h H C ` E Ah M
` B h
!D
` B 2h
C % > 5123", 0;[email protected] )1 A @3 A28 FH15 3 *C [email protected] 6)H+ 7$8*"F) %$S'"212 ",#
}
k L= !H 1 :5 !12U H> 1' >
G` E 5hHN M G ` E # h H C 6` E Ah M
M
/ [= Q4 ` B T h / 12 H1J 5 12 w= 2 > ESD
G12
`
E hG
CQ R
` B h
,% 0U! L5 4= 5 E5 "!D
'CM E 5h `
E 5hHN
> 1'H>@16 >@aU1' SG125 9 |`
>12Fb0 1' 7
512"H 5G != B
'*M E h ` = 9" 12_` B "h5hBGe4G>
E
C E
>1'>(16 > $$aS1 VG125X9" ` h / |` h B % / 12
h / 124!U12 1J "X w>D;v= 9 12 ` B h / M ` 5h K>
4712=c9" M ` C
4= !5 "LM 7` 5hFN B . 12 ! M 6` 5hFN |` h BGe% / > >12L % 1' 12EH' 5G !"= B
, `
E#
E
E
E
E
L> F 0<! H12 += (12E50^
= >D;ZK>! 12$ <10G >aS1 S10G >@Nb0 1' RB
# *V
#
% *V$
#
,.!5124H> H $! L' 8A ECD
'
M
` E 5h E 5h<N
>1'><106 >U3b 1'4qG125w9 ?`
aS1 S 5H 5G != B
'M 7` E h
E 5h
>1'[>106 >..b 1'4G1259BM 7`
D
` E h
M
N
C ` E h M ` E 5h M M N
BqeT / ><10G >U
E 5h
?`
BM4 '9"
` B h
1r
M 4N
C* E h ,%12 ! 6`
, MK` M
M
h
N
Q4 > E
` B h
B
C ` E 5h N
E h
|`
L> F 0<! / 12 += (12E50^
= > 0= ! '19B
124! > H!12 12 MNaS1 S
12E 8 = 'QM= B
> -50123", 0;@ )1 A @3 A28 $ ",-2,-57
}
3.1.3 Écriture en équations des lois de Signorini-Moreau
ir [!12 12 j%b 1'4 IdaU1' ] 12"q"5 8 q qT[= 9" 12 B 12 q 1G12512 j L> ^= !'
512 ] 1N6;v= 9 12 / 1 l
"%>D;ZTE510 ! 12U <!& 4%' L+= QK L'L> 12
G15 E >>!12E !L
QSRB O %!'& KN %>T"= H > ; 4 "5 E( %> 8 > Iz
B
ky12L! > 12 $= J4>T512 H> = > L' 8A E9D
Proposition 3.2
' % T HN #
! M K
` ! M N N
W h
! C N N N * *
( 0$#V * HQ!R
H Q!R
#
T
B
W
HQ R
H Q!R
H Q!R
`! M h
! C h
` B
h
` B > h
` B "h
` B }2h
` B
F V # V @ # V #* F $# ir% A . ++= 9 12 0 = 9 12 +5.[= 12"5 %> h
L>@!'& K5
`m5 !12
#
! M ! 7M '
'
ky12
12 !
12 !
# / 0<! L5
'!M >1' C
'M N
N
12 !
N
N
M<
N
/ 712 F512 +
N
h5hL
/ > H! 8A "9D
`m6; ` B > hFN` B
0= E5 "!D
`{ ` B
6;W 5F H!12E5 EB
D'%9
B }B hB
O 12 ! E> = ! 109 / 12 F!12 '[= 12
ky12 N
D
< x h5hBGe% /
`m6; ^` B > hM` B
UN
h5hB7e% /
M
M4
N
`m6; ^` B "h5hB %J4 8 2
1 H9 / B4,.5 12 F> H 0$! 8A E
D
N
B
B
H!^! /
}}
C % > 5123", 0;[email protected] )1 A @3 A28 FH15 3 *C [email protected] 6)H+ 7$8*"F) %$S'"212 ",#
! C 9 ! C '
>1' 12< 4!1
N
! C
'j "5 "
+= 4 512 F X` B }2hF9 N
`{ .` B }2h5hj
B . (
T C L
>1' N
B . L
`{ .` B "h5hB L12 [[= 12 jN` B }2h
/ N B
C*
C
`{ X` B }2h5h /
C C
C
N
N
`{ X` B "h5hB 12
B
O @9 !'&6 8 > 4 8 @@> 0= ! 109"B
3.1.4 Formulation faible-forte du problème de contact dynamique
Y = "H12 += j
9 > !12E !L 5X= !'F 8 5 5 / 12 = !' 8 124 G>D;v= 9 >J H 8 5 5 4 8 "
U! > aU1 S
_`{B B W} / h / U>T "M>@kj'4!G Hk[ ! Mt% >> @`mk[[email protected]
O !G.;d= !'L!12(( 9D
A EC!&
ky12L512
?E
S 8 5 '5 (TJ4> E`{LG12M512L5 4H
`BC E A h K
& `BC EA h H M !
A
M7Q N
5
5h
` B
h
1r
B` C EA Hh N H J
K
`BC EA Hh N HKJ
BC
8 ! ` h N
] 155 " # C A
` hH @` B
hB
M G
`BC h
` B 2
h
BA h `
M7G
[= Q4 ` B
` B h
h` B }2hB .J45 8 12 %9 12 += >T 12 .>
= !' 8A E+ "5 "[> [>12q L!12E !F5124 ] 1' m] J4>H512q 2
' " 512 ] 1'
$ q+= 4> ! E ! >T 8 5 /
] 15Lj :512 EM>Hk[kjt [>D;v= 9 12X > "j>L!& Yq
> -50123", 0;@ )1 A @3 A28 $ ",-2,-57
12 H1J 5 12 H> 1J4>D @' 8A E / G12F512
}
#`m! ] BKB B
h-D
BC E E : E h 5 7L9 879 ` <` Q!R'h5h : `BA E h 5 7L9 ` B h
& `BC E A h ! H M A M7Q N K `BC EA h ` B0 h
! HKM ! ` ! C h M7Q N ` E h*N ` E C# h H C ` E h M
G #
` B h
N ` ! M h
' Q!R
` B T h
N ! C h
QSR
` B "h
` B h
` E # hHN # ` h C ` E C# hHNC # ` h G #
1r
79 / 7L9 512EX> 5 ! X!+= 9 " JK> N (!'& (4( 8 5 5 / += 4> ! ( E. .N4>4>T! 5 'H^!12" ! / 5G ! 8 " / .16
5 3= >y 12c >|
X 5L U= >r55'!5 ( E712' ] B
f|12 8 `
>?; A %6;Z ] 1 ' > 123!12ET" / E
& lJ 'T / !12E5 TE5 / = !'5N 8 5 5 B7b % ! 12 512"L> 8 5 / > H+= 4> ! E / > H >4>! 5 [@!12" ! %L> H!& 4H B
Remarque 3.1
#&# % # * W * 2 #& ( Q
* $ B ! # # *% HKM C A * Q
* ( 0$#V # %
@
% W
M>Q
` B }2h
% 2 !$ @ $# ! %%*" % * * % *
Q # # $* %
#
*
( # W
(
Formulation pénalisée en vitesse
, > >5= ' / [email protected]>D; 8 124 47 > = !'& 45 / T>g 5
!12" !% cl0 (912L5 "
= c> EL> _= &10 M= >T 12V`m! ] BKB > 10G >@Nb0 1' IdaU1 $+= 8 >1 M
= ! / > ] 1'N4> 12wM= >T"= 5L> 8
= 9 E9" >
]0
B / h BGky12
"5 / G12M512
}
C % > 5123", 0;[email protected] )1 A @3 A28 FH15 3 *C [email protected] 6)H+ 7$8*"F) %$S'"212 ",#
#?D
BC E : E h 8
5 79
`BC EA h & `BC
K
f|12 8 7L9 ` ` Q R h5h : A 5 7L9 EA h H M C A M>Q N !
`
` E h*N ` E #Ch H C ` E h M
N
N
` ! M 0h
! C ` E # hHN # ` h
1r
` B
QSR
h
` B 2
h
Q R
'
C ` E #ChHNC # ` h
G #
` B
h
` B h
G #
` B 0 h
5L4w <5 += > 12 `m55'T!5 EL712' ] 'J45 ' E KhB
3.1.5 Prise en compte du frottement
ir 4&+= 12w ] 155 " ] 12E^"5 8 > 8 5 '5 ' > 8 " >> 8 V
Q!RB7eT / > (] 1'N4> 12U41G12"=N 5% ]m 5 " " = 3 >R;ZE= 12c N! %4&+= 12< B
n 47 >12 ?> y>12| ] 155 "g O 12 >12NJP= !'5 g12 ] 1' +6;d= 9" 12 *Q!R
/ 1 l
"
>D;Z"510 [email protected] 0w!& [email protected] / += QK F QSR
-D
I
1r
Yq^5
` ! h ! ` N `! M h
N # ` C h
N
L w0= >? 12U"4> /
Ej!1245H *4 ' !
C 2h `
C
! 2 h C
` B
(N
` B T h
h
` B "h
` B
512"L H= >L55!5 "HG12 ] B
8 >> y+= 8 >1 M= [ j> y G1j ] 155 " / >M '1J4>m h
> -50123", 0;@ )1 A @3 A28 $ ",-2,-57
}
l0 (9 @!12" ! ] 15 "L;v= !'%!12(' 9D
BC E EE : E E 2 h 5 79 879 ` ` Q!R'h5h :
`BA E E h
5879 Q T *$ `BC EA h ! H M A M>Q
K `BC EA h C
! H M ` ! 2h C A M7Q N % W J )! ! H2M ! ! C %M>Q N # ! '! H M ! ! ` C 2h ` ! 2h CC
H2M ` ! h $ $# $ W ` E 5hJN ` E C# h H C ` E Ah M
4*G #
% @
# *
! ` ! M hHN JQ!R
! ` ! C h*N *Q!R
! # C B N Q!R
' ` E C# h*N # ` h C ` E C# hHNC # ` h
JG #
f|12 8 M`
T
` B }2h
1r
` B
M>Q N
M7Q
h
` B 2
h
` B
h
` B
h
` B
h
` B
h
` B h
+= >D;W 5 ! H!& 4H ( Iz >4>T! 5 '+ ] 155 " B
ir%5' ] 155 "M >12F @!12E !` B hB
` B }2h+[= 7 4$T4>!5 E[
'!%9 $ w[= G
w +>
i @] 1' > 12( '1J4>m Ll0 (T9"F!12E ! ] '15 E ` B }2hzI` B h[G Fx 5 J4>=L C % > 5123", 0;[email protected] )1 A @3 A28 FH15 3 *C [email protected] 6)H+ 7$8*"F) %$S'"212 ",#
C
> < x 10!4= H9H 0= !K= EB0Yq( 7 / j>H! q16 N
N / 12UG :5125 Uk[kjt / += Q4 G N` B }2h / > 5'4 8 "9D
1r
! H M *C A
`mDBWBE!12" ! &[= "h /
M>QSR
` B "h
L w <5'G12 ] B
i ] 1'N4> 12N = !4= E5L 5|1 >+! g!A;W 5g ] 1' > 12!12"" = += >T Ej> ] 1'0I
> 124r> A r!> '9* ""= B k >> / > r>126 q!12E ! IzT ! / = !'5 ?r;Z D' = 9 8A > E5
512 ] 16;v= 9 12 / 5G !5 " !5 " > L!12 412 b0 1' I
aS1 6BAir *120Iz>T[= '"= | +!12" !g512EyQ = g y> y!'& (4| [
/ += Q44|r;Z q 4m
1 ' > 12w 1G12"[email protected] 5L @ ]m 4
J >I ] 1 5B
] 15X`m>10! > hB4,%! ]m / > ^] O 12((N12 >D; 8 2
1 4+v= : ( E124[= / > ] 1' > 12U[= 8 >1 M= ^ 5.> A B4,; 5 ] 15I
> 12 %> >T5 "= ^>^512E / / T B6f?12TLG12"L512" 12 > CBKir L 5 9
15 ] 1N > 12 5w!12ET"3G'X3! ]{ $U !12'(w6; >l5$> !& x c
' 1 12 L" = 9" B i !12 @ 5%9"r;W >> 5<= [email protected] U5 !'& KL 8 5 /
= 8 E *
Y Q4 / >>' >m 8 ] 1! s A Sr;Z !12 12P 5G' !U`{B B / hBr[
> HT[= 2 >"= L4&[= E5 > = ! [email protected] <!12" !% L F!& 4HT !12 / !> 'Q E 4G>
X! >"= H> = $4&+= 12w @!12E !%G12F U(4>Q M> 5 5 E" = 9" B
3.1.6 Résultats de conservation
YqS12 F4> ! s EL L>@! U!12" ! .l
G12M> F512>12 [4 '1J4>m [= Q4 7 @
` B 8 "L>T[= . > 512"H!12 5 8 = 9 (
] 155 " / 12 >>124H12"59
hzI` B h / >D;d= 4K9 > M9 E"= L 120I
U!12 'H w5 KB O F '1 ' = "= L5 8 EL!12
M712 H>D;d= > JG1 12U AJG12 !'&+= . = '9 F6;ZTE= 12S <5 4 B
ij;v= _51 >V4 5l05 _= ! 9V!12 += 0V
= 5c[= Q4 !12(V> !+=9 FL715 " >> H H $512>T B4b12CD
N
1r
HJ
# `BC h M G H J
M7G
5L> 4"
= >10! > @6;v= @+= ] 1' 123 <512> 512 p= ` B
B
h
> -50123", 0;@ )1 A @3 A28 $ ",-2,-57
YqS[= ' 8A EF L G1 <5 KH>D;d= 9" 12_` B
N
H J # C C M7G H J
Yq
>T E>3kj'4!G$ Xkq T 1J 5 124CD
N H2M C
`
` h h / 12 M1J 5
BC h `
h
M7G
12
D
` B E}2h
` h N `BC h^
! t%' >> U` B h> ]{ X9"
M7Q
12
` B
h
i ! 1245 8A 12 >D;v= 5 '= 6 @>1 9">D;W12-5 5 !5 ( E>(!12E !N
l0 (
9 ! [email protected] ]{ 12 F ` B 0 hB
:
"%!12( 5 L4&+= 12w ] 155 "(`m O 12 > 12NJ7h / >D;v= N!12 5 8 c10
6; &+= ' !U` N
4N
hF += !1 x % -10 T> 5 "` N
<N h BGY[
G
D
Yq35
NIH2M C M7Q H M C M>Q H M ` ! 2h CC C M>Q
` B 2
h
, L>@! L1r
8 "BK,%12 !
N
, %>! 16
BK,%12 !
