Contribution à la modélisation mécanique et numérique des problèmes de contact-impact.
148
0
0
Texte intégral
(2)
(3)
(4) " !# $ &%' ( )* &%' / ! 0 1 ! ,23 + 1! + , " - . 4'57698;:=<>@?BACA?BD,E FGFHFGFGFHFGFHFGFGI.
(5)
(6) Remerciements
(7)
(8) #% !'&
(9) (*)+ -,.&
(10) 0/ 12 3 4 !5 .
(11) ! " $ &6 5 /7 8 !9"
(12) 0:; 6 3
(13) <0= >?4> @ 5 8A >>CB4 > ' '!@ !
(14) >D ' EFG12 H
(15) 5G1AI
(16) JK>= / 5 L!12
(17) > ML> 9 >"N = @ 512 <
(18) !
(19) "B O 553&G 5 = "P = !1AIRQ4
(20) !4= 3 $ >S(T
(21) 5 U 4
(22) V 3>D;W
(23) 5
(24) EX M= '
(25) ( $>
(26) ' !12 < !& '!'&
(27) N 6;ZY[,%\+BK2
(28) N > H>
(29) H 512 <QK
(30) ! B O 5 8A > = "X = [= / > ] 12 /G -> JG1 512^ ^_ = !
(31) 9 U`Dacb
(32) bIdaSegfh ^ij; Y[= !12> O E5 > k ' g (+ = 5 ( EMe >l j_ = !
(33) 9 j+e!12 4 59 `me%a3e hg L>
(34) 8 12 ^no@,p 6;ZYq,.\+B ^ 4 ('! !'& ! 4 P > ' % NJ
(35) Br^ ('! !
(36) >Tm ".\
(37) *!s 12 .tg12>
(38) 12 / k 5'!'u<a T / b14&
(39) .i '!& @L>D;v = 9
(40) G@e% 55HG12 H> !12 Q
(41) ! 9 6;Z> F<;W12 "H = 12 += B %
(42) . ' '! 8 8 EMkqe%> [Ma3!& >72 w9"
(43) 412 EM!12
(44) !0@ = 4 (5 4 @ >D;W . @!@5 8A >6 !!
(45) = 6;W Px 5> F 471'5 ' B ;W0
(46) 'T<= 2 > "M512 5 ' !12
(47)
(48) ! aST!'& >7ky15Iz\ 'lX9
(49) !!
(50) "@ = L4 = M> 512 5
(51) ! @ &G 5B aS H
(52) !A H '! ( E H ; 5 "w= 2 > ,%
(53) Fe% J
(54) l(^ k 5!uUa wG12 8 12 = ]{ @
(55) @12 : lw &G 5B !!
(56) ^ Yq
(57) Q4 / :5[' '![512
(58) |> | NJ
(59) y +>D;d = 9"
(60) 7Fb5'
(61) !
(62) q
(63) > JG1 512Macb4b"IdaSegf_* [>D;v = 9
(64) G fL} 4 < [= 5 "e.aSe~G12 H>
(65) FJ712 4 &"
(66) ( B.
(67)
(68)
(69)
(70). . ` Table des matieres ´ ´ ´ 1 Formalismes et schemas d’integration du contact en dynamique. Un bref etat de l’art. 11. 2 Contact entre solides : du local au global. 37. " !$#&%')(+*,# -.*/-10*325476- #&%"*98;: ( <"0 76- ="!3%>0$( <?0$-7# @A !3-7' ')-BCCCCC CCCCCCC DFE G !3# %')(+*$#H-.*/-7<I#JK6- 2K%>< (+LNM - 8 MO27 <N03%"2P0QCCCCCCC CCCCCCC DFE RE ST254/76- #&%*/8;: ( <"0 76- ="!3%>0$( <&03-7# @A !3-7' ')-UC CCCCCCC CCCCCCC D.V REXWY76- 0$4 T8T-.*/<ZM #J76- !3( LNM -.*@[ "M !\')-]27 <N03%"2P0\-7<O8T^Z<G%#H(+LNM -_CCCC CCCCCCC D.` RET a<I@G!$ "b 'd-7c # - 8 -C! 76- e.76- !3-7<[21-gf ()# @G%"2P0h8T-]8T-KMTijb[%>!3!$-.*hCCC CCCCCCC D.` RETRE G !3# %')(+*$#H-C8TMk27 <N03%"2P0\@G%>!/@/76- <G%')(+*$%>0$( <j-10h( <"0 76- ="!3%>0$( <l-K<I0$-7# @G*HCCCCCC mEn RET o G !3# %')(+*$#H-'+%>="!3%< =( -7<?8 MO27 <N03%"2P0\-10h( <N0 K6- =!5%p03() "<l-7<I0$-K# @G*U CCCCCCC mENq o r9 <[21' MG*,( < CCCC CCCCCC CCCCCCC CCCCCCC o"s. ET tu "*$()0$( <O8TMO@ !$ "b 'd-7c # - -10\< >05%p03() "<G*vCCC CCCCCCC CCCCCCC o"V ETRE r9( <76- #&%>0$(+L"MG-8 -K*\="!3%<G8T-K*903!3%<G*wex "!$#&%>0$( <G*yC CCCCCCC CCCCCCC o"` ET o z 6 LNMG%>0$( <G*\8{:w.6- LNM ()' ( b !$-?CC CCCCCC CCCCCCC CCCCCCC v|ZE ET o z 6 L"M[%p0$( <[*/') T27%')-.*/8;:wK6- LNM ( ')( b !3-mCC CCCCCCC CCCCCCC v|ZE ET o RE t!