Mécanique Des Structures Tom3 Thermique Des Structures ,Dynamique Des Structures

Mécanique des structures


Tome 3

Tome 3

Thermique des structures Dynamique des structures


Serge L AROZE

Serge L AROZE

CÉPADUÈS-ÉDITIONS


111, rue Nicolas-Vauquelin 31100 TOULOUSE – France Tél. : 05 61 40 57 36 – Fax : 05 61 41 79 89


(de l’étranger ) + 33 5 61 40 57 36 – Fax : + 33 5 61 41 79 89

(de l’étranger ) + 33 5 61 40 57 36 – Fax : + 33 5 61 41 79 89

www.cepadues.com Courriel : [email protected]




Tome 3

Tome 3



Serge L AROZE

Serge L AROZE





(de l’étranger ) + 33 5 61 40 57 36 – Fax : + 33 5 61 41 79 89

(de l’étranger ) + 33 5 61 40 57 36 – Fax : + 33 5 61 41 79 89



Chez le même éditeur Solides Elastiques / Plaques et Coques / Exercices �� Barrau J�-J�, Laroze S� Tourbillons Instabilité Décollement �� Béguier Cl� et al� Les Milieux continus multiphysiques �� Borghi R� Thermo-mécanique et modélisation par systèmes �� Borghi R� Introduction à la Dynamique des Gaz Réactifs �� Brun R� L ’Art de la formule expliqué aux scientifiques �� Bruneau M�, Potel C� Turbulence en Mécanique des Fluides �� Chassaing P� Mécanique des Fluides ��Chassaing P� Combustion dans les moteurs fusées �� CNES Maths/Méca. Que Savez-vous de l'Outil Mathématique? 6 fascicules �� Collectif Précis de Résistance des Matériaux �� Datas J-M� Vibration des structures par analyse modale. Test modal et identification modale �� Diaby M'P� Introduction à la Dynamique des Structures �� Gourinat Y� Mécanique Générale �� Laroze S� Thermique et Dynamique des Structures �� Laroze S� Poutres �� Laroze S� Solides Elastiques / Plaques et Coques �� Laroze S� Thermique et Dynamique des Structures / Exercices �� Laroze S�, Lorrain M Poutres / Exercices �� Laroze S�, Lorrain M Calculs thermomécaniques des structures par les éléments finis ��Marcelin J�-L� Optimisation des Structures et d'Éléments Mécaniques �� Marcelin J�-L� Optimisation des Vibrations des Structures Mécaniques �� Marcelin J�-L� Conception optimale des Engrenages Cylindriques ��Marcelin J�-L� Fuite et Rupture des Tubes endommagés �� Pluvinage G� 120 exercices de Mécanique Élastoplastique �� Pluvinage G� La Rupture du Bois et Composites �� Pluvinage G� Mécanique Élastoplastique de la Rupture �� Pluvinage G� Prévision statistique de la résistance �� Pluvinage G�, Sapunov V�T� Résistance des Matériaux �� Pluvinage G�, Sapunov V�T� Principes et applications de Mécanique Analytique �� Potel C� Leçons sur les Grandes Déformations �� Souchet R� Des Ondes et des Fluides �� Thual O� Introduction à la Mécanique des Milieux Continus Déformables �� Thual O� Contrôle des Décollements - Optimisation des performances et nouveaux actionneurs, GDR 2502�� Collectif Contrôle des décollements - du développement des actionneurs à l'amélioration des performances, GDR 2502 �� Collectif © CEPAD 2005

ISBN : 2.85428.714.2

CHEZ LE MÊME ÉDITEUR Le GRAFCET ...............................................................................................................................................ADEPA/AFCET Optimisations en fabrication ............................................................................................................................. Agullo M. Robustesse et commande optimale ......................................................................................................... Alazard D. et al. Cours de mécanique générale ...............................................................................................................................Bellet D. Problèmes de mécanique rationnelle ...................................................................................................................Bellet D. Problèmes de mécanique des solides ....................................................................................................................Bellet D. Problèmes d’élasticité ............................................................................................................................................Bellet D. Cours d’élasticité ................................................................................................................................Bellet D., Barrau J.-J. Comprendre, maîtriser et appliquer le GRAFCET .......................................................................................Blanchard M. Tables de détente ou compression isentropique de choc m = 1,400 ...............................................Bonnet A., Luneau J. Vous avez dit « Résistance des matériaux ” ? Qu’en savez-vous ?..................................................Boudet R., Stephan P. Que faut-il savoir en mécanique ? .....................................................................................................Boudet R., Sudre M. La stratégie productique ............................................................................................. Brzakowski S., Delamalmaison R. Produits et analyse de la valeur ..................................................................................................................... Chevallier J. Conduite et gestion de projets...............................................................................................Chvidchenko I., Chevallier J. Elasticité linéaire .................................................................................................................................................Dartus D. Précis de résistance des matériaux .................................................................................................................. Datas J.-M. 7 facettes du GRAFCET .............................................................................................................................Gendreau et al. Introduction à la dynamique des structures .................................................................................................. Gourinat Y. Le GRAFCET : de nouveaux concepts .....................................................................................................GREPA (ADEPA) Concepts et outils pour les systèmes de production.............................................................................. Hennet J.-C. et al. Optimisation des vibrations des structures mécaniques............................................................................Marcelin J.-L.. Conception optimale des engrenages cylindriques ....................................................................................Marcelin J.-L.. Mécanique élastoplastique de la rupture...................................................................................................... Pluvinage G. 120 exercices de Mécanique élastoplastique de la rupture ........................................................................... Pluvinage G La rupture du bois et de ses composites ....................................................................................................... Pluvinage G.. Fuite et rupture des tubes endommagés.............................................................................. Pluvinage G.., Sapunov V.-T. Ingénierie & Ergonomie...........................................................................................Pomian J.-L., Pradère T., Gaillard I. Leçons sur les grandes déformations................................................................................................................Souchet R. Les Nouvelles rationalisations de la production .................................................................de Terssac G., Dubois P. et al.

© CEPAD 2005

1er

ISBN : 2.85428.714.2 1er

Le code de la propriété intellectuelle du juillet 1992 interdit expressément la photocopie à usage collectif sans autorisation des ayants-droit. Or, cette pratique en se généralisant provoquerait une baisse brutale des achats de livres, au point que la possibilité même pour les auteurs de créer des œuvres nouvelles et de les faire éditer correctement est aujourd’hui menacée.

Le code de la propriété intellectuelle du juillet 1992 interdit expressément la photocopie à usage collectif sans autorisation des ayants-droit. Or, cette pratique en se généralisant provoquerait une baisse brutale des achats de livres, au point que la possibilité même pour les auteurs de créer des œuvres nouvelles et de les faire éditer correctement est aujourd’hui menacée.

Nous rappelons donc que toute reproduction, partielle ou totale, du présent ouvrage est interdite sans autorisation de l’Éditeur ou du Centre français d’exploitation du droit de copie (CFC – 3, rue d’Hautefeuille – 75006 Paris).


Dépôt légal : septembre 2005

N° éditeur : 714


CHEZ LE MÊME ÉDITEUR

N° éditeur : 714

CHEZ LE MÊME ÉDITEUR

Le GRAFCET ...............................................................................................................................................ADEPA/AFCET Optimisations en fabrication ............................................................................................................................. Agullo M. Robustesse et commande optimale ......................................................................................................... Alazard D. et al. Cours de mécanique générale ...............................................................................................................................Bellet D. Problèmes de mécanique rationnelle ...................................................................................................................Bellet D. Problèmes de mécanique des solides ....................................................................................................................Bellet D. Problèmes d’élasticité ............................................................................................................................................Bellet D. Cours d’élasticité ................................................................................................................................Bellet D., Barrau J.-J. Comprendre, maîtriser et appliquer le GRAFCET .......................................................................................Blanchard M. Tables de détente ou compression isentropique de choc m = 1,400 ...............................................Bonnet A., Luneau J. Vous avez dit « Résistance des matériaux ” ? Qu’en savez-vous ?..................................................Boudet R., Stephan P. Que faut-il savoir en mécanique ? .....................................................................................................Boudet R., Sudre M. La stratégie productique ............................................................................................. Brzakowski S., Delamalmaison R. Produits et analyse de la valeur ..................................................................................................................... Chevallier J. Conduite et gestion de projets...............................................................................................Chvidchenko I., Chevallier J. Elasticité linéaire .................................................................................................................................................Dartus D. Précis de résistance des matériaux .................................................................................................................. Datas J.-M. 7 facettes du GRAFCET .............................................................................................................................Gendreau et al. Introduction à la dynamique des structures .................................................................................................. Gourinat Y. Le GRAFCET : de nouveaux concepts .....................................................................................................GREPA (ADEPA) Concepts et outils pour les systèmes de production.............................................................................. Hennet J.-C. et al. Optimisation des vibrations des structures mécaniques............................................................................Marcelin J.-L.. Conception optimale des engrenages cylindriques ....................................................................................Marcelin J.-L.. Mécanique élastoplastique de la rupture...................................................................................................... Pluvinage G. 120 exercices de Mécanique élastoplastique de la rupture ........................................................................... Pluvinage G La rupture du bois et de ses composites ....................................................................................................... Pluvinage G.. Fuite et rupture des tubes endommagés.............................................................................. Pluvinage G.., Sapunov V.-T. Ingénierie & Ergonomie...........................................................................................Pomian J.-L., Pradère T., Gaillard I. Leçons sur les grandes déformations................................................................................................................Souchet R. Les Nouvelles rationalisations de la production .................................................................de Terssac G., Dubois P. et al.

Le GRAFCET ...............................................................................................................................................ADEPA/AFCET Optimisations en fabrication ............................................................................................................................. Agullo M. Robustesse et commande optimale ......................................................................................................... Alazard D. et al. Cours de mécanique générale ...............................................................................................................................Bellet D. Problèmes de mécanique rationnelle ...................................................................................................................Bellet D. Problèmes de mécanique des solides ....................................................................................................................Bellet D. Problèmes d’élasticité ............................................................................................................................................Bellet D. Cours d’élasticité ................................................................................................................................Bellet D., Barrau J.-J. Comprendre, maîtriser et appliquer le GRAFCET .......................................................................................Blanchard M. Tables de détente ou compression isentropique de choc m = 1,400 ...............................................Bonnet A., Luneau J. Vous avez dit « Résistance des matériaux ” ? Qu’en savez-vous ?..................................................Boudet R., Stephan P. Que faut-il savoir en mécanique ? .....................................................................................................Boudet R., Sudre M. La stratégie productique ............................................................................................. Brzakowski S., Delamalmaison R. Produits et analyse de la valeur ..................................................................................................................... Chevallier J. Conduite et gestion de projets...............................................................................................Chvidchenko I., Chevallier J. Elasticité linéaire .................................................................................................................................................Dartus D. Précis de résistance des matériaux .................................................................................................................. Datas J.-M. 7 facettes du GRAFCET .............................................................................................................................Gendreau et al. Introduction à la dynamique des structures .................................................................................................. Gourinat Y. Le GRAFCET : de nouveaux concepts .....................................................................................................GREPA (ADEPA) Concepts et outils pour les systèmes de production.............................................................................. Hennet J.-C. et al. Optimisation des vibrations des structures mécaniques............................................................................Marcelin J.-L.. Conception optimale des engrenages cylindriques ....................................................................................Marcelin J.-L.. Mécanique élastoplastique de la rupture...................................................................................................... Pluvinage G. 120 exercices de Mécanique élastoplastique de la rupture ........................................................................... Pluvinage G La rupture du bois et de ses composites ....................................................................................................... Pluvinage G.. Fuite et rupture des tubes endommagés.............................................................................. Pluvinage G.., Sapunov V.-T. Ingénierie & Ergonomie...........................................................................................Pomian J.-L., Pradère T., Gaillard I. Leçons sur les grandes déformations................................................................................................................Souchet R. Les Nouvelles rationalisations de la production .................................................................de Terssac G., Dubois P. et al.

© CEPAD 2005

© CEPAD 2005

ISBN : 2.85428.714.2

ISBN : 2.85428.714.2

Le code de la propriété intellectuelle du 1er juillet 1992 interdit expressément la photocopie à usage collectif sans autorisation des ayants-droit. Or, cette pratique en se généralisant provoquerait une baisse brutale des achats de livres, au point que la possibilité même pour les auteurs de créer des œuvres nouvelles et de les faire éditer correctement est aujourd’hui menacée.

Le code de la propriété intellectuelle du 1er juillet 1992 interdit expressément la photocopie à usage collectif sans autorisation des ayants-droit. Or, cette pratique en se généralisant provoquerait une baisse brutale des achats de livres, au point que la possibilité même pour les auteurs de créer des œuvres nouvelles et de les faire éditer correctement est aujourd’hui menacée.




N° éditeur : 714


N° éditeur : 714

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- 28 -

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ρ= T

λ

∫

λ

T ′λ ρλ Gλ α = T

λ

∫

λ

T ′λ αλ Gλ τ = T

λ

∫ T′λ τλ Gλ

λ

ρ= T

λ

∫

λ

T ′λ ρλ Gλ α = T

λ

∫

λ

T ′λ αλ Gλ τ = T

λ

∫ T′λ τλ Gλ

λ



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∞

∞

∫

T = T ′λ Gλ DYHF

∫


T ′λ =

π K F

λ

KF . H λ

−

⎧K = ⋅ − - ⋅ V FRQVWDQWHGH3ODQFN ⎪ 6RLWHQFRUHDYHF ⎨F = ⋅ PV FpOpULWpGHODOXPLqUH ⎪. = ⋅ − -GHJUp. FRQVWDQWHGH%ROW]PDQQ ⎩

T ′λ =

& λ

& Hλ

T ′λ =

π K F λ

KF . H λ

−


⎧& = π K F = ⋅ − : ⋅ P ⎪ KF ⎨ − ⎪⎩& = . = ⋅ P ⋅ °. −

T ′λ =

& λ

& Hλ

⎧& = π K F = ⋅ − : ⋅ P ⎪ KF ⎨ − ⎪⎩& = . = ⋅ P ⋅ °. −

&HVIRUPXOHV HW H[SULPHQWODORLGH3ODQFNSRXUOHUD\RQQHPHQWGXFRUSVQRLU2Q UDSSHOOH TXH OD FRQVWDQWH GH %ROW]PDQQ HVW OH TXRWLHQW GH OD FRQVWDQWH PRODLUH 5 GHV JD] SDUIDLWVSDUOHQRPEUH1G¶$YRJDGUR


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- 29 -

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∞

∞

∫


∫


T ′λ =

π K F

λ

KF H .λ

−


T ′λ =

& λ

& Hλ

T ′λ =

π K F

λ

KF H .λ

−


⎧& = π K F = ⋅ − : ⋅ P ⎪ KF ⎨ − ⎪⎩& = . = ⋅ P ⋅ °. −

T ′λ =

& λ

& Hλ

⎧& = π K F = ⋅ − : ⋅ P ⎪ KF ⎨ − ⎪⎩& = . = ⋅ P ⋅ °. −







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T = ε 7 DYHF ε = ⋅ − : ⋅ P − ⋅ GHJ . −

T = ε 7 DYHF ε = ⋅ − : ⋅ P − ⋅ GHJ . −

&HUpVXOWDWFRQVWLWXHODORLGH6WpIDQ ε HVWODFRQVWDQWHGH6WpIDQ


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λP =

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λP =

µP × GHJUp. 7

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∞

∞

∫

T = ελ T′λ Gλ = ε ⋅ ε 7

∫

T = ελ T′λ Gλ = ε ⋅ ε 7

T′λ HVWGRQQpSDUODORLGH3ODQFN /HVFRHIILFLHQWVVDQVGLPHQVLRQV ελ HW ε FRPSULV HQWUHHWVHQRPPHQWpPLVVLYLWpVSRXUODORQJXHXUG¶RQGH λ HWUpVXOWDQWHUHVS ,OVVRQW pJDX[jGDQVOHFDVGXFRUSVQRLU,OVVRQWpJDX[jGDQVOHFDVGHVFRUSVQRQDEVRUEDQWV FRUSVEODQF α = τ = HWFRUSVWUDQVSDUHQW α = ρ = G¶DSUqVOHVHFRQGSULQFLSHGHOD WKHUPRG\QDPLTXH (Q HIIHW HQIHUPRQV XQ WHO VROLGH QRQ DEVRUEDQW GDQV XQH FDYLWp YLGH IHUPpHGRQWODSDURLDWKHUPDQHHVWXQHFRTXHQRLUHjWHPSpUDWXUHSOXVpOHYpHTXHFHOOHGX


- 30 -

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±/DORLGH6WpIDQ

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T = ε 7 DYHF ε = ⋅ − : ⋅ P − ⋅ GHJ . −

T = ε 7 DYHF ε = ⋅ − : ⋅ P − ⋅ GHJ . −



±/DORLGH:LHQ

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λP =

µP × GHJUp. 7

λP =

µP × GHJUp. 7











∞

∞

∫

T = ελ T′λ Gλ = ε ⋅ ε 7

∫

T = ελ T′λ Gλ = ε ⋅ ε 7



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$=

7 − 7 7 /RJ 5 − 7 /RJ 5 HW % = 5 5 /RJ /RJ 5 5

- 31 -

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∆7 =

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G ⎛ G7 ⎞ ⎟ = ⎜U U GU ⎝ GU ⎠

7 − 7 7 /RJ 5 − 7 /RJ 5 HW % = 5 5 /RJ /RJ 5 5

- 31 -

G7 $ = HW 7 = $ /RJ U + % GU U


7 − 7 7 /RJ 5 − 7 /RJ 5 HW % = 5 5 /RJ /RJ 5 5

- 31 -

3RXUXQHORQJXHXUXQLWDLUHGXWXEHRQD


± OHIOX[GHFKDOHXUUDGLDO P = T U ⋅ π U = −N

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G7 πN (7 − 7 ) ⋅ π U VRLW P = 5 GU /RJ 5

7 − 7 5 = /RJ 3 Nπ 5



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G ⎛ G7 ⎞ ⎟ = ⎜U U GU ⎝ GU ⎠


∆7 =

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5 5 (7 − 7 ) 5 − 5

7 − 7

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=

$ = (7 − 7 )

7 5 −7 5 5 5 HW % = 5 − 5 5 − 5

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$ G7 $ = HW 7 = % − U GU U

7 5 −7 5 5 5 HW % = 5 − 5 5 − 5


VRLW

G ⎛ G7 ⎞ ⎟ = ⎜U U GU ⎝ GU ⎠

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5 − 5 N π 5 5

5=

7 − 7

P

- 32 -

=

5 − 5 N π 5 5

- 32 -





G7 ⋅ π U GU

G7 πN (7 − 7 ) ⋅ π U VRLW P = 5 GU /RJ 5

7 − 7 5 = /RJ 3 Nπ 5



G7 πN (7 − 7 ) ⋅ π U VRLW P = 5 GU /RJ 5

7 − 7 5 = /RJ 3 Nπ 5







∆7 =

G ⎛ G7 ⎞ ⎟ = ⎜U U GU ⎝ GU ⎠


∆7 =

2QHQGpGXLWSDULQWpJUDWLRQV $ G7 $ = HW 7 = % − U GU U

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P=Nπ


G7 ⋅ π U GU

5 5 (7 − 7 ) 5 − 5

7 − 7

P

=

5 − 5 N π 5 5

- 32 -

$ = (7 − 7 )

7 5 −7 5 5 5 HW % = 5 − 5 5 − 5



$ G7 $ = HW 7 = % − U GU U

7 5 −7 5 5 5 HW % = 5 − 5 5 − 5


VRLW

G ⎛ G7 ⎞ ⎟ = ⎜U U GU ⎝ GU ⎠

P = TU ⋅ π U = −N VRLW

P=Nπ

G7 ⋅ π U GU

5 5 (7 − 7 ) 5 − 5


7 − 7

P

=

5 − 5 N π 5 5

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5=


H H = 6 = π D π D N N6

5=


H H = 6 = π D π D N N6


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±/DORLGH+RRNH

±/DORLGH+RRNH

(WDEOLHDXFKDSLWUH,,,GXWRPH,GDQVOHFDVLVRWKHUPH 7 = 7 HOOHV¶pFULW


E=

+ ν ν Σ − V , RX Σ = λ H , + µ E ( (

E=

+ ν ν Σ − V , RX Σ = λ H , + µ E ( (

DYHF ± ( = PRGXOHG¶
/HVFRHIILFLHQWVGH/DPp λ HW µ V¶H[SULPHQWHQIRQFWLRQGH(HW ν SDUOHVIRUPXOHV


λ=

(ν ( HW µ = * = ( + ν )( − ν ) ( + ν )