+ = !1 xB
N
N
N
/ 12 0
/ 6; N` B hHN` B 2h /
/ !9 = J4>L> ! 12 5 8 [email protected]>D;d= B
N
M
N C
C N
C
N
/ 5G ! I
N
h ^` B 2h / N N
/ 12 0/ r; ` B L
M7QjB .G12L4S!12" !% G ! ]v/ 3 %+= 2 ]v/ 12 !>R;d= 51 > Conservation de la quantité de mouvement linéaire
C
Yqw G12 E+9" 6Q N
= 12 8 EF>T[= % 5F!12 5 8 = B4Yq$ G / !55
/ >1'M> 9" [email protected]
9 E"N
= 5L+= Q44. D
N H J # C M
G
` B
h
C % > 5123", 0;[email protected] )1 A @3 A28 FH15 3 *C [email protected] 6)H+ 7$8*"F) %$S'"212 ",#
b12 / = 5 " / N ` h*N
N
,N>D;d= 9" 12_` B
hH12<1J "CD
M G
! H2J `
N
O 12((
H2J # C B
G
` B h
` B h / 12 F1J45 12 H>D;v= 2
= 8A "5HD
>"
,; %>@kqkqt
G
8 !5 .!12 5 " J45 :
N
h
HKM M G
! 5L
M7Q
` B0 h
8 !5 .!12 5 " /
12 M512 8 12
!D
` B
h
6;W1rU> 1 ' = = !12 5 A8 12U @> 9" E
= @12 8 ">T[= B
Conservation de la quantité de mouvement angulaire
Yq G12 "j9" Q N
[96;W (! +L!12" ![ G ! ] > T 5 ! M^[= Q4 F ` B q
h 5q >>
`m! !* [email protected] &"l"G1&6 5^ > 8 P'! &12 c> ' ]m !G15 " >>^!J4>12P x 5' h / > 9 ""=
@12 8 E > . 5L!12 5 8 = S!12'H U5 4B O 55 9" [email protected]
= L[= Q4 D
;
1r
N N
HKJ # C M
` h
G
` B T h
`8 2
1 !& 45'
h+
+= w '1 4 8 !51 >?!> '9B
.p= J4>%!55^!1245 8A 126BKY[U
G
8 !5 %!12 5 " J45 ;
N ,; %>@kqkqt
;
12<
1r
N
HKJ # ` C h D
M G
H2J # C ` h M7G
H2M ` N
! 2H J `
h
D M7G
H M ` 5L> 5'! " Iz5l0_= 5'9+= Q44 ` B "h
h
M7Q
` B
h
51' >D
!1' / UT> "+ F 1 = "= % <! >!4>65
;
B
M G N ` B h / 12<1J "D
! H2J '1!K= " > w x m^9"N 0= !4= (
" / 512
G B ijv; = 9 12V` B T hF 8 "F L+= ' 8A 12LD
UN
|B
h
M>QSR
` B }2h
> 1ŠF#9%.123",# " !' " 5.23", ,-5 ' =.57 Z
O 12((U> 5 !3r;Z V 10 ^6;Z
>> / 12<1J EPD
;
N
H M ` h
w 5'!S5l0 = 5'9< X U 5'T! E Izl = 5'T9"S 5
M7Q!R
` B
h
,% 0U! L5 4= 5 E5 "!D
b
b
3.2
N
N
;
= @12 8 E
>1'H> 9 ""
> 1'*>F!12" !j 5g G ! ] T @12 8 " 4> B
5!12 5 8 >T> "B
N
/ 6;W1r^> !12 5 8 12 M> 9 ""=
Approximations et résolution numériques
, F!555 !12 / 12 JG1'12 M> T != 12w > ] 1' > 12w!12"" % 8 5 ` B hzI
` B hH H 0= !5124H> 55 = >T"= 7
12 H> = 512>12U F '1 J4>m H T!B
12
|!12 *s!12 y *
> 4!0= 125 G1 >T>q44r12 y+= >>12 |> ? L= "5 = G g+> '!= 12 5 !M
G'5 "g++= 8 g> @= > = ( Ej M!12" ! Iz ! B L12 g!12 E5124
T! >D "H>D;Z""= 123" = 9" . F5' @!12E !B
3.2.1 Formulation semi-discrétisée du problème de contact dynamique
i c
<<!12" !l0 (9$ 5_= ! '5< 8 5 5Bj,%U! ]m /
] 1' > 12V!12"" U 1J4>D >>N 5r;W1' U 35 4% S!&[= 6;W1''N S712 >D; 41 123 U5
6;Z B
O 12 '[= 12 G512|6; JG1' >D;Z"5 A8 > >gg5 4
N5 4 / DBWB / FN B L
1 5124 H ` h
# E
HN
N
>R; 1C0 12< <!'& _` h >R;Z 5 "
12
#*E HN
!12(q
! j!12>> [email protected] j512 Iz"5 A8 >>
> 5 4% !0= 12
B
|!12 *s!12 y> 4
!0= 125 G1 >T>qq> H] 1'N4> 12!12"[[email protected] 1JK>m [+!12E !
] 155 "L H 10!'& 51 > "K>!5B L12 L !5124[K> M>12<6; 5 MzlG
'!= 12 +G'5 "F (T"+>.5 4M ! >! >G [= F 10!'& %51 >I
"LK>!5B
C % > 5123", 0;[email protected] )1 A @3 A28 FH15 3 *C [email protected] 6)H+ 7$8*"F) %$S'"212 ",#
b0471212 |9H> g+= 8 = j j!'& (4g 8 5 5 *j> g+= K> ! "q512 "j! >! > = % >D;Z 5 "
!12(( 9D
C C
N
N
! C
!
`
` B 2
h
! !"h C ! C 1r L w <5'0= > /
` B
12U!12E5 EL712 H>@12 "B
C
/ 0= 5 "9(> !'& 4 12E!124"
'! & '!'&124 12 >D;ZT 5 " / 5L>@' 8A E9D
b0471212
BC E : E ! `BC h A
2H J # `BC h
f|12 8 .`
! ` h
1r
/ >^ 1J4>D ( ( Iz '! 9"( 12
E h 5 7 9 879 HKJ ` h `BA
! H M HKM ! ! C N ` h ` ! "! h`BC h 4! `BC h ! ` ! M hHN E Q R
! ! C N QSR
5 w <512U >G
N
A `
h
` QSR'h5h
: B` A E 7h 5 79 N
` B}
!G #
` B}
h
` B}
h
` B} h
` B } "h
5'@ <555'!5 ( EHG12 ] B
!
B%Yq~!'&12' "
k P512 ! T4>QK! 12 / 1512
'!= 12< L>@!&[= 6;ZY[ >F51 > ( E%4>T!5B
h
ir <5 < 5< 0= 5 "4P= > 8 = 'Q "N !12 12P5 J4T>"(
= T> [= S` 1r " 5L> K > 12w > @ <5l05 Nl0 (T9" /08 2
1 0
! !
h
!
N
/
!S
hB
12 U5'12 8 12 -> i +
= G !H 8 !5 +1' > g G1 5 4jr;W 5g j T!"= N !L5 B L12 jI
1246?! 5G ![`m120Iz>T[= '= = 12 = 5'T9" h 12 "r1r12 r+= !'12 ? 15 q55 = 8 0= 512>T126B
> 1ŠF#9%.123",# " !' " 5.23", ,-5 ' =.57 Formulation en vitesse semi-explicite
i 0
= 512>T12 `{ ( w = &
E ` B } hzI` B } "h
"5 Hl0 (9 +G F9".> 1U $dl"G H512Kh ^ '1J4>m 5 (Iz ! / += Q44U 512
/ += ! '5%>! >! >4+> ]{ !51 12$ + 5'! +L = q 0I
!
<5' N M4 L= "F = '13`m! ! NT4>!5L $!'&+= hB
ir%!16x [ L! [! >T! >j+L! ]m !51' 124jG Hx 5%5 (= > 8 = / w
x .
T4>!5M>R;W12(G 5G'55^6;Z>T5L 4 $ N5 KB7ky12 !!K= > = > 0= 12>12 / 12cG T>5
""= 12 ( IR04>!5 5K= 4-5 8A >g O G "5 Dr> % [email protected]!12" !N512"
5 = (r;Z < D <(4>!5XG12 ^ !5 "X> > 512 > "= > (> ^ 71'
"5' (12E(5 "= (04>!5 ( EBjky12(JM= +=QK!X3>D;Z""= x $ ] 1 ' > 12 ^04>T!5 / > 5'[email protected] @[email protected] 5!124 [email protected] 1JK>m 51 > "(4>!5` B } hzI` B } "h% 8 "CD
BC E : E E h 5 79 879 ` ! `BC h A ! H M A N ! H J # `BC h
! H M ! ! C N ` h
N ` h ` ! !"h`BC h !4`BC h G #
! ` ! M hHN E ' Q!R
! ! C N QSR
f|12 8 .`
`
`
Q R
h5h
: B` A E
h h
5 ` 7 9 E ` B}
h
` B }}2h
h
` B}
h
` B} 2
h
` B}
h
%J4 8 12
r512 5 ] 12|9" q
w x + !12 4 Ey> 5!qq 5 / > ? 5'T! r "5 *[= ' 8 = 6;Z @ !0= 12p= > = "%Q4 F4U 1K
J >m += Q4 r ` B } hzI` B } [email protected]" ? A 12 > += Q4 @G12 8 B % 512> 8 XzlGX!124> = ( E^wb!'& 12x 5w>T"=(G1204>125> G12JK>.
= @!12 [email protected]> 5'[email protected] 5B
3.2.2 Discrétisations en espace
ir 3= 9 12 ` B } h%(` B } "h.(12E5 " "
9 > .!& 4 ^ 512".5 @0= >T'B
>@$5 x 5 "^12 ! < 4 '1 12P N> _= &1 X P= >= EQ44Brky12^! > 0/
12 j 1G12512 jr;Z>j> _= &10% = > = "[QK *G12 41!&q> [!'& (4 / j
C
` 469" U!12 '[= "L> ] 155 "[email protected]_= &[email protected]!12>>1! 12_`{712TEFQK h
G12>D; '1 12 !'& 4 ` 4M9" V! (w 'X !1245U
/ 4&+= 12< % ] 155 "hBG,S= >>12 .! 7 ! / 3125 E > = ] = 4!3 > !0= 12
<5 KFG12HK>Q M>D;d= !''B
C % > 5123", 0;[email protected] )1 A @3 A28 FH15 3 *C [email protected] 6)H+ 7$8*"F) %$S'"212 ",#
}
Approximation éléments finis
C
ky12 10!'&H> H !12 + 'T >
[email protected]> ] 1'N4> 12 A8 ' 12 >> / 12
Maillage
!12( gs!12 H H>@ >> A @ F 0<512> F F> H!12Q 12UT > G G B .<'I
G125(9(! '.512" X ( " = >.712 @9"(>R;W12P4 '5> 10!'& 0
12 T FG12>l 12 0w
LG12>l0&[= '9 M SB4, %> 45 / 12 L1512 L! %12 T /
'1!& E>
e~!'& 9N 12 (G
G N
! 1r
G
8 H12 T /
G
5 510!= N .5' > 12
#
D
8M F>@5 4F16LD
` B h
> 12
+= Q4 G D
N
# B
+>%12NJ r;d= > = ( EH > 5' #
D
ky12L>@ >> A @(Q
R
`
h E
5 G125
! N
>5' 8 D
b012H < >T> A @ += G
B
5M>L [.!55.5' `{Q B ` h5h
l E.512< '1 . `{Q B `{J7h5hB
/ %
> 12
4
` B h
b012H < >T> A @&[= 'N
= @> 5 ! = 12 = 5'T9"
E
"F
[1r
& * > 1ŠF#9%.123",# " !' " 5.23", ,-5 ' =.57 , L> F 0<! / 12S 15
Q R N
1r
RM
#
> 5' RJN
D
R
/ RBWB /
8M h
7 B
5L>@ 12J '@6;d= >= E.
> 12w(Q
` B
ir [email protected]!55N5' 12
7 = R / 5L+= Q44G D
> 12 / 1"
D ` hE
R
` B Z h
L G12512 FG12L> [email protected]> F5' > 12
!> '9" .4<5'`m! ] B O '> hB
7 512E= >D
S
Remarque 3.2
CV % % * # )! $* $ #* 7 # * ( C* $ $* * W
8$QF " % * $ $% * Q$*
J ! $#V # * 4 % .2$# $* # T 2 * %
% % & # * * * J 2 J
% 7 % B : 2 # #
#
* $ % B
@ %
# * 7LV J # ( * $* $ # %
* # %
J # *
# * LV % ! $ $# $ %
&% * '
#
% i?L! ! ( 05N`m&"l"J hyL> ] 1N > 12!12ET"%!12E5 E
Espaces d’approximation
>c!'&12 ( 5 ! wc !0= T 12 !& 4X ' 0p`m+= K> ! "w 8 5 5 h^w
!'& 4. 0V`mN >4>! 5 'L^!12" [email protected] 5 ( Iz >K>! 5 ' ] 155 ( Eh > !124 I
12
W / / B0Yq 1 "+> [= '!& 8 F M)+ X,.& >RB / 9 5?: 5QM= !12 '[= 12 .5 JK>= = J4> /
/ / 124% >5124L> 5 !(>D;W 5 !(6; I
1C0T 12V !'& 4^ 'T 0 G12 10!'& (> ^!'& K 0 `m< x (SM6; 5 !&12 512"HG12JK> `{B B h5hB
C % > 5123", 0;[email protected] )1 A @3 A28 FH15 3 *C [email protected] 6)H+ 7$8*"F) %$S'"212 ",#
12
1512 >U512 RI 5 !S
Espace
des champs primaux
G'd’approximation
5 "L6; 4 10!'&L> F!'& KH 'T 0 JK> B v>G 5[= Q4 6!12(N CD
79 2` G h*N A 7 9
5
`
G
K h
A: 5 ` h K
!
/
A M N
` B "h
E^>D;W 5 ! w !'& (
4N!12" :G >R;W 5 !w G12>l0|x12 N
0
= / 7 ! 8 EBky12*> !> "M
= +> 0= 5 " 12 / 12 y!12 += '12 | @= > = "j>T[= U!& 4F4' 0P` (N
hB
1r
`
G
79
BA 1512
`
h
A N A 1rS` h
h.
+= K
` B
I K
5j> J 5! 12 9F
Q4 Th[! > L!> 9 B
K
/ `
N C N 12 h*
A 8 !5 'N
7L9
M4N
ir !'& K ([= 4> ! "@N 8 5 '5
D
12Q4 hB2Yq>>F;v= !'q!12(L' 9D
> J LH!j 5 ! 8 !51' >y`mH (
512"q
h
] 12 !12 gFJ 5` = > = ( E
;d= !' 8 "N 4 !55J 5(!12
A C A ` B }2h
` B
h
12 *!12 T[= 12 *> [
= '!'& M' 8 q *)+
Espace d’approximation des champs duaux
,.& 12 BK, >! 16S> ]m ! L !12E ! 512"LK> / >N!'&12 UN> 5 !^
L
Q R !12(( 5 !(6; 41 12- !& 4 4 0S 5+: 5Q+
= / / / B
79
O G
" / F> ! F1r<! L m] ! F512" 2 !'& / 12 2 ' 12 F> w x @dl"G 6;W 5 !N6; I
1C0T 12S F c> "H> !'& K LJ 5 %>10! > H S!12 T[= E. !'& KL '!
- >4>T! 5 .(!12E ! -5 ( Iz >4>! 5 %
] 155 ">+= ' 3= > = ( EN
!12""4HF5125>D;Z"5 ]m !([email protected]`m! ] B
G12F4>T
4> H+= 8 >1 G E hB
ij;W !4!0= 12< U >4>T! 5 M! > @ !12E ! 5L1=
` Q!R'hHN
5
` Q!R'h
: ! 5
`
h /
R L+= Q446 CD
` B 2
h
> 1ŠF#9%.123",# " !' " 5.23", ,-5 ' =.57 ir+!'& ( 8 ! 5 *+5 (
Iz >K>! 5 6 '!? ] 155 ( E
5[+= !12G125My> J [>10! >
!1 8A ' "5
{` 8 " >> ">[email protected]!12(N = N
= 4= !4= (
"H
h N> [email protected]' A8 E5HD
( K
N 5
8 !
`
Q R
` B
h
h
,% ! {] / 12<!12 55 [>R;W 5 ! 6
; '1 12w $5 ( Iz >4>! 5 8 !51' >6 ] 155 "
!12(N 9D
K
N ` QSR'hHN
: 5
ir
` Q!R'h /
!