$( <G21( @A-8 -K*h}~!3%Fp%MTil( !,03M -7'+* CC CCCCCCC CCCCCCC v|No ET | { "(8T-C21 "# @[ "!,03-7# -7<N0CC CCCCCC CCCCCCC CCCCCCC v|"| ETR <N0$-K!3%"2P0$( <[*\8T-C21 <N05%2P0jC CCCCCC CCCCCCC CCCCCCC v|Z ETRT t%!3%#J16- 0$!3(+*$%>0$( <&-10h!3-7@{-7c !3-K*9') T2K%>MTiOC CCCCCCC CCCCCCC v|Z ETRTRE @ @G%>!3( -7# -7<N0CC CCCCCC CCCCCCC CCCCCCC v|Tq ETRT o r9 "<G8T()0$( <k8T-< <O( <N0$-7!3@/K6- </76- 0$!5%p0$( <&-70hZ(03-K*3*,-.*/!$-K' %>0$( -K*C CCCCCCC v|NV ETRT | t!$( <G21( @A-8 -C'd: %210$( <I-108T-C'+%H! .6- %210$( < CCCCCCC CCCCCCC v|N` ETRTR (+*\8T-]21 "<"05%210/-70h8T-ex!$ 0,03-7# -7<N0C CCCCCCC CCCCCCC v|N` ET sD "!$# -.*/') T27%')-.*?K6- LNM ( p%>' -7<N0$-.*/8T-K*/' (+*\8T-]21 "<N03%210\-108T-ex!3 >0$0$-7# -K<"0C CCCCCCC m"E ET s t!$-K#H(-Kc !$-.*&.6- 21!3(03M !3-K*&K6- LNM ( p%>' -7<N0$-.*\CC CCCCCCC CCCCCCC m"E ET s RE -7MTiT(d-7c # -.*?K6- 27!$()0$M !3-K*l.6- L"MG()p%>' -7<N03-K* C CCCCCCC CCCCCCC mo ET s o G !3#M '+%p03() "<j'+%>="!3%< =( -7< <G-8TMO@ !3 b '-Kc #H-8T^Z<G%#H(+LNM -C8T-]21 "<N03%210/ex!$ 0,03%<N0CCC ms ETq "!$#M '+%p0$( <O*,03%>bG()' ( * K6- -fZM < ()[27%>0$( <k8T-.*ex !3# %')(+*$#H-.*/' %=!5%>< ="()-K<G*C CCCCCCC mV ET V r9 <[21' MG*,( < CCCC CCCCCC CCCCCCC CCCCCCC s"n.
(71) {H
(72)
(73) ` ´ 3 Formulation hybride faible-forte pour les problemes d’impact et discretisations. 61. ´ ` 4 Approches multi-echelles pour les problemes de contact. 87. o " !$#M '+%p0$( <I4Z^Zb !$(+8T- e %>( b ' -ex !$0$-C21 "<N0$( <NMG-DC CCCCCCC CCCCCCC sNE o ()# ()0$-K*8TMO#H T8;-7c ' -8 -CST()="< !3()<G(;-7<k8T^Z<G%># (+LNM -?CCCC CCCCCCC sNE o RE (+*\8T-]21 "<"05%210\8 -CST()="< !3()<G( Wk !3-K%M CCCCCCC CCCCCCC sNE o o z 6 21!3(03M !$--K< .6- LNMG%p03() "<G*/8T-K*\' (+*\8T-]SZ( =< "!$( < ( W !3-K%>M CCC CCCCCCC sN o | G !3#M '+%p03() "<?e %()b ' -ex !$0$-C8TMO@ !3 bG'd-Kc # - 8T-C27 <N03%"2P0h8T^Z<G%># (+LNM -CCCCCCC s"s o R t!$(+*,--K<I27 # @T0$-]8 Mjex!3 >0$0$-K#H-K<N0yCC CCCCCCC CCCCCCC s"V o s IK6- *$M '05%p05*\8T-C21 <[*,-K!$p%p03() "< CCCC CCCCCCC CCCCCCC q>n o REXh@G@ !$ FiT( #&%p0$( <[*/-10h! K6- *$ ' MT0$( <?<NMG#J76- !3( LNM -.* C CCCCCCC CCCCCCC q>o o RET G !3#M '+%p03() "<I*$-7# ( 8T(+*321! 16- 03( *T76- -8TMO@ !$ "b 'd-7c # -8T-]21 "<"05%210\8 ^N<[%># ( LNM - CCCCC q>o o RETRE ( *321!G76- 0$(+*$%>0$( <G*-7<I-K*$@G%"21-]CCCC CCCCCCC CCCCCCC q o RET o ( *321!G76- 0$(+*$%>0$( <j8 Mj03-7!3# -8 -]*w05%>b ( ')(+*3%p0$( <DCCCCCC CCCCCCC VG o o SN03!3%>0 K6- =( - 8T-C!G.6- *, "')MT03() "< C CCCCCC CCCCCCC CCCCCCC VNE o | r9 <[21' MG*,( < CCCC CCCCCC CCCCCCC CCCCCCC VN. |G Wk T8;-Kc ')-C8;: ( <"03-7!$e %27- #M ')0$(< ( -.%>M CCCC CCCCCCC CCCCCCC V"V |G Wk 0$( p%p0$( < CC CCCCCC CCCCCCC CCCCCCC V"V |G RE Wk T8;-7c ' - =76- "#J16- 03!$(+LNM -8T-.*\()<N03-7!$e %21-.* C CCCCCCC CCCCCCC V"V |G o z 6 L"M[%p0$( <[*/') T27%')-.