λ=

(ν ( HW µ = * = ( + ν )( − ν ) ( + ν )

* = µ V¶DSSHOOHDXVVLPRGXOHG¶pODVWLFLWpGHJOLVVHPHQWRXWUDQVYHUVDO


HHWVGpVLJQHQWOHVWUDFHVGHVWHQVHXUV E HW Σ ,GpVLJQHOHWHQVHXUXQLWp


±/DORLGHODGLODWDWLRQWKHUPLTXH


/RUVTX¶XQVROLGHKRPRJqQHLVRWURSHHWOLEUHGHVHGpIRUPHUGRQFQRQK\SHUVWDWLTXH YRLWVD WHPSpUDWXUH SDVVHU GH VD YDOHXU XQLIRUPH 7 j OD YDOHXU XQLIRUPH 7 WRXWHV VHV GLVWDQFHV LQWHUSDUWLFXODLUHVVXELVVHQWODYDULDWLRQUHODWLYH α (7 − 7 ) /HFRHIILFLHQW α FDUDFWpULVWLTXH GXPDWpULDXVHQRPPHFRHIILFLHQWGHGLODWDWLRQWKHUPLTXHOLQpLTXH


/D WUDQVIRUPDWLRQ SRQFWXHOOH VXELH SDU OH VROLGH HVW GRQF XQH KRPRWKpWLH GH UDSSRUW [ + α (7 − 7 )]'HSOXVFHWWHYDULDWLRQGHWHPSpUDWXUHQ¶LQWURGXLWDXFXQHFRQWUDLQWH


7RXWHFKDvQHGHSDUWLFXOHVLVROpHDXVHLQGXVROLGHYRLWVDORQJXHXUSDVVHUGH A j A DYHF A − A ε= = α (7 − 7 ) /HWHQVHXUGHGpIRUPDWLRQDVVRFLpGLW©WKHUPLTXHªHWQRWp EWK HVW A GRQFXQLIRUPHHWVSKpULTXHLVRWURSH DYHF EWK = α (7 − 7 ) ,


/HVYDOHXUVGH α jODWHPSpUDWXUHGH&VRQWGRQQpHVSDUOHWDEOHDXGXFKDSLWUH,,,SDJH WRPH,


- 37 -

- 37 -

±5(/$7,216(175(&2175$,17(6(7'e)250$7,216

±5(/$7,216(175(&2175$,17(6(7'e)250$7,216



±/DORLGH+RRNH

±/DORLGH+RRNH



E=

+ ν ν Σ − V , RX Σ = λ H , + µ E ( (

E=

+ ν ν Σ − V , RX Σ = λ H , + µ E ( (



λ=

(ν

( + ν )( − ν )

HW µ = * =

( ( + ν )

λ=

(ν

( + ν )( − ν )

HW µ = * =

( ( + ν )















- 37 -

- 37 -

7RXWH QDSSH GH SDUWLFXOHV GX VROLGH VXELW OD YDULDWLRQ UHODWLYH GH VXUIDFH 6 − 6 = α (7 − 7 )WRXWGRPDLQH DDXVHLQGXVROLGHVXELWODYDULDWLRQUHODWLYHGHYROXPH 6 RXGLODWDWLRQYROXPLTXHUHODWLYH 9 − 9 = α (7 − 7 ) 9




3RXUpWDEOLUODORLGXFRPSRUWHPHQWWKHUPRpODVWLTXHOLQpDLUHLVRORQVDXVHLQG¶XQVROLGHXQ SDYppOpPHQWDLUHILJXUH


± GDQVO¶pWDW LOHVWQRQGpIRUPp E = QRQFRQWUDLQW Σ = HWODWHPSpUDWXUH 7


± GDQV O¶pWDW ©DFWXHOª W VHV pWDWV GH GpIRUPDWLRQ HW FRQWUDLQWH VRQW FDUDFWpULVp SDU OHV WHQVHXUV E HW Σ VDWHPSpUDWXUHHVW7


7

Σ = = E

W 7 Σ E

L 7 Σ = E = EWK

7

Σ = = E

W 7 Σ E

L 7 Σ = E = EWK

)LJXUH

)LJXUH

3RXUSDVVHUGHO¶pWDW jO¶pWDWW SDVVRQVSDUXQpWDWLQWHUPpGLDLUHL GDQVOHTXHOOHSDYpVH WURXYHjODWHPSpUDWXUH7VDQVFRQWUDLQWHVVXUVHVIDFHWWHV


(Q SDVVDQW GH j L RQ SDVVH GH 7 j 7 HW GH E = j EWK = α (7 − 7 ) , OHV FRQWUDLQWHV


UHVWDQWQXOOHV

UHVWDQWQXOOHV

(Q SDVVDQW GH L j W RQ DMRXWH j EWK OHV GpIRUPDWLRQV GXHV DX[ FRQWUDLQWHV F¶HVW j GLUH


+ ν ν Σ − V , GRQQpHVSDUODORLGH+RRNHVRLW ( (

GRQQpHVSDUODORLGH+RRNHVRLW

)LQDOHPHQWHQSDVVDQWGH jW RQLQWURGXLWODGpIRUPDWLRQ


E=

+ ν ν Σ − V , + α (7 − 7 ) ,

( (

GpIRUPDWLRQV PpFDQLTXHV

+ ν ν Σ − V , ( (

E=

GpIRUPDWLRQV WKHUPLTXHV

+ ν ν Σ − V , + α (7 − 7 ) ,

( (



- 38 -

- 38 -











7

Σ = = E

W 7 Σ E

L 7 Σ = E = EWK

7

Σ = = E

W 7 Σ E

L 7 Σ = E = EWK

)LJXUH

)LJXUH





UHVWDQWQXOOHV

UHVWDQWQXOOHV




+ ν ν Σ − V , ( (



E=

+ ν ν Σ − V , + α (7 − 7 ) ,

( (


- 38 -


+ ν ν Σ − V , ( (


E=

+ ν ν Σ − V , + α (7 − 7 ) ,

( (


- 38 -


&HWWH IRUPXOH WUDGXLW OD ORL GH +RRNH'XKDPHO HOOH OLH HQ WRXW SRLQW 3 OH WHQVHXU GHV FRQWUDLQWHV Σ OH WHQVHXU GHV GpIRUPDWLRQV E HW OD WHPSpUDWXUH 7 DX PR\HQ GH WURLV FRHIILFLHQWFDUDFWpULVWLTXHVGXPDWpULDX ( ν α


6LO¶RQLQYHUVHODIRUPXOH SRXUH[SULPHUOHVFRQWUDLQWHVHQIRQFWLRQGHVGpIRUPDWLRQVHWGH ODWHPSpUDWXUHRQWURXYH Σ = λ H , + µ E − . α (7 − 7 ) , ( FRHIILFLHQWGHFRPSUHVVLELOLWp DYHF . = λ + µ = − ν


5HPDUTXHOHVUHODWLRQVHQWUHWUDFHVHHWVGH EHW Σ V¶pFULYHQW


V = . [H − α (7 − 7 )]HW H =

V + α (7 − 7 ) .

V = . [H − α (7 − 7 )]HW H =

V + α (7 − 7 ) .

5HPDUTXHFRPPHWRXVOHVFRHIILFLHQWVFDUDFWpULVWLTXHVGHVPDWpULDX[RXSUHVTXHWRXV ( ν α GRQF λ µ . YDULHQWDYHFODWHPSpUDWXUH


/H PRGXOH G¶
/H PRGXOH G¶
/HV FRHIILFLHQWV GH 3RLVVRQ GHV PDWpULDX[ XVXHOV YDULHQW WUqV SHX SRXU OHV DFLHUV LQR[\GDEOHV YRLVLQ GH j & Y SDVVH SDU XQ PD[LPXP GH YHUV SRXU GpFURvWUHHQVXLWH


5HPDUTXHODORLGH+RRNH'XKDPHO V¶pFULWHQWHUPHVGHFRPSRVDQWHVFDUWpVLHQQHV


(

)

(

)

(

)

+ ν ⎧ ⎧ ⎪ε [ = H σ [ − ν σ \ + σ ] + α (7 − 7 ) ⎪ γ \] = H τ \] ⎪ ⎪ + ν ⎪ ⎪ τ][ ⎨ε \ = σ \ − ν σ ] + σ [ + α (7 − 7 ) ⎨ γ ][ = H H ⎪ ⎪ ⎪ ⎪ + ν τ [\ ⎪ε ] = σ ] − ν σ [ + σ \ + α (7 − 7 ) ⎪γ [\ = H H ⎩ ⎩

(

)

(

)

(

)


±/(352%/Ê0(*e1e5$/'(7+(502e/$67,&,7e

±/(352%/Ê0(*e1e5$/'(7+(502e/$67,&,7e

2QVHGRQQH ± ODJpRPpWULHGHVVROLGHVFRQVWLWXDQWODVWUXFWXUH ± OHVFDUDFWpULVWLTXHV ( ν α GHVPDWpULDX[


± OHVOLDLVRQVLQWHUQHVHWH[WHUQHV


± OHVFKDUJHPHQWVPpFDQLTXHHWWKHUPLTXH


± OHVD[HV {R [\]}FKRLVLVOLpVjODVWUXFWXUH


- 39 -

- 39 -







V = . [H − α (7 − 7 )]HW H =

V + α (7 − 7 ) .

V = . [H − α (7 − 7 )]HW H =

V + α (7 − 7 ) .



/H PRGXOH G¶
/H PRGXOH G¶




(

)

(

)

(

)


(

)

(

)

(

)


±/(352%/Ê0(*e1e5$/'(7+(502e/$67,&,7e

±/(352%/Ê0(*e1e5$/'(7+(502e/$67,&,7e








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2QYHXWFDOFXOHUHQFKDTXHSRLQW3GHODVWUXFWXUH


±OHVFRPSRVDQWVGXYHFWHXUWUDQVODWLRQ 3R3


± OHVFRPSRVDQWVGXSVHXGRYHFWHXUURWDWLRQ ± OHVVL[FRPSRVDQWHVGXWHQVHXUGpIRUPDWLRQ E ± OHVVL[FRPSRVDQWHVGXWHQVHXUFRQWUDLQWH Σ


VRLWIRQFWLRQVGH[\]HWW


/DWHPSpUDWXUH7[\]W D\DQWpWpGpWHUPLQpHSUpDODEOHPHQWRQGLVSRVHSRXUFHFDOFXOGH pTXDWLRQVLQGpSHQGDQWHVOLDQWFHVSDUDPqWUHVjVDYRLU


±OHVUHODWLRQVHQWUHURWDWLRQVHWWUDQVODWLRQVVRLW


±OHVUHODWLRQVHQWUHGpIRUPDWLRQVHWWUDQVODWLRQVVRLW


E = JUDG 3R3 + JUDG W 3R3 pTXDWLRQVVFDODLUHV RXOHVpTXDWLRQVGHFRPSDWLELOLWp URW ⋅ URW E =


±OHVpTXDWLRQVGHO¶pTXLOLEUHORFDOVRLW G G I + GLY Σ = R pTXDWLRQVVFDODLUHV


±OHVUHODWLRQVGH+RRNHHW'XKDPHOVRLW + ν ν E= Σ − V , + α (7 − 7 ) , pTXDWLRQVVFDODLUHV ( (


UHPpWKRGHGHUpVROXWLRQPpWKRGHGHVGpSODFHPHQWV


(OOH FRQVLVWH j SUHQGUH FRPPH LQFRQQXHV SULYLOpJLpHV OHV WURLV FRPSRVDQWHV GX YHFWHXU WUDQVODWLRQ 3R3 QRWpHVXYZ2QREWLHQWWURLVpTXDWLRQVOLDQWFHVWURLVIRQFWLRQVLQFRQQXHV G G HQ UHPSODoDQW GH O¶pTXLOLEUH ORFDO I + GLY Σ = R Σ SDU λ GLY 3R3 , + µ JUDG 3R3


+ JUDG W 3R3 − . α 7 − 7 , HQYHUWXGHODORLGH+RRNH'XKDPHOVDFKDQWTXH H = GLY 3R3


HW E = JUDG 3R3 + JUDG W 3R3 2QREWLHQWWRXVFDOFXOVIDLWVO¶pTXDWLRQYHFWRULHOOHGH1DYLHU 'XKDPHO G λ + µ JUDG ⋅ GLY 3R3 + µ ∆ 3R3 = λ + µ α JUDG 7 − I


TXH O¶RQ SURMHWWH HQVXLWH VL EHVRLQ HVW VXU OHV WURLV D[HV FH TXL GRQQH WURLV pTXDWLRQV VFDODLUHV DX[ GpULYpHV SDUWLHOOHV VHFRQGHV /¶LQWpJUDWLRQ GRQQH OH YHFWHXU WUDQVODWLRQ 3R3 DYHFGHVFRQVWDQWHVG¶LQWpJUDWLRQ


3DUGpULYDWLRQRQFDOFXOHHQVXLWH SXLV E HWHQILQDXPR\HQGHODORLGH+RRNH'XKDPHO Σ


/HVFRQVWDQWHVG¶LQWpJUDWLRQVHGpWHUPLQHQWJUkFHDX[FRQGLWLRQVDX[OLPLWHVHWGDQVOHVFDV QRQVWDWLRQQDLUHVJUkFHDX[FRQGLWLRQVLQLWLDOHV


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5HPDUTXHXQHIRLVOHFKDPSGHWHPSpUDWXUH7GpWHUPLQpOHSUREOqPHHVWIRUPHOOHPHQW LGHQWLTXH j VRQ KRPRORJXH GG¶pODVWLFLWp LVRWKHUPH H[SRVp DX FKDSLWUH ,9 GX WRPH , RQ G UHPSODFHODIRUFHYROXPLTXH I SDU I − λ + µ α JUDG 7 /HVLQGLFDWLRQVSUDWLTXHVGRQQpHV jFHWWHRFFDVLRQUHVWHQWGRQFYDODEOHVLFL


G 5HPDUTXHFRPSWHWHQXGHO¶LGHQWLWpHQWUHRSpUDWHXUV ∆ = JUDG ⋅ GLY − GH1DYLHU'XKDPHODO¶H[SUHVVLRQpTXLYDOHQWHVXLYDQWH


λ + µ JUDG ⋅ GLY 3R3 − µ

G ⋅ 3R3 = λ + µ α JUDG 7 − I

O¶pTXDWLRQ


G ⋅ 3R3 = λ + µ α JUDG 7 − I

O¶pTXDWLRQ

5HPDUTXHODVROXWLRQJpQpUDOHGHO¶pTXDWLRQGH1DYLHU'XKDPHO RX HVWODVRPPH GHODVROXWLRQJpQpUDOHGHO¶pTXDWLRQVDQVVHFRQGPHPEUHHWG¶XQHVROXWLRQSDUWLFXOLqUH TXHOFRQTXHGHO¶pTXDWLRQFRPSOqWH


5HPDUTXH OH YHFWHXU WUDQVODWLRQ 3R3 FRPPH WRXW FKDPS GH YHFWHXUV SHXW G¶DSUqV OH WKpRUqPHGH&OHEVFKrWUHFRQVLGpUpFRPPHODVRPPHGHGHX[FKDPSVVRLW


3R3 = 8 [ \ ] W + 8 [ \ ] W

3R3 = 8 [ \ ] W + 8 [ \ ] W

R 8 GpULYHG¶XQSRWHQWLHOVFDODLUHG¶R

8 = HW 8 GpULYHG¶XQSRWHQWLHOYHFWHXU

G¶R GLY 8 =



G¶R GLY 8 =

2QSHXWDLQVLWRXMRXUVFKHUFKHUVpSDUpPHQWOHVGHX[FKDPSVYHFWRULHOV 8 HW 8 LOVXIILW


TXHOHFKDPS 3R3 VRLWGHFODVVH &


5HPDUTXH HQ DSSOLTXDQW O¶RSpUDWHXU GLYHUJHQFH DX[ GHX[ PHPEUHV GH O¶pTXDWLRQ REWLHQW G λ + µ ∆ H = λ + µ α ∆ 7 − GLY I


'HPrPHHQDSSOLTXDQWj O¶RSpUDWHXUURWDWLRQQHORQREWLHQW


HPpWKRGHGHUpVROXWLRQPpWKRGHGHVFRQWUDLQWHV


2QSUHQGDORUVFRPPHLQFRQQXHVSULYLOpJLpHVOHVVL[FRPSRVDQWHVGXWHQVHXUFRQWUDLQWH Σ 2Q REWLHQW XQ V\VWqPH GH pTXDWLRQV OLDQW FHV IRQFWLRQV LQFRQQXHV HQ UHPSODoDQW GDQV E = E SDU VRQ H[SUHVVLRQ HQ IRQFWLRQ GHV O¶pTXDWLRQ GH FRPSDWLELOLWp FRQWUDLQWHV ORL GH +RRNH'XKDPHO G RQ REWLHQW WRXV FDOFXOV IDLWV HW FRPSWH WHQX GH O¶pTXDWLRQG¶pTXLOLEUHORFDO GLY Σ = − I O¶pTXDWLRQWHQVRULHOOHGH%HOWUDPL'XKDPHO


∆Σ +

+

G G JUDG ⋅ JUDG V = − ⎡⎢JUDG I + JUDG W I + ν ⎣

G ν (α (α ⎤ GLY I ⋅ , + ⋅ ∆7 ⋅ , + JUDG ⋅ JUDG 7 ⎥ + ν − ν + ν ⎦

∆Σ +

+



- 41 -

- 41 -






G ⋅ 3R3 = λ + µ α JUDG 7 − I

O¶pTXDWLRQ


G ⋅ 3R3 = λ + µ α JUDG 7 − I

O¶pTXDWLRQ





3R3 = 8 [ \ ] W + 8 [ \ ] W

3R3 = 8 [ \ ] W + 8 [ \ ] W



G¶R GLY 8 =



G¶R GLY 8 =













∆Σ +

+



- 41 -

∆Σ +

+



- 41 -

&HWWHpTXDWLRQWHQVRULHOOHpTXLYDXWjVL[pTXDWLRQVVFDODLUHVDX[GpULYpHVSDUWLHOOHVVHFRQGHV /¶LQWpJUDWLRQGRQQHOHVFRQWUDLQWHVDYHFGHVFRQVWDQWHVG¶LQWpJUDWLRQ


2Q HQ GpGXLW HQVXLWH OHV GpIRUPDWLRQV DX PR\HQ GH OD ORL GH +RRNH'XKDPHO SXLV SDU GH QRXYHOOHV LQWpJUDWLRQV GHV UHODWLRQV HQWUH GpIRUPDWLRQV HW WUDQVODWLRQV OH YHFWHXU WUDQVODWLRQ 3R3 G¶ROHSVHXGRYHFWHXUURWDWLRQ


/HV FRQVWDQWHV G¶LQWpJUDWLRQ VH GpWHUPLQHQW SDU OHV FRQGLWLRQV DX[ OLPLWHV HW OHV FRQGLWLRQV LQLWLDOHV


5HPDUTXHXQHIRLVGpWHUPLQpOHFKDPSGHWHPSpUDWXUH7OHSUREOqPHHVWIRUPHOOHPHQW LGHQWLTXH DX SUREOqPH KRPRORJXH G¶pODVWLFLWp LVRWKHUPH H[SRVp DX WRPH , FKDSLWUH ,9 OHV JUDGLHQWVGHWHPSpUDWXUHDJLVVDQWFRPPHGHVIRUFHVGHYROXPH


5HPDUTXH OD VROXWLRQ JpQpUDOH GH O¶pTXDWLRQ GH %HOWUDPL'XKDPHO HVW OD VRPPH GH OD VROXWLRQJpQpUDOHGHO¶pTXDWLRQVDQVVHFRQGPHPEUHHWG¶XQHVROXWLRQSDUWLFXOLqUHTXHOFRQTXH GHO¶pTXDWLRQFRPSOqWH


5HPDUTXHHQpJDODQWOHVWUDFHVGHVGHX[PHPEUHVGH RQREWLHQWO¶pTXDWLRQ G (α + ν ∆7 GLY I ⋅ − ∆V = − − ν − ν


HPpWKRGHGHUpVROXWLRQPpWKRGHGHV©HVVDLVª


/HV pTXDWLRQV GH 1DYLHU'XKDPHO FRPPH FHOOHV GH %HOWUDPL'XKDPHO VRQW FRPSOH[HV HW LO Q¶H[LVWHSDVO¶DOJRULWKPHSRXUOHVLQWpJUHUGDQVOHFDVJpQpUDOELHQpYLGHPPHQW2QQHVDLWHQ WURXYHUOHVVROXWLRQVJpQpUDOHVTXHGDQVTXHOTXHVFDVUHODWLYHPHQWVLPSOHVTXHQRXVYHUURQV HQH[HUFLFHV