M
` B
h
!& 4 > K>! 5 '!12E ! @ ] 155 ( [email protected] T!.512".+= 8 >1 += 4%> '
J 5 N` h
N`
h
5G ! 8 !12(' 9D
N N K
` B
h
` B
h
e q T!= 124j q!& 4qL+= 4> ! ( E /"8 5 5 j[N >4>! 5 '*H!12E ! / >0' 55
% 10!'&g> y!'& 4*M ` '"9 H ^! *+ 'q !12 5F ] '155 E h
+= Q4 T[>10! > "+MTE5 8 EF +> + L= EM5 M%! 12" !H +> N] 1N > 12X ( I
'!= "=$` B }
hzI` B } "h / @ = &[email protected]!12>T>1! 126B
Collocation
i = & 1( (!12>>10! 12P!12 5U< 4 10!'&@>
> !12S G12EF Q R B
!& 4 (`m'= >' h / P4!12> I
V # HG12H! > ` h
@!12>> !12<Q4 G12"F Q R / ` h ` h >
B Yq! q F '5M (!12( 5
"
A8 > 'g q!'& K G12"H` h % >D;Z45 E FK&[= 12w % ] '155 E / 12 4 10!'&12 $= 2 > E ^@> <
x (N D 'B
1512
Remarque 3.3
' * " *
$
" " J*
# C % > 5123", 0;[email protected] )1 A @3 A28 FH15 3 *C [email protected] 6)H+ 7$8*"F) %$S'"212 ",#
% J #&# $ $# $* # % JQ R * * # * 4 # * # *#* $ ( * !$ * %
* * E
* ' E % # & $* . ! $#V % *
Problèmes discrets
i !12JK 512X> = &10. [4 L= 4! [QK g X5 4[+ [ = &10 M = >= EMQK N!12>>10! 12S U 5 !^12 6>T 0U
'1J4>m L' 8A E^ 0= 512 / '! & 9J >5
5 K B
C
YqX G12 "j9> [!'& (
4 ` 4 96;d= 8 E >T> "+ +!'& 4q LzlG 8A ' 4
J > q0I
5' y ^! gM!12G15 " [= > 59 h|!12 / > y 1J4>D y '!*+!12" !jl0 (9
`m ] 155 ( EhF7 8 EL ;d= !!12(( 9D
C E : TE ` h ET` h 5 79 879 E : B` A E
`
K h ` & h ! ` h ` h ` h A ` hHN ! ` h ! ` h ` h ` h ! C ` h ` h*N ` h
N ` h ` ! "! h `BC h ! `BC h ?N E
` h N ! `M h ` h
N E R
` h
N ` h ! C* ` h5h
N E R
`BA hM512"L> $= > = ( E%> J [email protected] L 7L9 /
`BC h `BC h MG
!
HKJ #
`
N
A
K h
` h N H J `BA h M G
f|12 8 1r
BC h `
q1r
N N BC h A `
`N
E
h
` B h
` B "h
` B
` B }2h
h
` B
` B 2
h
h
` B
` B h
h
h
` B0 h
0= E5.>LG124 7 != X712TEq L!12>>10! 12
`{G12
N
EE
R
hB O [712T j512"
> 1ŠF#9%.123",# " !' " 5.23", ,-5 ' =.57 5 !&[= L !12>>10! 12
M!.5 C 8 > 0X712T q ] 1'N4> M6;Z""= 12 F _= '9 +> E+> qG12"M
!12(712TEF6;ZTE= 123"4_= '[email protected] F5'( [email protected]!12" ! B
3.2.3 Discrétisation du terme de stabilisation
M C A M7Q G 12
nH 8 12 H "5 " > 4!0= 12< $5' 5 J4>T 12
M M7Q G12 .> ] 1' > 123 -[= 4> ! " /r 8 !
> X] 1N > 12S 8 5 5^12cJ4
0w <
5' L0= >LG12 ] B
M A Y = "w12 +=39S> 0 5 '( Xc5 J4>T 12 += ! '5 E<>3w x 3dl"Gc6; 1C0T 12
j
Yq\|Iz!12>>1! 2
1 / 12 FN[= 8 >1 G12 [email protected]> T!= 12S <5 !12 S5' B
n 47 >12 9->S5'(- 35 J4T> 12 ; :5125 0
5 C8 0
8 ' > X G1w-!12E !
M>QqB 1512 >1'9" X>X (N ]m NT E5 8 4N9X N!& 4 N ' 0 >1'
[email protected]>@5 !12 ]{ HE5 8 H> '1 4+r;Z <!'& (U ' >K L 5'@ >DB
M A , %>! 16S> L5 ! . L!& 4+= 4> ! ( E.>D;Z"5 ]{ !!12" !.>!& -
4>T! 5 %
^i A (
512" 10!'&+= @ S= >= EQ4 G12>l0|x12( 0_`{ @1'! 04h
ww x < 0= / >5$ $5 JK > 12 ! 5 ( E5<> N5 8A 0 8 > ] 1'! !12" ! %6;Z4 "
9 "N
= 9 6
G JG L> 512>12< ! 54< 1J4>D (
@@!12E !B
Yq ' 8 !& / > 19" c> >4>! 5 wP!12" !3 '!w 5S <4 ! YHBZ\+BL - =
55'T!5 E ] = ' X ! > j N!'& (45 ! N += 4> ! ( E^ '[email protected]>D;Z"5 ]{ !$
!12" !L`m! ] BAB B 712 | | (4> h / >+5'M+ J4> 12
@!12" !% " 126B v>r 5H < G [email protected]!12 8 T
'!
L!12 5 "H <= > = ( E RF Q R / >D;d= !''@ U!12"
H! M
M7Q N
R
5
`
M
h
A M>Q
` B
h
RB
eQ439> J4> 12< 55!12&+= "5 8 ! > 1JK>m > A
5'@ @5 J4>T 126B L124F 1G12512 F .> {] '% U 4> ! s
!9 / 4H>@! H1rS>@!'& (
!L G ! ] 6;ZG125 >1'L9"HD
Q R
[email protected]"5M= +=5 12S >>. U1 l
HKM
GJGq> 512> 12N4 '1J4>m c !' / > {] .10 Q %>
M M>Q_ CD
"H> 5'(
M A ` B T h
C % > 5123", 0;[email protected] )1 A @3 A28 FH15 3 *C [email protected] 6)H+ 7$8*"F) %$S'"212 ",#
5^ P1M= 5 ( 1 :5 !12_ 5 ! N$!'& (4$[= 4> ! "N' Q R >R;W I
!
N >K>! 5 ' X! 12" !B 12
= 8 >T 12 N!55w!1' !12 N> J 5$5 _= '9 `m!& 45 hB
1r
3.3
Stratégie de résolution
ir 1J4>D @[email protected]` B hzI` B h% 5 ] 15 ( EL 120Iz>+= 'B b0 H120Iz>T[= '"= L512"L> = D
>R; ' ( EH H712TEF9 6512"L '! J4> F E5'H U!12" !
> _= !12
!N L5 L @!12E ! ] 155 " Ul0 (9
> 120Iz>+= "
= @5 T>6 ] '155 E^`m10G > O 124>12JGh
> >[email protected]!12G15 ( EL4< "= ' S!124+= 0=
>
H5 ] 1' 124
O .120Iz>T[= ' = .512"%0= 12> . > 1'&4 ". .55 "
= @G12"LQ0 % = &[email protected](1 4>. E^` LL512 = [= >=<`m! ] B / 5h hB
n 47 >12 96;Z>y556; 5' 55 "= 0= 12>12 5'^9! >>^5 !^5 C8 T>DB
Yq>T> ^5 ] 12 E>$ 'T !G<> > += 12p` = [= >= ! 5U '= > '= hBgi?
5'T! E5 512" >1'%123l = 5'T9" ^`{ c 0= 5 ! ] '155 EhB O = & 10 %G5I
55 "[512 5 ] 12[H ! 12 8 '`{G12q! 4q!12G15 "hj> jG ] 1 ' ! q q = &10 +
H512S6 9"@> H5 @!12E !.512"L5 J4T>= @`m! ] B B B /
/ / } / / hB
.
%
i 55 = ! &12 >R; 8A E A [= !124> N> + L= "5 + 2
1 0Iz>+= '"= / $12 "[> G12 I
JK>
= $> N5 5 X 4B ._l < ^ 5'! N "5 (5l0_= 5'9 w`m[> >12+
!12G15 "L L "= ' 0U!12 5 [email protected]> H45l0_= 5'5. h B
ij; > 1'& L>>455.
= H>D;W1 2
(@N> Q B B
12
^ G12512 N9"r; A8 E(!& 9" XJG12 !>$$712TENQ / > !124" N12" = S
= 4 >= w A8 > '|: ! 5 "F!'&12' B
@9>R; > 1'& (w`{Q B h%6;W 5 [= !> 0
= !12 8 "9" ^>1'9N512 5 @> %JG12 !> .
G12"MQ HL>D; > 1'& @H512U12"L!12 8 = ' !! 8 ( EB
1512
> > C 1./' >3 )1 S' " 5.23",
a. Boucle sur tous les pas de temps
k
u = uk , v = vk , λ = λ et
Λ=Λ
k
b. Boucle d’appariement
et τ = τ
g
p =pg , n = n
g
c. Boucle de seuil de frottement
g
λ =λ
(dans Rτ )
d. Boucle des statuts
g
Su = Su ,
g
Sv = Sv
g
et S f = S f
e. Boucle de Newton
(generalise)
Fin boucle e
Fin boucle d
Fin boucle c
Fin boucle b
Fin boucle a
B
% #
,U= !' 8 12 * !!T !5 "y!& 9" qJG12 !>B k|12 |4>T rq+= T>|'|! T
0w= ] = 4! B
) - 0
7 ! / 124r 8 1Cl12
D
)+12 !> H> F H @5 4D
Yq>T>LG'M>D;v= 8 12>12$4X5 (4
M6;Z X @ 4 5'B0ky12F+= 5'F> M !12 @! >!4
> / 124H G125124+96;W >> F512E!12 " >D;Z45 E B
>D;Z 5 "
) - 0 +*D4)+12
!>@L> ^ = 12 = 5' ' (
ED
O 55JG12 !> G'L> !'& '!'&^ 1' > / 8 !5 '% E`m>T"= HG12L!' L! h
L H712TE '= L <!12" !B L12 L 8 12 F> [= '!'& N D
1r
`
hHN
`
`
h5h
`
`
h5h
5L>D;ZT [email protected]> [email protected]= 1236; "B
` B "h
C % > 5123", 0;[email protected] )1 A @3 A28 FH15 3 *C [email protected] 6)H+ 7$8*"F) %$S'"212 ",#
) - 0
.D4)+12 !>'H>@5 T>6 ] '155 ED
`m! ] B B B
) - 0
D4)+12 !>@L> F!& [email protected] , [email protected] !> / 12 M5 512 F> 120Iz>+= "@
= U5 >G ] 155 "BKk >6 < >G
>12JP = 0= `m"4_= '9 "h%!12(( 4!! 12-(!''
] 155 " / >!'( O 124
f? '! / 3!124"= 9" 4! / > 5'! @
= E5 ] 155 ( E 8 "L5l0_= 5'9
hB
D
Yq>T>qG'*M5 5g> 120Iz>T[= 'F
= > = > = !124 4!M g5 g !12E ! ] 155 "
l 9" B 12 = 512> 8 12 X!55c12Iz>+= 'U
= U55 "= c= 8 ' E= > = &10H j!12"5 "5 ! 8 E} B2Yq>T>+!12455N 4 >r4 : 55L`{ [ *5 5h*> g5 I
%!12" ! ! ] 155 " 0UG12"Lr;ZE= 123 = '9 H5'( !12E ! /
512^ 712TE / 512 9"( $G12"B 1512 (9U> ^512^dl"G U G 8 "wx 5'
5 = 5 NJ4> L @< x @JG12 !> 12U"= 0= "B
) - 060
D
)+12 !>[email protected] >. "D
, !55-JG12 !> / 12 (5 512 > 120Iz>+= "= 4= X> 41 : !12 X> JG12 >
"= / . 8 E >> ". > >12y^!12G15 " 3 "= 6B6ky12 ! m] / 12 %>T512
>^10 >@ E 9 ? .405 12-^> _= &1 ^ H512-5 ] 124 E.%> 112-
512 5Iz L= ' E >T> B
Remarque 3.4
% % # % $* * V J % %J W
$# * # !.W. % $* ! 5 E Q$ J % # #W
# J % * # 8 $ # @ # 2 # &# # Q # #
% # * ! N # % *
! N 4 Q # V J ! # !.W. % * % !.W $ $# 4
*
% $# $
e @>+= 12- @(1 4>N zlG 8 "9D
9 E 7
7
77
E
E
E / 12- 5 +=c > = 512>12P (5l ( >+= EJE
N
` B
h
> *
",9$ 5 ",
1r
[= > 5'! 9 /7
E / E
/
512"%>
= l 9" /
"5^ ' 4".
7L7
12EL> F 5'! ' 4"= F!12E ! /
EE
12EL> F 5'! ' 4"=
] 155 ( E /
' = 5 "5 E.> F !0= ( [email protected]> 512> 12<&ElJ 'T w 1J4>D (@ ! /
8 !5 '%0= 4 >D;Z"= 12c S(1 4> "`{.! >T>
(12E.G12"
Q hB
3.4
Conclusion
, S!_!'& 4
5 / ' 8A EU> U5 8A 0 _aS1 / pc> '<!12>> J71 5 ' `{B B /
} / / } / / h12 8 12 1 "U
= G12^><16 >USb0 1'T IdaS1 _!12U10G >U
!12" !X l 9" B*e X>D; 8 12V= !6;Z U D = 9 8A > "5S512 ] 1'Sr;d= 9 124 /
12
= ] 1' > 12w&"l"J4' ]{ JK> ] 1'5 "= 0w 1J4>D Fr;Z !B 12
C 8 12 $= > JG10
1 44= 5 E(
= > !0= T 12cN!55 ] 1'N4> 12U1rc12 8 12 4!N
= > '!= 12c 8 2
5'U $5 JK> 126B %<55 = 3w= 512>12 J =3>$ ' !7Xw712TEQ0 = "=
= 2
> "F 1G12=B0, 4[>%!'& K5' 8A E /
F41J4>D H @!12E ! Iz( ! B
12 [ 12
] 1! >T512 [[>
5G !MN4> I = !'& >T>
}
C % > 5123", 0;[email protected] )1 A @3 A28 FH15 3 *C [email protected] 6)H+ 7$8*"F) %$S'"212 ",#
Chapitre 4
Approches multi-échelles pour les
problèmes de contact
=<
.O
> L+ 5 +, 5/0$! *# AE $ /E ! A);:;;:;:;;:;:;;:;:;;:;
|G
|GRE
|G o
|G |
|GR‚
Wk0$(F%>0$(< C CCCCCCC CCCCCC CCCCC
WkT8;-Kc ')-=†76- "#J16- 0$!3(+L"MG-8T-K*h()<N0$-K!,e %"21-.*ˆ CCCCCC CCCCC
z 6 LNMG%>0$(<G*/'T27%')-.*8 MI# T8;-Kc ')-C8;: (<"03-7!$e %27- #M ')0$(Ž< (-.%>MC CCCCC
"!$#M '+%p0$(<je %>(b '- Ž[email protected] !3b 'Ž-Kc #H-8T-]27<N03%"2P0/#M ')0$(Ž<G()"-K%>MˆCCC
t !3-7# (-7!5*! .6- *,MG'‰05%p03*<ZM #K6- !$(+LNM -K*9CC CCCCCC CCCCC
, 47E7E&1#%E
|GRET
|GRETRE
|GRET o
|GRET |
|GRETR‚
|GRET s
|GRET‘q
OK6- 21-.*$*$()0
DBA5/67-3A 5!;7E (+EG# /0$ #H ;;:;:;;:;
V"V
"V V
"` n
G` `NE
N
6- 8T-K*\%@ @ !3T254 -K*#MG'‰03(K6- 254G-7'')-.*CCCCC CCCCC `|
‡[email protected]! 21 Mk*$M !\'+% #J16- 0$4GZ8 -‡!$'-KLNM (<„C CCCCCC CCCCC `N‚
 : %>@[email protected] !$T254 -C‡!$'-KLNM (<j-K<k8T^Z<G%># (+L"MG-k CCCCCC CCCCC `"s
 : %>@[email protected] !$T254 -C‡!$'-KLNM (<[email protected]!h')-.*/@ !3b 'd-7c # -.*8 -27<N03%"2P0\-7<k8T^Z<G%# (LNM -XC?.nn
(*321! 16- 03(*3%p03()"<G*/[email protected] !3b 'd-7c # - # Z8;-7c '- CCCCCC CCCCC?.n"E
SN03!3%>0 K6- =(- 8T-C!G.6- *,"')MT03()"<ŠCCCC CCCCCC CCCCC?.n"E
r9<[*,-K!$p%p03()"<I8 - 'Ž:w76- < -K!$="()-%F-.2'Ž: %>@ @ !3T254 -C‡h!3')-.LNM ()< C CCCCC?.no
P:/#@AE$53/
:;":;;:;:;;:;:;;:;:;;:;:;;:;:;;:; HC F#9%1$& 5 2 .'8$ - 3 *H15 3 J9 [email protected] L)1 $ ", 7$8
ir '1J4>m ^w!12E ! zI ( ! 512" = += > "N4> I = !'& >T> V 5 !w^ _5 (4B O ! !^N4> I = !'& >T>c= {] H9"HD
[> ]m ! (S!12" !X!12(71'5 E / V= !'& >> ((T!12!149 / >R;W ' 12wG x 5 9[= @ L> L16 > H !12!1K9" B
12 @!12" K>
5M= '=
W} 12"
!%512"L512 8 EL>1! >T"= B
r>D;ZT !%0!5(>1! > ( E .4&+= 12w @& 5
TG1 E5 G12H>R; >l05 [email protected]@> ] 0= 9
1 !"= B
2
!