*/8TMO#H T8;-7c ' -8{: ( <N0$-7!$e %27-#M ')0$( <G()"-K%>MCC CCCCCCC `"n |G | G !3#M '+%p03() "<?e %()b ' - ex !$0$-C8TMO@ !3 bG'd-Kc # - 8T-C27 <N03%"2P0\#M ')0$( < ()"-K%M CCCCCC `G |G R t!$-K#H( -7!5*! K6- *$M '05%p05*9<ZM #K6- !$(+LNM -K*9CC CCCCCCC CCCCCCC `NE |GREX : %>@G@ !$ T254 -C!$' -KLNM ( <j@A MG!h')-.*/@ !3 b 'd-7c # -.*8 -27 <N03%"2P0 CCCCC CCCCCCC `| |GRET OK6- 27-K*3*,()0 6- 8T-.*\%>@ @ !3 T254 -K*#M '03( ".6- 254 -7' ')-.*QCCCCCCC CCCCCCC `| |GRETRE @[-K! 27 MO*$M !\'+%H#J16- 0$4G Z8 -!$' -KLNM ( <XC CCCCCCC CCCCCCC `N |GRET o : %@ @ !3 Z254G- !$' -KLNM ( <O-7<k8T^Z<G%># (+LNM -OC CCCCCCC CCCCCCC `"s |GRET | : %@ @ !3 Z254G- !$' -KLNM ( <O@[ "M !\')-.*\@ !3 b '-Kc #H-.*98 -]21 <N05%2P0\-K<O8 ^N<[%># ( LNM -XCCCC UKn"n |GRETR ( *321!G76- 0$(+*$%>0$( <G*/8TMO@ !3 b '-Kc #H- # T8 -7c ' - CCCCCCC CCCCCCC UKnNE |GRET s SN03!3%>0 K6- =( - 8 -C! .6- *, "')M 0$( < CCCCC CCCCCCC CCCCCCC UKnNE |GRETq r9 "<G*,-K!$p%>0$( <j8 -C'd:w76- <G-7!3=( - %F"-K2'd: %>@ @G!$ T254 -h!3')-.LNM ()< CCC CCCCCCC UKn"o |G o r9 <[21' MG*,( < CCCC CCCCCC CCCCCCC CCCCCCC UKnN. ´ 5 Exemples numeriques. 107. T mSN05%>b ( ')(+*3%p0$( <fGLNM -7';( <N0 K6- ! -7 0
(74) CCCCCC CCCCCCC CCCCCCC UKn"V T r9 "<"05%210\8 - h-7!$0J CCCCCC CCCCCCC CCCCCCC UKn"V T RE t%>0$( <?ex!3 >0$03%><N0\-7<k8T^Z<G%# ( LNM - CC CCCCCCC CCCCCCC U.n T o t%>03254 0$-.*w0\8T-]}g%F^N' ! CCCCC CCCCCCC CCCCCCC U" TREX "!$#M '+%p0$( <I-7<OZ()0$-K*3*$- C CCCCCC CCCCCCC CCCCCCC U" TRET # @G%"2P0h-7<N03!$-bG%!$!3-K*CCCCC CCCCCCC CCCCCCC U.o TRETRE # @G%"2P08;: M <k21^Z' ()<G8 !$-C*$M !\M <Ob ' Z2!3( =(+8T- CCCCCC CCCCCCC UF.
(75)
(76)
(77) T RET o # @G%"2P08T-]}g%F^Z') "!k CCCCCC CCCCCCC CCCCCCC Upq T oDWkM ')0$(K6- 254 -K')' -vCCCC CCCCCC CCCCCCC CCCCCCC U.E T o Wk T8;-7c ' - #M ')0$(< ( -.%>M CCCCC CCCCCCC CCCCCCC U.E T o RE Wk T876- ' ( *3%p03() "<?@G%!\'d: %>@ @G!$ T254 -h!3')-.LNM ()< CCCCCCC CCCCCCC U.E T | z iT-K#H@G')-.*/()<G8 MG*w03!$( -7'+* CC CCCCCC CCCCCCC CCCCCCC U.Eo T |G !5%>@ @A-C8T-]21 "# # %<G8T- CCCCC CCCCCCC CCCCCCC U.E>| T |GRE r9!5%F^ "<j8T-C21 "#b M[*w03()b ' -C-10h="!$( ' ')-8T-C#&%>( <N0$( -7<Cf M <G- @G!$-K#H(-Kc !$- @[-K!3*$@A-K2P03()"- CCC U.E>| T |G o <N0 76- ! -1 0\8T-'d: %>@G@ !$ T254 -C!$' -KLNM ( <j-K<O@ ! K6- *$-7<G27- 8T-4G%MT0$-.*/ex! K6- LNM -K<G21-.*CCCCCC U.E>| TR r9 <[21' MG*,( < CCCC CCCCCC CCCCCCC CCCCCCC U.E` .
(78) }. {H
(79)
(80).
(81) O/ ]. Introduction ir 4&+ = 12< N !12 " ! Iz ! ^ 512 "@12(
(82)
(83) = 5 " @
(84) @> 8 = ! 9" 1T
(85) P (!& !
(86) / = 2 > " 4 F
(87) 8 ' F 5 !5
(88) ' H
(89) =
(90) @! 8 >712 UT
(91)
(92)
(93) 55 >DB
(94) > F 5@ 4 ] 5 E / F0 4> / >1' > '!& /
(95) S12
(96) > A N 8 = &
(97) !
(98) > / LT E5 !12
(99) L 512> L ] 12
(100) 12
(101) /
(102) L = ( / F41!K = + = % @( ] 1 / y:512 " H T
(103) M12 U
(104) !1' H! '&
(105) H 8 = &
(106) !
(107) > F!ABBB ky12 ^> ^! > !