3RXUFHVUDLVRQVODPpWKRGHODSOXVHPSOR\pHSUDWLTXHPHQWFRQVLVWHjHVVD\HUHWDMXVWHUGHV VROXWLRQVIRQFWLRQVSRO\QRPLDOHVWULJRQRPpWULTXHV FKRLVLHVFRPSWHWHQXGHODJpRPpWULH GHODVWUXFWXUHGXFKDUJHPHQWGHVFRQGLWLRQVDX[OLPLWHVGHVpYHQWXHOOHVV\PpWULH


(WDQW GRQQp O¶XQLFLWp GH OD VROXWLRQ GX SUREOqPH HW OH FKDPS GH WHPSpUDWXUH D\DQW pWp GpWHUPLQpRQSHXWpQRQFHUOHWKpRUqPHIRQGDPHQWDO


8Q HQVHPEOH GH IRQFWLRQV UHSUpVHQWDQW OHV FRPSRVDQWHV GX YHFWHXU WUDQVODWLRQ GX SVHXGRYHFWHXUURWDWLRQ GXWHQVHXUGpIRUPDWLRQ HWGHWHQVHXUFRQWUDLQWH FRQVWLWXH ODVROXWLRQGXSUREOqPHGHWKHUPRpODVWLFLWpVLHWVHXOHPHQWVLLOYpULILH ± OHVUHODWLRQVGpIRUPDWLRQVWUDQVODWLRQV


E = JUDG 3R3 + JUDG W 3R3


± OHVUHODWLRQVURWDWLRQVWUDQVODWLRQV


- 42 -

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- 42 -

- 42 -

± OHVpTXDWLRQVGHO¶pTXLOLEUHORFDO G G I + GLY Σ =


± ODORLGH+RRNH'XKDPHO


+ ν ν Σ = λ H , + µ E − . α 7 − 7 , RX E = Σ − V , + α 7 − 7 , ( ( ± OHVFRQGLWLRQVDX[OLPLWHVHWOHVFRQGLWLRQLQLWLDOHV

+ ν ν Σ − V , + α 7 − 7 , ( ( ± OHVFRQGLWLRQVDX[OLPLWHVHWOHVFRQGLWLRQLQLWLDOHV

Σ = λ H , + µ E − . α 7 − 7 , RX E =

$SSOLFDWLRQLPSRUWDQWHFKDPSVGHWHPSpUDWXUHVDQVFRQWUDLQWHV


/DORLGHODGLODWDWLRQWKHUPLTXHH[SRVpHDXDIILUPHTXHVLO¶RQVRXPHWjXQHYDULDWLRQ XQLIRUPHGHWHPSpUDWXUHXQVROLGHKRPRJqQHLVRWURSHHWOLEUHGHVHGpIRUPHURQQ¶LQWURGXLW DXFXQHFRQWUDLQWH Σ =


2Q LQWURGXLW SDU FRQWUH XQ FKDPS GH GpIRUPDWLRQ XQLIRUPH HW LVRWURSH EWK = α 7 − 7 ,


3DULQWpJUDWLRQRQREWLHQWOHYHFWHXUWUDQVODWLRQ


⎧ X = α 7 − 7 [ + $ ⎪ ⎪⎪ 3R3 ⎨ Y = α 7 − 7 \ + % G¶R ⎪ ⎪ ⎪⎩Z = α 7 − 7 ] + &

⎧ X = α 7 − 7 [ + $ ⎪ ⎪⎪ 3R3 ⎨ Y = α 7 − 7 \ + % G¶R ⎪ ⎪ ⎪⎩Z = α 7 − 7 ] + &

/HVFRQVWDQWHVG¶LQWpJUDWLRQ$%&VRQWQXOOHVVLO¶RQFRQVLGqUHFRPPHIL[HODSDUWLFXOH RULJLQHGXUHSqUH { [\]}


([LVWHWLOG¶DXWUHVFKDPSVGHWHPSpUDWXUHTXLGDQVOHVPrPHVFRQGLWLRQVQ¶LQWURGXLVHQWSDV GHFRQWUDLQWHV"/DUpSRQVHHVWGRQQpHSDUXQWKpRUqPHLPSRUWDQWTXHQRXVDOORQVGpPRQWUHU PDLQWHQDQWHQVXSSRVDQWOHUpJLPHSHUPDQHQW


7KpRUqPH pWDQW GRQQp XQ VROLGH KRPRJqQH LVRWURSH HW OLEUH GH VH GpIRUPHU SRXU TX¶XQ FKDUJHPHQW WKHUPLTXH Q¶LQWURGXLVH DXFXQH FRQWUDLQWH LO IDXW HW LO VXIILW TXH FH FKDPS GH WHPSpUDWXUHVRLWOLQpDLUHF¶HVWjGLUHGHODIRUPH 7 − 7 = D[ + E\ + F] + G


7KpRUqPHGLUHFWVLOHFKDPSGHWHPSpUDWXUHQ¶LQWURGXLWDXFXQHFRQWUDLQWHO¶pTXDWLRQ VH UpGXLWj JUDG ⋅ JUDG 7 = F¶HVWjGLUHjODQXOOLWpGHVVL[GpULYpHVSDUWLHOOHVVHFRQGHVGH7 SXLVTX¶RQDDORUV ⎧Σ = V = SDVGHFRQWUDLQWHV ⎪ ⎪⎪G ⎨I = SDVGHFKDUJHPHQWPpFDQLTXH ⎪ ⎪ ⎪⎩∆ 7 = HQYHUWXGHOpTXDWLRQ

7KpRUqPHGLUHFWVLOHFKDPSGHWHPSpUDWXUHQ¶LQWURGXLWDXFXQHFRQWUDLQWHO¶pTXDWLRQ VH UpGXLWj JUDG ⋅ JUDG 7 = F¶HVWjGLUHjODQXOOLWpGHVVL[GpULYpHVSDUWLHOOHVVHFRQGHVGH7 SXLVTX¶RQDDORUV ⎧Σ = V = SDVGHFRQWUDLQWHV ⎪ ⎪⎪G ⎨I = SDVGHFKDUJHPHQWPpFDQLTXH ⎪ ⎪ ⎪⎩∆ 7 = HQYHUWXGHOpTXDWLRQ

/D QXOOLWp GH FHV GpULYpHV LPSOLTXH TXH OD WHPSpUDWXUH 7 HQ 3 HVW XQH IRQFWLRQ OLQpDLUH GHV FRRUGRQQpHVGH3


- 43 -

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+ ν ν Σ = λ H , + µ E − . α 7 − 7 , RX E = Σ − V , + α 7 − 7 , ( ( ± OHVFRQGLWLRQVDX[OLPLWHVHWOHVFRQGLWLRQLQLWLDOHV

+ ν ν Σ − V , + α 7 − 7 , ( ( ± OHVFRQGLWLRQVDX[OLPLWHVHWOHVFRQGLWLRQLQLWLDOHV

Σ = λ H , + µ E − . α 7 − 7 , RX E =









⎧ X = α 7 − 7 [ + $ ⎪ ⎪⎪ 3R3 ⎨ Y = α 7 − 7 \ + % G¶R ⎪ ⎪ ⎩⎪Z = α 7 − 7 ] + &

⎧ X = α 7 − 7 [ + $ ⎪ ⎪⎪ 3R3 ⎨ Y = α 7 − 7 \ + % G¶R ⎪ ⎪ ⎩⎪Z = α 7 − 7 ] + &







7KpRUqPHGLUHFWVLOHFKDPSGHWHPSpUDWXUHQ¶LQWURGXLWDXFXQHFRQWUDLQWHO¶pTXDWLRQ VH UpGXLWj JUDG ⋅ JUDG 7 = F¶HVWjGLUHjODQXOOLWpGHVVL[GpULYpHVSDUWLHOOHVVHFRQGHVGH7 SXLVTX¶RQDDORUV ⎧Σ = V = SDVGHFRQWUDLQWHV ⎪ ⎪⎪G ⎨I = SDVGHFKDUJHPHQWPpFDQLTXH ⎪ ⎪ ⎩⎪∆ 7 = HQYHUWXGHOpTXDWLRQ

7KpRUqPHGLUHFWVLOHFKDPSGHWHPSpUDWXUHQ¶LQWURGXLWDXFXQHFRQWUDLQWHO¶pTXDWLRQ VH UpGXLWj JUDG ⋅ JUDG 7 = F¶HVWjGLUHjODQXOOLWpGHVVL[GpULYpHVSDUWLHOOHVVHFRQGHVGH7 SXLVTX¶RQDDORUV ⎧Σ = V = SDVGHFRQWUDLQWHV ⎪ ⎪⎪G ⎨I = SDVGHFKDUJHPHQWPpFDQLTXH ⎪ ⎪ ⎩⎪∆ 7 = HQYHUWXGHOpTXDWLRQ



- 43 -

- 43 -

/HWHQVHXUGHVGpIRUPDWLRQVSXUHPHQWWKHUPLTXHHVWLVRWURSHHWV¶pFULW E = EWK = α 7 − 7 ,


DYHF 7 − 7 = D[ + E\ + F] + G

DYHF 7 − 7 = D[ + E\ + F] + G

/¶LQWpJUDWLRQGHVUHODWLRQVGpIRUPDWLRQVWUDQVODWLRQVGRQQHDORUVOHYHFWHXUWUDQVODWLRQ


⎧ ⎪ X = α D [ − \ − ] + α E\ + F] + G [ + $ ⎪ ⎪ 3R3 ⎨ Y = α E \ − ] − [ + α F] + D[ + G \ + % ⎪ ⎪ ⎪Z = α F ] − [ − \ + α D[ + E\ + G ] + & ⎩

2QFDOFXOHHQILQOHSVHXGRYHFWHXUURWDWLRQ

⎧ω[ = α E] − F\ ⎪ ⎪⎪ ⎨ω\ = α F[ − D] ⎪ ⎪ ⎪⎩ω] = α D\ − E[



⎧ω[ = α E] − F\ ⎪ ⎪⎪ ⎨ω\ = α F[ − D] ⎪ ⎪ ⎪⎩ω] = α D\ − E[

7KpRUqPH UpFLSURTXH VL OD YDULDWLRQ GH WHPSpUDWXUH HQ 3[\] HVW GH OD IRUPH 7 − 7 = D[ + E\ + F] + G RQYpULILHDLVpPHQWTXHODVROXWLRQHVWHIIHFWLYHPHQWFRQVWLWXpHSDU OHVGL[KXLWIRQFWLRQVVXLYDQWHV ± OHVWURLVIRQFWLRQV SRXUOHVWUDQVODWLRQV ± OHVWURLVIRQFWLRQV SRXUOHVURWDWLRQV


± ε [ = ε \ = ε ] = α 7 − 7 γ \] = γ ][ = γ [\ = SRXUOHVGpIRUPDWLRQV


± σ [ = σ \ = σ ] = τ \] = τ][ = τ [\ = SRXUOHVFRQWUDLQWHV


/HVFRQVWDQWHVG¶LQWpJUDWLRQ$%&VRQWQXOOHVVLO¶RQFRQVLGqUHFRPPHQXOOHGpSODFHPHQW GHODSDUWLFXOHRULJLQHGXUHSqUHF¶HVWjGLUHVLO¶RQFRQVLGqUHTX¶HOOHUHVWHRULJLQHGXUHSqUH DXFRXUVGHODGpIRUPDWLRQ


(WXGLRQVPDLQWHQDQWOHVWURLVSULQFLSDX[SUREOqPHVGHWKHUPRpODVWLFLWpELGLPHQVLRQQHOOH


±'e)250$7,2163/$1(6±'3

±'e)250$7,2163/$1(6±'3

8QVROLGHVHWURXYHHQpWDWGHGpIRUPDWLRQSODQHV¶LOH[LVWHXQUHSqUHRUWKRQRUPpOLpDXVROLGH SDUUDSSRUWDXTXHOOHYHFWHXUWUDQVODWLRQDGHVFRPSRVDQWHVGHODIRUPH


⎧X = X [ \ W ⎪⎪ 3R3 ⎨ Y = Y [ \ WR&W GpVLJQHXQHIRQFWLRQGHWGRQQpH ⎪ ⎪⎩ Z = & W ⋅ ]


- 44 -

- 44 -



DYHF 7 − 7 = D[ + E\ + F] + G

DYHF 7 − 7 = D[ + E\ + F] + G





⎧ω[ = α E] − F\ ⎪ ⎪⎪ ⎨ω\ = α F[ − D] ⎪ ⎪ ⎩⎪ω] = α D\ − E[



⎧ω[ = α E] − F\ ⎪ ⎪⎪ ⎨ω\ = α F[ − D] ⎪ ⎪ ⎩⎪ω] = α D\ − E[











±'e)250$7,2163/$1(6±'3

±'e)250$7,2163/$1(6±'3





- 44 -

- 44 -

2QHQGpGXLWOHVFRPSRVDQWHVGXSVHXGRYHFWHXUURWDWLRQ HWFHOOHVGXWHQVHXUGpIRUPDWLRQ E WRXWHVLQGpSHQGDQWHVGH]


⎧ ⎧ ⎧ ∂X ⎪ω[ = ⎪τ \] = ⎪ε [ = ∂ [ ⎪ ⎪ ⎪ ⎪ ⎪ ∂Y ⎪ ⎨ω\ = E ⎨ε \ = ⎨τ][ = ∂ \ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ε = & ⎛ ∂Y ∂X ⎞ ⎛ ∂X ∂Y ⎞ ⎟⎟ ⎟ − + ⎪ω] = ⎜⎜ ⎪ ] ⎪τ [\ = ⎜⎜ ⎩ ⎝∂[ ∂\⎠ ⎝ ∂ \ ∂ [ ⎟⎠ ⎩ ⎩

⎧ ⎧ ⎧ ∂X ⎪ω[ = ⎪τ \] = ⎪ε [ = ∂ [ ⎪ ⎪ ⎪ ⎪ ⎪ ∂Y ⎪ ⎨ω\ = E ⎨ε \ = ⎨τ][ = ∂ \ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ε = & ⎛ ∂Y ∂X ⎞ ⎛ ∂X ∂Y ⎞ ⎟⎟ ⎟ − + ⎪ω] = ⎜⎜ ⎪ ] ⎪τ [\ = ⎜⎜ ⎩ ⎝∂[ ∂\⎠ ⎝ ∂ \ ∂ [ ⎟⎠ ⎩ ⎩

8QHVHXOHpTXDWLRQGHFRPSWDELOLWpQ¶HVWSDVLGHQWLTXHPHQWYpULILpHHOOHV¶pFULW


∂ γ [\ ∂[ ∂\

=

∂ ε\ ∂ [

+

∂ ε[ ∂ \

/DORLGH+RRNH'XKDPHOGRQQHHQVXLWHOHVUHODWLRQV

SXLV

RXHQFRUH

]

[

]

+ ν ⎧ ⎪ε [ = ( − ν σ [ − ν σ \ − ν & + + ν α 7 − 7 ⎪ + ν ⎪ − ν σ \ − ν σ [ − ν & + + ν α 7 − 7 ⎨ε \ = ( ⎪ ⎪ + ν τ [\ ⎪γ [\ = ( ⎩

[

]

[

]

( ⎧ ⎪σ [ = + ν − ν − ν ε [ + ν ε \ + F − ( α 7 − 7 ⎪ ( ⎪σ = ⎪⎪ \ + ν − ν − ν ε \ + ν ε [ + F − ( α 7 − 7 ⎨ ( (α ⎪σ ] = ν ε [ + ε \ + − ν & − 7 − 7 ⎪ + ν − ν − ν ⎪ ⎪τ [\ = ( γ [\ ⎪⎩ + ν

[

∂ ε\ ∂ [

+

∂ ε[ ∂ \

]

σ ] = (& + ν σ [ + σ \ − ( α 7 − 7

'HSOXVODUHODWLRQ GHYLHQW

∂[ ∂\

=


σ ] = (& + ν σ [ + σ \ − ( α 7 − 7

[

∂ γ [\

SXLV

RXHQFRUH

[

]

[

]


[

]

[

]

( ⎧ ⎪σ [ = + ν − ν − ν ε [ + ν ε \ + F − ( α 7 − 7 ⎪ ( ⎪σ = ⎪⎪ \ + ν − ν − ν ε \ + ν ε [ + F − ( α 7 − 7 ⎨ ( (α ⎪σ ] = ν ε [ + ε \ + − ν & − 7 − 7 ⎪ + ν − ν − ν ⎪ ⎪τ [\ = ( γ [\ ⎪⎩ + ν

[

]


∆ σ [ + σ \ = −

G GLY I + ( α ∆ 7 − ν

∆ σ [ + σ \ = −

G GLY I + ( α ∆ 7 − ν

- 45 -

- 45 -



⎧ ⎧ ⎧ ∂X ⎪ω[ = ⎪τ \] = ⎪ε [ = ∂ [ ⎪ ⎪ ⎪ ⎪ ⎪ ∂Y ⎪ ⎨ω\ = E ⎨ε \ = ⎨τ][ = ∂\ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ε = & ⎛ ∂Y ∂X ⎞ ⎛ ∂X ∂Y ⎞ ⎟⎟ ⎟ − + ⎪ω] = ⎜⎜ ⎪ ] ⎪τ [\ = ⎜⎜ ⎩ ⎝∂[ ∂\⎠ ⎝ ∂ \ ∂ [ ⎟⎠ ⎩ ⎩

⎧ ⎧ ⎧ ∂X ⎪ω[ = ⎪τ \] = ⎪ε [ = ∂ [ ⎪ ⎪ ⎪ ⎪ ⎪ ∂Y ⎪ ⎨ω\ = E ⎨ε \ = ⎨τ][ = ∂\ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ε = & ⎛ ∂Y ∂X ⎞ ⎛ ∂X ∂Y ⎞ ⎟⎟ ⎟ − + ⎪ω] = ⎜⎜ ⎪ ] ⎪τ [\ = ⎜⎜ ⎩ ⎝∂[ ∂\⎠ ⎝ ∂ \ ∂ [ ⎟⎠ ⎩ ⎩



∂ γ [\ ∂[ ∂\

=

∂ ε\ ∂ [

+

∂ ε[ ∂ \


SXLV

RXHQFRUH

]

[

]


[

]

[

]

[

]

( ⎧ ⎪σ [ = + ν − ν − ν ε [ + ν ε \ + F − ( α 7 − 7 ⎪ ( ⎪σ = ⎪⎪ \ + ν − ν − ν ε \ + ν ε [ + F − ( α 7 − 7 ⎨ ( (α ⎪σ ] = ν ε [ + ε \ + − ν & − 7 − 7 ⎪ + ν − ν − ν ⎪ ⎪τ [\ = ( γ [\ + ν ⎩⎪

∆ σ [ + σ \ = −

∂ ε\ ∂ [

+

∂ ε[ ∂ \

σ ] = (& + ν σ [ + σ \ − ( α 7 − 7


∂[ ∂\

=


σ ] = (& + ν σ [ + σ \ − ( α 7 − 7

[

∂ γ [\

SXLV

RXHQFRUH

[

]

[

]


[

]

[

]

[

]

( ⎧ ⎪σ [ = + ν − ν − ν ε [ + ν ε \ + F − ( α 7 − 7 ⎪ ( ⎪σ = ⎪⎪ \ + ν − ν − ν ε \ + ν ε [ + F − ( α 7 − 7 ⎨ ( (α ⎪σ ] = ν ε [ + ε \ + − ν & − 7 − 7 ⎪ + ν − ν − ν ⎪ ⎪τ [\ = ( γ [\ + ν ⎩⎪

'HSOXVODUHODWLRQ GHYLHQW G GLY I + ( α ∆ 7 − ν

- 45 -

∆ σ [ + σ \ = −

G GLY I + ( α ∆ 7 − ν

- 45 -

5pVROXWLRQG¶XQSUREOqPHGHGpIRUPDWLRQVSODQHV UHPpWKRGHPpWKRGHGHVGpSODFHPHQWV


/HV pTXDWLRQV GH 1DYLHU'XKDPHO VRQW DX QRPEUH GH GHX[ HOOHV WUDGXLVHQW HQ WHUPHV GH WUDQVODWLRQVO¶pTXLOLEUHORFDOVXLYDQWOHVGLUHFWLRQV[HW\HWV¶pFULYHQW


⎧ ⎪λ + µ ⎪ ⎨ ⎪ ⎪λ + µ ⎩

∂ ∂[

⎛ ∂X ∂Y ⎞ ∂ ⎛ ∂Y ∂X ⎞ ∂7 ⎜⎜ ⎟⎟ − µ ⎜⎜ ⎟⎟ = λ + µ α + − − I[ ∂\ ⎝∂[ ∂\⎠ ∂[ ⎝∂[ ∂\⎠