12"@> 'N P!1245 5
, L!N!'& 45 / 12 L12 H>T(512 0 5G !L >I = !& >> L < 5 !w`{ 0= 5 !6; 5M= '"=
N>1! >T 12- !$
12 !12" !hB*, 4 (D( / [email protected] [email protected]
10G >H6;Z"5 ]m !L > Iz 8 } / G'5 "j F 5M (!12 5 j!12G15 "q>1! 0
>1J 0
( j"5 ]m ! qL!12" !B L124j12"512 j512 j !4>m "q>D;Z""= x [!L10G >
G12?> r 1J4>D |q!12" !* l0 ( 9B , | q 0D q / 12 r12"512 |!12( "
> _= &1 +e'> 9 G qx 5M>T"=q!12(+12>""4_= '9qG12r5 5y 5G !* > I
= !'& >T>
4.1
> Iz106 > F 1JK>m [email protected]!12" ! Sl (9B
Modèle d’interface multi-niveau
4.1.1 Motivation
ir ?>126[b0 1' 12E? r>126 !'12!149 r1 l
"|> GK&[= 12w y(!1AIR4&"l09
6;Z"5 ]m ! BCY[>> 6 4 5 "K ?0= > "= ? r!& 4
?_= ! 9 [email protected]!12&+= ' E5 8 ![>D;Z""= 12
5 G1 >> H F!'&+= !> T9" + w5 [email protected]`{ = 5 !r;W12!>> 12 $12 "H F!&10!
} hB O 55>5^7 ^
x 5' >>T= >T ". @>12 6;ZTE5 ]m ! AK> .4&"l'9" / zlG
}
!124>T !B <= 12T / > 10[= > 12$ _= '9. X
!12" ! L 4> "
= 7
> F
! F>12M%G'
NX x 5'[email protected]> N"5+= [=5 12 ^ !'12!149 @G > 1 J4>D N !0= T"=
"L5 >r!12 412 += B
eQ4VX5 "5$ 1Q45^ ( 8 ' 8A E A X" = 9" 4 (16 >wSb0 1' g
10[= > 12 > ^>12TN<!12K> ! / 12 1G125124N
10G ><6;Z"5 ]m ! w > Iz 8 /
= } B
] 1' >> ( EH"510 M L+= 8 >1 M
4.1.2 Modèle géométrique des interfaces
ir N m] ! @0= T> > (!12" ! (512" m] 5 ">'5 B6Yq>> 0
= 5 "5 E / -= !'& >>
(!'12!149 / += '5 (12"> 5w ! 12 5U (> 106 > ( U!12E !$
G $x 5'
. ) 3 )+ ,- A 7$ 6 5 2 ,#8%.5
G1 "5 ` 8 12?Q ' B hB i >>j(1Cl [ 5M= "= |+= G @
q> 'q = ' r
B
k L0 4
> / >T>. [email protected]>D;W1 @
9 >9 F(!1AIz<5'G12H> F = ' 0U_= >>9 H ' >B
Ecrasement
des asperites
Phase d’interactions
macroscopiques
Phase d’interactions
microscopiques
UT
B
* JW #
# .
i {] !MG15 " >>MF!12E
6; 5M= "= JK55 `{Q
!+6;Z 512> ++= ] 1' J4>+G 44x 51+= >T"=M j M!12 !'& = _!12E ! ]m ' B h(`m! ] B?B B hB?i?X512>T^ 5N!12 '[= 9 @>[J455 r 5 55 TE @= ! 5 "yq5125 *> 5M= '5 ?G "?> 4& 5q q!12E !B
irN!12G15 "%>10! >rN! 5M= "= G @x 5'!12 '[= N
= !12(( = > 5T9"12p= > 551K> 59
By, 4!5 8A > / [email protected] G125124 9"6;ZT>* 5P= > 5T9" / 12 += > 12 dl"G$!12^I
4> !B
substrat
couche d’asperites
a
dn
dn
n
Corps rigide
B % #
*
* % * eQ45
M!12 5. +!12G15 "+>1! 0
( 5M= '"= M >1J ['J455 / > @
] 12 /
12 % '171212 . -16 >(6;Z"5 ]{ ! > Iz4 8 / 1 J 5 "- @GG1212 P16 > I
!12!1K9" S$b 1'4|G12 8A " 0= E5^>106 >$!12" !^4 JK55 / 6;Z4 10G >
(!'12!149 10 EM`{M= 'T E > Ehg>D; ! 121' >+!12!14T9"+ 5M= '"=
6;Z"5 ]m !B O @106 > / zl7 >1! > >1J > / F 0
= 5 ""N
= > 4&@ > Q ' B D
ir + w5
5M= '5 ^(> 45 D
!12" ! 1! = M
.! 10G >' = 5 "5 E / 7 ! 8 E / !w +=< "5< 0_'J455 y
B i? !'& K
0U16 > H>10! >6 >1J > / 5G ! 8 "B
>T>! != 'T59 512E> ^ 4"= HC F#9%1$& 5 2 .'8$ - 3 *H15 3 J9 [email protected] L)1 $ ", 7$8
−Rn
−λ
−Pn
dn
dn
dn
−a
−a
Multi−niveau
Signorini
Compliance
*V * %
B #& W %J4 8 12
H9> 'GG1212$ U106 > [email protected]' 8 "N A' 4> ! > 16 >c-!124>T !_`m! >>c9 HG125c w !4>"= 6;Z EQK! 12 0+= ' " > -w
1JK>m = 'T9" / > (] 12hM %! >4?b0 1'T DB O ;W 5%6; >> 'H5124! >N 9
[email protected]>@10G >+= 8 >14+N
= !6124H5 NJ4>^x 5N> 4> M""= "B
4.1.3 Équations locales du modèle d’interface multi-niveau
ir w= 9" 12 H>10! > H S16 >@[= 7 4 E H L= "5 H4& H <!12" !B7Y[U>D;W10!!
I! D
M !
D
M M
'
'
'
!HD
>1'H> H ]m ! FG15 " >> H @!12E !.512"L>J '
>1'H>
5M= '= 8 I I 8 H512" ! 8 >1'H> H'J455 M' E5 " '6 U!12" !G125 "9M
N
kq>T y = !5 " / > '"F
= 51 >L * G1j1 0NF!12" !JI
5^= !5H!12(H> 512
6;Z % 4"= 6;W 71'F "F!12 [email protected]
> !124>T ! >D;Z"5 ]m !. .!12" !HH6;Z4 5
'"= / N
4>4>T! 5 + i A I
N
/ 512CD
` B h
1rLD
N
H1rLD
N
N N
! ` M D
`M D h
SQ!R
h
Q
R
`
! M QSR ` N h
! M !Q!R ` h
` B h
` B h
` B "h
h
` B h
. ) 3 )+ ,- A 7$ 6 5 2 ,#8%.5
D
0
% <5' _= '9 %12P "QM= .0+= ' " > " / } B6nH 5I
8 !
/
9 12 *9" + 4|>M! *1r
:N / 12512 8 H>[106 >F++= > 12^>+= +9"4 / 4 7 >12 Iz> /
6;W 5L H <106 > @!124> 4! M712 H> H"5 ]m ! L = >>T9" B
ir $!'& 4$3 = Q44( P` B (
+
h -` B h / 5G ! 8 ( E / G'55 "$3! I
!= '> F5'12F4& [email protected]>D; 4 1!&@ HTE5 ]m ! B4Y[w 7 SD
N
'
N
N
'
'
`{ ] 1!4= ( E
N
>1'
N
I :N
N
I
h >1'
N
BKi {] [email protected] !12E ! 5 46>J '
:
$L>@!12" ! 5L5 "@= L4<16 >@!124>T ! :
h / > '"
= 6;W G
1N !12E ! N!1' = $ `{ ] 1'!4= "
4 12<>T= U10G >Nb 1'4DB
Remarque 4.1
4 * *
% * 7 # V @
&
*
& #
#
$V
* D % * % % V
J
%
*V
#
#& W ir-10G >c 1G12"=3 w = 'T9" ( E = 0 '1J4>m w-!12E ! < l0 (9BqY[
G / > 4& 5S 3!124>T !<G'r; 2
1 ! $> !&1!XXr; J451JG / ! 12 = 9
! / N 12!>> 12 H = '9 % %!'& K = ! T9" / 5 >y9" N! > | 8 5 '5 BKY[S12 5 /
12 G12 8 12 % .!55(< x ^4& 5
@!12K> [email protected];v= !' >1'H!12 -D
/ 1r
N
! ` M D
512" !12454&+= 12< ! ` M D
h
h C
6; 1'5 "BGi?N5
12 8 03 <5 @4c(16 >6;ZTE5 ]m ! ..16
8 5 H1' > / U5
` B }2h
C 5 >!'& (- L> += Q4412 ` B 2
hB
Remarque 4.2
* W
* J $# *B Q * * % # $# $ % [email protected] * $# $* F Q # # $* * # *V % # % J $# * # $V
# JF * J
( # 4.1.4 Formulation faible-forte du problème de contact multi-niveau
12
M12 12 [> N
] 1' > 12 ]m J4>%>D;v= 9 >J L6;Z4512>T+= ] 1' J4> $!12E ! IzT ! 8 !
$512> ' B"i?%(16 > (T!1AIz !1 .!12" !N` B hzI` B hM 5$= 2 > ( Ew= !'M % 4m
HC F#9%1$& 5 2 .'8$ - 3 *H15 3 J9 [email protected] L)1 $ ", 7$8
L>@ >4>T! 5 ]m J4>Bir <= 9" 12 F >T EH> H!'& ([email protected]+= K> ! "
0w!& 4
N'= >' 12E 8 4!'& = !'5 ^w D ] 15B O > 12 X>X '1J4>m 8 "9D
E : E h 5 >9" / G12F512 `BA E h E
` E A h ` EA h ! H M ! ` M D h A HKM ! ! M M>Q N ` M D h Q!R ! 7 M !Q R f|12 8 K
! N
N
1r
`
M>Q N
!"# `BA.h
` B h
` B 2h
P w5X12 " > /
5 P w5X5'!5 "G12 ] 1r / ` B h
` B h
@>(! 6;v= > 5T!"N
= >+= D
` E A h*N 2H J ABM7G
A h*N H J ` h B` A.h>M
` E.
` B h
K
!%"# `BA
h
N
HKJ L BA M7G
G
` B
h
` B > h
O N 1J4>D N 5 5B6b 4
L= 4!N E >T>@ % G1 U 1J4>D L !12"5= >
!'& 45
5ww > : 12^ _5'ww!124> 4! 5l (U6;d= 9" T>J S` B hB
O 12 ! E%>!& 3 / 12S"510 !12< L 5 !&[= 5 " >> " U!'& 12 > I
1'&4(9B
4.1.5 Premiers résultats numériques
ij;ZTE= x "4_= '9> (] 1'N4> 12< '1712=N 5.>> 5= w0 4> [email protected]>D !12" ! >j r N= M9" ' Iz5 9j*>q5 !124 l0 (
9B12 |(12E5124 / 5 > ' (M (4> / >D;v= 8 12>12 A= >D X F>12F @!12E !B7, 4L> 0D 0 4> /
:
%
8 '
12
5512 %>D; !! "%'L>D; G1 U10G >Nr;ZE5 ]{ !N > Iz 8 _ > 0= 4!12S L12!>> 12
_= '9 F H!& 4l0 (9 0< 8 0< H ]{ ! [email protected]!12E !BGir L= > H1J 5 "4
512"L!12 0= 6; 5' H1J 5 F >@10G > 4 ( EH !'12!149Nb 1'4DB
. ) 3 )+ ,- A 7$ 6 5 2 ,#8%.5
T
Description des exemples
irg4 (60 4>g' = 5 "5[>D;v= ! 5 ( E / ? .JK>1!*9 IR / 6
;Z .J4>1!N= > 5T9"j(1+= >T"= /
( 8 H> ]m !HH!12" ! / j J45 * q F!12 !&L6; 5M= "= qF >T> 1Cl
B
`{Q B 4` [email protected] 0m 0 4> >>T 55%>D;Z !F H wJ4>10!%' .6;Z %
J = > 59w512 (T5c 8 5 $4 > / 0= E " 5+= '= $w x w >>(1Cl <9
L>D;W (4>. 0= !4= "^`{Q B 4`{J7h5hB
Ud
V=100 m/s
substrat
Barre:
asperites
L = 0.1 m
Taille des asperites
=500µm
l = 0.005 m
11
E = 2.10 Pa
ρ = 7800 kg/m3
Bloc rigide
ν=0
Modele:
κn =
Modele:
κn =
mn
=
15
Bloc rigide
5.10
2
a. Contact statique
B
1012
mn = 1
4
cn = 2.10
m’n = 0
b. Contact dynamique
$#
Résultats numériques
ir M0= ' > H = '9 M512"H12 += F M> MQ B B }B 12 Ml4= 5 E5124H> $= 8 2
1 >12
!12 'j A5 (4 @ [= 4> ! "[(712= K / [= 4> ! "[6;Z4^712TEj!12E ! R*+
!12"5 "5 H !12E !L $! G12" / 12 += M HT> 12X w10G > b 1'4 F w0
1 G >
> Iz 8 6BEi Q 'B 12E5'% = 8 12>124q4>T j0= >Tm M [!& 4+%[= 4> H!12"5 "5 N` 3
8 U @> ]m [email protected]!12" !hL12 += F >@10G > > Iz 8 i Q ' B }$12"5^J4 3>D;Z""= x @ = 'T9"^^!^zl7^^10G >N -!12" [email protected] ! "M
B
6
(9BKY[
G / > %12!>> 12 . 8 5 5 .1J 5 @4N0= 12>12P c
'1J4>m !12E !N> E
>%16 >. >Iz 8 12EM Q4! 8 " 5= [= H + G1 40= 12>12X M> M>12
!> '9" NSb0 1' RB?i 4& $w
M= >T 12V"510 5c >Dv; = !& >>U 5M= "= G'(
6; 124!H>@!'& 1! @512 8 L U!12G15 ( E >1J >G4> M0= 4> B
HC F#9%1$& 5 2 .'8$ - 3 *H15 3 J9 [email protected] L)1 $ ", 7$8
(a)
(b)
* W *$* B
! impact d une barre sur un plan rigide
signorini
multiniveaux sans amortissement
multiniveaux avec amortissement
100
V(m/s)
50
0
−50
−100
0
1
2
3
4
5
6
t(s)
B }
# @ ! % 7
−5
x 10
%
! O y 'r0= ' > *"4_= '9 ?= 12 EgM>D; G1y (16 >M > Iz 8 @ 1G12=B 12
!12 !'12 M> 0D % .!%!& 45'^ >D;Z>T 12(%> _= &1 %
e'> 9 (!12.12>
6; 4.2
>[email protected] >I = !& >> L F 1JK>m H6;Z( !B
L’approche Arlequin pour les problèmes de contact
4.2.1 Nécessité des approches multi-échelles
Yq _= ! 9X P!12" ! / 12 5N512 8 "N!12 ] 12""P
= @ 1J4>D 16 >(!12"
! >=S`mT " 12 `{Q ' B h / 4
Q 12 / !BBB hB O zlG((>10! > 12 !12 ! "5 12 /j8 12< T > '"= ^U!12"5 "5 1r w= 71245 ] 1'5 ] = 9" 4! 9S> ((16 >
12(<G 8 "! 5 BqeQ4
! 55 >1AI
= 6$ ] 1 '5
EX!'& = X
r; 8 12X4> ^U4= !12 > 1Š6+ #9 .$ -H 8=.5,:H15 *9 [email protected] 6)1 $ ",- 7$8
512> 123 = 'T9" 8 12T A !