(108) > / > 41J4>D X!12 " ! Iz !
(109) = 5 "5 E^4> 4 '
(110) !
(111) >" = N
(112) &+= "5 U > ! ! < 12 Iz> += / 0=
(113) >T' ($N 4> I = !'& >T> / '
(114) "X> X = 512> 12 >l9 512 8 "@G12 'J4> ! >>^
(115) _= '9 '
(116) BGi ! 12(
(117) ( = !& '!& ' % =
(118) ' = !
(119) !
(120) ;W 5 ""= = 4m E5 4 5< > 0 = 512>T 12 - !^zl7 ^
(121) '1J4>m .
(122) " > + '
(123) D ' + + = ! 4
(124) B0ky12 F M 512> 12 4 >l9 [12 $ ( I >l9 / 12 XG12 ' !12
(125) 4>5 >D;W12 8 A (12&4
(126) 512 W} G12 >^!12 E ! N P 5 9 ^12 P9" ' Iz 5 9 / ! >
(127) * %12>
(128) '(& 712
(129) N>D;Z( !B?ij;ZTG12 J4T>" = ! > !
(130) > >l"9 " 512>
(131) 12
(132) 0 41J4>D !12 " ! Iz ! / 6;Z
(133)
(134) $ (X>U [= 8 >1
(135) G "X S = &
(136) 1 X 6;W12
(137) >
(138) _= '9 / 6;
(139) 5 / 2 1 "g!12
(140)
(141)
(142) ! >!
(143) >
(144) 41!&[F = F! *
(145) 1J4>D . B% F>5=
(146) J712 4 E5 = "L = !12
(147) !0= 0 ] 1'
(148) > 12
(149) /0 >l0 5 /
(150)
(151) 1C0T 12
(152) +F0 = 512>
(153) 12
(154) F
(155) _= '9 M M
(156) 1JK>m F !12 E ! 9"
(157) 12 9 Iz 5 9 / !X 4 4 @> += JK
(158) += Byky12
(159) !12 8 ' X>
(160) 12 3 &
(161) 8 / 12
(162) % 8 1Cl12
(163) w >D;W12 8 A ^ = ! E P' ' } Brky12 @> .
(164) 1JK>m . ! 12 " !c l0 ( 9 / 12 4 < 8 1Cl12
(165) G12 S
(166) PJKJ4>1 4&
(167) - " 4_= '9 P = ! E5 (>U :5 B L12 4 8 1Cl12
(168) = 2 > " 0V5 C8 0 SaS1 / X> ' ^!12>> JG1 5
(169) ' `{B B / } / / } / / / hLG12 %
(170) S = !> A @
(171) %> 5G ! '=
(172) >' %
(173) &+= ' E ( > = !
(174) 9 4 <!12 E !% U l0 ( 9 B. O (5 8A >q ;ZT
(175) !'@!12!12 "5'J4 12 ! > ! 4>q
(176) _= '9 (
(177) 1J4>D N !12 " ! Iz ! "5@ 512>
(178) + += ] 1' JK> M
(179) M5
(180) ] 1' 12
(181) B >G E5'.
(182) F> ! 6;Z
(183)
(184) .!12>T> JG1 12 "5 >@> JG1 512@acb
(185) bIda L
(186) @>D; Yq= !12> O E5 > k ' ML>@ += 5 "e% >l0 5 Ha = !
(187) T9" .
(188) k/ ]/ . . Fe!12 4 59 ^`me.a3e.h+ .>
(189) 8 12 ( !'& !'& .F [= 8 >1
(190) G "F 6;ZYq,.\ @` O > 'hB K > 712
(191) '
(192) F5 8A 0U @!12>> JG1 12 E= '
(193) ' / += 8 >14+ = 4 F
(194) U! @9 Iz 5 9 / B % H
(195) m H1. A8 12 ( 4
(196)
(197) 55' >>[
(198) H!H5 8A >4 5q>H! >! 4> 6; 5 NJ4> A [!12 J4 4 5J4> g r;Z
(199) ! 8 $ 0= !5
(200) !> = Byky12 !$ 5l0 5 / > 512>>T! 12
(201) @= '
(202) /
(203) 4
(204)
(205) (! >1AI G15 /
(206)
(207) 4 5 "H !& 10! c 5 c %> 12
(208) F "5> ! l12
(209) !12NJ4
(210) 5JK> H.> '>> . . " p`{Q ' hB O !'& 10! U`m>10! > = ^ V5 4 ^ 5 ! h^!12
(211) 5
(212) E
(213) w 512 !U 6;W 0I 12( A EFG12 F> J4> A B
(214) . !12
(215) 1 8A 2 1 w 5F> = X > ! "5@ .>
(216) G !12( 4 Bri 5 @& 50
(217) !( 2 x ( = T
(218) 5 >> 12
(219)
(220) !> = '
(221) -M= 8 (> !12
(222) T
(223) !
(224) 0 = !T 5 @! T
(225) H!'& 4 L = ! 49"
(226) / T!
(227) >D "H ] 1'! H
(228) !12 E ! B O P 10!
(229) "U 5P!12G12 5 ! 49. !'& K5 BHi?c4 (<!12G15
(230) J
(231) ' ] = S P>D; ' U . = & 10 ?
(232) = 'T9" 6 + = 4= 5 5 "? 6
(233) '1J4>m r j!12 " !y @ l0 (9 B O r = & 10 . !1 712 4 E !12 5 "? ] 1' > G j> _ = !
