Contribution à la modélisation mécanique et numérique des problèmes de contact-impact.

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(1)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. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

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(5)

(6) Remerciements

(7)

(8) #% !'&

(9) (*)+ -,.&

(10) 0/ 12 34 !5 .

(11) ! " $ &6 5 /7 8 !9"

(12) 0:; 6 3

(13) <0= >?4> @ 5 8A >>CB4> ' '!@ !

(14) >D ' EFG12H

(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 U4

(22) V3>D;W

(23) 5

(24) EX M= '

(25) ( $>

(26) ' !12 < !&'!'&

(27) N6;ZY[,%\+BK2

(28) N >H>

(29) H 512 <QK

(30) ! B O 5 8A > = "X = [= / > ] 12 /G -> JG1 512^^_ = !

(31) 9U`Dacb

(32) bIdaSegfh^ij; Y[= !12> O E5 > k ' g (+ = 5 ( EMe >l j_ = !

(33) 9 j+e!124 59`me%a3e hgL>

(34) 8 12 ^no@,p6;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% 55HG12H> !12 Q

(41) !96;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 > "M5125 ' !12

(47)

(48) ! aST!'& >7ky15Iz\'lX9

(49) !!

(50) "@ = L4 = M> 5125

(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~G12H>

(65) FJ712 4 &"

(66) ( B.

(67)

(68)

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(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. ETtu"*$()0$(<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@ !3b '-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#HT8;-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@ !3bG'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. |GWkT8;-Kc ')-C8;: (<"03-7!$e %27- #M ')0$(< (-.%>M CCCC CCCCCCC CCCCCCC V"V |G Wk0$(p%p0$(< CC CCCCCC CCCCCCC CCCCCCC V"V |GRE WkT8;-7c '- =76- "#J16- 03!$(+LNM -8T-.*\()<N03-7!$e %21-.* C CCCCCCC CCCCCCC V"V |G o z 6 L"M[%p0$(<[*/')T27%')-.*/8TMO#HT8;-7c '-8{: (<N0$-7!$e %27-#M ')0$( <G()"-K%>MCC CCCCCCC `"n |G | G!3#M '+%p03()"<?e %()b '- ex!$0$-C8TMO@ !3bG'd-Kc # - 8T-C27<N03%"2P0\#M ')0$( < ()"-K%M CCCCCC `G |GR t!$-K#H(-7!5*! K6- *$M '05%p05*9<ZM #K6- !$(+LNM -K*9CC CCCCCCC CCCCCCC `NE |GREX : %>@G@ !$T254 -C!$'-KLNM (<j@AMG!h')-.*/@ !3b 'd-7c # -.*8 -27<N03%"2P0 CCCCC CCCCCCC `| |GRET OK6- 27-K*3*,()0 6- 8T-.*\%>@ @ !3T254 -K*#M '03( ".6- 254 -7'')-.*QCCCCCCC CCCCCCC `| |GRETRE @[-K! 27 MO*$M !\'+%H#J16- 0$4GZ8 -!$'-KLNM (<XC CCCCCCC CCCCCCC `N |GRET o : %@ @ !3Z254G- !$'-KLNM (<O-7<k8T^Z<G%># (+LNM -OC CCCCCCC CCCCCCC `"s |GRET | : %@ @ !3Z254G- !$'-KLNM (<O@["M !\')-.*\@ !3b '-Kc #H-.*98 -]21<N05%2P0\-K<O8 ^N<[%># (LNM -XCCCC UKn"n |GRETR (*321!G76- 0$(+*$%>0$(<G*/8TMO@ !3b '-Kc #H- # T8 -7c '- CCCCCCC CCCCCCC UKnNE |GRET s SN03!3%>0 K6- =(- 8 -C! .6- *,"')M 0$(< CCCCC CCCCCCC CCCCCCC UKnNE |GRETq r9"<G*,-K!$p%>0$(<j8 -C'd:w76- <G-7!3=(- %F"-K2'd: %>@ @G!$T254 -h!3')-.LNM ()< CCC CCCCCCC UKn"o |G o r9<[21'MG*,(< CCCC CCCCCC CCCCCCC CCCCCCC UKnN. ´ 5 Exemples numeriques. 107. TmSN05%>b (')(+*3%p0$(<fGLNM -7';(<N0 K6- ! -7 0