12 / 12-G % >5%> ._= &10 8 . - I
!<` 8 12 S5 (4h / J "= % L 5 '%6;W' 'N`{B B W}} / / hB <= 12T / !
= &10 %G 8 E.+= ! 5. L >> A %>12'4H6;Z p= ( 5 / "> 0= 512> 12
_= '9(5 !1rx 5 45 _5 4X!124! 126Byk >T> ' / >T> T>5 "@
(2
1 1AIz10G >
12" >!12G15 "% @>< x (7 4 E.> 10!4= ^6; 4 8 "= B % >5' 8 3 !
= &10 $ 5<> = & 1Pe'> 9 / / B O 55P_= &10PN > I = !'& >> 1 G-
! !12 8 J4>$G12N> GG1212 `m >T> A h $10G > ^ L= ' E 5 6;Z < x (
55'4!BK, .!^9"4? / 12 0= 5 "512 %J 'D 8 ".> T = " N!55 10!'& G12
>D; K>9[ 4 5 0w 1 JK>m H > IA= !'& >> HN!12E ! Ul0 (
9B
u0
R
ν
E
B
$
4.2.2 Aperçu sur la méthode Arlequin
i = &10weL'> 9" T P 5^ X = & 1w$GG1212P $106 > NG'5 "6;Z >5N
10G > r L= " @ 6;Z j< x (q55' ! B irq! jeL> 9"4%765126dl"G rr;W1+= 12
@10+= > 126D
I
D
>HQK^ 5qGG12=N 416 > 12j0T5 E / [!HG12g Xq> 512>T12
12 y!'9 B O qzlGq6;W1M= 12 5| 9jG12|> r '1J4>m ? q!2
1 E !
0 8 12T A | `m! ] B Q B 2h
#
%(106
B #
%
HC }
F#9%1$& 5 2 .'8$ - 3 *H15 3 J9 [email protected] L)1 $ ", 7$8
:D O zl7N6;W1+= [email protected] !!1 w106 > `[email protected] '5 FLN!12G15I
" L
= "h / 5 !12 8 E >> E(`{Q B hB %5 >>(1+= >T 12 ; 4>9U
I
[55' !' = > !K= / !12((> +! l124[!12NJ4 5J4> / > G [!12( 5' BKYq>T>@G' / 512 '! >D ( E / @0= N>N12J451 >* =
HM> j l 0I
>JG= / 512
<1M= "> " 41 ' = d> 1r3>7 5L+= ! B
B
I !
:D
O 55<1+= 12 !12455 GG125NG12J45>1! > (
ES V10G >
12'*0 Eg 10G >M4> |Q46BM5 >>[1M= 12^ 5g>[712 y"510 ' 8 !H4 J4>T"%
= <+= ]m N`{QK. L K> h+ F <(16 >@ `{Q B hB L2
1 4H>D;Z>T512 +!
G12H> F41J4>D H @!12E ! Iz( ! / T B
Sc
Ssc
Ω sain
B &
! 4.2.3 L’approche Arlequin en dynamique
ky12= 4
J >g> ] 1N > 12N +> = &10FeL> 9"4 ^l0 (9 / 12 *!12 += 124y4 '1J4>m 10G > = > 5!_
= >T[= BHky12S4 512> -+= ] 1' JK>~= > 59 E {` Q ' B h / 10!! "
>D; &+= ' !$r;Z
12 8 ':
G I K ` MPN
/ G12N'4>Q > M= 5 " 12Kh / !$ 1JK>m X;v= !'D
`{G12F512
5
/ < ' EH> H1 12 L U!'& 45
h
1Š6+ #9 .$ -H 8=.5,:H15 *9 [email protected] 6)1 $ ",- 7$8
f
Ω
B 5 7 9 `2GLh
K ` E A h ` EA
f|12 8 A 5879
5 >?9 G12F512
!%"# `BA
h*N
$ h
`2GLh
` B "h
`2Gh+ 5L>D;W ! !'& KH+= 4> ! E!+= 9 " 4(J4> .` >D;Z45 E h
+16
K / & !"# 4 = 5 E5 "F> [5 8A 0X ] 1'! M6;Z /2] 1! [""= ' ' q ] 1 !
0"= / 5G ! 8 " /K U< x ( 5 " B
1r
79
b0471212
/ "5 E / [email protected] F!'& E / > F!& 4H = ! 49" H6;Z . '
12 T G JK5 "j A8 ' 124jG1 "5 L`{Q 'B h
B eQ4( H! 5+! 8 12 /
8 12 F> H L= "5 <= G L @>D; 10!'& @eL'> 9" T D
4>9M> H!'& 4H = ! T9" 12 F> !12K>H> H = . f
12
/
12 @GG1212 "5> L L= ' Ew= B
F> 12 12U F @5124 I 12 RF `{Q ' B
0= 'L> $= L 4H> hB
S Sc
Ssc
Ω
% F B &
ir!12K> A E5'^>
B
*V$ J
#
L= ' E 106 > .G x 5 ]m . 10!'& .+= >=
12-> A 0I
HC F#9%1$& 5 2 .'8$ - 3 *H15 3 J9 [email protected] L)1 $ ", 7$8
> = "w
= 12"50
= 4 9( >(! 16P> @> 512 .e'> 9 -512" 5 "= @ @ > I
4>T! 5 'Ni A / >| 5%TG1 E%N!124+= %> 12 LNzlG
4>T 1x 9^
> T512 Udl"G
/ ! X! $'4m 4 5 " / (!' ! -`{ XS '!= 12 5 Gh / 8A +!12 12 "` 8 12 hB % 5U= !> A .= ! EF+! 5G !
G .x 5N!124 >
= 4 0 B
irS 1J4>D U(16 ><4= !4= EP` B "h( ;d= ! / 5 E$!12 5 S L= ' EX!12G12 EX
>D; 41!&Se'> 9 !12 += E$
!!1 +
= K> ! " XN >4>! 5 ' F
i A / !12%' `{1r$> [G12 " 8 "h-D
= ] " 0 4 ' >1J > [+>10! > / 5G ! I
E E @h
`2GLh
` h
5 >H9 G12F512+`BA EA E
5 L7 9
879 :
K ` E EA EA : yh '` E EA EA : h ` EA E A h N !%"#`BA :
` E E h N f|12 8 `
h JK>
h
` B
` B }2h
8 !
` E EA EA : yh N H J A M7G H ` ! *h A M ` E EA EA : h*N H J ` h `BA h M G H ` ! h ` h `BA
!%"# `BA : h*N H2J L A M7G H2J ` ! h L A M EA ! A `5` E h 10 +! > ` h712<512L
` EA EA hHN
K
h
h
M 1r
/
512"q
N
N
ir H 5 !
79 `2GLh
7 9 ` h
N
N
`
N
`
`
7L9 ` `2GLh5h
` h5h h+
G
`2Gh /
79
N I
512E[= Q4 M !D
` B h
` B h
` R'h5h `
h
h
` B 2
h
` B 0 h
9D
` B
i >12HS!12(71'5 E$ 106 >3Q4 G
wx 5- L= ' E5cS! >>3 ;v= !' / F>@! 6;v= > 5T!"N
= >+= / ! 12(' 9D
10 (= 9" 8 > "h
] 12 !12 gHG12 += 12( = q9 8 = 'Q E[> j!12 12 g 8 "5
N
E E 10G > 12' BqY[>>
` B
h
1Š6+ #9 .$ -H 8=.5,:H15 *9 [email protected] 6)1 $ ",- 7$8
# n 47 > 12 q9. M>! 5 9M M>%! M16
N
/ > 512> 12X4 '1J4>m e '> 9 55L[ 5[4 9`m! ] B hB0Yq!12" / >>H 5[ 12 [=`m5124[! T [&"l71&6 (4> @
N
R
SG
N
12
N
]{ ' h[ D
R
y 8 1 l12 R*N Rh
` B T h
` B "h
! h
`
`
R
` B
h
12 y y0= 4> g*> !12 5 4!+g> G 4![M> _= &1 B
/ 0 7
/
! / 12 H 12 12 F w0= ' > [email protected]!1245 8A 12S6;v= B
Proposition 4.1
' % J %
# $ E % 4 ( >$#V
*
# O 12 T[= 12 >X! '$!12" 5 4N^ !B L
12 ^ G12512 9$>
] 1'! H0"= ' 512"L"4>> Bij;v= @51 >eL'> 9" TX L[= Q4 . !D
$#
N H J `BC h M7G H
C C
! *h `BC h M H J `
M7G H
`
! h
M ` B }2h
512"> ^!'& (
4^ 8 5 '5 5 "U` 5 12 5 " h 0 5 !
2` GLh
` hF1r
512"%> "
= %>10! > Lr;d= .N+= ] 1' 12 `C= > 9" h > 1J >
L>10! > / 7 ! 8 EB
1r
7L9
79
YqS[= ' 8A EF L G1 <5 K` B 2
} h /
N H J C 7M G H
`
12 F1J 5 124CD
! *h C M H J ` ` h `BC h5h M7G H ` ! h ` ` h `BC h5h M ` B
h
,; %>@kqfFt
N
!
` B
h / >D;d= 9" 12 ` B
EC ! C h `
[email protected] @
D
` B 2
h
1 / > F> 5124FeL'> 9" T$TG125 E9D
`
*E ! h
N
` B
h
HC F#9%1$& 5 2 .'8$ - 3 *H15 3 J9 [email protected] L)1 $ ", 7$8
12<
!1' / H+= ' 8 12U H G1 U5 4`
*EC ! C h `
N
EC ! C h N
D
` B h
YjL ! >Tm " / 712 `
@[= 7 4< H w5 (4h
N
/ 12<1J ED
` B 0 h
O @9 `m6; ^` B 2
h5hH> !12 5 A
8 [email protected]>R;d= @51 >B
%J4 8 12
512 8 12
hB
"g9"F>M ' A L
@` B hH ` B hj6;W 5g 'F
= !'y ^5 (4B 12
T q 1J4> = T9" / !> T9" j712 q> j ! !1' + m] !9 g l 9" `{B B
g!G
4.2.4 L’approche Arlequin pour les problèmes de contact en dynamique
, g!F A 4& / 12 4>T9" 12 ?> = &10He'> 9 ^ 1J4>D FF!12" ! !Bky12
! m] / 12 . G12512496;Z -512>TN+= ] 1' J4>:E E5' P!12" ! C8 !( -512> N' BKi 12%G15 E >T>H!12" !F 12> EV 5+1"= LQ R B"e% 8 12T A %L!55 "5 ]{ ! / 12 [G5I
G12512
106 > (9 >QM
= >1J > / [= Q4 g 4 >12 <G / 0
1 !!M(
= (E / -10G >(>10! >
P
> 9(Q!R ]m 5 @@> ^] 12"D {` Q B > hB
/ !'&12T65 r
Ω
Sc
S
B >&
W.
@ # Γc
& * k @512 ! *!> = / 124%!12 *s!12 . . 47 >%> $] 1' > 12V`{4 ( E.> A / += 4> ! ( [email protected] P 1JK>m (!12" ! !^ ] '155 EN GG1212-e'> 9 6B
Yq>T> !12 55X |
D `m! ] BK!'& 45
h
1Š6+ #9 .$ -H 8=.5,:H15 *9 [email protected] 6)1 $ ",- 7$8
E : h 5 >TL9 G12F512 `BA E h
K ` E A h ` E A h R ` EA.hJN !#" `BA.h
!
R ` E E hHN f|12 8 1rLD
`
` B h
` B T h
R '` EA.h*N
! H2M A 7M7Q
!
R ` E EChHN ! H M ! ` ! M h %M>Q
N ` ! M h
` h
Y TE510 T E*> ] 12 !124gFG12 [= 12 / q
/ [= Q4
`
N h
` B "h
j L` B hgL` B 2
hjg (4 8 "
> $= G
%@> = &10e'> 9 / 12 H1J 5 12 H> ^] 1' > 12< 8A E5HD
f|12 8 `
K ` E EA
!
R 1r
`
E EE h 5 >TH9" `BA EA E*E h
EA yh : ` E EA E A h R ` EA
:
: `
E
A
E
A
h*N
!"#`BA :
`
E E hHN
E E h N
h
` B h
` B }2h
` B ` E EA EA : yh N H J A M7G H ` ! *h A M ` E EA EA : h*NIH J ` h `BA h M G H ` ! h ` h `BA
R ` EA hHN ! H2M A M>Q
`
E
A
E
A
H
h
N
`
EA ! A h !%"# `BA : h*N H2J L A M7G H ` ! h L A M ! ` N h
R ` E E hHN ! HKM ! ` ! M h %M>Q
N ` ! M h
` h
K
h
M 12
h
M1J45 8 2
1 4+9 > M"5 !12 M !12" !L512"M ED ( EH4'5 [ w!12 5 F>.10G >
>10! >DB
h
HC F#9%1$& 5 2 .'8$ - 3 *H15 3 J9 [email protected] L)1 $ ", 7$8
4.2.5 Discrétisations du problème modèle
, [email protected] !12 / 12 q4= 5 E5124H> !0= 12 w !.+ $5 KF 41J4>D .10G >
` B hzI` B hH+124[+= !' 8 12 M !! !5 (
EM> 5 "= @%= 512>12$ 1J4>D . _= '9B
Discrétisation en espace
5 !c 1J4>D ( ` B hzI` B h< m] 8 "> [= '!'& -+= !5
4712(9"$>512>T3`C= > 5T9" [email protected] 5N712>l 12 >q96;Z>j 5N
jr;Z G'5 "@> !12455' !12P4c!& T! 12
/ += QK *(G.B
2` GLhM>D;W !c= >= "QK 510! w
= > 5' > 12
@ 2B 12
i !0= T 12 N>X!& 45' B
5' 12
N
> 12c1"= N
L15124H 79
9 6; > 12 5 510! = H M5' / >> / = > = "QK M9"46> 7 5 1! N
= 5L1=
` hB
471212
N
> 12
+ N
Bij;W 5
!