(234) 9 q
(235) !12 " !|| r !&[ = | r;Z E= 12. . <5 K H 12 "H 12
(236) F 47 >12
(237) M9" >T9" X = > = " BGir H 4 L= "5 L = &
(238) 1 / [= 8 >1
(239) M= %
(240) " = (9 L 6;Z( !j H 0 !55L&6
(241) / 512 E >l = > > 4(m F 6;Z
(242) ! ! [ J ' B.
(243)
(244) . . . ir. 5 !12
(245) $!'& 45%
(246) 1G12 5L
(247) % 12 8 >> = ! [ | = 9 12
(248) [>12 q>10! > + !12 " ! FG5I 5 "
(249) X += ' 8
(250) ] 1'
(251) > 12 _> A
(252) (!12 " "
(253) / 9 >QM= $ 5 J4T> = /? = += >T " ! >> .> A
(254) ^!> '9"
(255) N(`m9 Ivh> A
(256) ""= Byi 12 8 >T> ] 1'
(257) > 12 31 G. .
(258) '
(259)
(260) (! 'LG12 q
(261) 4Q
(262) j> jT4> = " 12
(263) [ "
(264) = 9"
(265) q [_= & 1
(266) [> A
(267) j !5 B ,
(268) ^>
(269) (D w ' X
(270) V!& 45 / V
(271) 8A EN 5
(272) !<> 5 C8 0V <aU1 V(2 / 12
(273) +
(274) '1712 12
(275) [ 4 ] 1 '
(276) > 12 w!12 " &ElJ
(277) '
(278) ]m J4>I ] 15 /0
(279) "= 041J4>D F 6;ZT !B Yq>T>L += ' 8 6;Z 4 < = !'
(280) `C = 9
(281) 8A > E5 h[ ? = 9" 12
(282) ^ M>12 [ @b0 1'T
(283) IdaS1 / 1 l
(284) ". . . . >D;Z "510
(285)
(286) !12 <
(287) !'& (4
(288) @dl"GN> 8 > Iz 5 / [= Q4
(289) H ' > 12 G15 " >>
(290) !12 E !B ir L>12 M ] 155 "L O 12
(291) >12NJU 512 "$= 2 > "< = !'5 H ? = 9" 12
(292) L 12 EH ""= =
(293) > w x ] 1'
(294) > 12 6B T E= x ( '!
(295) > <!55 ] 1'N 4> 12 _ 5(9"
(296) / ^ 4 !0= 12
(297) /. . . >> 7L 6; ' >@ 5G !L %!12
(298)
(299) 12
(300) H! + = 9"
(301) L !12 E !. UK> ! " L% 8 I 5 5 [ > ] / > ] 12 B2,
(302) +
(303) H 0D H 4 !'& 45 / ^ = > = " M L!12 " ! Iz ! 512 "% += 8 = (
(304) N> ] 1'
(305) > 12 S!12 " "
(306) B 12
(307) % > 12
(308) F
(309) 8 ' E5
(310) AIz '!'&+ = G12 . "!. >
(311) T != 12 - 5 4 Bri
(312) !0= T 12 - P 5 !( 5 ]m 5 >J4 . (> = &
(313) 1 X . #%$'*&( ),+.-0/2143,-.35/ +7& 68+:981;%<8-.+ $ =$>@? 68+ +.& ) +.& AB+.</4CEDF<F3,CG/4HF+014A ? I A +0& -J;%<83LKM8+N6O (QP 9.
(314) O/ ]. = > = " XQK
(315) ^ 6;Z 4 U_= & 10 3 3!12>>10! 12 6Bji
(316) (D 3 5(G12
(317) 41!& > X!'& (4
(318) 'T 0w
(319) w%> 0D NG12
(320)
(321) '1!& > %!'& 4 L
(322) N
(323) N`m0= 4>' F S5 4 $ 5 ! hB % > 1&
(324) %( EM + 55 "= H G12 E [Q 0 qF
(325)
(326) = & 10 Ldl"G L L512 = G12 H> 0 = 512>T 12 U
(327) = ' 9 @ H
(328) 1J4>D L
(329) ! B = [= > = 5L
(330) > . ij; 7 !g
(331) > IA = !& >> `{ 5 ! h| y
(332) 1J4>D ( y
(333) +!12 " ! Iz !q 5 JG1' +F = 4 |>M!'& 45 B 12
(334) y
(335) 1G12 512
(336) / 512 * T!
(337) >D " /
(338) N(1 6 >F 6;ZT E5 ]m ! MN
(339) > Iz
(340) 8 NG'5 "*
(341) [ 4 !12
(342) 5N H!12G15 " L>1! 0w >1J 0w
(343) F E5 ]{ ! @!12 E !B O @10 6 >@4 = 5 E5
(344) ""= x @ "
(345) = 9"
(346) @ ! 4>HG12 %> 0 = 512>T 12 c L41J4>D . !'& 10! / 5 " = . . ] 1' 0I > 12 4 !> 9 B*ij;
(347) K>! 12 V <> _ = & 10 UeL> 9" 4 _
(348) 1JK>m N 4> I = !'& >T> . . !12 " !F 5[4 = 5 E= @ 4 [
(349)
(350) 5 !12
(351)
(352) 'B"i G12 J4>H = 9 6;W1 G%!55 = & 10 %G12 q5 5 q4&[ = 12< +>10! 0S`m& 5 ] 0 = 9
(353) ! hj
(354) q> 12 +!'9 g 6;Z( !j q q4&+ = 12< . > 1J `m10 @ r;d= J4
(355) > "h.>12T P (> FG12 " F '9 " L @!N!'& 45B. . 12 !9 . / !