∂7 ∂ ⎛ ∂X ∂Y ⎞ ∂ ⎛ ∂X ∂Y ⎞ ⎜ ⎟ = λ + µ α ⎜ ⎟−µ − I\ − + ∂\ ∂ [ ⎜⎝ ∂ \ ∂ \ ⎟⎠ ∂ \ ⎜⎝ ∂ [ ∂ \ ⎟⎠

⎧ ⎪λ + µ ⎪ ⎨ ⎪ ⎪λ + µ ⎩

∂ ∂[

⎛ ∂X ∂Y ⎞ ∂ ⎛ ∂Y ∂X ⎞ ∂7 ⎜⎜ ⎟⎟ − µ ⎜⎜ ⎟⎟ = λ + µ α + − − I[ ∂\ ⎝∂[ ∂\⎠ ∂[ ⎝∂[ ∂\⎠

∂7 ∂ ⎛ ∂X ∂Y ⎞ ∂ ⎛ ∂X ∂Y ⎞ ⎜ ⎟ = λ + µ α ⎜ ⎟−µ − I\ − + ∂\ ∂ [ ⎜⎝ ∂ \ ∂ \ ⎟⎠ ∂ \ ⎜⎝ ∂ [ ∂ \ ⎟⎠

HPpWKRGHPpWKRGHGHVFRQWUDLQWHV


2QSHXWGpWHUPLQHUOHVWURLVFRQWUDLQWHV σ [ σ [ τ [\ SDULQWpJUDWLRQGXV\VWqPH G ∂ τ [\ ∂ σ \ ∂ σ [ ∂ τ [\ GLY I + ( α ∆ 7 + = − I [ = − I \ ∆ σ [ + σ \ = − + ∂[ ∂\ − ν ∂\ ∂[


HPpWKRGHPpWKRGHGHODIRQFWLRQG¶$LU\


G ,O V¶DJLW G¶XQH YDULDQWH GH OD SUpFpGHQWH GDQV OH FDV R I = JUDG 9 SUDWLTXHPHQW WRXMRXUV G G G UpDOLVpSXLVTXHOHVIRUFHVYROXPLTXHV I VRQWGHVIRUFHVGHSHVDQWHXU ρ J RXG¶LQHUWLH − ρ Γ 2QSRVHDORUV ∂ φ ∂ φ ∂ φ − 9 τ [\ = − σ[ = − 9 σ \ = ∂[ ∂\ ∂[ ∂\


/D IRQFWLRQ GH FRQWUDLQWH φ [ \ RX IRQFWLRQ G¶$LU\ VDWLVIDLW DXWRPDWLTXHPHQW DX[ GHX[ SUHPLqUHVpTXDWLRQV pTXLOLEUHORFDO ODWURLVLqPHV¶pFULW


∆ ∆ φ =

− ν ∆ 9 − ( α ∆ 7 − ν

∆ ∆ φ =

− ν ∆ 9 − ( α ∆ 7 − ν

5HPDUTXH OHV GpIRUPDWLRQV SODQHV VH UHQFRQWUHQW GDQV OHV F\OLQGUHV RX SULVPHV ORQJV VRXPLVjGHVHIIRUWVSHUSHQGLFXODLUHVjO¶D[H]GLUHFWLRQGHVJpQpUDWULFHV HWLQGpSHQGDQWHVGH ]DLQVLTX¶jXQFKDPSGHWHPSpUDWXUHLQGpSHQGDQWGH]


)RUPXODLUHHQFRRUGRQQpHVF\OLQGULTXHV U θ ]


⎧X = X U θ W UDGLDO ⎪⎪ 3R3 ⎨Y = Y U θ W FLUFRQIpUHQWLHO ⎪ ⎩⎪Z = & W ⋅ ]D[LDO

⎧ ⎪ωU = ⎪⎪ ⎨ωθ = ⎪ ⎛ Y ∂Y ∂X ⎞ ⎟ − ⎪ω] = ⎜⎜ + ⎝ U ∂ U U ∂ θ ⎟⎠ ⎩⎪

⎧X = X U θ W UDGLDO ⎪⎪ 3R3 ⎨Y = Y U θ W FLUFRQIpUHQWLHO ⎪ ⎩⎪Z = & W ⋅ ]D[LDO

- 46 -

⎧ ⎪ωU = ⎪⎪ ⎨ωθ = ⎪ ⎛ Y ∂Y ∂X ⎞ ⎟ − ⎪ω] = ⎜⎜ + ⎝ U ∂ U U ∂ θ ⎟⎠ ⎩⎪

- 46 -





⎧ ⎪λ + µ ⎪ ⎨ ⎪ ⎪λ + µ ⎩

∂ ∂[

⎛ ∂X ∂Y ⎞ ∂ ⎛ ∂Y ∂X ⎞ ∂7 ⎜⎜ ⎟⎟ − µ ⎜⎜ ⎟⎟ = λ + µ α + − − I[ [ \ \ [ \ ∂ ∂ ∂ ∂ ∂ ∂[ ⎝ ⎠ ⎝ ⎠

∂7 ∂ ⎛ ∂X ∂Y ⎞ ∂ ⎛ ∂X ∂Y ⎞ ⎜ ⎟ = λ + µ α ⎜⎜ ⎟⎟ − µ − I\ − + ∂\ ∂ [ ⎜⎝ ∂ \ ∂ \ ⎟⎠ ∂\ ⎝∂[ ∂\⎠

⎧ ⎪λ + µ ⎪ ⎨ ⎪ ⎪λ + µ ⎩

∂ ∂[

⎛ ∂X ∂Y ⎞ ∂ ⎛ ∂Y ∂X ⎞ ∂7 ⎜⎜ ⎟⎟ − µ ⎜⎜ ⎟⎟ = λ + µ α + − − I[ [ \ \ [ \ ∂ ∂ ∂ ∂ ∂ ∂[ ⎝ ⎠ ⎝ ⎠

∂7 ∂ ⎛ ∂X ∂Y ⎞ ∂ ⎛ ∂X ∂Y ⎞ ⎜ ⎟ = λ + µ α ⎜⎜ ⎟⎟ − µ − I\ − + ∂\ ∂ [ ⎜⎝ ∂ \ ∂ \ ⎟⎠ ∂\ ⎝∂[ ∂\⎠











∆ ∆ φ =

− ν ∆ 9 − ( α ∆ 7 − ν

∆ ∆ φ =

− ν ∆ 9 − ( α ∆ 7 − ν





⎧X = X U θ W UDGLDO ⎪⎪ 3R3 ⎨Y = Y U θ W FLUFRQIpUHQWLHO ⎪ ⎪⎩Z = & W ⋅ ]D[LDO

- 46 -

⎧ ⎪ωU = ⎪⎪ ⎨ωθ = ⎪ ⎛ Y ∂Y ∂X ⎞ ⎟ − ⎪ω] = ⎜⎜ + ⎝ U ∂ U U ∂ θ ⎟⎠ ⎪⎩

⎧X = X U θ W UDGLDO ⎪⎪ 3R3 ⎨Y = Y U θ W FLUFRQIpUHQWLHO ⎪ ⎪⎩Z = & W ⋅ ]D[LDO

- 46 -

⎧ ⎪ωU = ⎪⎪ ⎨ωθ = ⎪ ⎛ Y ∂Y ∂X ⎞ ⎟ − ⎪ω] = ⎜⎜ + ⎝ U ∂ U U ∂ θ ⎟⎠ ⎪⎩

⎧ ⎪γ θ ] = ⎪ ⎪ ⎪ ⎨γ ] U = ⎪ ⎪ ⎪γ = ⎛⎜ ∂ X − Y + ∂ Y ⎞⎟ ⎪ Uθ ⎜ U ∂θ U ∂ U ⎟ ⎝ ⎠ ⎩

⎧ ∂X ⎪ε U = ∂ U ⎪ ⎪ X ∂Y ⎪ ⎨εθ = + U U ∂θ ⎪ ⎪ ⎪ε = & W ⎪ ] ⎩

⎧ ∂X X ∂Y + + +& ⎪H = GLY 3R3 = WU E = ∂U U U ∂θ ⎪ ⎨ ⎪ ∂ ∂ ⎛ ∂ ⎞ ∂ + ⎪∆ = U ∂ U ⎜⎜ U ∂ U ⎟⎟ + ∂] ⎝ ⎠ U ∂θ ⎩

(TXDWLRQVGH1DYLHU


(TXDWLRQVGH1DYLHU

∂H ∂7 ∂ ω] ⎧ ⎪λ + µ ∂ U − µ U ∂ θ = λ + µ α ∂ U − I U ⎪ ⎨ ∂ ω] ∂H ∂7 ⎪ ⎪λ + µ U ∂ θ + µ ∂ U = λ + µ α U ∂ θ − I θ ⎩

(TXDWLRQGHFRPSDWLELOLWp

∂ εθ ∂ ε U ∂ γ U θ ∂ ε U ∂ εθ ∂ γ U θ + − − + − = ∂ U U ∂ θ U ∂ U ∂ θ U ∂ U U ∂ U U ∂θ

(TXDWLRQVDX[FRQWUDLQWHV ⎧ ∂ σ U σ U − σθ ∂ τU θ + + = − IU ⎪ U U ∂θ ⎪ ∂U ⎪ ∂ σθ ⎪ ∂ τU θ + τU θ + = − Iθ ⎨ ∂ U U U ∂θ ⎪ G ⎪ ⎪∆ σ + σ = − GLY I + ( α ∆ 7 U θ ⎪ − ν ⎩

)RQFWLRQG¶$LU\

⎧ ∂ σ U σ U − σθ ∂ τU θ + + = − IU ⎪ U U ∂θ ⎪ ∂U ⎪ ∂ σθ ⎪ ∂ τU θ + τU θ + = − Iθ ⎨ ∂ U U U ∂θ ⎪ G ⎪ ⎪∆ σ + σ = − GLY I + ( α ∆ 7 U θ ⎪ − ν ⎩

σU =

∂ φ ∂ φ ∂ ⎛ ∂φ⎞ ∂ φ ⎜ ⎟ + − 9 σθ = − 9 τU θ = − ∂ U ⎜⎝ U ∂ θ ⎟⎠ U ∂U U ∂θ ∂U

- 47 -

- 47 -

⎧ ⎪γ θ ] = ⎪ ⎪ ⎪ ⎨γ ] U = ⎪ ⎪ ⎪γ = ⎛⎜ ∂ X − Y + ∂ Y ⎞⎟ ⎪ Uθ ⎜ U ∂θ U ∂ U ⎟ ⎝ ⎠ ⎩

⎧ ∂X ⎪ε U = ∂ U ⎪ ⎪ X ∂Y ⎪ ⎨εθ = + U U ∂θ ⎪ ⎪ ⎪ε = & W ⎪ ] ⎩


(TXDWLRQVGH1DYLHU







(TXDWLRQVDX[FRQWUDLQWHV

(TXDWLRQVDX[FRQWUDLQWHV ⎧ ∂ σ U σ U − σθ ∂ τU θ + + = − IU ⎪ U U ∂θ ⎪ ∂U ⎪ ∂ σθ ⎪ ∂ τU θ + τU θ + = − Iθ ⎨ ∂ U U U ∂θ ⎪ G ⎪ ⎪∆ σ + σ = − GLY I + ( α ∆ 7 U θ ⎪ − ν ⎩

)RQFWLRQG¶$LU\

σU =

⎧ ⎪γ θ ] = ⎪ ⎪ ⎪ ⎨γ ] U = ⎪ ⎪ ⎪γ = ⎛⎜ ∂ X − Y + ∂ Y ⎞⎟ ⎪ Uθ ⎜ U ∂θ U ∂ U ⎟ ⎝ ⎠ ⎩

⎧ ∂X ⎪ε U = ∂ U ⎪ ⎪ X ∂Y ⎪ ⎨εθ = + U U ∂θ ⎪ ⎪ ⎪ε = & W ⎪ ] ⎩

(TXDWLRQVGH1DYLHU


)RQFWLRQG¶$LU\

∂ φ ∂ φ ∂ ⎛ ∂φ⎞ ∂ φ ⎜ ⎟ + − 9 σθ = − 9 τU θ = − ∂ U ⎜⎝ U ∂ θ ⎟⎠ U ∂U U ∂θ ∂U



σU =



⎧ ⎪γ θ ] = ⎪ ⎪ ⎪ ⎨γ ] U = ⎪ ⎪ ⎪γ = ⎛⎜ ∂ X − Y + ∂ Y ⎞⎟ ⎪ Uθ ⎜ U ∂θ U ∂ U ⎟ ⎝ ⎠ ⎩

⎧ ∂X ⎪ε U = ∂ U ⎪ ⎪ X ∂Y ⎪ ⎨εθ = + U U ∂θ ⎪ ⎪ ⎪ε = & W ⎪ ] ⎩

⎧ ∂ σ U σ U − σθ ∂ τU θ + + = − IU ⎪ U U ∂θ ⎪ ∂U ⎪ ∂ σθ ⎪ ∂ τU θ + τU θ + = − Iθ ⎨ ∂ U U U ∂θ ⎪ G ⎪ ⎪∆ σ + σ = − GLY I + ( α ∆ 7 U θ ⎪ − ν ⎩

)RQFWLRQG¶$LU\

∂ φ ∂ φ ∂ ⎛ ∂φ⎞ ∂ φ ⎜ ⎟ σ = − 9 τU θ = − + − 9 θ ∂ U ⎜⎝ U ∂ θ ⎟⎠ U ∂U U ∂θ ∂U

- 47 -

σU =


- 47 -

([HPSOH

([HPSOH

&RQGXLWH F\OLQGULTXH G¶XQ IOXLGH FKDXG 2Q QRWH 5 HW 5 OHV UD\RQV LQWpULHXUV HW H[WpULHXU OHVSDURLVLQWHUQHHWH[WHUQHVRQWVXSSRVpHVPDLQWHQXHVDX[WHPSpUDWXUHV 7 HW 7 UHVSHFWLYHPHQW/HVVHFWLRQVH[WUrPHVVRQWPDLQWHQXHVjODGLVWDQFHFRQVWDQWH/


,O V¶DJLW G¶XQ SUREOqPH GH GpIRUPDWLRQ SODQH D[LV\PpWULTXH DYHF ε ] = TXH QRXV DOORQV UpVRXGUHHQFRRUGRQQpHVF\OLQGULTXHVG¶D[H]D[HGHODFRTXH


/HFKDPSGHWHPSpUDWXUHDpWpFDOFXOpDXFKDSLWUHSUHPLHU


2QDWURXYp

2QDWURXYp

7 = $ /RJ U + % DYHF $ =

7 − 7 7 /RJ 5 − 7 /RJ 5 HW % = /RJ 5 − /RJ 5 /RJ 5 − /RJ 5

7 = $ /RJ U + % DYHF $ =

7 − 7 7 /RJ 5 − 7 /RJ 5 HW % = /RJ 5 − /RJ 5 /RJ 5 − /RJ 5

)LJXUH

)LJXUH

/DSUHPLqUHpTXDWLRQGH1DYLHUV¶pFULW

G GU


⎛ GX X ⎞ + ν G7 ⎜⎜ + ⎟⎟ = α ⎝ GU U ⎠ − ν GU

G GU

⎛ GX X ⎞ + ν G7 ⎜⎜ + ⎟⎟ = α ⎝ GU U ⎠ − ν GU

2XU GpVLJQHOHGpSODFHPHQWGH3SXUHPHQWUDGLDO


8QHSUHPLqUHLQWpJUDWLRQGRQQH


GX X + ν + ν + = α 7 + & = α $ /RJ U + % + & GU U − ν − ν


8QHVHFRQGHLQWpJUDWLRQGRQQH X=

8QHVHFRQGHLQWpJUDWLRQGRQQH

+ ν ⎡$ ⎛ $⎞ '⎤ α ⎢ U /RJ U + ⎜ % + & − ⎟ U + ⎥ − ν ⎣ ⎝ ⎠ U⎦

X=

+ ν ⎡$ ⎛ $⎞ '⎤ α ⎢ U /RJ U + ⎜ % + & − ⎟ U + ⎥ − ν ⎣ ⎝ ⎠ U⎦

- 48 -

- 48 -

([HPSOH

([HPSOH







2QDWURXYp

2QDWURXYp

7 = $ /RJ U + % DYHF $ =

7 − 7 7 /RJ 5 − 7 /RJ 5 HW % = /RJ 5 − /RJ 5 /RJ 5 − /RJ 5

7 = $ /RJ U + % DYHF $ =

7 − 7 7 /RJ 5 − 7 /RJ 5 HW % = /RJ 5 − /RJ 5 /RJ 5 − /RJ 5

)LJXUH


G GU

)LJXUH


⎛ GX X ⎞ + ν G7 ⎜⎜ + ⎟⎟ = α ⎝ GU U ⎠ − ν GU

G GU

⎛ GX X ⎞ + ν G7 ⎜⎜ + ⎟⎟ = α ⎝ GU U ⎠ − ν GU






8QHVHFRQGHLQWpJUDWLRQGRQQH X=


8QHVHFRQGHLQWpJUDWLRQGRQQH

+ ν ⎡$ ⎛ $⎞ '⎤ α U /RJ U + ⎜ % + & − ⎟ U + ⎥ − ν ⎢⎣ ⎝ ⎠ U⎦

- 48 -

X=

+ ν ⎡$ ⎛ $⎞ '⎤ α U /RJ U + ⎜ % + & − ⎟ U + ⎥ − ν ⎢⎣ ⎝ ⎠ U⎦

- 48 -

2QHQGpGXLW

2QHQGpGXLW ⎧ GX + ν ⎡$ ⎛ $⎞ '⎤ /RJ U + ⎜ % + & + ⎟ − ⎥ = α ⎪ε U = G U − ν ⎢⎣ ⎝ ⎠ U ⎦ ⎪ ⎨ ⎪ε = X = + ν α ⎡ $ /RJ U + ⎛⎜ % + & − $ ⎞⎟ + ' ⎤ ⎪⎩ θ U − ν ⎢⎣ ⎝ ⎠ U ⎥⎦

⎧ GX + ν ⎡$ ⎛ $⎞ '⎤ /RJ U + ⎜ % + & + ⎟ − ⎥ = α ⎪ε U = G U − ν ⎢⎣ ⎝ ⎠ U ⎦ ⎪ ⎨ ⎪ε = X = + ν α ⎡ $ /RJ U + ⎛⎜ % + & − $ ⎞⎟ + ' ⎤ ⎪⎩ θ U − ν ⎢⎣ ⎝ ⎠ U ⎥⎦

SXLVSDUOHVUHODWLRQV GH+RRNH'XKDPHO


⎧ (α ⎡$ ' (α ⎤ ⎪⎪σ U = − ν ⎢ − − (7 + & )⎥ + − ν (7 + & ) U ⎣ ⎦ ⎨ ⎪σθ = ( α ⎡− $ + ' − (7 + & )⎤ + ( α (7 + & ) ⎥⎦ − ν − ν ⎢⎣ U ⎩⎪

⎧ (α ⎡$ ' (α ⎤ ⎪⎪σ U = − ν ⎢ − − (7 + & )⎥ + − ν (7 + & ) U ⎣ ⎦ ⎨ ⎪σθ = ( α ⎡− $ + ' − (7 + & )⎤ + ( α (7 + & ) ⎥⎦ − ν − ν ⎢⎣ U ⎩⎪

/HV FRQGLWLRQV DX[ OLPLWHV σ U = SRXU U = 5 HW U = 5 GRQQHQW OHV FRQVWDQWHV G¶LQWpJUDWLRQ ⎧ 7 5 − 7 5 $ − − ν − − ν 7 ⎪& = − ν 5 − 5 ⎪ ⎨ ⎪' = 5 5 (7 − 7 ) ⎪ 5 − 5 ⎩


±&2175$,17(63/$1(6&3

±&2175$,17(63/$1(6&3

8QVROLGHVHWURXYHHQpWDWGHFRQWUDLQWHSODQHV¶LOH[LVWHXQUHSqUHRUWKRQRUPpOLpjFHVROLGH SDUUDSSRUWDXTXHO ± G¶XQHSDUWODWHPSpUDWXUH7[\W QHGpSHQGSDVGH] ± G¶DXWUHSDUWODPDWULFHGHVFRQWUDLQWHVLQGpSHQGDQWHGH]HVWGHODIRUPH