79
ir< >K>! 5 <!12E ! ` Q!RhN><N4>4>T! 5 N !!1
ES >D;W 5 !
e'> 9 5 D
' EN >D;W 5 ! = 4
/ += Q44G N
`
79
Nh
12
N
`
79
h ` B 2
h
Discrétisation en temps
i S
Iz ! 5X 5 ! X 5N4!0= =$ 5 K > <]m (>>^ N!&[= ^
] 1N > 12 5 (
L u7B ijZ; ""= 12< w5 (4F G 12 G %x [email protected]= >T"=5 >12w 0w!'&+= L L= "B
eQ4( ] 1! >T5q j>D; 5G !q > IA= !'& >>F 5 ! / 12 q!12 += 124j qF5 4qQH > N 0 12 E B
G
B?, !w9" q /
12 4124 >
8A > N!'& K 0
4.2.6 Stratégie de résolution
Y = "H12 [email protected] D> ;W12w!12 [email protected] $ 1J4>D (. .!12" !L 4 ] 155 ( E / = > !"
j
= >+= "= /
> >(120Iz>T[= ' = 5 5 5! >>X6;Z > "= >"= B?ky120= 12 $>( 1JK>m 120Iz>T[= 0= !K= "$`m T! P5 K P 5 ! h / 12 T>512 %
> 1& G12 "@@> < x (
55 = 9" ! >>@+= 8 >14+= N L>@!'& K5 B
1Š6+ #9 .$ -H 8=.5,:H15 *9 [email protected] 6)1 $ ",- 7$8
ir 1J4>D @4!M U [email protected] <5 (4 > ^] [email protected] 8A "5HD
h TE ` h E E TE ` h EA EA ` h
: R EA ! ` h E E N TE ` h N TE ` h N ` h ! G ` M h f|12 8 K
R h `
h 1r `
`
K
1rLD
I
I
R R
9
I
I
I
K
9
/
7
R
`
9
7
R
`
9 9
h
` B ` B
h
`
R
h
` B
h
` B
h
B
`
`
h R
R
h
R
R
R
R
N
R
R
` B h
/
/
L 5'! H @!12E !( >D;Z= 12
512"L> H 5'! [email protected]>R; 10!'&@e'> 9 >R;Z"= 12
/
0= E5 "> 12>12 `m12 J4> "h+( [email protected] < '1J4>m /
8 !5 'L= >D;Z"= [email protected]!12E !
F96; Q4U6; >T>= L> H1 12
'!` hB
/ 12
B
8 12 L12([email protected] "12
H> = ] = 4! <5 (4
4.2.7 Conservation de l’énergie avec l’approche Arlequin
12
@=4 12
h
12 [= h / 12U0= 12L>@5l05 >+= 4 8 "CD
R R h G12F512
+= ' "L> H 5! L = l0 (9 >D;v= %@!12E !
/
12EL>
%J4 8 12
9 K 9 9 R 9 9
R K51 2"L>
9 / 9
R
9 N R
` 5 @!12>> !12<Q4 G12"F(Q
e~>D;Z"= [email protected]!12E !
BA E A E E
h E A EA : E A EA B
N !"# A E : h :
` h ET`
`
/ ! / D> ;W 7g !!1'^e'> 9 '|> !12 5 8 12M>D;d= +51 >F '!5 / 4H w!12 5 + G1M.!12E !B4ky12F! ]{ / 2
1 [!124+= 12 M 512> [ !!1'+=
>D; 10!'& eL> 9"4 `{Q B "hB 12 NE= 124N> l (9( N!'& 9X12> >
!&[= !1245 8A ] L u `mf| 6 5 SD
N >D;v= @51 >@ S = 5 ! 4<!124> A e'> 9 rB
/
N
hH 12
>l0512 L>D;d= 8 12>12c
HC F#9%1$& 5 2 .'8$ - 3 *H15 3 J9 [email protected] L)1 $ ", 7$8
Ω
Ω1
2
raccord
Arlequin
J @.
B
ir w= 9" 12 H '!5 L> l (9% D D 1r
` N
N ,E F h512E .
N
H!'& 9"N12 @12ED
` B "h
N
` B
` B E}2h
h
^ 5 % ' "= `{G12 += 0= h% !& 9" ^12>
5 ! 512E> F 5'! <!124> A eL> 9"4 6B
g
ij;v= $ >D;ZT 5 "
N
5H+= Q4 D
BC h C ` h `
` B
ky12<= 4L> !12 8 [email protected]>D;v= @ <5 4L ! / 123! >! >
! BC C h `
N `BC ! C h `
/ 512CD
! h ` i (8 ' E5 !12 8 8 X <'!'&+= L u<12 > H > 12 L 8A E5
N
YqS1 EL . D D
D C
NC D
C ! N > 12(
BC `
"
"
h D
h
h
` B 2
h
9D
` B
` B h
h
hL 8 ECD
/ >D;d= 9" 12_` B 2
`BC h ` B 0 h
> *
",9$ 5 ",
h / ij;v= 9 12 ` B 0 hH 8 "CD
YqS> EM> w= 9 12 N` B "hFN` B
! ! N
`
C YqU "
! ! `BC h h C N
] 12 !12U
! N
`
N
` B
/ 12 F1J 5
12
h ! < 124D
N
h
` B T h
CD
` B "h
ij;v= 9 12 ` B E}2h 0< E
` B
h
,;W16S> 41 ' = = N!12 5 A8 [email protected]>R;d= B
Remarque 4.3
' J"$
# % #&#
*
' $# % C N
Illustration numérique
NJ 'N c5 !120Iz!12 '12cG12"=^ .>N12 4G N
[email protected][= >=^ 0c16 > / >D;Z cQ4c LG
N 7 >D; 5' '12 LG N /
!!1'+= @>D; 41!&eL> 9"43' N B|i J ^ [email protected] (5U -+= 4> ! ( ET >
L= "y = 1B2i l (9j 5*""= 0= F y> L8 E5M!12 5 8 8 M+> H]m (>>q y!'&+= 12
E
.!12 += 124%4S0 4>^6;Z
E
E
E
L 'uKB 12
8 ' >5124y> [= ] 1_= L> J 'HG12j9" >9" j 5 "q L! > ! >?`{Q B h.12 0= 5 "512 >D;v= N5
1 > 4> J ' -!12'. S5 4`{Q B 2
} hB 12
9"124F9"N> !!1'Ue'> 9 w!12 8 N>D;v= B
4.3
Conclusion
%(16
>F6;ZTE5 m] !FN4> Iz 8 NG'5 "j+ 'q !12 5F g!12(71'5 Eg>1! 0
5M= '= < >1J wJ455 = "P
= 41G12"=BMi = &10Pe'> 9 = "~
= = 2 > "
= 1"12
4>T9"+= w41J4>D N > I = !'& >> L!12" ! Iz ! / 12"5 "> 712'J4>T"%
>
12 !9 H!12E ! Iz( !.( 0L L(16 > !&[= . L= E c5 6;Z
.< x (
12r;Z
@55'[email protected] 5 F [email protected]!12E ! T !B
, U> ' U!& 45
12 <12 124< U>>T 55 124X = '9 < S L= ' E5
+= 8 >1 M= % L> F!'& K5 H4= !4= EB
%
G1 /
HC }
F#9%1$& 5 2 .'8$ - 3 *H15 3 J9 [email protected] L)1 $ ", 7$8
1
0.5
0
1
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0
0.5
1
1.5
2
2.5
x
3
3.5
4
4.5
5
0.5
0
1
0.5
0
1
0.5
0
1
0.5
0
deplacement
1
0.5
0
B # 8
! J Q * Q * 2
1.8
1.6
1.4
Energie totale
1.2
1
0.8
0.6
0.4
0.2
0
0
0.5
B }
1
1.5
2
temps
@ % 4 2.5
3
Q
3.5
! %
4
& Chapitre 5
Exemples numériques
L
L=<
L.O
L
L
L$ E54/ EDBA
/01'@! ;;:;:;;:;:;;:;:;;:;:;;:; ‚T
‚TRE
‚T o
r <N05%2P0h8T--7!$0 CCCCCC C
9
CCCCC C
CCCC?.nV
t†%p03()<jex!3>0,05%><N0h-7<O8T^Z<G%#H(+LNM - CC C
CCCCC C
CCCC?"Kn
t†%p05254 Ž0$-K*,0h8T-C}~%F^Z'! CCCCC C
CCCCC CC
CCC?"
‚TRET
‚TRETRE
‚TRET o
ƒ # [email protected] %210h-7<N0$!3-bG%>!3!3-K*ŒCCCCC C
CCCCC C
CCCC?"Ko
ƒ # @G%2108;: M <k21^Z')(<G8T!3-C*,MG!\M <Ob ')T2!3()="(8 - CCCCC CC
CCC?".‚
ƒ # @G%2108T-]}g%F^N'!kCCCCCC C
CCCCC CC
CCC?"Fq
‚T o ‚T o R E
WkT8;-Kc ')-#M '‰03( < (-.%>M C
CCCC C
CCCCC C
CCCC?FET
WkT8K6- ')(+*3%p0$(<[email protected]%>!\'Ž: %>@ @ 3! T254 -C‡h!3')-.LNM ()< CCCCC CC
CCC?FET
‚T |G
‚T |GRE
‚T |G o
5! %>@ @A-]8T-C21"#H#&%<G8T-_CCCC CCCCCC CCCCC?FEp|
r9!5%F^<I8T-]21"#b M[*w03()b '--10h="!$(')'-C8T-#&%>(<"03()-K<Cf M < [email protected]!$-K#H(Ž-Kc [email protected][-K!3*[email protected][-.2P03()"- ?FEp|
ƒ <N0 K6- ! -10\8T-'d: %>@ @G!$T254 -C‡h!3'-KLNM (<I-K<[email protected]! .6- *,-K<G21-8T-C4G%MT0$-.*ex! K6- LNM -7<[21-K* CC?FEp|
A54$53/6/ L !
;:;;:;:;;:;:;;:;:;;:;:;;:; > A$ ' #%E5 ;":;;:;:;;:;:;;:;:;;:;:;;:;:;;:; *<L
L87
(5/+EA$ 5!5
:;:;;:;:;;:;:;;:;:;;:;:;;:; *<3O
P:/#@AE$53/
:;":;;:;:;;:;:;;:;:;;:;:;;:;:;;:; *<3N
> # 3 ,-50 ' =#5> I
] 1' > 124++= 8 >1 += L 4+> +!& 45'
/ / 4G9".> F = &10 / '!'&+= 6; 1C0T 12 | > 1'& ( ? '1712= / 12"N= "H
= T4> = ""= g [` O 10(= > = ( EgQK &'(12_= ! 96;ZY[,%\jhB
ir
O ^!'& K
5 8 < "= 12 @ > G 4!N .12 8 c!12G12 " 9" 124 C8 12 @"51AI
r |>[! '[+!55[&6 5 / y! 5 8 '* |5 5y"4_= '9 ! +
= (T9" y12 455' > B
, ( < (Dw ' / 12 124(""= ' 512 3 > c] 1' > 12 > A 4<5 J4>T"=U
1JK>m @(!12" ! B L12 c= 4 12 ! >T%>R;ZE= x -5'(( J4> 12 `{12
6; ( E 127h '9">D;Z 4 !FL512X1 LH '!= 12Xq> j= > +"4_= '9 B
i 0D w 'X ^!124 != > 3] 1N > 12V!12Ew 8 5 5wS 8 ' E5 Bgi?
0= > H4[ !55 ] 1' > 12w512"H!12 0
= 6; 5 M1J 5 M H ] 1' > 124+!> I
9 `{ 1 ( = c= 2 > [email protected] B iq; 5G !
/ "!55&6 5 h } / / 0 / 7
> IA= !'& >>c 5 J71[= w S } / / T B 12 $l 0= E512 / 512 $ I
! >D' E / > 8 4F" = 9" +4X106 > 6;Z"5 ]m ! NIz 8 $F >D; 10!'& %e'> 9 L> 5 5 "L H 1J4>D [email protected]!12E ! IzT !B
5.1
Stabilisation: quel intérêt?
n 47 > 12 w9" > _] 1' > 12p> A - J4>=-G'U = += >5c> ] 1N > 12
> A N!> 9( ""= B L12 P= 124 / ! / >D;Z K ! L= ' EN <5
!55 ] 1' > 12 / 1 ( "%! >4rN J4> 12 `{12 E 12KhH'L> L7 ] 1' !
.>D; > 1'& (9H.!12E ! / r;Z L qF+> !12 5 4!%%> 512>12X"4_= '9 / 6; 5
BGky12.! ]m / 12 %!12 += '12 L5'12L0 4> CD7>N!12" [email protected] # W} / >N ] 15 "
$l (9HM>L !&0IR5 5H q.f l>1CBi? ' (
q 8A > > 5 JK>L
= %> ! 12 8 !
U>D; > 1'& 0 L= "( <5' B[i?U 0m :5 U>D;ZTE= x X U> 5 J4>T 12
G12L> L 1J4>D ( L!12E ! . S l 9" B ir5'12 ^5 8 G12<= 4>D;ZT 4
5 J4T> 12w H> !12 ! @> 12>12< = '9B
! > 5.1.1 Contact de Hertz
12
H!12 T[= 12 L>@!12E ! % ] 155 "L U 9" @ E5' 0< ( Iz4&G H10+= >= L
5l0_= 5'9B4ir 0U (Iz54&6 H12E% "9" F G12"= x 5^!12 5+= %6;Z U< x (
= ' = > 59(X14 >^ j12 N (!1 ! "NG12'512 PN B
D
,% 0([= 4> ! "MG12= / = 2 0Xq1 G12= / 12E 4>9+= j q> q 0 ]m ! [4> j q 0
> C [email protected] 123", *=.5> !,#' "
U0
σyy
Deformee
Maillage
U0
Distribution de λ
* B B
#
(Iz54&6 F A8 > ' 8A ' "F . F !'& B tjU>D;Z&"l71&6 @6; 0Iz5l = 5 /
5 >Ty 09 gF T9" y12Ej T>>= BirF >T> A F! 12G15 1 / = >= "qfHn ve /
V= > = " %e%, T = > = "%bY `m"5 ]m [email protected] !12E !h B7ir $= > = "L.!12" !512"
""= =
w1 F X= >= [email protected]
12
@> @G15 E >T>@@!12" ! B
12
12 124%' > Q B > += ] 1'_= 039 '9" 4|9> . '"=
1' > [ ] 1'! M.!12E !F
1 J45 " + F '1!&K "[> A ^` <N hBk <5H6; E 12
*>[ 12J '[6;Z= 12 F`m
>> ' / 12 = 412 y>D;Z 4 !q
H512KhM 4> = " F F G1N ! >4G1J 5 "U L 4
10!'& 4 "H> A 4B
!
B 4rB
12
B
J
J
}
J
# ! < 9"124<9"-G12U!P '1J4>m `m += ! ' E ! P5 JK> 12Kh / >R; : 2
1 Ur;Z 5 J4T> 12p`
! ! X 5 J4T> 12rByi?
h<G 9" U[= "= 1w> ^G ] 1 <5' -`m! ] B45 !12 B hF6;W12"H Hr;Z 4 4!. F!! B
(a) Sans stabilisation NO OK!
F
F
> # 3 ,-50 ' =#5> Matrice tangente non inversible
(b) Avec stabilisation OK!
B&
* !$ Remarque 5.1
& V $ $# $ * * # V " V %
# J# $* % % [email protected] E E # % $* '! V # #* %* % # ' #
!$ # ! #& $# $ % % @B # %" % $* ( 0$#V
$ $ % )! * ! $# $* * $# * % ! 5.1.2 Patin frottant en dynamique
O .! +5 5M7[q: +>R;ZE= x H%> !12G15^ 3J4>10!+= ] 1' J4><`{ P+= ] 1'
' !12< <12 8 ( E / U!12E ! ] 15
5 J4>T 12 $0= l0 (T9"B"ir.5l (3= 4 =
= 3 51.^ " c 8A E%> 12c4> h / >T<
E%H wJK>1!. `m! ] B4Q B hB
ir< / U!E1x "=
N
= 6;Z V = ' = > 59UU(1 4>$ j12 N
/ 5!12 [S
!1! "N(G12'512 N N =
N B v>g >> = S
/ %e,
Bjb 12<$!12" !S`m!'& 12w!12(< ]{ !$ !> C8 hN 5 >> = $ = >= "
= >= ".b0Y B ir ' [email protected] TE $ 51+= >T"[email protected] + H
= > = " T!F
J D
N
12
B
Bi >127 ] '155 EF M! >T>% O 12 >12NJ C8 [email protected] $!1"! EH ] 155 "
8 >T512 6 ?> Q B >R;d= 8 12>T12 ?!& 4|[+= 4> ! E / 8 5 5 / [
= ] 1!