(356) @4= A N r;W12
(357) / 5N>D;Z
(358) . ir% '
(359) j!'& 45H 5+ [ =
(360) =
(361) K>! 12
(362) q "
(363) = 9"
(364) ! + = (9 [ 0
(365) ' (m 44> I ! 12
(366)
(367)
(368)
(369) 5' >>. B ^ [= 8 > 1
(370) G " @ ] 1' T9" . ] 1'
(371) > 12 4 41G12 "= . = T E= 0= /
(372) F>@!
(373) @!55N&G 5 / 4 B = ".
(374)
(375) .
(376).
(377) J
(378).
(379) .
(380) "!.
(381) $#.
(382) #. ! % &. ! . '!
(383).
(384) . N ( % #. ! . )!
(385).
(386) . k/ ]/.
(387) . Chapitre 1. ´ ´ Formalismes et schemas d’integration du contact en dynamique. Un bref ´ etat de l’art
(388) =<. .O.
(389)
(390) !"$#&%(!' )*"+-,./01' 234$53/6$ 87
(391) 5 9:;;:;:;;:;. *<. >?@' %1+ ;/BAC!' DBAE ;7-3AF G#@3/0 *#HI !/J+KL/
(392) DBAE M;:;:;;:;. 4N. P:/#@AE$53/. OQ. RE. " !$#&%')(+*,# -.*/-7<I#JK6- 27%< (+L"MG-8 MO27 <N03%"2P0QCCCCC CCCCC ST254/76- #&%*/8;: ()<N0 76- ="!3%>0$( <l03-7# @A !3-7' ')-C CCCCCC CCCCC. RET RETRE RET o. ah<O@ !3 bG'd-Kc # - 8T-C!G76- e.76- !3-7<G27-~fG()# @G%"2P0h8T-]8T-KMTijbG%!$!3-K*CC CCCCC "!$#&%')(+*,# -C8TMk21 "<N03%210\@G%>!\@/76- <G%')(+*$%>0$( <l-70h()<N0 76- ="!3%>0$( <l-K<j03-7# @G* CC "!$#&%')(+*,# -'+%>=!5%><G=( -7<j8 MI27 <N03%"2P0\-10h( <N0 K6- =!5%p0$( <?-7<I0$-K#H@[*CCCC. :;":;;:;:;;:;:;;:;:;;:;:;;:;:;;:;. FE .V. . ` En NE q.
(393)
(394)
(395) !
(396) "# %$&('
(397) *)+ ,-.'
(398) /0 123",4)-56$ ", 7$89
(399) ,:);#,< =#5>
(400) " ? ,@
(401) AB"'
(402) 1C)1
(403) ? + " 8 . ir = &
(404) 1
(405) = 'T9" + =
(406) = V5 5 "^
(407) N41J4>D ^ <!12 " !( V l (9 12 " ]m H>D;W1J0:5H 5 L
(408) 12 J
(409) '
(410) 5 F4
(411) J4>! 12 4 %`{B B W} / / / hB4i J4J4>1 K&
(412) ' H>@ :5 = 6; ".4>
(413) J712 4 E59 4>
(414) ' ] 1' >T ! .
(415) 1J4>D %12 "3= "X = !12 = 8 ! = "$
(416)
(417) A8 ' = = !&[ = . " 4_= '9 H 6;Z ""= 12 3 U5 K D4!& 9"
(418) N!12 E% 12
(419) " > 4 @_= & 10 @ T!
(420) >D B ,
(421) %!!& 45 / 12
(422) L SJ > l12
(423) .> L4>
(424) H!12
(425) E5. B 12
(426) %> >l 12
(427) H 8
(428) N % 12 !12 "5'J4 12
(429) B eQ4 - ^ >12 ' 4%
(430) 15^G12 = / 12 4 C8 12
(431) !& 12 * ^
(432) 0 = E5
(433) .
(434) ^
(435) (D ' > ] 1' > ' [!> 9 M F> J 5 6Z;
(436) $
(437) '1J4>m . !12 E !L 9"
(438) %H
(439) ] 155 ( EB
(440) i?. !12 E q F !'&+ = + 6;Z ""= 12 ^5 (4 8 !> ] 1' > ' * 512 Ej>T 5" = / 5 5 = j >l0 " = > >T
(441) (D 6;Z
(442) w0 4>@ 4>?D
(443) ! >T
(444) 6 @>D;Z( !H @ 0wJ ' X = > 59 B. 1.1. ´ ´ Formalismes et schemas d’integration temporelle. ´ 1.1.1 Formalismes en mecanique du contact J >FE / 10!! " y>D; 4&[=
(445) !F 6;Z
(446) 12 8 HG_
(447) JILK.` M:N 12 h / ] 1' 4
(448) '!
(449) J4>U
(450) 3 "5$ !12 " ! 8 !P
(451) 1J4 5 ! >PO
(452) 8 6;Z
(453)
(454) S ]m !SG15 " >>c !12 " !BQSR@` 8 12HQ ' B hB 12
(455) *!12
(456) += 12 4 |
(457) 512>T q +=. g Γu. f. Γg. Ω Γc. O . Yq. B. UT. g0. n. J !
(458) V #
(459) W # *
(460) $V
(461)
(462) & * . '
(463) G12 " ^9 U>U 12> w 5V = > 9"
(464) /
(465) 8A E (> >12F # 11uSV= 8 12>T EX VG5 G5I J 12
(466) / > P = 9 12
(467) >10! > 6;v = 9
(468) >J
(469)
(470) 12> ( _ 5 9 / > >12j
(471) X!12G15 " / >.
(472) -
(473)
(474) " %$&('
(475) )+ ,-.'
(476) /0 123",
(477) H
(478) 3
(479). >. = 9" 12
(480) L
(481) @!12 J4T>"= L> H!12 4
(482) 12
(483) w>T(5 M ;d = !' 8 "!12((N
(484) -D. N N D 4 8. N. . 1r . ` B h ` B h. LN
(485) `
(486) h N . G.