(74) CCCCCC CCCCCCC CCCCCCC UKn"V T r9"<"05%210\8 - h-7!$0J CCCCCC CCCCCCC CCCCCCC UKn"V TRE t%>0$(<?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$(K6- 254 -K')'-vCCCC CCCCCC CCCCCCC CCCCCCC U.E T o WkT8;-7c '- #M ')0$(< (-.%>M CCCCC CCCCCCC CCCCCCC U.E T o RE WkT876- '(*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 >712UT

(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) M12U

(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' ($N4> 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>T12 -!^zl7^

(121) '1J4>m .

(122) " > +'

(123) D ' ++ = ! 4

(124) B0ky12F 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 9^12P9" ' 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 "XS = &

(136) 1 X6;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 / !X44 @>+= JK

(158) += Byky12

(159) !12 8 'X>

(160) 12 3&

(161) 8 / 12

(162) % 8 1Cl12

(163) w >D;W12 8 A ^ = ! EP' ' } Brky12@> .

(164) 1JK>m . ! 12 " !c l0 ( 9 / 124 < 8 1Cl12

(165) G12S

(166) PJKJ4>1 4&

(167) - "4_= '9P = ! E5 (>U :5 B L124 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) 94<!12 E !% Ul0 ( 9B. O (5 8A >q ;ZT

(175) !'@!12!12 "5'J412 ! > !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!124 59^`me.a3e.h+.>

(189) 8 12 ( !'&!'&.F[= 8 >1

(190) G "F6;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 [!12J44 5J4> gr;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!U6;W 0I 12( A EFG12F> J4> A B

(214) . !12

(215) 1 8A 2 1 w 5F> = X > ! "5@.>

(216) G !12( 4Bri 5 @& 50

(217) !( 2x ( = 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 P10!

(229) "U 5P!12G12 5 ! 49. !'& K5 BHi?c4 (<!12G15

(230) J

(231) ' ] = SP>D; ' U. = &10 ?

(232) = 'T9" 6+ = 4= 5 5 "? 6

(233) '1J4>m rj!12 " !y @l0 (9B O r = &10. !1 712 4 E !12 5 "? ] 1' > Gj> _ = !

(234) 9q

(235) !12 " !|| r !&[ = |r;Z E= 12. . <5 K H12 "H 12

(236) F 47 >12

(237) M9" >T9" X = > = " BGir H4 L= "5 L = &

(238) 1 / [= 8 >1

(239) M= %

(240) " = (9L6;Z( !jH 0 !55L&6

(241) / 512 E >l = > >4(m F6;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) (! 'LG12q

(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 F6;ZT !B Yq>T>L+= ' 8 6;Z4 < = !'

(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'N4> 12 _ 5(9"

(296) / ^4 !0= 12

(297) /. . . >>7L6; '>@ 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 H4!'& 45 / ^ = > = " ML!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) 1X. #%$'*&( ),+.-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;Z4 U_= &1033!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 "= HG12 E [Q0 qF

(325)

(326) = &10 Ldl"G L L512 = G12H> 0 = 512>T12 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(16 >F6;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 @106 >@4 = 5 E5

(344) ""= x @ "

(345) = 9"

(346) @ !4>HG12%> 0 = 512>T12 c L41J4>D .!'&10! / 5 " = . . ] 1'0I > 12 4 !> 9 B*ij;

(347) K>! 12 V<> _ = &10UeL> 9"4 _

(348) 1JK>m N4> I = !'& >T> . . !12 " !F 5[4 = 5 E= @ 4 [

(349)

(350) 5 !12

(351)

(352) 'B"i G12 J4>H = 96;W1 G%!55 = &10%G12q5 5 q4&[ = 12< +>10! 0S`m& 5 ] 0 = 9

(353) ! hj

(354) q> 12 +!'9 g6;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 Nr;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).