> 1 -50123",
, " P
Ressorts
k
F
7
2
B
UT
%J 2 103
*
$
( # F!12" ! qg5 ( Iz >4>! 5 * ] 155 "g 9" >T9"
12 M.!12E !`m9 7512"M [ 15%! M M1 + X
^> Q B hBjir 0= > ^ = 5 ""= 512E^1J 5 yG12"*6;Z""= 12( _= '9FF> >> A .%> 12 %.!12" !H 9+=
"N> A 0I
] 1N > 12 K
B r
i H< x 0= > HG 8 " x [email protected] 5 H L ] 1'N4> 12U> A 5 JK>=B
Yj= " 12 [=N9N> 5 '!N' @
= l0 ([email protected] 35l05 N 8 'JK> / >@5'N J4 I
> 12<6; ! SE= x % <0= Nl (9B
5.1.3 Patch-test de Taylor
>H; A r;Z V 1J4>D (SU!12" ! $ 4 ] 155 "c`{ 9" D'[ !12( J4> C8 !+ %= >= "*Q4 GJ4T>+= e%, ^`m!
> H+= K> ! "L % $= > = "%Q4 Fzl7
`m>F12E
h E5'S 0VJ4>10!( >T>= (
] BCQ B hB L12 10!'& 12
L>D;Z"5 m] [email protected]!12" ! hH>
AB 12 8 12 $! >! > = > 512>T12
#
>4>T! 5 '3!12E !w $ = > = "wQ4 TSdl"G
^ C8 !U5 J4T> 12rB 12 !12 5 512 X9" w>X5'(U<5 J4>T 12_7'JGX> 512>T12
_= '[email protected] w '1J4>m @ L5 J4>T 12P`m! ] B4Q B hB
5.2
Formulation en vitesse
i (
q0 4> D4>D;Z ! F E5NJ
] 1' > 12< 1G12= H5 5"= H4>T !l0> 'P12 6
;Z4 J4T>> c
J >1! ' c
K
>D;ZT !3 _> J f
= 2 > " / L= "5 8 ' E5 M.> N] 1' > 12 '1712=*
D 41!&.> A L""= 12S51 > "%(4>!512S5 Iz4>T!5B
/ >D;Z !H6;Z
l0>1 B L124S5 55124
4L12$M= >T"=
0.5
> # 3 ,-50 ' =#5> 3
N2
0.4
2
0.3
1
Vx(m/s)
Ux(m)
N2
0.2
0.1
0
0
−1
0
0.5
1
t(s)
1.5
−2
2
0
0.5
1
t(s)
1.5
2
5
0
1
−2
0.5
Lambda
lambda
x 10
−4
−6
−10
B
. 2# $*
$
−0.5
N2
N7
N103
−8
0
0.5
1
t(s)
0
N2
N7
N103
−1
1.5
2
0
0.5
* %$# * # Q * * E & 1
t(s)
* #
#
U
d
st
o
ti
ns
sa lisa
i
ab
U
λ
P1
P0
st
B
T
ab av
il ec
is
at
i
. )>W
n
( on
!$ 1.5
2
W # > 1 -50123",
, " >
5.2.1 Impact entre barres
12
!12 '[= 12
/ 6; JG1' / > S! ! [= (9-6;Z ! "5c 0VJ V= > 59 +v= : 8 J 12E G12= x5'$>J Iz>J4 / 512 ( X 8 5 '5
_ (N!'& 45B|i? N > = 2 > +M5 4q1 G12= Birq: X > E5'> + 0J ' q 5+
M> 8 5 '5
> F512"U= 2 > (
> H' 8A E5 CD
i?12 %N
b0 !12N
'"= N
,%
O 1"! EL
EBKi? L '1 ' = "= ._= ! 9 L = 125'9 . 0UJ ' L512"
N < J D
@k|12T512N B
aS10 >@ j124 ir L 0wJ F512"L T>>= M L $= > = ( E # Y%e
'1!&[= H L $= >= "LJ4T>+= B
0B
i? H
'"= H1' > [email protected]!12E !.512"
12
F 0= 5 "512 H> Q B }(> F!& 4H @[= 4> ! " /48 5 5 @
= @!12" !.6;Z
G12"@X> ]m !!12" ! / 1 J 5 @ > <] 1' > 12 8 5 5B O 0= 4> N12E!12^I
0= w ! 0 12 += ] 1'
> 12 . += 4> ! ( E12EN> l0 (9^ 5NTE= 0= $ P!&[= L 'u C8 !
N N L> <
5' 5!'&12'= 2 +
> !
N
B
}
`m12 ! T ] hBKir @5 4. !
eQ4X L0= >D; (1( E+" = 9" 4 g M
> Iz!'&+= 0
"q> 4& 5.
G
/ 6;W1
+= !12>> ( E / 12 M 12 _= 2 >w ir L= > 512E12 += H L> Q ' B Z B
B!
Autres variantes de la formulation en vitesse
irXw x X5 X`m !( 0PJ ' h% 5N >= 8 $
! > <8A ' "5(M= >T"=XX> <] 1'N4> 12
8 5 5 8 !(4TE= 12 5 G
1 >>4>T!5X`{Q ' B 2hB 12 .5'12 8 12 %0= > 10!'& c! 0 1J 5 X> P] 1' > 12 8 5 5B %J45 8 12 (512$ T! >D "9
>c!'& p 8 5 5- G12"(
!12" !<715 " >%S4= 5 E5- w6;W12'!>> 124("4_= '9
'5 BEi?L0= > q
5+6; "q4>4* 10!'&L L> 512>T12(124[=L + ] 1'N4> 12(> A 0I
+9M>M <5H+M= >T 12 5* > 8 ( E rB , g15'M! gT>E 5N= 2 >| B
12
1 ^= 2 > "q5 5
= >R;ZE= 12(5 (
IzT4>!5qF> %] 1 N > 12^> A 4M 8 5 '5
8 2
` ] 1'! | j!12E ! gK>!5 7 ] 1'! ?TE= ' r04>!5 hB ir ?= > |512Ey12 += |?> Q (b) Formulation en vitesse
(θ=1)
(a) Formulation en deplacement
Newmark (β=0.3025;γ=0.6)
B}
> # 3 ,-50 ' =#5> # B $
! %
Vitesse en fonction du temps pour deux valeurs de theta
theta=1
theta=0.5
10
v(m/S)
5
0
−5
−10
0
B
1
2
3
t(s)
4
5
6
−4
x 10
*$ # V J ! * % # J > 1 -50123",
, " −3
1
x 10
Penalisation
Lagrange
0.5
d(m)
0
−0.5
−1
−1.5
0
2
4
6
t(s)
−4
x 10
10
Penalisation
Lagrange
v(m/s)
5
0
−5
−10
0
2
4
6
t(s)
@.
B
$ −4
x 10
#& $
%
B B O Ndl"GN T!5= 12c!12 5 8 > L 1 = "= .6;ZTE= 12c 35 4%N> (] 1'N4> 12
8 5 5-G "X> 4& 5--!12E ! ` J45 !P6;W12!T>> 12 $ _= '9 h $> w10 Q > = "G EH> K& [email protected]@ 8 12>|>J '%9 6
M>@[= !12>> "B
5.2.2 Impact d’un cylindre sur un bloc rigide
O 3 K> G'3 A8 >3> ] 1' > 12 1G12"=_ 3
! 31r~> 12 _ G ! 8 !12" ! 5 8 ' J4>.G "H> 4& [email protected]
;Z ! B >r; A @>D;Z !H6;Z <!l0> 'S= > 9" `{
+= ] 1' 1234> hHL4UJ4>10! ' `{5 = / K>N 4 hB7ir %12 += . = ' >T>
= 12 = 5'9 . <!l0> %12EL> H 8A "5
n l12N
,%
'"= N
CD
N < J D
@k|12T512N B
aS10 >@ j124 O 1"! EL
,.5 "M [ G
1 $J4>10!H Lr;Z :5 w > / > %!l0> H 5M512 (T. 8 5 '5
>H
EBir + 0(512>T gM> 'j' ]m ! qG15 " > > [!12" !F512E+ >> = q M
%e, $ .b0Y / 5G ! 8 ( EBGir!12E !. 5%E5 0= 0S1 LN> 12 !12E !
!
w!l0> 'B"ir F.5 4M 5H EBir M0= > F1J 5 M H15 41!& 8 !
N
`{ X125> Q B h N> <] 1N > 12 _[= 4> ! " C8 !w w5 ^
L u
N $ N `{ 2 !'&^> Q B hH12E%!12 0= B 12 H 0= E512 /
}
−4
−4
x 10
5
5
0
0
d(m)
d(m)
x 10
−5
−10
−15
0
2
4
−15
6
0
0
−10
0
8
x 10
2
4
t(s)
−20
6
−4
x 10
2
Syy(Pa)
Syy(Pa)
−2
2
0
2
4
6
t(s)
−4
x 10
−2
−4
0
2
4
t(s)
#& 0
8
x 10
0
−4
B
6
−4
x 10
−10
0
−6
4
t(s)
10
2
2
x 10
10
−20
0
−4
v(m/s)
v(m/s)
−5
−10
t(s)
> # 3 ,-50 ' =#5> −6
6
4
x 10
$ $#
$ %
W #
6
t(s)
−4
V % −4
x 10
#
V
. > 1 -50123",
, " −3
5
−3
x 10
5
−5
−10
−15
0
2
4
−15
6
2
4
t(s)
x 10
6
−5
x 10
v(m/s)
500
0
−500
0
theta=1
theta=0.5
−500
0
10
x 10
2
4
6
t(s)
0
10
x 10
−5
x 10
2
4
t(s)
6
−5
x 10
0
lambda
0
lambda
0
−5
500
v(m/s)
−5
−10
t(s)
−2
−4
−6
theta=1
theta=0.5
0
d(m)
d(m)
0
x 10
0
2
4
−4
−6
6
t(s)
−2
theta=1
theta=0.5
0
2
x 10
# B B &
4
t(s)
−5
( 512* T! >D " / > *!'& (4gF[= 4> ! " /8 5 5Mj
>N4> FJ 3!l>T B i? .[= ] 1' = 4S!l0> ' A8 E / G '69"@> H!12E5 E5 8 120Ida3 % H>@!l0>4.512EL =
%
6
−5
x 10
'"M
= 1' >MM!12E !+ G12"
"L .> 4& 5N!12E !
5 ""= . H> Q B B
5.2.3 Impact de Taylor
O .0 4>G' 5 [email protected]> ] 1' > 123 8 5 5^ .512> H 0= 5 " " .!12G15I
5 " = > 5514> 5T9" 512 ($ ] 1'! ] 155 "B >j; A N>R;Z [email protected]
f l0>16;[email protected] ' S+= ] 1' 1234> @'L S4> w' TB i J 5%512 (T5X 8 5 '5
BKb H
i?12 %N i N ,% '"= N T >
C#
N
N @k|12T512N
aS10 >@ j124 O 1"! EL
1 ' = = %512"D
<
D
> # 3 ,-50 ' =#5> t=34µs
t=22µs
t=0s
t=46µs
t=70µs
t=51µs
σ. VMIS
B
i|(5(r;d= > 5!=
N
D
B
N
#
* F CV
Q
& ( / @G "5> 125(4 !12G15 "4> 59^ 5
D
#W
[email protected]>! 8 ! ] 155 ( E / 12 .!12 += 124 -!1"! [email protected] ] 155 "
N B?i l0 I
(9 5%""= 0= ^ > ] 1'N4> 2
1 < 8 5 '[email protected] L> ] 1' > 12U!> 9 S+= K> ! "
L L u
N N `m12 "H <!&[= T ] hB
8 !N H w5 L ir M!& 4+.[= 4> ! " /08 5 '5F > K>! 5 q !12" ! k 12N>(! 8 ! ] '155
] 155 "N512"12 += N> Q B > B*y
Q ' B >R;d= 8 12>T12 !12'| @5 (4? 5 ( Iz >K>! 5 r XK .> J . ! F E / 12 @ = 5 "512 > Ey 9 >9 rG12"
] 155 (
(> 12 ((!12E !B*i += ] 1' = X> @!12"5 "5 8 2
1 IdaS N > J / ! >! > = jT 5 "q!'& 12 / 512"[ 12 [= q q> gQ B
m` ] 155 "hg B w` 8 ! ] '155 EhB
12 9"124(9" U> (!'& 4(3[= 4> ! " /q8 5 8 S> 123U!12E !
5G !5 "@J4 P> !12 124 P!12E !G' E.9(> ' 12P(!12" !1J45 " 15 ] 1 ' > 12w F4> +0= >D'@9"@! >T>.1J 5 " @ H> ] 1'N4> 12w U+= 4> ! ELH ""= 12U S5 4F L>@!&[= L 'uKB
ir j512j 'y0 4> g>T> 55 "yJ4 >D;Z""= x [H> .] 1' > 12N 8 5 '5L g>R;ZE= 12
6;Z +l0 (T9"[ = 5 !H6;ZT !BAij; C8 " A M!55 ] 1' > 12N * G1 0 ] 1'0I
> 1 -50123",
, " V0
12 34 56
t = 2 µs
t = 0s
CV # B
! %
t = 10 µs
t = 5 µs
( %J 2$# * #
W $ %
F #W
λ(N/m2)
v(m/s)
u(m)
Newmark ( β=0.49; γ=0.9)
methode proposee
t(s)
J
( .2$# $* % $# * % % * B # # B >
! ! > # 3 ,-50 ' =#5> V0
1 2 34 5 6
t= 2 µs
t= 0 s
# B
! J
( t= 10 µs
t= 5 µs
%J 2#* @ # 8 % * *
F
#W
||Λ||
Ν1
+ + Ν2
∗ ∗ Ν3
∗ ∗ Ν4
+ + Ν5
Ν6
t(s)
B
% ( % J 2#* * & $ $# *
# B & . 2# $*
! $
%
$#
%
# W &
CV
> > 5 2 1'8$& 3
0
> 124H!> T9" M L9" > H!& 4_= ! 9 U
4 8 U> [email protected]!12" [email protected] 0= 5 "5 "
4>46;W12'!>> 124 = '9 `m12c4&"l'9" hMG ".> 4& 5^6;Z [email protected]! .' !12'(
F5 5 " F0= ' > L = 'T9" B
F 5.3
Multi-échelle
O 55(5 !12- [email protected]!12 !0= P >R;Z>> 5 12S ^>R;ZE= x > 5 P!12 5(>D; 5G ! > I
= !'& >T> > . 1J4>D ( @(!12" ! Iz !B*,% 0cG12" 512"c= !> 0= SDr>(10G >(6;Z"5 ]m !
> Iz 8 wL>D; '1!& eL'> 9" TX 3l0 (9B
5.3.1 Modèle multi-niveau
, (!U A 4& / 2
1 8 124((><106 >Sr;ZE5 ]{ !U > Iz4 8 6B L124(!12 += '12
>D;Z( [email protected];Z J4>><`{ [
= ] 1' 12 Iz5l0 = 5'9(< x X 1 = "= = ! 9 9" >
!l0> ' / 0= 5 ""
= = !4= ( "h%% SK> U' X`{Q B }2hBre 5 C8 ' !. (4> / 12
12"512 @>R; [email protected] P10G >( '1712=- >D;Z""= 12 r
;Z l0 (9^ P 0= 5 4!X6;Z( !B
ir ^0= ' > (124[= ^ (!U10G ><512E(!12( = c ! 0 12 += ^ (>$106 >U
Sb0 1' `{Q ' B hB
ir H <5 % <10G >6;ZTE5 ]m !X`{G12H
N
N
HN
<
N
>> 6
; 5M= "= hF512ECD
U 5.3.2 Modélisation par l’approche Arlequin
ky12>T> 55 >D;ZTE= x N >D; '1!&eL'> 9" TP >(5 5 "
l0 (9 / 12 H !12 += '12 H>D;W0 4> @>R;Z [email protected] 0
l >1 B
12
'1J4>m @X!12E !
8 124 8 (9H> j!& 4[_= ! 9 8 12T A L H> 12L!12E
>1FJK5 "[ ] 15 8A ' 12 B0i? F >> A 12''F%G'55 "F
! 8A ' 124U`{Q B ` h5hB L1241 512 G12 w(1+= >T 12V 12
Q
! +L> J ' HHf l"I
+512 F5 >T+ .! 45
(>D; 4 10!'&Ue'> 9 6B
'GG12512 F U(16 >NQ4 4
1 ^!12" ! / S16 > 12' B4i?