⎡ σ[ ⎢ [Σ] = ⎢τ[\ ⎢ ⎢⎣

τ [\ σ\

⎤ ⎥ ⎥ ⎥ ⎥⎦

⎡ σ[ ⎢ [Σ] = ⎢τ[\ ⎢ ⎢⎣

τ [\ σ\

⎤ ⎥ ⎥ ⎥ ⎥⎦

G ,O HQ UpVXOWH G¶DSUqV OHV pTXDWLRQV GH O¶pTXLOLEUH ORFDO TXH OHV IRUFHV YROXPLTXHV I VRQW LQGpSHQGDQWHV GH ] HW QRUPDOHV j O¶D[H ]′] ,O HQ UpVXOWH DXVVL TXH OD PDWULFH GpIRUPDWLRQ LQGpSHQGDQWHGH]HVWGHODIRUPH ⎧ ⎪ε [ = ( σ [ − νσ \ + α 7 − 7 ⎪ ⎡ ε [ γ [\ ⎤ ⎪ε = σ − νσ + α 7 − 7 ⎢ ⎥ [ ⎪ \ ( \ [E] = ⎢⎢γ [\ ε \ ⎥⎥ DYHF ⎪⎨ ⎪ε = ν σ + σ + α 7 − 7 ⎢ ⎥ \ ⎪ ] ( [ ⎢⎣ ε ] ⎥⎦ ⎪ ⎪τ [\ = ( γ [\ + ν ⎩⎪

G ,O HQ UpVXOWH G¶DSUqV OHV pTXDWLRQV GH O¶pTXLOLEUH ORFDO TXH OHV IRUFHV YROXPLTXHV I VRQW LQGpSHQGDQWHV GH ] HW QRUPDOHV j O¶D[H ]′] ,O HQ UpVXOWH DXVVL TXH OD PDWULFH GpIRUPDWLRQ LQGpSHQGDQWHGH]HVWGHODIRUPH ⎧ ⎪ε [ = ( σ [ − νσ \ + α 7 − 7 ⎪ ⎡ ε [ γ [\ ⎤ ⎪ε = σ − νσ + α 7 − 7 ⎢ ⎥ [ ⎪ \ ( \ [E] = ⎢⎢γ [\ ε \ ⎥⎥ DYHF ⎪⎨ ⎪ε = ν σ + σ + α 7 − 7 ⎢ ⎥ \ ⎪ ] ( [ ⎢⎣ ε ] ⎥⎦ ⎪ ⎪τ [\ = ( γ [\ + ν ⎩⎪

- 49 -

- 49 -

2QHQGpGXLW

2QHQGpGXLW ⎧ GX + ν ⎡$ ⎛ $⎞ '⎤ /RJ U + ⎜ % + & + ⎟ − ⎥ = α ⎪ε U = G U − ν ⎢⎣ ⎝ ⎠ U ⎦ ⎪ ⎨ ⎡ X $ $ + ν ⎛ ⎞ '⎤ ⎪ε = = /RJ U + ⎜ % + & − ⎟ + ⎥ α ⎪⎩ θ U − ν ⎢⎣ ⎝ ⎠ U ⎦

⎧ GX + ν ⎡$ ⎛ $⎞ '⎤ /RJ U + ⎜ % + & + ⎟ − ⎥ = α ⎪ε U = G U − ν ⎢⎣ ⎝ ⎠ U ⎦ ⎪ ⎨ ⎡ X $ $ + ν ⎛ ⎞ '⎤ ⎪ε = = /RJ U + ⎜ % + & − ⎟ + ⎥ α ⎪⎩ θ U − ν ⎢⎣ ⎝ ⎠ U ⎦



⎧ (α ⎡$ ' (α ⎤ ⎪⎪σ U = − ν ⎢ − − (7 + & )⎥ + − ν (7 + & ) U ⎣ ⎦ ⎨ ⎪σθ = ( α ⎡− $ + ' − (7 + & )⎤ + ( α (7 + & ) ⎥⎦ − ν ⎪⎩ − ν ⎢⎣ U

⎧ (α ⎡$ ' (α ⎤ ⎪⎪σ U = − ν ⎢ − − (7 + & )⎥ + − ν (7 + & ) U ⎣ ⎦ ⎨ ⎪σθ = ( α ⎡− $ + ' − (7 + & )⎤ + ( α (7 + & ) ⎥⎦ − ν ⎪⎩ − ν ⎢⎣ U



±&2175$,17(63/$1(6&3

±&2175$,17(63/$1(6&3



⎡ σ[ ⎢ [Σ] = ⎢τ[\ ⎢ ⎣⎢

τ [\ σ\

⎤ ⎥ ⎥ ⎥ ⎦⎥

⎡ σ[ ⎢ [Σ] = ⎢τ[\ ⎢ ⎣⎢

τ [\ σ\

⎤ ⎥ ⎥ ⎥ ⎦⎥

G ,O HQ UpVXOWH G¶DSUqV OHV pTXDWLRQV GH O¶pTXLOLEUH ORFDO TXH OHV IRUFHV YROXPLTXHV I VRQW LQGpSHQGDQWHV GH ] HW QRUPDOHV j O¶D[H ]′] ,O HQ UpVXOWH DXVVL TXH OD PDWULFH GpIRUPDWLRQ LQGpSHQGDQWHGH]HVWGHODIRUPH ⎧ ⎪ε [ = ( σ [ − νσ \ + α 7 − 7 ⎪ ⎡ ε [ γ [\ ⎤ ⎪ε = σ − νσ + α 7 − 7 ⎢ ⎥ [ ⎪ \ ( \ [E] = ⎢⎢γ [\ ε \ ⎥⎥ DYHF ⎪⎨ ⎪ε = ν σ + σ + α 7 − 7 ⎢ ⎥ \ ⎪ ] ( [ ε ] ⎦⎥ ⎣⎢ ⎪ ⎪τ [\ = ( γ [\ ⎪⎩ + ν

G ,O HQ UpVXOWH G¶DSUqV OHV pTXDWLRQV GH O¶pTXLOLEUH ORFDO TXH OHV IRUFHV YROXPLTXHV I VRQW LQGpSHQGDQWHV GH ] HW QRUPDOHV j O¶D[H ]′] ,O HQ UpVXOWH DXVVL TXH OD PDWULFH GpIRUPDWLRQ LQGpSHQGDQWHGH]HVWGHODIRUPH ⎧ ⎪ε [ = ( σ [ − νσ \ + α 7 − 7 ⎪ ⎡ ε [ γ [\ ⎤ ⎪ε = σ − νσ + α 7 − 7 ⎢ ⎥ [ ⎪ \ ( \ [E] = ⎢⎢γ [\ ε \ ⎥⎥ DYHF ⎪⎨ ⎪ε = ν σ + σ + α 7 − 7 ⎢ ⎥ \ ⎪ ] ( [ ε ] ⎦⎥ ⎣⎢ ⎪ ⎪τ [\ = ( γ [\ ⎪⎩ + ν

- 49 -

- 49 -

RXLQYHUVHPHQW

RXLQYHUVHPHQW

( (α ⎧ ⎪σ [ = − ν ε [ + νε \ − − ν 7 − 7 ⎪ ( (α ⎪ ε \ + νε [ − 7 − 7 ⎨σ \ = −ν − ν ⎪ ⎪ ( γ [\ ⎪τ [\ = + ν ⎩

DYHF

ε] = −

+ ν ν ε [ + ε \ + α 7 − 7 − ν − ν


DYHF

ε] = −

+ ν ν ε [ + ε \ + α 7 − 7 − ν − ν

5HPDUTXHVOHVFRQWUDLQWHVSODQHVVHUHQFRQWUHQWGDQVOHVSODTXHVPLQFHVSHUSHQGLFXODLUHVj O¶D[H ] VRXPLVHV j GHV HIIRUWV SHUSHQGLFXODLUHV j ] HW LQGpSHQGDQWV GH ] DLQVL TX¶j GHV FKDPSVGHWHPSpUDWXUHLQGpSHQGDQWVGH]O¶pTXDWLRQ SRXUL M GRQQH G ν GLY I + ( α ∆ 7 =


8Q VROLGH QH SHXWGGRQF VH WURXYHU ULJRXUHXVHPHQW HQ pWDW GH FRQWUDLQWHV SODQHV TXH VL OHV IRUFHVYROXPLTXHV I HWOHVWHPSpUDWXUHV7VDWLVIRQWjFHWWHUHODWLRQ


1RXVSUHQGURQVWRXMRXUV[\ GDQVOHSODQPR\HQGHODSODTXH


5pVROXWLRQVG¶XQSUREOqPHGHFRQWUDLQWHVSODQHV UHPpWKRGHPpWKRGHGHVGpSODFHPHQWV


⎧X = X [ \ ] W ⎪⎪ 3R3 ⎨Y = Y [ \ ] W ⎪ ⎩⎪Z = ] ⋅ ε [ [ \ W

H = GLY 3R3 = WU E = ε [ + ε \ + ε ] =

⎧ ⎛∂Z ∂Y⎞ ⎟ − ⎪ω[ = ⎜⎜ ⎝ ∂ \ ∂ ] ⎟⎠ ⎪ ⎪ ⎛ ∂X ∂Z ⎞ ⎪ ⎟ − ⎨ω\ = ⎜⎜ ⎝ ∂ ] ∂ [ ⎟⎠ ⎪ ⎪ ⎪ω = ⎛⎜ ∂ Y − ∂ X ⎞⎟ ⎪ ] ⎜∂[ ∂\⎟ ⎝ ⎠ ⎩

⎧X = X [ \ ] W ⎪⎪ 3R3 ⎨Y = Y [ \ ] W ⎪ ⎩⎪Z = ] ⋅ ε [ [ \ W

− ν + ν ε [ + ε \ + α 7 − 7 − ν − ν

H = GLY 3R3 = WU E = ε [ + ε \ + ε ] =

/HVpTXDWLRQVGH1DYLHU'XKDPHOV¶pFULYHQWVXUOHVD[HV[HW\ ⎧ ∂ ⎪ ⎪∂ [ ⎨ ⎪ ∂ ⎪∂ \ ⎩

⎛ ∂X ∂Y ⎞ ∂7 − ν ⎜⎜ ⎟⎟ + − ν ∆ X = + ν α + − I[ ∂[ ( ⎝∂[ ∂\⎠ ⎛ ∂X ∂Y ⎞ ∂7 − ν ⎜⎜ ⎟⎟ + − ν ∆ X = + ν α + − I\ ∂\ ( ⎝∂[ ∂\⎠

⎧ ⎛∂Z ∂Y⎞ ⎟ − ⎪ω[ = ⎜⎜ ⎝ ∂ \ ∂ ] ⎟⎠ ⎪ ⎪ ⎛ ∂X ∂Z ⎞ ⎪ ⎟ − ⎨ω\ = ⎜⎜ ⎝ ∂ ] ∂ [ ⎟⎠ ⎪ ⎪ ⎪ω = ⎛⎜ ∂ Y − ∂ X ⎞⎟ ⎪ ] ⎜∂[ ∂\⎟ ⎝ ⎠ ⎩

− ν + ν ε [ + ε \ + α 7 − 7 − ν − ν

/HVpTXDWLRQVGH1DYLHU'XKDPHOV¶pFULYHQWVXUOHVD[HV[HW\

⎧ ∂ ⎪ ⎪∂ [ ⎨ ⎪ ∂ ⎪∂ \ ⎩


- 50 -

- 50 -

RXLQYHUVHPHQW

RXLQYHUVHPHQW


DYHF

ε] = −

+ ν ν ε [ + ε \ + α 7 − 7 − ν − ν


DYHF

ε] = −

+ ν ν ε [ + ε \ + α 7 − 7 − ν − ν









⎧X = X [ \ ] W ⎪⎪ 3R3 ⎨Y = Y [ \ ] W ⎪ ⎪⎩Z = ] ⋅ ε [ [ \ W

H = GLY 3R3 = WU E = ε [ + ε \ + ε ] =

⎧ ⎛∂Z ∂Y⎞ ⎟ − ⎪ω[ = ⎜⎜ ⎝ ∂ \ ∂ ] ⎟⎠ ⎪ ⎪ ⎛ ∂X ∂Z ⎞ ⎪ ⎟ − ⎨ω\ = ⎜⎜ ⎝ ∂ ] ∂ [ ⎟⎠ ⎪ ⎪ ⎪ω = ⎛⎜ ∂ Y − ∂ X ⎞⎟ ⎪ ] ⎜∂[ ∂\⎟ ⎝ ⎠ ⎩

⎧X = X [ \ ] W ⎪⎪ 3R3 ⎨Y = Y [ \ ] W ⎪ ⎪⎩Z = ] ⋅ ε [ [ \ W

− ν + ν ε [ + ε \ + α 7 − 7 − ν − ν

H = GLY 3R3 = WU E = ε [ + ε \ + ε ] =


⎧ ∂ ⎪ ⎪∂ [ ⎨ ⎪ ∂ ⎪∂ \ ⎩


- 50 -

⎧ ⎛∂Z ∂Y⎞ ⎟ − ⎪ω[ = ⎜⎜ ⎝ ∂ \ ∂ ] ⎟⎠ ⎪ ⎪ ⎛ ∂X ∂Z ⎞ ⎪ ⎟ − ⎨ω\ = ⎜⎜ ⎝ ∂ ] ∂ [ ⎟⎠ ⎪ ⎪ ⎪ω = ⎛⎜ ∂ Y − ∂ X ⎞⎟ ⎪ ] ⎜∂[ ∂\⎟ ⎝ ⎠ ⎩

− ν + ν ε [ + ε \ + α 7 − 7 − ν − ν


⎧ ∂ ⎪ ⎪∂ [ ⎨ ⎪ ∂ ⎪∂ \ ⎩


- 50 -



/HV WURLV FRQWUDLQWHV LQFRQQXHV σ [ σ [ τ [\ SHXYHQW rWUH GpWHUPLQpHV SDU LQWpJUDWLRQ GX


V\VWqPH

V\VWqPH ∂ τ [\ ∂ σ \ ⎧ ∂ σ [ ∂ τ [\ + = − I[ + = − I\ ⎪ ∂ ∂ ∂[ ∂\ [ \ ⎪ ⎨ G ⎪ + ν GLY I + ( α ∆ 7 ⎪∆ σ [ + σ \ = − − ν ⎩

∂ τ [\ ∂ σ \ ⎧ ∂ σ [ ∂ τ [\ + = − I[ + = − I\ ⎪ ∂ ∂ ∂[ ∂\ [ \ ⎪ ⎨ G ⎪ + ν GLY I + ( α ∆ 7 ⎪∆ σ [ + σ \ = − − ν ⎩

HPpWKRGHPpWKRGHG¶$LU\


G 3RXU I = JUDG 9 RQSRVH


σ[ =

−∂ φ ∂ φ ∂ φ − 9 σ = − 9 τ = [\ \ ∂[ ∂\ ∂ \ ∂ [

σ[ =

/D IRQFWLRQ GH FRQWUDLQWH φ [ \ RX IRQFWLRQ G¶$LU\ VDWLVIDLW DXWRPDWLTXHPHQW DX[ GHX[ SUHPLqUHVpTXDWLRQV pTXLOLEUHORFDO ODWURLVLqPHGHYLHQW ∆ ∆ φ =

− ν ∆ 9 − ( α ∆ 7 − ν

)RUPXODLUHHQFRRUGRQQpHVF\OLQGULTXHV

/D IRQFWLRQ GH FRQWUDLQWH φ [ \ RX IRQFWLRQ G¶$LU\ VDWLVIDLW DXWRPDWLTXHPHQW DX[ GHX[ SUHPLqUHVpTXDWLRQV pTXLOLEUHORFDO ODWURLVLqPHGHYLHQW ∆ ∆ φ =

− ν ∆ 9 − ( α ∆ 7 − ν


⎧X = U θ ] W ⎪⎪ 3R3 ⎨Y = U θ ] W ⎪ ⎩⎪Z = ] ⋅ ε ] U θ W

−∂ φ ∂ φ ∂ φ − 9 σ = − 9 τ = [\ \ ∂[ ∂\ ∂ \ ∂ [

⎧ ⎛ ∂Z ∂Y ⎞ ⎟ − ⎪ωU = ⎜⎜ ⎝ U ∂ θ ∂ ] ⎟⎠ ⎪ ⎪ ⎛∂X ∂Z ⎞ ⎪ ⎟ − ⎨ωθ = ⎜⎜ ⎝ ∂ ] ∂ U ⎟⎠ ⎪ ⎪ ⎪ω = ⎛⎜ ∂ Y + Y − ∂ X ⎞⎟ ⎪ ] ⎜ ∂U U U ∂θ ⎟ ⎝ ⎠ ⎩

⎧ ∂X ⎪ε U = ∂ U ⎪ ⎪ X ∂Y ⎪ + ⎨ε θ = U U ∂θ ⎪ ⎪ ⎪ε = − ν ε + ε + + ν α 7 − 7 [ \ ⎪ ] − ν − ν ⎩

⎧X = U θ ] W ⎪⎪ 3R3 ⎨Y = U θ ] W ⎪ ⎩⎪Z = ] ⋅ ε ] U θ W

⎧ ⎪γ θ ] = ⎪ ⎪ ⎪ ⎨γ ] U = ⎪ ⎪ ⎪γ = ⎛⎜ ∂ Y + Y − ∂ X ⎞⎟ ⎪ Uθ ⎜ ∂ U U U ∂θ ⎟ ⎝ ⎠ ⎩



- 51 -

⎧ ⎪γ θ ] = ⎪ ⎪ ⎪ ⎨γ ] U = ⎪ ⎪ ⎪γ = ⎛⎜ ∂ Y + Y − ∂ X ⎞⎟ ⎪ Uθ ⎜ ∂ U U U ∂θ ⎟ ⎝ ⎠ ⎩

- 51 -





V\VWqPH

V\VWqPH ∂ τ [\ ∂ σ \ ⎧ ∂ σ [ ∂ τ [\ + = − I[ + = − I\ ⎪ ∂\ ∂[ ∂\ ⎪ ∂[ ⎨ G ⎪ + ν GLY I + ( α ∆ 7 ⎪∆ σ [ + σ \ = − − ν ⎩

∂ τ [\ ∂ σ \ ⎧ ∂ σ [ ∂ τ [\ + = − I[ + = − I\ ⎪ ∂\ ∂[ ∂\ ⎪ ∂[ ⎨ G ⎪ + ν GLY I + ( α ∆ 7 ⎪∆ σ [ + σ \ = − − ν ⎩





σ[ =

−∂ φ ∂ φ ∂ φ − 9 σ = − 9 τ = [\ \ ∂[ ∂\ ∂ \ ∂ [

σ[ =

/D IRQFWLRQ GH FRQWUDLQWH φ [ \ RX IRQFWLRQ G¶$LU\ VDWLVIDLW DXWRPDWLTXHPHQW DX[ GHX[ SUHPLqUHVpTXDWLRQV pTXLOLEUHORFDO ODWURLVLqPHGHYLHQW

∆ ∆ φ =

− ν ∆ 9 − ( α ∆ 7 − ν


/D IRQFWLRQ GH FRQWUDLQWH φ [ \ RX IRQFWLRQ G¶$LU\ VDWLVIDLW DXWRPDWLTXHPHQW DX[ GHX[ SUHPLqUHVpTXDWLRQV pTXLOLEUHORFDO ODWURLVLqPHGHYLHQW

∆ ∆ φ =

− ν ∆ 9 − ( α ∆ 7 − ν


⎧X = U θ ] W ⎪⎪ 3R3 ⎨Y = U θ ] W ⎪ ⎪⎩Z = ] ⋅ ε ] U θ W

−∂ φ ∂ φ ∂ φ − 9 σ = − 9 τ = [\ \ ∂[ ∂\ ∂ \ ∂ [



- 51 -

⎧ ⎪γ θ ] = ⎪ ⎪ ⎪ ⎨γ ] U = ⎪ ⎪ ⎪γ = ⎛⎜ ∂ Y + Y − ∂ X ⎞⎟ ⎪ Uθ ⎜ ∂ U U U ∂θ ⎟ ⎝ ⎠ ⎩

⎧X = U θ ] W ⎪⎪ 3R3 ⎨Y = U θ ] W ⎪ ⎪⎩Z = ] ⋅ ε ] U θ W



- 51 -

⎧ ⎪γ θ ] = ⎪ ⎪ ⎪ ⎨γ ] U = ⎪ ⎪ ⎪γ = ⎛⎜ ∂ Y + Y − ∂ X ⎞⎟ ⎪ Uθ ⎜ ∂ U U U ∂θ ⎟ ⎝ ⎠ ⎩

H = GLY 3R3 = WU E = ε U + εθ + ε ] =

∆=

− ν + ν ε U + ε θ + α 7 − 7 − ν − ν

∂ ⎛ ∂ ⎞ ∂ ∂ ⎜⎜ U ⎟+ + ⎟ U ∂ U ⎝ ∂ U ⎠ U ∂ θ ∂ ]