(487) . ` B h
(488)
(489) . . Q . ` B "h. Q . ` B h. . . 5M>H5
(490) +
(491) [!12 "5 "5 / +>5
(492) 5 + + [= ] 1 ' 12
(493) +> += ' = /. 5+>L5
(494) [
(495) ' ^ = > 5T9" 8 = Q "+> [
(496) 14'= " = +!> 9 / +>%!'& ($ + += 4> ! ( E / 5. . . . . . > 4 " = 6;W G1 8 12> 4(9 M X L>
(497) '" = 6;W G1 H ]{ !9 F 12 4 [= B. . ! 5l0 5 / T > ]m . :512 5@> .>12 . !12 " ! LQ R B6ky12 @! > 0/ 12
(498) . 1512 4 LM S>
(499) T 5
(500) ! 1' ""= @>
(501) {] ! G15 " >>. @!12 " !^ >D;W1JK 5 !>B %J4 8 12
(502) M9" r;W # k[k `m&"l"G1&6 5 . G5 FG 'J 12
(503) h / 12 4 8 12
(504) SD e. . M 1r . . "! $# N. . ` B }2h. % #. 5F> 1' >% ""= ' '^ >D;W1J4 5 !> Q ' B hB. 5F>j: w
(505) > "5 > 512>T LM>D;W1J4 !>^` 8 12. i X ] 1'N 4> 12 0S [= 4> ! " ]m J4>
(506) S41J4>D _ = !
(507) 9 [ = !'.! Iz 4 / - 4712 E 9 @>@!12 " ! % 6;W H ! ] ` N / Q!R'h / L>
(508) 8 " 5HD. . &('*),+.- 02/ 13034 f|12 8 65879: 1r . 79. ;q` . h. N. (. <=?>?@. ;q`BA h. L>D;W 5 !N H += 4> ! ( E %! + = 9 "
(509) JK> qL16L D. ;q`BC h*N D B` C,EC h !GF `BC0h. 8 !. D ` E C h*NIH2J ` hD `BC0h M7G F `B C h N HKJ L C M7G HK*M N C7M>Q. ` B h.
(510)
(511)
(512) !
(513) "# %$&('
(514) *)+ ,-.'
(515) /0 123",4)-56$ ", 7$89
(516) ,:);#,< =#5>
(517) " ? ,@
(518) AB"'
(519) 1C)1
(520) ? + " 8 . . ( j9"
(521) H!L
(522) 1JK>m
(523) L = " J4>(9"
(524) 7L9. ;. q 4 H 512> 12
(525)
(526) 9 B O 2 1 (H>R;d =
(527) HG15 " >> U 5 ( x 5 0I
(528) 5
(529) P 5 !X # >JG(`m! ] B? B B h /
(530) 12
(531) @ C8 12
(532) ^9" X!55$ 512>T 12 ]{ H>R;Z ""@ =
(533) 5@ 6;ZY[
(534) > B kq>T
(535) M
(536) 0 = ! 5 ( E / >
(537) 1J4>D 5< = 9
(538) 8A > E S
(539) 8A ECD. &('*),+.- 02/ 130 f|12 8 65879: A 5879 1r . ;. j; ` h A N. D ` E A.h ! F B` A. h*N. ` B 2h . ;. + = > += ' 8 = S 5
(540) ( x 5 S gB. Yq U5 E!12
(541) 5$ 4 = 5 E
(542) U!12 " !
(543)
(544) T> "= >6 ! / >
(545) '1J4>m . 8 "CD. &('*),+.- 02/ 130 f|12 8 65879 : ;q` . N h. . <=?>?@
(546) . 1r . . N. ;q`BA.h. A : M B` A.h. ` B h. . QSR. ` B h. BA. !
(547) 5 N> ! 12 "5 "5
(548) 4> "= >X` M ` h h / 12
(549) G12 8 12
(550) L4>
(551) L > H>D;Z ""N = 6;ZY[ 0I >
(552) B 12
(553) q712 8 2 1
(554) * >d x !& [> !12 "5 "5 r;Z
(555)
(556) > = >T"F = 712
(557) g (5
(558) j!12(
(559) 5 4 g>D;d = B O ! 12
(560) H w <41J4>D '
(561) 8A E9D e. &('*),+.- 02/ 130 f|12 8 65879: ;q` h. N. (. <=?>?@. q ; `BA.h `BA.h . 1r 5> ^] 12 4!12
(562) >>.! !"= 59 . `BA h `BA h. N . N " !. . A 5. 12. . . ` B h += Q4
(563) D.
(564) -
(565)
(566) " %$&('
(567) )+ ,-.'
(568) /0 123",
(569) H
(570) 3
(571) a P 12
(572) P VG12 8 12
(573) -512 A: 12 - P >T 5c>R;Z ""= 6 ;ZYq
(574) >3G12 ! != 'T 5_> 512>T 12
(575) V
(576) '1J4>m X! !
(577) 53 U>D;Z0= 4> 'U = B O 55 ] 12
(578) !12
(579) >T>< 6;W 5 ( x 5 0I. . . .