/ 8 12T A > 2
0(16 > +512"[ !!1'[= M M>D; 10!'& e'> 9 > 12T @
12 M!'9 %`m>LG12 q 5M(
> # 3 ,-50 ' =#5> # B @
B }
!$ $
W
* ( #
%
! −3
0
x 10
600
Signorini
multi−niveau
400
v(m/s)
d(m)
−2
−4
200
0
−200
−6
Signorini
multi−niveau
−400
−8
0
2
4
t(s)
−600
6
0
2
4
t(s)
−5
x 10
6
−5
x 10
9
10
x 10
3
x 10
0
2
1
−4
a(m/s2)
lambda
−2
−6
−8
−1
−10
−2
−12
0
2
4
t(s)
B 0
#
−3
6
0
2
x 10
@
4
t(s)
−5
! *V$ * #
6
−5
x 10
#& W > Š # C3 ,9)95 3 T
Zone de raccord
(b)
(a)
B
(d)
(c)
# * &#&
+>(16 >%>10! > Q4KhB L124qG'55124 0 (16 > + 0= !K= " / 512+ +!12G15 "
> ^` = > 5514> 59 h^`{Q ' B `mC! h5h%12 !12G15 "@ L= E Dr! 12G15 "
= > 5514> 59FG12g>H10G >FQ4q!12G15 "^= > 5T9"FG12g>F106 > 12'L`{Q ' B `mKh5hB L12 8 12 M>H!&12 (6;ZG125j>H!12E !F512 0^(16 > [512q4 9 " X
10G > Q4 `m!&12 U> 4>4+TE= E / U T9" hB
ir (= > 1J 5 "4( (! X1+= >T 12 3`m[= ] 1' 12 4
> 59<! N4>= h^512"!12 0= P
! 01J 5 [ q 1+= >T 12((121AIz10G >U= > 514> 59H q >> A 12'%`{Q B S` h5hFL < T >> A Q4 {` Q B S
`{J7h5hF F5125N> J B
i (1+= >T 12 U
>D; 4 10!'&Pe'> 9 `{Q B `m!ChX `mKh5h$12 P> << x ( U0= > 96;Z (10+= > 12 12 1AIz16 >V= > 5514> 59
T>> A Q4_ 5125 w> J B %
@5 C8 " A 5H9"r;W >>.G'F 0= > 12J4.51 >? F,.,.i / 512 F <G'5 "H 0J4>@
= N1+= >T 126DK12UG L!? A 9"N>N106 > P= > 59 _= '9
0= IR5 ^ D> ;Z( !B .<0= 4 6 12U5 > ( EL> 5 4H !& / H 512H>@5 (4
&12((B
5 5.4
Exemples industriels
e .> L5 ! += 9" / 12 L 12 TE= 12 0U (4> % 5' >B 12 .!12 += '12
0S! .9 r12" 1 8 = %>> 4! E.!55&6 B v>|; A N> +
= ! "5( G > [JG 4 [+>%!12E ! IzT ! w5 $ +> 5124g E5'.> [! l12 F%!12J445J4> [M>
> # 3 ,-50 ' =#5> '>T> M " 6B
5.4.1 Grappe de commande
O ! 455' 7
> G'H6;v= 4L> L4&[= 12< %@!12" ! Iz !.>1'Hr;Z @+= ! "5r;Z GHH!12 H 4g ^JG @`{Q B hB"i? ] 1'N4> 12 *!> '9" * 1 ( = L = 5 "5 E 12'!>> 124 = '9 .G1 "5 . .!'& (4 8 5 '5 .
!12" !BEij;ZT> 12[> L] 1N > [email protected] 8 5 5[7|M> |Q4>5 / 12 " T ] 1! y [
F ] 1' 12 +4> M5 JK> MG12F>D;d= 8 > 12U >D;Z '. ! L!12G12 "H 4 55' >TBEe 5
6;Z>T> 55 12 / 12 8 ' >5124j +!'& K[ = ! T9" + $9" >9" jG12"q
@!12" !%r;Z [email protected]@!12( X
`{Q B hB
12 +G15 E >>
5.4.2 Crayon de combustible et grille de maintien : une première perspective
, H!55N 1J4> = T9" / > H4&[= 12< @!12E ! IzT [email protected] ] 55 " U
512 "543! l 12PN!12NJ4 J4>@.> ' >> L " 6B Yq= E.12 +=^>! = !'& >T>X 5 !X 5 4X!$ 1J4>D ( / 124 8 124>=X 0 10G >
8 < L> I
! > I
= >= E^QK L= "B2ir10! > " 2
6;Z '>T>[H E / 12 8 12 q>=F +10+= > 12
/ 8 12T A F
Yq\ ,B*ir12 !55 2
1 X!'9" / 12 C8 12 ^>T"=( 10+= > 12P P= > = ( EQ4 [email protected]
G125q qfH12'!'& u1B ir ? [email protected] > r12"@= "M
= !!1'+= / ? @ !!1' ]m !9 ,IR712 5
B, M!& "++= '10 9 g12"(= "= 4>T9"+= ! l12 B 12 >l0512 q>R;d= 8 12>T12$
!'& 4^!+= 9 V9" >9" @G12"N >T> A , V!12E !G15 " > / 12 [= ^ ^>
] 1'N4> 12 % -[= 4> ! " @> $] 1'N4> 123 8 5 '5B O .= > 12E 4= 5 E= N > Q '
B0 B
O 12"5 ( EX > w] 1' > 12c P+= 4> ! (
E / ! >T>^
H> 12!12" ! F12!>T> 12 F = '9 B
8 5 512 !& 4 8 5 '5
[email protected]! H4 55' > G1 " w x G12 M6; C8 12 F F!12(4>m5 "HK>12"@
= > M0= > _= '9 / > 12 +5 N4
J >L9%>L !!1 8 12>T (9 / 4&[= [email protected] > = &10.eL'> 9" T / 8 G'55^6;v= 8 5 >4 = A ( 124 & 5 ] 0= 9 ! . %> 12T !"= B O G12"
:
5w= !> N
= L> ' D'@5 !12U9 H4B
5.4.3 Intérêt de l’approche Arlequin en présence de hautes fréquences
ir [ 1JK>m [ 8 J 124q512 [!12E ! = 6 "H [= G12 5 [5 M'!'& j ] 0= 9
ij;Z > [email protected] y !! 1' *!> '9 +` N
8 0
yTE5 ]m ! h ] 12Ey4 = g> *& 5
!
` # ZB \+B hB
= 9" 4!
]
> Š # C3 ,9)95 3 t=0s
B
t=0.3s
%$ E % t=0.9s
t=0.6s
#&#
"
t=1.2s
!
% t=1.4s
# 8 * }
> # 3 ,-50 ' =#5> DX
0.5
0
−0.5
0
0.5
1
1.5
0
0.5
1
1.5
0
0.5
1
1.5
0x 107
0.5
1
1.5
DY
0
−10
−20
VX
5
0
−5
VY
−9.5
−10
−10.5
VMIS
4
Formulation en vitesse
Newmark beta=0.4,gamma=0.7
2
0
0
0.5
1
1.5
t(s)
B # % # * $#
$ "# *
$
&%
$
* * #W C V E
> Š # C3 ,9)95 3 liaison crayon bossette
liaison crayon ressort
B0
N ( &% #
! )! ' # > # 3 ,-50 ' =#5> <> <16 > wQ4 <10 Q " %> 512> 12 0 8 12T A <
12 U != B%ky12
>>T 55 j!4&[= 12< / 12 [!12 += '12 [4 1J4>D (10G >r;Z HJ 'N`{Q '
B hg512 5
L%5 +50= "= H0! 12$'!'& L $& 5 ] 0= 9 ! `m! ] B / hB0e% 8 12T A 12
$!55
/ 12 >(
106 > XQ4 `{G12! 5> # BZ\+B h !!1[$
= K> @>12T _10G >
= 5 "512 (> ^= > 1J45 " ^ ( 10!'& w!> 9wG12 "^'(
12' B L124 0
!!1'- ]{ !T9"w`{ !!1'-G12 ! >| %15! hB O %0= > %(12E5 " > %>(5 ^!
'1!& B4Y[< 7 / S > "F < !!1S ]{ !9 / > # ZB \+B7512EL4= = % L>@(16 >NQ4
`{Q ' B
`m!Ch5hBKY[< 8A !'& / x !w 4< !!1'3eL'> 9" T-`{
8 12> Eh / 1 7 "L> G12'J4>"=
"
"
A > (& 5 ] = 9" ! X>1'(S> 5 (T12 8 12>T _`{Q B
`{JGh5h / > # BZ\+B
512"LQ4>5= 8 2
1 % '! 12U>@10G >QK6B , L>@! 0= 5 " / 12 8 124% >=(`m > 12$( !!1'[email protected]
!'&+= c%
!'&+= LL uP!12 5 8A ] G12X # # f >T> ' Q4>5'L> HJ 5 ] 0= 9 ! BGir L= > ^`{Q B
`mKh5hH12"5 E. 5 ('12 A '1 @>D;ZT ] 1 12w U106 > Q4 U10G > 12 B
(a) Deplacement initial
modele fin
(c) Couplage ponctuel
modele grossier
Raccord ponctuel
modele
fin
modele grossier
Raccord
Arlequin
(b) Couplage de modeles
(d) Raccord Arlequin
B
%. * "
$
$ > > *
",9$ 5 ",
5.5
Conclusion
ir ^0 4> = 'T9" 0= E= ^!w!& 4
5'w 12 ^5 JK> E_= !> '(> GT !w
L= "[G12"qTE510 j +>! % %!%5 A8 >*`{M= 7124 E >R;W0 4!%.9 >
= hB,; 5' [0 4> q 4>T! 12 jT 55 >> gT> "q> ] 1N > 12 j41G12"=
12"w= "N
= 0= >T"= H12U512"H U!12 'L 0
= T> 126B
>
> # 3 ,-50 ' =#5> / /A
& H H /
>0
Conclusions et perspectives
O 1! ( EH L @!12"5'J4 12 S! >T! >r _= '[email protected] L 1J4>D L!12" ! Iz !B 12
l 8 12 JG1'[
= > MG12"H9 6 8 "B
B e
M6;Z S= !'% ?= 9 124 F>12+>10! > M !12E ! / J >m M!12" ! = @
= += ' 8 = B O 55
J4>=G12[> [ 1K
w!> 9$(> A 4
= += >T5X> ] 1' > 124> A ] 1' > 12X> A 0I
] 1' > 12G'+
E= / !> 'Q "
T G>D;Z""= x H M <5 L6; " 12<H 4Q "[> F(4>= " 12 M"4_= '9 B
B %
] 12 > 12S!12E / &ElJ '
/]{ JK>I ] 1'5 / 5K= @ F5 C8 SaS1 3 3
1JK>m r6;Z( ! / = "[
= '1712=B Yq>>y'712jr = !'j |= 9 12
"= 0.
>12 $b0 1 IdaS1 / 1 l
"N>D;Z A (!'& 4 ( Bri += ' 8A 12_
= > = ( Ej M!12" ! Iz ! 7(y[ >>*> * !4>"= yr;ZE= 12 5 4 5 !
] 1N > 12 F!> '9" B
]m (>>w w(16 > XU!12" !X > Iz 8 <zl73>1! > >1J > /*] 12 [= S(> 0I
B %
GG1212_r;Z V16 >w Ub 1'4% V10G ><<!124> ! g
= E5'1 45B|i?
/ = c
4 ('N5 5 = '9 ^12"5 "^9w!<16 >$G'( $! T 4 10!'&<w> 0= >T"= 4&El09% X!12" ! / 512+ 5"= " "L Q4! 8 ( EM = ! >+ _= '9 M
dl"[email protected]'!>> 124[ 5 L12U 8A F!12 12 "B
BLiq; 10!'& Ne'> 9 = "= 44>9[= 0U41J4>D %N4> I = !'& >T> `{ 3 5 ! h!12E !
P l 9" BKi J4T>"
= !55 '1!&^. E= x . G1 '1!&
> "H !!1' % ]{ !9 F12"U= = = lj= B O 512 H (4>512 8A " A 3(]m 5 5 124 @ "5X(16 > 6= !& >> ^ L= " / @K= >D;v= 8 _
(16 > 6= !& >> *QK r | 12 ? 5 / ! 5G [email protected]
= "*TG1 E?712 |> 'N4> 12
= 'T9"@ H55' ! H512 L !B
>
/ \A
& H /
]/
,; 5 712TE / 12 JG1'+= (!U10! " /j 5 "(!12(S ^75G ! 8 B 12
"12 124M512H T ! >D "CD
L> 4'5X
!12 5<<>D; 5G !^N > I = !'& >>$ V5 (4^ ^>w5 5 " ^ 1JK>m r;Z !CD> 12<`m q>L! %eL'> 9" TKh|!&[= M6;W1''F7 V= >= 8 = 0 8 12 T A 12 F6;Z ![H6; 5 F6;W1''% += ' M>12$ !
12 F!'9" / = 8 "H> I
G''12<"4_= '9B
L>R;Z
A (( = &10 @ zl7(k+aSi
40124 / F>@! e'> 9 6B
L>R;W5
712 @ c5 5 " 8A
!KX
= 5 '12S L[= 8 >1 G "L0= >T"= j: 96; 0= E E5> L12>
41J4>D M.!12E ! IzT !H "5 55' ! M&[= "= '1 :5 ! ] H6;Z &G 5 U!12 'B
124 4.I
Iz,
12 Iz,
B O G12"[!12 D> ;Z X +1JI
H
>T
Annexe 1
Impact de deux barres
, !55 0 / 124([= >>124(>U! >! >LU> 512>T12 >l9U V '1J4>m S6;ZT !
"5 0XJ (= > 5 9 MF "9 / 512 ( 8 5 5 F > j14712= H $ M
= 2 > N P14 > BGi 0
= 512>T12P 5 ]m 5 @4 10!'& ^ ] 12 [email protected] @> 1 A2 12 12 B
Y = ".12 +=N> 5l0_= 5'^ < 1JK>m N!12 '[= = /
j
5 > J @ H w4> w' (`{Q B T hB
12 H0= 512> 8 12 > 1JK>m >D;Z !Hr;Z V0
g0
A
L,S,ρ,E
x
O
12
B T
# B @ ! F 10!4= 12 > = 512>[email protected][email protected] '1J4>m <= G
Yq G
Yq G
D
8 12>r>J D05125> J
D > J ' 5H U!12"
4
%
D
5 4 8 5 5
5L12 (
!%75 "B
C$#AB
Yq G 4
D > J '@[= !12>>B4Y[>> 6;W 5L4> + U!12" !B
b12>D;d= 9" 12c4S12 8 ( E%6;Z N712 [email protected] U5 !120Iz!12 ' 12 / 512 (T5$ ] 1'!
! >= UG12"MeNB
>1AI
>
! b12 `
h?> 5 N`m! ] B T 9
h D
` 5h
E Gh*N
` Gh !12&
` B h
> !M ] 1'_= HFi 4
`
1r
N
!
R
`
R
' &
Gh
` hB2i 5 4 ] 1_= HFi 4
> !FF> 0= G12 5M 5j 12 [=
` B h
5> ^8 5 [email protected] '1 A2 12S F12 H @!12 '12 / 12 += CD
N
` B h
YqU ! >T / > ^8 5 [email protected] <G12"
E 5h5h
` `
N ` Gh
N R YqU712 E
E 5h5h
` `
N
` B "h
/ 12<1J4 E!D
` Gh ! ` B h
Yq ]{ EH <+= 8 >1 G E%>("= E 5h5h
` `
i 5 > ! D
] 1' = @i 4
!1&
L12 [email protected] S5 N ` Gh
`
h
!
/ 12 M1J45 12
> !T 8 '@12 HD
] 1' = @i 4
` E 5hJN ` h
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