H = GLY 3R3 = WU E = ε U + εθ + ε ] =

∆=

− ν + ν ε U + ε θ + α 7 − 7 − ν − ν

∂ ⎛ ∂ ⎞ ∂ ∂ ⎜⎜ U ⎟+ + ⎟ U ∂ U ⎝ ∂ U ⎠ U ∂ θ ∂ ]

(TXDWLRQVGH1DYLHU'XKDPHO


⎧ ∂ ⎛ ∂X X ∂Y⎞ ⎛ ∂7 X ∂Y⎞ − ν ⎟⎟ + − ν ⎜⎜ ∆ X − − ⎟⎟ = + ν α + + − IU ⎪ ⎜⎜ ∂U ( U U ∂θ ⎠ ⎝ ⎪∂ U ⎝ ∂ U U U ∂ θ ⎠ ⎨ ⎛ ⎪ ∂ ⎛ ∂ X X ∂ Y ⎞ Y ∂X ⎞ ∂7 − ν ⎪ U ∂ θ ⎜⎜ ∂ U + U + U ∂ θ ⎟⎟ + − ν ⎜⎜ ∆ Y − − ∂ θ ⎟⎟ = + ν α U ∂ U − ( I θ U U ⎝ ⎠ ⎝ ⎠ ⎩


(TXDWLRQGHFRPSDWLELOLWpLGHQWLTXHj'3






⎧ ∂ σ U σ U − σθ ∂ τU θ + + = − IU ⎪ U U ∂θ ⎪ ∂U ⎪ ∂ σθ ⎪ ∂ τU θ + τU θ + = − Iθ ⎨ ∂ U U U ∂θ ⎪ G ⎪ ⎪∆ σ + σ = − + ν GLY I + ( α ∆ 7 U θ ⎪ − ν ⎩

)RQFWLRQG¶$LU\LGHQWLTXHj'3

σU =



H = GLY 3R3 = WU E = ε U + εθ + ε ] =

∆=

σU =


- 52 -

− ν + ν ε U + ε θ + α 7 − 7 − ν − ν

∂ ⎛ ∂ ⎞ ∂ ∂ ⎜⎜ U ⎟⎟ + + U ∂U ⎝ ∂U ⎠ U ∂θ ∂]

H = GLY 3R3 = WU E = ε U + εθ + ε ] =

∆=

− ν + ν ε U + ε θ + α 7 − 7 − ν − ν

∂ ⎛ ∂ ⎞ ∂ ∂ ⎜⎜ U ⎟⎟ + + U ∂U ⎝ ∂U ⎠ U ∂θ ∂]













σU =


- 52 -



∂ φ ∂ φ ∂ ⎛ ∂φ⎞ ∂ φ ⎜ ⎟ σ = − 9 τU θ = − + − 9 θ U ⎜⎝ U ∂ θ ⎟⎠ ∂ U ∂ U U ∂ θ ∂U

- 52 -

σU =

∂ φ ∂ φ ∂ ⎛ ∂φ⎞ ∂ φ ⎜ ⎟ σ = − 9 τU θ = − + − 9 θ U ⎜⎝ U ∂ θ ⎟⎠ ∂ U ∂ U U ∂ θ ∂U

- 52 -

([HPSOHFRQWUDLQWHVWKHUPLTXHVGDQVXQHSODTXHUHFWDQJXODLUH


)LJXUH

)LJXUH

/D ILJXUH GRQQH OD JpRPpWULH OHV D[HV HW OHV OLDLVRQV O¶DVVHPEODJH HVW UpDOLVp j OD E⎞ ⎛ WHPSpUDWXUH 7 VDQVMHXQLVHUUDJH2QSRUWHOHERUG ⎜ \ = − ⎟ jODWHPSpUDWXUH 7 OHERUG ⎠ ⎝ E⎞ D⎞ K⎞ ⎛ ⎛ ⎛ ⎜ \ = + ⎟ j OD WHPSpUDWXUH 7 OHV IDFHV ⎜ [ = ± ⎟ HW ⎜ ] = ± ⎟ VRQW UHFRXYHUWHV G¶XQ ⎠ ⎠ ⎠ ⎝ ⎝ ⎝ LVRODQWSDUIDLWVDQVULJLGLWp

/D ILJXUH GRQQH OD JpRPpWULH OHV D[HV HW OHV OLDLVRQV O¶DVVHPEODJH HVW UpDOLVp j OD E⎞ ⎛ WHPSpUDWXUH 7 VDQVMHXQLVHUUDJH2QSRUWHOHERUG ⎜ \ = − ⎟ jODWHPSpUDWXUH 7 OHERUG ⎠ ⎝ E⎞ D⎞ K⎞ ⎛ ⎛ ⎛ ⎜ \ = + ⎟ j OD WHPSpUDWXUH 7 OHV IDFHV ⎜ [ = ± ⎟ HW ⎜ ] = ± ⎟ VRQW UHFRXYHUWHV G¶XQ ⎠ ⎠ ⎠ ⎝ ⎝ ⎝ LVRODQWSDUIDLWVDQVULJLGLWp

2Q YHXW FDOFXOHU HQ UpJLPH VWDWLRQQDLUH OHV WHPSpUDWXUHV 7 WUDQVODWLRQV 3R3 URWDWLRQV GpIRUPDWLRQV E HWFRQWUDLQWHV Σ


/D SODTXH VH WURXYH HQ pWDW GH FRQWUDLQWHV SODQHV HW O¶RQ WURXYH OD VROXWLRQ SDU O¶XQH TXHOFRQTXHGHVPpWKRGHSURSRVpHV


7 + 7 7 − 7 + \ E

7=

⎧X = ⎪ ⎪⎪ ⎡⎛ 7 + 7 7 − 7 \ − ] ⎤ ⎞ 3R3 ⎨Y = + ν α ⎢⎜ − 7 ⎟ \ + ⎥ E ⎦⎥ ⎠ ⎪ ⎣⎢⎝ ⎪ ⎪⎩Z = + ν α 7 − 7 ]

7 − 7 ⎧ ⎪ω[ = + ν α E ] ⎪ ⎨ω\ = ⎪ ⎪ω] = ⎩

7 + 7 7 − 7 + \ E

7=

⎧X = ⎪ ⎪⎪ ⎡⎛ 7 + 7 7 − 7 \ − ] ⎤ ⎞ 3R3 ⎨Y = + ν α ⎢⎜ − 7 ⎟ \ + ⎥ E ⎦⎥ ⎠ ⎪ ⎣⎢⎝ ⎪ ⎪⎩Z = + ν α 7 − 7 ]

7 − 7 ⎧ ⎪ω[ = + ν α E ] ⎪ ⎨ω\ = ⎪ ⎪ω] = ⎩

- 53 -

- 53 -



)LJXUH

)LJXUH

/D ILJXUH GRQQH OD JpRPpWULH OHV D[HV HW OHV OLDLVRQV O¶DVVHPEODJH HVW UpDOLVp j OD E⎞ ⎛ WHPSpUDWXUH 7 VDQVMHXQLVHUUDJH2QSRUWHOHERUG ⎜ \ = − ⎟ jODWHPSpUDWXUH 7 OHERUG ⎠ ⎝ E D K⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎜ \ = + ⎟ j OD WHPSpUDWXUH 7 OHV IDFHV ⎜ [ = ± ⎟ HW ⎜ ] = ± ⎟ VRQW UHFRXYHUWHV G¶XQ ⎠ ⎠ ⎠ ⎝ ⎝ ⎝ LVRODQWSDUIDLWVDQVULJLGLWp

/D ILJXUH GRQQH OD JpRPpWULH OHV D[HV HW OHV OLDLVRQV O¶DVVHPEODJH HVW UpDOLVp j OD E⎞ ⎛ WHPSpUDWXUH 7 VDQVMHXQLVHUUDJH2QSRUWHOHERUG ⎜ \ = − ⎟ jODWHPSpUDWXUH 7 OHERUG ⎠ ⎝ E D K⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎜ \ = + ⎟ j OD WHPSpUDWXUH 7 OHV IDFHV ⎜ [ = ± ⎟ HW ⎜ ] = ± ⎟ VRQW UHFRXYHUWHV G¶XQ ⎠ ⎠ ⎠ ⎝ ⎝ ⎝ LVRODQWSDUIDLWVDQVULJLGLWp





7 + 7 7 − 7 + \ E

7=

⎧X = ⎪ ⎪⎪ ⎡⎛ 7 + 7 7 − 7 \ − ] ⎤ ⎞ 3R3 ⎨Y = + ν α ⎢⎜ − 7 ⎟ \ + ⎥ E ⎥⎦ ⎠ ⎢⎣⎝ ⎪ ⎪ ⎪⎩Z = + ν α 7 − 7 ]

7 − 7 ⎧ ⎪ω[ = + ν α E ] ⎪ ⎨ω\ = ⎪ ⎪ω] = ⎩

- 53 -

7 + 7 7 − 7 + \ E

7=

⎧X = ⎪ ⎪⎪ ⎡⎛ 7 + 7 7 − 7 \ − ] ⎤ ⎞ 3R3 ⎨Y = + ν α ⎢⎜ − 7 ⎟ \ + ⎥ E ⎥⎦ ⎠ ⎢⎣⎝ ⎪ ⎪ ⎪⎩Z = + ν α 7 − 7 ]

7 − 7 ⎧ ⎪ω[ = + ν α E ] ⎪ ⎨ω\ = ⎪ ⎪ω] = ⎩

- 53 -

⎧ γ \] = ⎪ ⎪ ⎨ γ ][ = ⎪ ⎪⎩γ [\ =

⎧ε [ = ⎪⎪ ⎨ε \ = + ν α 7 − 7 ⎪ ⎪⎩ε ] = + ν α 7 − 7

σ [ = −( α 7 − 7 σ \ = σ ] = = τ \] = τ][ = τ [\

⎧ γ \] = ⎪ ⎪ ⎨ γ ][ = ⎪ ⎪⎩γ [\ =

⎧ε [ = ⎪⎪ ⎨ε \ = + ν α 7 − 7 ⎪ ⎪⎩ε ] = + ν α 7 − 7

σ [ = −( α 7 − 7 σ \ = σ ] = = τ \] = τ][ = τ [\

±e7$7$;,6<0e75,48(0e5,',(1

±e7$7$;,6<0e75,48(0e5,',(1

&¶HVW OH FDV SDU GpILQLWLRQ R HQ FRRUGRQQpHV F\OLQGULTXHV U θ ] G¶D[H ]′] OH YHFWHXU WUDQVODWLRQHVWGHODIRUPH ⎧X = X U ] W UDGLDO ⎪⎪ 3R3 ⎨Y = FLUFRQIpUHQWLHO ⎪ ⎩⎪Z = Z U ] W D[LDO

&¶HVW OH FDV SDU GpILQLWLRQ R HQ FRRUGRQQpHV F\OLQGULTXHV U θ ] G¶D[H ]′] OH YHFWHXU WUDQVODWLRQHVWGHODIRUPH ⎧X = X U ] W UDGLDO ⎪⎪ 3R3 ⎨Y = FLUFRQIpUHQWLHO ⎪ ⎩⎪Z = Z U ] W D[LDO

&HFLDOLHXORUVTXHR]HVWD[HGHUpYROXWLRQSRXUODVWUXFWXUHSRXUOHFKDPSGHWHPSpUDWXUH SRXU OHV IRUFHV DSSOLTXpHV GRQQpHV HW GH OLDLVRQ TXL Q¶RQW GH SOXV SDV GH FRPSRVDQWH FLUFRQIpUHQWLHOOH


2QDDORUVOHVIRUPXOHVVXLYDQWHVHQFRRUGRQQpHVF\OLQGULTXHV


⎧ ⎪ωU = ⎪ ⎪ ⎛ ∂X ∂Z ⎞ ⎟ − ⎨ωθ = ⎜⎜ ⎝ ∂ ] ∂ U ⎟⎠ ⎪ ⎪ ⎪ω] = ⎩

⎧ ⎪γ θ ] = ⎪ ⎪ ⎛ ∂X ∂Z ⎞ ⎟ − ⎨γ ]U = ⎜⎜ ⎝ ∂ ] ∂ U ⎟⎠ ⎪ ⎪ ⎪γ U θ = ⎩

⎧ ∂X ⎪ εU = ∂ U ⎪ X ⎪ ⎨ εθ = U ⎪ ⎪ ∂Z ⎪ε ] = ∂] ⎩

⎧ ⎪ε U = ( [σ U − ν σθ + σ ] ]+ α 7 − 7 ⎪ ⎪ ⎨ε θ = [σθ − ν σ ] + σ U ]+ α 7 − 7 ( ⎪ ⎪ ⎪ε ] = [σ ] − ν σ U + σθ ]+ α 7 − 7 ( ⎩

H = GLY 3R3 = WU E = ε U + εθ + ε ] =

∆=

⎧ ⎪τ θ ] = ⎪ + ν ⎪ γ ]U ⎨τ]U = ( ⎪ ⎪ ⎪τ U θ = ⎩

∂X X ∂Z + + ∂U U ∂]

∂ ⎛ ∂ ⎞ ∂ ⎜U ⎟+ U ∂ U ⎜⎝ ∂ U ⎟⎠ ∂ ]

⎧ ⎪ωU = ⎪ ⎪ ⎛ ∂X ∂Z ⎞ ⎟ − ⎨ωθ = ⎜⎜ ⎝ ∂ ] ∂ U ⎟⎠ ⎪ ⎪ ⎪ω] = ⎩

⎧ ⎪γ θ ] = ⎪ ⎪ ⎛ ∂X ∂Z ⎞ ⎟ − ⎨γ ]U = ⎜⎜ ⎝ ∂ ] ∂ U ⎟⎠ ⎪ ⎪ ⎪γ U θ = ⎩

⎧ ∂X ⎪ εU = ∂ U ⎪ X ⎪ ⎨ εθ = U ⎪ ⎪ ∂Z ⎪ε ] = ∂] ⎩


H = GLY 3R3 = WU E = ε U + εθ + ε ] =

∆=

⎧ ⎪τ θ ] = ⎪ + ν ⎪ γ ]U ⎨τ]U = ( ⎪ ⎪ ⎪τ U θ = ⎩

∂X X ∂Z + + ∂U U ∂]

∂ ⎛ ∂ ⎞ ∂ ⎜U ⎟+ U ∂ U ⎜⎝ ∂ U ⎟⎠ ∂ ]

- 54 -

- 54 -

⎧ γ \] = ⎪ ⎪ ⎨ γ ][ = ⎪ ⎪⎩γ [\ =

⎧ε [ = ⎪⎪ ⎨ε \ = + ν α 7 − 7 ⎪ ⎪⎩ε ] = + ν α 7 − 7

σ [ = −( α 7 − 7 σ \ = σ ] = = τ \] = τ][ = τ [\

⎧ γ \] = ⎪ ⎪ ⎨ γ ][ = ⎪ ⎪⎩γ [\ =

⎧ε [ = ⎪⎪ ⎨ε \ = + ν α 7 − 7 ⎪ ⎪⎩ε ] = + ν α 7 − 7

σ [ = −( α 7 − 7 σ \ = σ ] = = τ \] = τ][ = τ [\

±e7$7$;,6<0e75,48(0e5,',(1

±e7$7$;,6<0e75,48(0e5,',(1

&¶HVW OH FDV SDU GpILQLWLRQ R HQ FRRUGRQQpHV F\OLQGULTXHV U θ ] G¶D[H ]′] OH YHFWHXU WUDQVODWLRQHVWGHODIRUPH ⎧X = X U ] W UDGLDO ⎪⎪ 3R3 ⎨Y = FLUFRQIpUHQWLHO ⎪ ⎪⎩Z = Z U ] W D[LDO

&¶HVW OH FDV SDU GpILQLWLRQ R HQ FRRUGRQQpHV F\OLQGULTXHV U θ ] G¶D[H ]′] OH YHFWHXU WUDQVODWLRQHVWGHODIRUPH ⎧X = X U ] W UDGLDO ⎪⎪ 3R3 ⎨Y = FLUFRQIpUHQWLHO ⎪ ⎪⎩Z = Z U ] W D[LDO





⎧ ⎪ωU = ⎪ ⎪ ⎛ ∂X ∂Z ⎞ ⎟ − ⎨ωθ = ⎜⎜ ⎝ ∂ ] ∂ U ⎟⎠ ⎪ ⎪ ⎪ω] = ⎩

⎧ ⎪γ θ ] = ⎪ ⎪ ⎛ ∂X ∂Z ⎞ ⎟ − ⎨γ ]U = ⎜⎜ ⎝ ∂ ] ∂ U ⎟⎠ ⎪ ⎪ ⎪γ U θ = ⎩

⎧ ∂X ⎪ εU = ∂ U ⎪ X ⎪ ⎨ εθ = U ⎪ ⎪ ∂Z ⎪ε ] = ∂] ⎩


H = GLY 3R3 = WU E = ε U + εθ + ε ] =

∆=

⎧ ⎪τ θ ] = ⎪ + ν ⎪ γ ]U ⎨τ]U = ( ⎪ ⎪ ⎪τ U θ = ⎩

∂X X ∂Z + + ∂U U ∂]

∂ ⎛ ∂ ⎞ ∂ ⎜U ⎟+ U ∂ U ⎜⎝ ∂ U ⎟⎠ ∂ ]

- 54 -

⎧ ⎪ωU = ⎪ ⎪ ⎛ ∂X ∂Z ⎞ ⎟ − ⎨ωθ = ⎜⎜ ⎝ ∂ ] ∂ U ⎟⎠ ⎪ ⎪ ⎪ω] = ⎩

⎧ ⎪γ θ ] = ⎪ ⎪ ⎛ ∂X ∂Z ⎞ ⎟ − ⎨γ ]U = ⎜⎜ ⎝ ∂ ] ∂ U ⎟⎠ ⎪ ⎪ ⎪γ U θ = ⎩

⎧ ∂X ⎪ εU = ∂ U ⎪ X ⎪ ⎨ εθ = U ⎪ ⎪ ∂Z ⎪ε ] = ∂] ⎩


H = GLY 3R3 = WU E = ε U + εθ + ε ] =

∆=

⎧ ⎪τ θ ] = ⎪ + ν ⎪ γ ]U ⎨τ]U = ( ⎪ ⎪ ⎪τ U θ = ⎩

∂X X ∂Z + + ∂U U ∂]

∂ ⎛ ∂ ⎞ ∂ ⎜U ⎟+ U ∂ U ⎜⎝ ∂ U ⎟⎠ ∂ ]

- 54 -


⎧ ⎪λ + µ ⎪ ⎨ ⎪ ⎪λ + µ ⎩


∂ ∂U

⎛ ∂X X ∂Z ⎞ ∂ ⎛∂Z ∂X ⎞ ∂7 ⎜⎜ ⎟⎟ − µ ⎜⎜ ⎟⎟ = λ + µ α + + − − IU U U ] ] U ] ∂ ∂ ∂ ∂ ∂ ∂U ⎝ ⎠ ⎝ ⎠

∂ ∂]

⎛ ∂ ⎞ ⎛ ∂X ∂ Z ⎞ ⎛ ∂X X ∂Z ⎞ ∂7 ⎟⎟ = λ + µ ⎜⎜ ⎟⎟ − µ ⎜⎜ + ⎟⎟ ⎜⎜ − I] − + + ∂] ⎝ ∂U U ⎠ ⎝ ∂] ∂U ⎠ ⎝ ∂U U ∂] ⎠

(TXDWLRQVGHO¶pTXLOLEUHORFDO

⎧ ⎪λ + µ ⎪ ⎨ ⎪ ⎪λ + µ ⎩

∂ ∂U


∂ ∂]

⎛ ∂ ⎞ ⎛ ∂X ∂ Z ⎞ ⎛ ∂X X ∂Z ⎞ ∂7 ⎟⎟ = λ + µ ⎜⎜ ⎟⎟ − µ ⎜⎜ + ⎟⎟ ⎜⎜ − I] − + + ∂] ⎝ ∂U U ⎠ ⎝ ∂] ∂U ⎠ ⎝ ∂U U ∂] ⎠


∂ σ U σ U − σθ ∂ τU] ∂τ ∂ σ] + + = − I U U] + τU] + = − I] ∂U U ∂] ∂] U ∂]


(TXDWLRQVGH%HOWUDPL'XKDPHO


∆ σU −

G (α ⎡ ∂I ν ∂ V ( α ∂ 7 ⎤ = − ⎢ U + ∆7 + σ U − σ θ − GLY I + ⎥ + ν ∂U − ν + ν ∂ U ⎥⎦ U ⎣⎢ ∂ U − ν