(580) L= " J4>B
(581) Y[>>. 5L 512 4 Iz
(582) L= E J4>`m! ] B
(583) B B / h+L 12 4 FG12 8 12
(584) $ = !'' >
(585) 1JK>m. 512
(586) ] 1'(@ 6;Z + = 9 12
(587) `{ 12 U 6;v = 9 12
(588) h 8 ' 2 1
(589) >> +!12N 4 D. &('*),+.- 02/ 130 f|12 8 65879 : A 5 7L9 ! F `BA ! h D ` E A ! h G. . ` B. . h. O H 5l0 5 L +4x 5L >? > 0 = 512>
(590) 12 " 4_= '9 B, ] 1' >
(591) 41!&[ = gq 6;
(592) 5 * ! 12 "w= " = += 8 >14+ = B 12
(593) > >12
(594) M
(595) G >FJ
(596) m 8 E> F4> 4 +!> 9 H 6;W "5 0GB. ´ Formalisme par penalisation. . . ij;v = !' `mkj1J4>D ( "hq
(597) X
(598) 1J4>D % !12 " ! F :512 5% ]m +
(599) AM = >"= ^ Q4
(600) N >D;v = 6 F9 6;Z
(601) $!'& 8 > % 5G !5. M> !2 1 E5 T E5 . 12 w "5+ = [= 5 12 6B
(602) i? ] 1' >T . A. FM = >T 12 P`{12 <> = & 10 @ .M = >T 12 Khq!> T9" .!12
(603) T 55 > 5 `{12 w0=
(604) > 5 h B O !r ]{ H 4> EM UT E510
(605)
(606) T EM> ^] 12
(607) !12
(608)
(609) >> M = >T 12 LD. 1r . . B` A.h*N H M M B` A h M7Q . ` B > h. . 5q 4 = >4 55!5 "qG12 ]v/
(610) G >H = < 5' FM = T> 12 [163 ` h G12 8 $` hB. . + = > . ir
(611) 1J4>D M = > = ;v = !' >1' CD. &('*),+.- 02/ 130
(612) f|12 8 5 7L9: ; ` . hHN. ; B` A.h N. 1r . 8 !. . (. <=?>?@. ; B` A.h. ;q`BA h B` A.h. += QK
(613) N` B > hB. ` B "h.
(614) }.
(615)
(616) !
(617) "# %$&('
(618) *)+ ,-.'
(619) /0 123",4)-56$ ", 7$89
(620) ,:);#,< =#5>
(621) " ? ,@
(622) AB"'
(623) 1C)1
(624) ? + " 8 . . >? ; A . 6;Z
(625) U
(626) 1J4>D 6;W1
(627) 12 U r;Z
(628) ] 12
(629) !12
(630)
(631) >> ( x 5 "Iz
(632) L= " J4>^ %
(633) < 5 ! # >J7'B L12 4 [G12 8 12 4 [ 12 4!L >T 5q>D;ZT E". = 6;ZY[
(634) >qG12 +! !"= ' 5H> 512>T 12 6BEY[ 12
(635) 5 /. G12 512 . . . = N> 512> 12 4
(636) 10!'&+= 5
(637) 8 ' / >
(638) 1JK>m }$ 5@J4 3G12 (
(639) 1JK>m h / 5 >12 <> 1' ' 51! = < 5
(640) 8 L>D;Z
(641) Q4
(642) *`{B B hB / >1 9"
(643) . 79. . . `m 512>
(644) 12 P
(645) . i > T(54 9 X < 5'$ X+ = >. =
(646) V!'& 12 V
(647) 5: < w> = & 10 < w+ = > 12 5>= >
(648) !
(649) >w 5X' > 8 " 4 += "= '1<>X!12
(650)
(651) 12 4 "
(652) 12 6B % w
(653) 1JK>m B % ( < 5'L > 8 E[G = G q "5M = += 5 12
(654) [
(655) 12 ! 8A J4> j "5%>. 0 512>T B|b0 >12
(656) . QK 9" r;Z>*0T 55X 6;
(657) 5 @dl"G
(658) (=
(659) > 'T 12 9 ( @+ = > 12 `{B B } hB. Formalismes lagrangiens. >G0 5@512 Fdl"G H
(660) ] 1' > F> A
(661) B. >F!12
(662) 5_ "51 4
(663) w
(664)
(665) >4>T! 5 ^ Ui A Formalisme purement lagrangien G'5 "? j!12 "5 12x >y> !12 E5 E5JM 7` h BCi = & 10 q ji A j5
(666) ] 1'j> !12 "5 "5 H>@!'& c @ += 4> ! ( E U
(667) @!12 "5 "5 H>@N
(668) >4>! 5
(669) `m! ] B hB. . ir
(670) 1J4>D w = 9" 4 8 > E <41J4>D '
(671) 8A E9D. &('*),+.- 02/ 130 f|12 8 N` E Kh 587L9 :. %` E7h 1r . N. . . ] = <=?>?@. `BA E
(672) |h. ` B. N
(673) 5 :
(674) ` U 5
(675) L @>
(676) >T"= h N ` Q!R'h `BA E |
(677) hJN ;q`BA.h !
(678) E M `BA h % E 0= ] = E P!10!'& N (
(679) >"= "5(>D;W ! . %J4 8 12
(680) @9 >. ` B }2h. . 8 ! >D;W 5 ! F5 ! H += 4> ! E % 1' 0w ' . Q!R. B. h. ` B. h. ` B 2 h
(681) >K>! 5 ' % (i A . . .
(682) -
(683)
(684) 1J4>D X ( !& '!'&
(685) X (G12 " Iz 5 >>^
(686) ] 1' > > A < > 9 >.> <
(687) >4>T! 5 ' X 512 EU 4
(688) > = > ( E<!12 "5 " `m> < += 4> ! ( E U
(689) >D;d= EUK>
(690) hB.
(691) -
(692)
(693) " %$&('
(694) )+ ,-.'
(695) /0 123",
(696) H
(697) 3
(698). E. ij;v = !' ( > 5 12 4 '( =
(699) -> A - P> 5 12> 12 V` Kh%
(700) ^<
Figure
Documents relatifs