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(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)

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(400) " ? ,@

(401) AB"'

(402) 1C)1

(403) ? + " 8 . ir = &

(404) 1

(405) = 'T9" + =

(406) = V5 5 "^

(407) N41J4>D ^<!12 " !( Vl (912 " ]m H>D;W1J0:5H 5 L

(408) 12J

(409) '

(410) 5 F4

(411) J4>! 12 4 %`{B B W} / / / hB4i J4J4>1 K&

(412) 'H>@ :5 = 6; ".4>

(413) J712 4 E594>

(414) ' ] 1' >T ! .

(415) 1J4>D %12 "3= "X = !12 = 8 ! = "$

(416)

(417) A8 ' = = !&[ = . "4_= '9 H6;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'J412

(429) B eQ4 - ^ >12'4%

(430) 15^G12 = / 124 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 qF !'&+ = +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

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(445) !F6;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>Tq+=. g Γu. f. Γg. Ω Γc. O . Yq. B. UT. g0. n. J !

(458) V #

(459) W # *

(460) $V

(461)

(462) & * . '

(463) G12 " ^9U>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 F12 4 [= B. . ! 5l0 5 / T > ]m . :5125@> .>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 / 124 8 12

(504) SD e. . M 1r . . "! $# N. . ` B }2h. % #. 5F> 1' >% ""= ' '^ >D;W1J4 5 !> Q ' B hB. 5F>j: w

(505) > "5 > 512>TLM>D;W1J4 !>^` 8 12. i X ] 1'N4> 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`BCh*N D B` C,ECh !GF `BC0h. 8 !. D ` E Ch*NIH2J ` hD `BC0h M7G F `BC h N HKJ L C M7G HK*M N C7M>Q. ` B h.

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(511)

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(519) 1C)1

(520) ? + " 8 . . ( j9"

(521) H!L

(522) 1JK>m

(523) L = " J4>(9"

(524) 7L9. ;. q4 H 512>12

(525)

(526) 9B 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>T12 ]{ H>R;Z ""@ =

(533) 5@6;ZY[

(534) > Bkq>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) 5N> ! 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 "5r;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) -512A: 12 - P>T 5c>R;Z ""= 6 ;ZYq

(574) >3G12 ! != 'T 5_> 512>T12

(575) V

(576) '1J4>m X! !

(577) 53U>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 5124 Iz

(582) L= E J4>`m! ] B

(583) B B / h+L 124 FG12 8 12

(584) $ = !''>

(585) 1JK>m. 512

(586) ] 1'(@6;Z + = 9 12

(587) `{ 12 U6;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_= '9B, ] 1' >

(591) 41!&[ = gq6;

(592) 5 * ! 12 "w= " = += 8 >14+ = B 12

(593) > >12

(594) M

(595) G >FJ

(596) m 8 E> F4>4 +!> 9 H6;W "5 0GB. ´ Formalisme par penalisation. . . ij;v = !'`mkj1J4>D ( "hq

(597) X

(598) 1J4>D %!12 " ! F :5125% ]m +

(599) AM = >"= ^ Q4

(600) N >D;v = 6 F96;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 `{12w0=

(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. . 5q4 = >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) !

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(625) U

(626) 1J4>D 6;W1

(627) 12 Ur;Z

(628) ] 12

(629) !12

(630)

(631) >> ( x 5 "Iz

(632) L= " J4>^ %

(633) < 5 ! # >J7'B L124 [G12 8 12 4 [12 4!L>T 5q>D;ZT E". = 6;ZY[

(634) >qG12+! !"= ' 5H> 512>T12 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 9X < 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 55X6;

(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_ "514

(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 = &10qji 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 EU4

(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) ^<