∆ σU −

G (α ⎡ ∂I ν ∂ V ( α ∂ 7 ⎤ = − ⎢ U + ∆7 + σ U − σ θ − GLY I + ⎥ + ν ∂U − ν + ν ∂ U ⎥⎦ U ⎣⎢ ∂ U − ν

∆ σθ −

G (α ⎡ ∂V (α ∂7⎤ ν σ U − σ θ − GLY I + = − ⎢ IU + ∆7 + + ν U ∂ U − ν − ν + ν U ∂ U ⎥⎦ U ⎣U

∆ σθ −


∆ σ] +

G (α ⎡ ∂ I] ∂ V ( α ∂ 7 ⎤ ν GLY I 7 = − + + ∆ + ⎢ ⎥ + ν ∂ ] − ν + ν ∂ ] ⎥⎦ ⎢⎣ ∂ ] − ν

∆ σ] +

G (α ⎡ ∂ I] ∂ V ( α ∂ 7 ⎤ ν GLY I 7 = − + + ∆ + ⎢ ⎥ + ν ∂ ] − ν + ν ∂ ] ⎥⎦ ⎢⎣ ∂ ] − ν

∆ τU] −

⎡∂ I ∂ V ∂I ( α ∂ 7 ⎤ τ + =−⎢ ] + U + ⎥ U] + ν ∂ U ∂ ] ∂ ] + ν ∂ U ∂ ] ⎥⎦ U ⎢⎣ ∂ U

∆ τU] −

⎡∂ I ∂ V ∂I ( α ∂ 7 ⎤ τ + =−⎢ ] + U + ⎥ U] + ν ∂ U ∂ ] ∂ ] + ν ∂ U ∂ ] ⎥⎦ U ⎢⎣ ∂ U

DYHF

∆V = −

G ∂I G (α + ν ∂I ∆ 7 HW GLY I = U + I U + ] GLY I − − ν ∂U U ∂] − ν

DYHF

∆V = −


±6<0e75,(63+e5,48(

±6<0e75,(63+e5,48(

8QSUREOqPHGHWKHUPRpODVWLFLWpSRVVqGHODV\PpWULHVSKpULTXHVLOHYHFWHXUWUDQVODWLRQ 3R3 HVWSXUHPHQWUDGLDOHWQHGpSHQGTXHGHODFRRUGRQQpH U = 23 HWGXWHPSVpYHQWXHOOHPHQW 2GpVLJQHXQSRLQWIL[HDSSHOpS{OH


,O V¶DJLW G¶XQ FDV D[LV\PpWULTXH PpULGLHQ SDUWLFXOLHU R WRXW D[H SDVVDQW SDU 2 HVW D[H GH UpYROXWLRQ


- 55 -

- 55 -


⎧ ⎪λ + µ ⎪ ⎨ ⎪ ⎪λ + µ ⎩


∂ ∂U


∂ ∂]

⎛ ∂ ⎞ ⎛ ∂X ∂ Z ⎞ ⎛ ∂X X ∂Z ⎞ ∂7 ⎟⎟ = λ + µ ⎜⎜ ⎟⎟ − µ ⎜⎜ + ⎟⎟ ⎜⎜ − I] − + + ∂] ⎝ ∂U U ⎠ ⎝ ∂] ∂U ⎠ ⎝ ∂U U ∂] ⎠


⎧ ⎪λ + µ ⎪ ⎨ ⎪ ⎪λ + µ ⎩

∂ ∂U


∂ ∂]

⎛ ∂ ⎞ ⎛ ∂X ∂ Z ⎞ ⎛ ∂X X ∂Z ⎞ ∂7 ⎟⎟ = λ + µ ⎜⎜ ⎟⎟ − µ ⎜⎜ + ⎟⎟ ⎜⎜ − I] − + + ∂] ⎝ ∂U U ⎠ ⎝ ∂] ∂U ⎠ ⎝ ∂U U ∂] ⎠






∆ σU −

G (α ⎡ ∂ IU ν ∂ V ( α ∂ 7 ⎤ σ − σ − = − + + ∆ + GLY I 7 ⎢ ⎥ θ U + ν ∂ U − ν + ν ∂ U ⎥⎦ U ⎢⎣ ∂ U − ν

∆ σU −

G (α ⎡ ∂ IU ν ∂ V ( α ∂ 7 ⎤ σ − σ − = − + + ∆ + GLY I 7 ⎢ ⎥ θ U + ν ∂ U − ν + ν ∂ U ⎥⎦ U ⎢⎣ ∂ U − ν

∆ σθ −


∆ σθ −


∆ σ] +

G (α ⎡ ∂ I] ∂ V ( α ∂ 7 ⎤ ν GLY I 7 = − + + ∆ + ⎢ ⎥ + ν ∂ ] − ν + ν ∂ ] ⎦⎥ ⎣⎢ ∂ ] − ν

∆ σ] +

G (α ⎡ ∂ I] ∂ V ( α ∂ 7 ⎤ ν GLY I 7 = − + + ∆ + ⎢ ⎥ + ν ∂ ] − ν + ν ∂ ] ⎦⎥ ⎣⎢ ∂ ] − ν

∆ τU] −

⎡ ∂ I] ∂ IU ( α ∂ 7 ⎤ ∂ V τ + = − + + ⎢ ⎥ U] + ν ∂U ∂] ∂ ] + ν ∂ U ∂ ] ⎥⎦ U ⎢⎣ ∂ U

∆ τU] −

⎡ ∂ I] ∂ IU ( α ∂ 7 ⎤ ∂ V τ + = − + + ⎢ ⎥ U] + ν ∂U ∂] ∂ ] + ν ∂ U ∂ ] ⎥⎦ U ⎢⎣ ∂ U

DYHF

∆V = −


DYHF

∆V = −


±6<0e75,(63+e5,48(

±6<0e75,(63+e5,48(





- 55 -

- 55 -

)LJXUH

G G G $SSHORQV U ϕ θ OHVFRRUGRQQpHVVSKpULTXHVGH 32 'DQVODEDVH 32 U ϕ θ OHVIRUPXOHV GHWKHUPRpODVWLFLWpV¶pFULYHQW

{

}

G ⎧X = X U W VXLYDQWU ⎪⎪ G 3R3 ⎨Y = VXLYDQWϕ G ⎪ ⎪⎩Z = VXLYDQWθ

∂X X ⎧ εϕ = εθ = εF = γθϕ = γϕ U = γU = ⎪⎪ε U = ∂ U U ⎨ ⎪εF GpVLJQHODGLODWDWLRQOLQpDLUHUHODWLYHFLUFRQIpUHQWLHOOH ⎪⎩

)LJXUH

⎧ ωU = ⎪⎪ ⎨ωϕ = ⎪ ⎪⎩ωθ =

/DVHXOHpTXDWLRQGHFRPSDWLELOLWpV¶pFULW


{


∂X X ⎧ εϕ = εθ = εF = γθϕ = γϕ U = γU = ⎪⎪ε U = ∂ U U ⎨ ⎪εF GpVLJQHODGLODWDWLRQOLQpDLUHUHODWLYHFLUFRQIpUHQWLHOOH ⎪⎩


∂ε ε U = εF + U F ∂U

ε U = εF + U

/HVUHODWLRQVGH+RRNH'XKDPHOV¶pFULYHQW

}

⎧ ωU = ⎪⎪ ⎨ωϕ = ⎪ ⎪⎩ωθ =

∂ εF ∂U


⎧ ⎪⎪ε U = ( σ U − ν σF + α 7 − 7 ⎨ ⎪ε = [ − ν σ − ν σ ]+ α 7 − 7 F U ⎪⎩ F (


- 56 -

- 56 -

)LJXUH


{


∂X X ⎧ εϕ = εθ = εF = γθϕ = γϕ U = γU = ⎪⎪ε U = ∂ U U ⎨ ⎪εF GpVLJQHODGLODWDWLRQOLQpDLUHUHODWLYHFLUFRQIpUHQWLHOOH ⎩⎪

⎧ ωU = ⎪⎪ ⎨ωϕ = ⎪ ⎪⎩ωθ =


∂ε ε U = εF + U F ∂U


)LJXUH

}


{


∂X X ⎧ εϕ = εθ = εF = γθϕ = γϕ U = γU = ⎪⎪ε U = ∂ U U ⎨ ⎪εF GpVLJQHODGLODWDWLRQOLQpDLUHUHODWLYHFLUFRQIpUHQWLHOOH ⎩⎪

/DVHXOHpTXDWLRQGHFRPSDWLELOLWpV¶pFULW ε U = εF + U

⎧ ωU = ⎪⎪ ⎨ωϕ = ⎪ ⎪⎩ωθ =

∂ εF ∂U




- 56 -

- 56 -

}

RXLQYHUVHPHQW

RXLQYHUVHPHQW ( (α ⎧ ⎪⎪σ U = + ν − ν [ − ν ε U − ν εF ]− − ν 7 − 7 ⎨ ( ⎪σF = [εF − ν εU ]− ( α 7 − 7 + ν − ν − ν ⎩⎪

( (α ⎧ ⎪⎪σ U = + ν − ν [ − ν ε U − ν εF ]− − ν 7 − 7 ⎨ ( ⎪σF = [εF − ν εU ]− ( α 7 − 7 + ν − ν − ν ⎩⎪

/DVHXOHpTXDWLRQG¶pTXLOLEUHORFDOV¶pFULWVXUO¶D[HUDGLDO

∂ σU + σ U − σ F = − I U ∂U U

2QDGHSOXV

H = GLY 3R3 = WU E = ε U + εθ + εϕ = ε U + εF =

HW

∆=


I θ = I ϕ = I F = ∂X X + ∂U U

∂ ⎛ ∂ ⎞ ⎜⎜ U ⎟⎟ U ∂ U ⎝ ∂ U ⎠

/HFKDPS 3R3 HVWLUURWDWLRQQHOHWO¶pTXDWLRQGH1DYLHUV¶pFULW

∂ σU + σ U − σ F = − I U ∂U U

2QDGHSOXV


HW

∆=

I θ = I ϕ = I F = ∂X X + ∂U U

∂ ⎛ ∂ ⎞ ⎜⎜ U ⎟⎟ U ∂ U ⎝ ∂ U ⎠


∂ ⎡ ∂ XU ⎤ + ν ∂ 7 + ν − ν α − IU ⎢ ⎥= ∂ U ⎣⎢ U ∂ U ⎦⎥ − ν ∂ U ( − ν

∂ ⎡ ∂ XU ⎤ + ν ∂ 7 + ν − ν α − IU ⎢ ⎥= ∂ U ⎣⎢ U ∂ U ⎦⎥ − ν ∂ U ( − ν



2QSHXWGpWHUPLQHUOHVFRQWUDLQWHV σ U HW σF SDULQWpJUDWLRQGXV\VWqPHG¶pTXDWLRQVpTXLOLEUH HWFRPSDWLELOLWp


∂ ∂ σU [σ U + σ F ] = − + ν I U − ( α ∂ 7 + σ U − σF = − I U ∂U − ν − ν ∂U ∂U U

(QVpSDUDQWOHVYDULDEOHVRQDUULYHDXV\VWqPHpTXLYDOHQW

/DSUHPLqUHVDQVVHFRQGPHPEUHDGPHWSRXUVROXWLRQJpQpUDOH σU =


⎧ ∂ ⎡ ∂ σU ⎤ ⎛ (α ∂7 ⎞ ∂I ⎟ IU + U U + + σ U ⎥ = −⎜⎜ ⎪ ⎢U U U U ∂ ∂ − ν ∂ − ν ∂ U ⎟⎠ ⎦ ⎝ ⎪ ⎣ ⎨ ⎞ ⎪ U ⎛ ∂ σU ⎪σF = σ U + ⎜⎜ ∂ U + I U ⎟⎟ ⎝ ⎠ ⎩

/DSUHPLqUHVDQVVHFRQGPHPEUHDGPHWSRXUVROXWLRQJpQpUDOH

$ + % U

σU =

- 57 -

$ + % U

- 57 -

RXLQYHUVHPHQW

RXLQYHUVHPHQW ( (α ⎧ ⎪⎪σ U = + ν − ν [ − ν ε U − ν εF ]− − ν 7 − 7 ⎨ ( (α ⎪σF = [εF − ν εU ]− 7 − 7 ⎪⎩ + ν − ν − ν

( (α ⎧ ⎪⎪σ U = + ν − ν [ − ν ε U − ν εF ]− − ν 7 − 7 ⎨ ( (α ⎪σF = [εF − ν εU ]− 7 − 7 ⎪⎩ + ν − ν − ν


∂ σU + σ U − σ F = − I U ∂U U

2QDGHSOXV


HW

∆=


I θ = I ϕ = I F = ∂X X + ∂U U

∂ ⎛ ∂ ⎞ ⎜⎜ U ⎟⎟ U ∂ U ⎝ ∂ U ⎠



⎧ ∂ ⎡ ∂ σU ⎤ ⎛ (α ∂7 ⎞ ∂I ⎟ IU + U U + + σ U ⎥ = −⎜⎜ ⎪ ⎢U U U U ∂ ∂ − ν ∂ − ν ∂ U ⎟⎠ ⎦ ⎝ ⎪ ⎣ ⎨ ⎞ ⎪ U ⎛ ∂ σU ⎪σF = σ U + ⎜⎜ ∂ U + I U ⎟⎟ ⎝ ⎠ ⎩

∂ ∂U

⎡ ∂ XU ⎤ + ν ∂ 7 + ν − ν α − IU ⎢ ⎥= ∂ U ⎦⎥ − ν ∂ U ( − ν ⎣⎢ U

∂ σU + σ U − σ F = − I U ∂U U

2QDGHSOXV


HW

∆=

I θ = I ϕ = I F = ∂X X + ∂U U

∂ ⎛ ∂ ⎞ ⎜⎜ U ⎟⎟ U ∂ U ⎝ ∂ U ⎠


∂ ∂U

⎡ ∂ XU ⎤ + ν ∂ 7 + ν − ν α − IU ⎢ ⎥= ∂ U ⎦⎥ − ν ∂ U ( − ν ⎣⎢ U







⎧ ∂ ⎡ ∂ σU ⎤ ⎛ (α ∂7 ⎞ ∂I ⎟ IU + U U + + σ U ⎥ = −⎜⎜ ⎪ ⎢U ∂ U − ν ∂ U ⎟⎠ ⎦ ⎝− ν ⎪∂ U ⎣ ∂ U ⎨ ⎞ ⎪ U ⎛ ∂ σU ⎪σF = σ U + ⎜⎜ ∂ U + I U ⎟⎟ ⎝ ⎠ ⎩


$ + % U

- 57 -



⎧ ∂ ⎡ ∂ σU ⎤ ⎛ (α ∂7 ⎞ ∂I ⎟ IU + U U + + σ U ⎥ = −⎜⎜ ⎪ ⎢U ∂ U − ν ∂ U ⎟⎠ ⎦ ⎝− ν ⎪∂ U ⎣ ∂ U ⎨ ⎞ ⎪ U ⎛ ∂ σU ⎪σF = σ U + ⎜⎜ ∂ U + I U ⎟⎟ ⎝ ⎠ ⎩


$ + % U

- 57 -

([HPSOHFRTXHVSKpULTXHDYHFJUDGLHQWGHWHPSpUDWXUHUDGLDO


8QH VSKqUH FUHXVH D SRXU UD\RQ 5 LQWpULHXU HW 5 H[WpULHXU OD SDURL LQWHUQH HVW PDLQWHQXHj 7 ODSDURLH[WpULHXUHHVWPDLQWHQXHj 7


$X FKDSLWUH SUpFpGHQW QRXV DYRQV WURXYp OH FKDPS GH WHPSpUDWXUH 7 = % − $=

5 5 7 5 75 7 − 7 HW % = 5 − 5 5 − 5

$ DYHF U


/¶LQWpJUDWLRQGXV\VWqPH GRQQH

5 5 7 5 75 7 − 7 HW % = 5 − 5 5 − 5

$ DYHF U


⎧ (α ⎪σ U = − − ν ⎪ ⎨ ⎪ (α ⎪σF = − − ν ⎩

⎡% + & $ ' ⎤ ⎢⎣ − U + U ⎥⎦ ⎡% + & $ ' ⎤ ⎢⎣ − U − U ⎥⎦

⎧ (α ⎪σ U = − − ν ⎪ ⎨ ⎪ (α ⎪σF = − − ν ⎩

⎡% + & $ ' ⎤ ⎢⎣ − U + U ⎥⎦ ⎡% + & $ ' ⎤ ⎢⎣ − U − U ⎥⎦

/HV FRQGLWLRQV DX[ OLPLWHV σ U = SRXU U = 5 HW U = 5 SHUPHW GH GpWHUPLQHU OHV FRQVWDQWHVG¶LQWpJUDWLRQ


% + & 5 5 5 + 5 7 − 7 5 5 7 − 7 = HW ' = 5 − 5 5 − 5

% + & 5 5 5 + 5 7 − 7 5 5 7 − 7 = HW ' = 5 − 5 5 − 5

'¶RILQDOHPHQW

'¶RILQDOHPHQW

⎧ ( α 7 − 7 5 5 ⎪σ U = − ν 5 − 5 ⎪ ⎨ ⎪ ( α 7 − 7 5 5 ⎪σF = − ν 5 − 5 ⎩

⎤ ⎡ 5 + 5 5 + 5 5 5 − 5 + 5 ⎥ ⎢ U U ⎦⎥ ⎣⎢ ⎤ ⎡ 5 + 5 5 + 5 5 5 − 5 + 5 ⎥ ⎢ U U ⎥⎦ ⎢⎣

⎧ ( α 7 − 7 5 5 ⎪σ U = − ν 5 − 5 ⎪ ⎨ ⎪ ( α 7 − 7 5 5 ⎪σF = − ν 5 − 5 ⎩

⎤ ⎡ 5 + 5 5 + 5 5 5 − 5 + 5 ⎥ ⎢ U U ⎦⎥ ⎣⎢ ⎤ ⎡ 5 + 5 5 + 5 5 5 − 5 + 5 ⎥ ⎢ U U ⎥⎦ ⎢⎣

- 58 -

- 58 -






5 5 7 5 75 7 − 7 HW % = 5 − 5 5 − 5

$ DYHF U



5 5 7 5 75 7 − 7 HW % = 5 − 5 5 − 5

$ DYHF U


⎧ (α ⎪σ U = − − ν ⎪ ⎨ ⎪ (α ⎪σF = − − ν ⎩

⎡% + & $ ' ⎤ ⎢⎣ − U + U ⎥⎦ ⎡% + & $ ' ⎤ ⎢⎣ − U − U ⎥⎦

⎧ (α ⎪σ U = − − ν ⎪ ⎨ ⎪ (α ⎪σF = − − ν ⎩

⎡% + & $ ' ⎤ ⎢⎣ − U + U ⎥⎦ ⎡% + & $ ' ⎤ ⎢⎣ − U − U ⎥⎦



% + & 5 5 5 + 5 7 − 7 5 5 7 − 7 = HW ' = 5 − 5 5 − 5

% + & 5 5 5 + 5 7 − 7 5 5 7 − 7 = HW ' = 5 − 5 5 − 5

'¶RILQDOHPHQW

⎧ ( α 7 − 7 5 5 ⎪σ U = − ν 5 − 5 ⎪ ⎨ ⎪ ( α 7 − 7 5 5 ⎪σF = − ν 5 − 5 ⎩

'¶RILQDOHPHQW

⎤ ⎡ 5 + 5 5 + 5 5 5 − 5 + 5 ⎥ ⎢ U U ⎥⎦ ⎢⎣ ⎤ ⎡ 5 + 5 5 + 5 5 5 − 5 + 5 ⎥ ⎢ U U ⎦⎥ ⎣⎢

⎧ ( α 7 − 7 5 5 ⎪σ U = − ν 5 − 5 ⎪ ⎨ ⎪ ( α 7 − 7 5 5 ⎪σF = − ν 5 − 5 ⎩

- 58 -

⎤ ⎡ 5 + 5 5 + 5 5 5 − 5 + 5 ⎥ ⎢ U U ⎥⎦ ⎢⎣ ⎤ ⎡ 5 + 5 5 + 5 5 5 − 5 + 5 ⎥ ⎢ U U ⎦⎥ ⎣⎢

- 58 -

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7+(50,48('(632875(6

7+(50,48('(632875(6

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/D WKpRULH GHV SRXWUHV D pWp GpGXLWH GH O¶pODVWLFLWp WULGLPHQVLRQQHOOH JUkFH j TXHOTXHV K\SRWKqVHVVLPSOLILFDWULFHVGXHVjOHXUJpRPpWULHVSpFLILTXHHWDX[pWXGHVGH6DLQW9HQDQWHQ WUDFWLRQ IOH[LRQ WRUVLRQ HW FLVDLOOHPHQW 1RXV DMRXWHURQV XQH K\SRWKqVH VXU OH FKDPS GH WHPSpUDWXUHFHOXLFLVHUDVXUFKDTXHVHFWLRQGURLWHOLQpDULVpF¶HVWjGLUHTXHO¶RQSUHQGUD VXU6G¶DEVFLVVHFXUYLOLJQHV 73 = 7* + $\ + %=

/D WKpRULH GHV SRXWUHV D pWp GpGXLWH GH O¶pODVWLFLWp WULGLPHQVLRQQHOOH JUkFH j TXHOTXHV K\SRWKqVHVVLPSOLILFDWULFHVGXHVjOHXUJpRPpWULHVSpFLILTXHHWDX[pWXGHVGH6DLQW9HQDQWHQ WUDFWLRQ IOH[LRQ WRUVLRQ HW FLVDLOOHPHQW 1RXV DMRXWHURQV XQH K\SRWKqVH VXU OH FKDPS GH WHPSpUDWXUHFHOXLFLVHUDVXUFKDTXHVHFWLRQGURLWHOLQpDULVpF¶HVWjGLUHTXHO¶RQSUHQGUD VXU6G¶DEVFLVVHFXUYLOLJQHV 7 3 = 7* + $\ + %=

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± G¶XQHFRQVWDQWH 7* WHPSpUDWXUHDXFHQWUH*GH6VLFLSRLQWHVWVXU6


± GHGHX[YDULDWLRQVGHWHPSpUDWXUHV$
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- 59 -

- 59 -

&+$3,75(,,,

&+$3,75(,,,

7+(50,48('(632875(6

7+(50,48('(632875(6













/D WKpRULH GHV SRXWUHV D pWp GpGXLWH GH O¶pODVWLFLWp WULGLPHQVLRQQHOOH JUkFH j TXHOTXHV K\SRWKqVHVVLPSOLILFDWULFHVGXHVjOHXUJpRPpWULHVSpFLILTXHHWDX[pWXGHVGH6DLQW9HQDQWHQ WUDFWLRQ IOH[LRQ WRUVLRQ HW FLVDLOOHPHQW 1RXV DMRXWHURQV XQH K\SRWKqVH VXU OH FKDPS GH WHPSpUDWXUHFHOXLFLVHUDVXUFKDTXHVHFWLRQGURLWHOLQpDULVpF¶HVWjGLUHTXHO¶RQSUHQGUD VXU6G¶DEVFLVVHFXUYLOLJQHV 73 = 7* + $\ + %=

/D WKpRULH GHV SRXWUHV D pWp GpGXLWH GH O¶pODVWLFLWp WULGLPHQVLRQQHOOH JUkFH j TXHOTXHV K\SRWKqVHVVLPSOLILFDWULFHVGXHVjOHXUJpRPpWULHVSpFLILTXHHWDX[pWXGHVGH6DLQW9HQDQWHQ WUDFWLRQ IOH[LRQ WRUVLRQ HW FLVDLOOHPHQW 1RXV DMRXWHURQV XQH K\SRWKqVH VXU OH FKDPS GH WHPSpUDWXUHFHOXLFLVHUDVXUFKDTXHVHFWLRQGURLWHOLQpDULVpF¶HVWjGLUHTXHO¶RQSUHQGUD VXU6G¶DEVFLVVHFXUYLOLJQHV 7 3 = 7* + $\ + %=







- 59 -

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UpIpUHQFH 7 SHXWGRQFrWUHFRQVLGpUpHFRPPHODVRPPHGHVWURLVWHUPHV 7* − 7 $<%=

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UpIpUHQFH 7 SHXWGRQFrWUHFRQVLGpUpHFRPPHODVRPPHGHVWURLVWHUPHV 7* − 7 $<%=

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([DPLQRQVPDLQWHQDQWFKDFXQGHFHVWURLVFDV


±())2571250$/1(7e&+$8))(0(1781,)250( 7* − 7

±())2571250$/1(7e&+$8))(0(1781,)250( 7* − 7

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± XQWHQVHXUFRQWUDLQWH Σ GHPDWULFH


⎡σ [ ⎢ [Σ] = ⎢ ⎢ ⎢⎣

⎤ ⎥ 1 ⎥ DYHF σ [ = 6 ⎥ ⎥⎦

⎡σ [ ⎢ [Σ] = ⎢ ⎢ ⎢⎣

⎤ ⎥ 1 ⎥ DYHF σ [ = 6 ⎥ ⎥⎦

- 60 -

- 60 -

&RQVLGpURQV XQH WUDQFKH pOpPHQWDLUH GH SRXWUH HQWUH OHV VHFWLRQV GURLWHV YRLVLQHV 6 HW 6′ GLVWDQWHVGHGVILJXUH /DYDULDWLRQGHWHPSpUDWXUH 73 − 7 SDUUDSSRUWjODWHPSpUDWXUHGH UpIpUHQFH 7 SHXWGRQFrWUHFRQVLGpUpHFRPPHODVRPPHGHVWURLVWHUPHV 7* − 7 $<%=

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)LJXUH

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±())2571250$/1(7e&+$8))(0(1781,)250( 7* − 7

±())2571250$/1(7e&+$8))(0(1781,)250( 7* − 7







⎡σ [ ⎢ [Σ] = ⎢ ⎢ ⎢⎣

⎤ ⎥ 1 ⎥ DYHF σ [ = 6 ⎥ ⎥⎦ - 60 -

⎡σ [ ⎢ [Σ] = ⎢ ⎢ ⎢⎣

⎤ ⎥ 1 ⎥ DYHF σ [ = 6 ⎥ ⎥⎦ - 60 -

)LJXUH

)LJXUH

± XQWHQVHXUGpIRUPDWLRQE GHPDWULFH


1 ⎧ ε[ = + α 7* 7 ⎪ (6 ⎡ε [ ⎤ ⎪ ⎥ ⎢ [E] = ⎢ ε \ ⎥ DYHF ⎪⎨ε \ = −ν 1 + α 7* 7 (6 ⎥ ⎢ ⎪ ⎢⎣ ε ] ⎥⎦ ⎪ 1 + α 7* 7 ⎪ε ] = −ν (6 ⎩ G¶DSUqVODORLGH+RRNH'XKDPHO

1 ⎧ ε[ = + α 7* 7 ⎪ (6 ⎡ε [ ⎤ ⎪ ⎥ ⎢ [E] = ⎢ ε \ ⎥ DYHF ⎪⎨ε \ = −ν 1 + α 7* 7 (6 ⎥ ⎢ ⎪ ⎢⎣ ε ] ⎥⎦ ⎪ 1 + α 7* 7 ⎪ε ] = −ν (6 ⎩ G¶DSUqVODORLGH+RRNH'XKDPHO

'HSOXV 6′ VXELWSDUUDSSRUWj6ODWUDQVODWLRQGHYHFWHXU


G ⎡1 G ⎤ + α 7* − 7 ⎥ GV ⋅ , GΛ = ⎢ ⎣ (6 ⎦

G ⎡1 G ⎤ + α 7* − 7 ⎥ GV ⋅ , GΛ = ⎢ ⎣ (6 ⎦

G R , GpVLJQHO¶XQLWDLUHGHO¶D[HGHV;


±020(17)/e&+,66$17 0 = (7*5$',(17'(7(03e5$785($

±020(17)/e&+,66$17 0 = (7*5$',(17'(7(03e5$785($

)LJXUH

)LJXUH

- 61 -

- 61 -

)LJXUH

)LJXUH



1 ⎧ ⎪ε [ = (6 + α 7* 7 ⎤ ⎪ ⎥ 1 ⎪ + α 7* 7 ε \ ⎥ DYHF ⎨ε \ = −ν (6 ⎥ ⎪ ε ] ⎥⎦ ⎪ 1 + α 7* 7 ⎪ε ] = −ν (6 ⎩ G¶DSUqVODORLGH+RRNH'XKDPHO

1 ⎧ ⎪ε [ = (6 + α 7* 7 ⎤ ⎪ ⎥ 1 ⎪ + α 7* 7 ε \ ⎥ DYHF ⎨ε \ = −ν (6 ⎥ ⎪ ε ] ⎥⎦ ⎪ 1 + α 7* 7 ⎪ε ] = −ν (6 ⎩ G¶DSUqVODORLGH+RRNH'XKDPHO

⎡ε [ ⎢ [E] = ⎢ ⎢ ⎢⎣


⎡ε [ ⎢ [E] = ⎢ ⎢ ⎢⎣


G ⎡1 G ⎤ + α 7* − 7 ⎥ GV ⋅ , GΛ = ⎢ ⎣ (6 ⎦

G ⎡1 G ⎤ + α 7* − 7 ⎥ GV ⋅ , GΛ = ⎢ ⎣ (6 ⎦



±020(17)/e&+,66$17 0 = (7*5$',(17'(7(03e5$785($

±020(17)/e&+,66$17 0 = (7*5$',(17'(7(03e5$785($

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- 61 -

- 61 -

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⎡σ [ ⎢ [Σ] = ⎢ ⎢ ⎣⎢

⎤ ⎥ −0 ⎥ DYHF σ [ = = ⋅ < ,= ⎥ ⎦⎥

± XQWHQVHXUGpIRUPDWLRQ E GHPDWULFH

⎡ε [ ⎢ [E] = ⎢ ⎢ ⎢⎣

ε\

⎡σ [ ⎢ [Σ] = ⎢ ⎢ ⎣⎢

⎤ ⎥ −0 ⎥ DYHF σ [ = = ⋅ < ,= ⎥ ⎦⎥


⎤ ⎥ ⎥ DYHF ⎥ ε ] ⎥⎦

⎧ ⎛ 0= ⎞ + α $ ⎟⎟ < ⎪ε [ = ⎜⎜ − (, = ⎝ ⎠ ⎪ ⎨ ⎛ ν 0= ⎞ ⎪ ⎪ε \ = ε ] = ⎜⎜ (, + α $ ⎟⎟ < = ⎝ ⎠ ⎩

⎡ε [ ⎢ [E] = ⎢ ⎢ ⎢⎣

ε\

⎤ ⎥ ⎥ DYHF ⎥ ε ] ⎥⎦

⎧ ⎛ 0= ⎞ + α $ ⎟⎟ < ⎪ε [ = ⎜⎜ − (, = ⎝ ⎠ ⎪ ⎨ ⎛ ν 0= ⎞ ⎪ ⎪ε \ = ε ] = ⎜⎜ (, + α $ ⎟⎟ < = ⎝ ⎠ ⎩

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±020(17)/e&+,66$17 0 < (7*5$',(17'(7(03e5$785(%

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⎡σ [ ⎢ [Σ] = ⎢ ⎢ ⎢⎣

⎤ ⎥ 0 ⎥ DYHF σ [ = < ⋅ = ,< ⎥ ⎥⎦


⎡ε [ ⎢ [E] = ⎢ ⎢ ⎢⎣

ε\

⎡σ [ ⎢ [Σ] = ⎢ ⎢ ⎢⎣

⎤ ⎥ 0 ⎥ DYHF σ [ = < ⋅ = ,< ⎥ ⎥⎦


⎧ ⎛ 0< ⎞ ⎤ + α % ⎟⎟ = ⎪ε [ = ⎜⎜ (, ⎥ ⎝ < ⎠ ⎪ ⎥ DYHF ⎨ ⎥ ⎛ ν 0< ⎞ ⎪ ε ] ⎥⎦ ⎪ε \ = ε ] = ⎜⎜ − (, + α % ⎟⎟ = < ⎝ ⎠ ⎩

G¶DSUqVODORLGH+RRNH'XKDPHO

⎡ε [ ⎢ [E] = ⎢ ⎢ ⎢⎣

ε\

⎧ ⎛ 0< ⎞ ⎤ + α % ⎟⎟ = ⎪ε [ = ⎜⎜ (, ⎥ ⎝ < ⎠ ⎪ ⎥ DYHF ⎨ ⎥ ⎛ ν 0< ⎞ ⎪ ε ] ⎥⎦ ⎪ε \ = ε ] = ⎜⎜ − (, + α % ⎟⎟ = < ⎝ ⎠ ⎩


- 62 -

- 62 -





⎡σ [ ⎢ [Σ] = ⎢ ⎢ ⎢⎣

⎤ ⎥ −0 ⎥ DYHF σ [ = = ⋅ < ,= ⎥ ⎥⎦


⎡ε [ ⎢ [E] = ⎢ ⎢ ⎢⎣

ε\

⎡σ [ ⎢ [Σ] = ⎢ ⎢ ⎢⎣

⎤ ⎥ −0 ⎥ DYHF σ [ = = ⋅ < ,= ⎥ ⎥⎦


⎤ ⎥ ⎥ DYHF ⎥ ε ] ⎥⎦

⎧ ⎛ 0= ⎞ + α $ ⎟⎟ < ⎪ε [ = ⎜⎜ − ⎝ (, = ⎠ ⎪ ⎨ ⎛ ν 0= ⎞ ⎪ ⎪ε \ = ε ] = ⎜⎜ (, + α $ ⎟⎟ < = ⎝ ⎠ ⎩

⎡ε [ ⎢ [E] = ⎢ ⎢ ⎢⎣

ε\

⎤ ⎥ ⎥ DYHF ⎥ ε ] ⎥⎦

⎧ ⎛ 0= ⎞ + α $ ⎟⎟ < ⎪ε [ = ⎜⎜ − ⎝ (, = ⎠ ⎪ ⎨ ⎛ ν 0= ⎞ ⎪ ⎪ε \ = ε ] = ⎜⎜ (, + α $ ⎟⎟ < = ⎝ ⎠ ⎩



±020(17)/e&+,66$17 0 < (7*5$',(17'(7(03e5$785(%

±020(17)/e&+,66$17 0 < (7*5$',(17'(7(03e5$785(%





⎡σ [ ⎢ [Σ] = ⎢ ⎢ ⎣⎢

⎤ ⎥ 0 ⎥ DYHF σ [ = < ⋅ = ,< ⎥ ⎦⎥


⎡ε [ ⎢ [E] = ⎢ ⎢ ⎢⎣

ε\

⎡σ [ ⎢ [Σ] = ⎢ ⎢ ⎣⎢

⎤ ⎥ 0 ⎥ DYHF σ [ = < ⋅ = ,< ⎥ ⎦⎥


⎧ ⎛ 0< ⎞ ⎤ + α % ⎟⎟ = ⎪ε [ = ⎜⎜ (, ⎥ ⎝ < ⎠ ⎪ ⎥ DYHF ⎨ ⎥ ⎛ ν 0< ⎞ ⎪ ε ] ⎥⎦ ⎪ε \ = ε ] = ⎜⎜ − (, + α % ⎟⎟ = < ⎝ ⎠ ⎩


⎡ε [ ⎢ [E] = ⎢ ⎢ ⎢⎣

ε\

⎧ ⎛ 0< ⎞ ⎤ + α % ⎟⎟ = ⎪ε [ = ⎜⎜ (, ⎥ ⎝ < ⎠ ⎪ ⎥ DYHF ⎨ ⎥ ⎛ ν 0< ⎞ ⎪ ε ] ⎥⎦ ⎪ε \ = ε ] = ⎜⎜ − (, + α % ⎟⎟ = < ⎝ ⎠ ⎩


- 62 -

- 62 -

'HSOXV 6′ WRXUQHSDUUDSSRUWj6DXWRXUGH
G R - GpVLJQHO¶XQLWDLUHGHO¶D[HGHV<

G ⎛0 ⎞ = ⎜⎜ < + α % ⎟⎟ GV ⋅ - ⎝ (, < ⎠


&RQVpTXHQFHV)RUPXOHVGH%UHVVH 6

∫

+

6

Λ = Λ +

** +

∫

6

6

+

∫

6

G ⎛0 ⎞ = ⎜⎜ < + α % ⎟⎟ GV ⋅ - ⎝ (, < ⎠

6

∫

+

6

⎡ 0W G ⎛ 0< ⎞G ⎛0 ⎞ G⎤ + α % ⎟⎟ - + ⎜⎜ = − α $ ⎟⎟ . ⎥ ⋅ GV , + ⎜⎜ ⎢ ⎢⎣ *⎝ (,< ⎠ ⎝ (, = ⎠ ⎥⎦ 6

7\ G ⎧⎡ 1 7 G⎫ ⎤G - + N ] ] . ⎬ GV + α (7* − 7 )⎥ , + N \ ⎨⎢ *6 *6 ⎭ ⎦ ⎩⎣ (6

Λ = Λ +

** +

∫

6

6

⎫ ⎧0W G ⎡⎛ 0 ⎞G ⎛0 ⎞ G⎤ ⎪ ⎪ ,Λ &* + ⎢⎜⎜ < + α % ⎟⎟ - + ⎜⎜ = − α $ ⎟⎟ . ⎥ ** ⎬ GV ⎨ ⎠ ⎝ (, = ⎠ ⎦Λ ⎣⎝ (, < ⎪⎭ ⎪⎩ *-

+

∫

6

±32875(&,5&8/$,5(¬3/$102<(1

7\ G ⎧⎡ 1 7 G⎫ ⎤G - + N ] ] . ⎬ GV + α (7* − 7 )⎥ , + N \ ⎨⎢ *6 *6 ⎭ ⎦ ⎩⎣ (6

⎫ ⎧0W G ⎡⎛ 0 ⎞G ⎛0 ⎞ G⎤ ⎪ ⎪ ,Λ &* + ⎢⎜⎜ < + α % ⎟⎟ - + ⎜⎜ = − α $ ⎟⎟ . ⎥ ** ⎬ GV ⎨ ⎠ ⎝ (, = ⎠ ⎦Λ ⎣⎝ (, < ⎪⎭ ⎪⎩ *-

±32875(&,5&8/$,5(¬3/$102<(1

)LJXUH

)LJXUH

- 63 -

- 63 -


G ⎛0 ⎞ = ⎜⎜ < + α % ⎟⎟ GV ⋅ - ⎝ (, < ⎠

+

∫

6

Λ = Λ +

** +

∫

6

6

+

∫

6

G ⎛0 ⎞ = ⎜⎜ < + α % ⎟⎟ GV ⋅ - ⎝ (, < ⎠


⎡ 0W G ⎛ 0< ⎞G ⎛0 ⎞ G⎤ + α % ⎟⎟ - + ⎜⎜ = − α $ ⎟⎟ . ⎥ ⋅ GV , + ⎜⎜ ⎢ ⎝ (,< ⎠ ⎝ (, = ⎠ ⎦⎥ ⎣⎢ *6



&RQVpTXHQFHV)RUPXOHVGH%UHVVH

⎡ 0W G ⎛ 0< ⎞G ⎛0 ⎞ G⎤ + α % ⎟⎟ - + ⎜⎜ = − α $ ⎟⎟ . ⎥ ⋅ GV , + ⎜⎜ ⎢ ⎢⎣ *⎝ (,< ⎠ ⎝ (, = ⎠ ⎥⎦ 6

+

∫

6

7\ G ⎧⎡ 1 7 G⎫ ⎤G - + N ] ] . ⎬ GV + α (7* − 7 )⎥ , + N \ ⎨⎢ *6 *6 ⎭ ⎦ ⎩⎣ (6

⎫ ⎧0W G ⎡⎛ 0 ⎞G ⎛0 ⎞ G⎤ ⎪ ⎪ ,Λ &* + ⎢⎜⎜ < + α % ⎟⎟ - + ⎜⎜ = − α $ ⎟⎟ . ⎥ ** ⎬ GV ⎨ *(, (, ⎠ ⎝ = ⎠ ⎦Λ ⎣⎝ < ⎪⎭ ⎪⎩

±32875(&,5&8/$,5(¬3/$102<(1

⎡ 0W G ⎛ 0< ⎞G ⎛0 ⎞ G⎤ + α % ⎟⎟ - + ⎜⎜ = − α $ ⎟⎟ . ⎥ ⋅ GV , + ⎜⎜ ⎢ ⎝ (,< ⎠ ⎝ (, = ⎠ ⎦⎥ ⎣⎢ *6

Λ = Λ +

** +

∫

6

6

+

∫

6

7\ G ⎧⎡ 1 7 G⎫ ⎤G - + N ] ] . ⎬ GV + α (7* − 7 )⎥ , + N \ ⎨⎢ *6 *6 ⎭ ⎦ ⎩⎣ (6

⎫ ⎧0W G ⎡⎛ 0 ⎞G ⎛0 ⎞ G⎤ ⎪ ⎪ ,Λ &* + ⎢⎜⎜ < + α % ⎟⎟ - + ⎜⎜ = − α $ ⎟⎟ . ⎥ ** ⎬ GV ⎨ *(, (, ⎠ ⎝ = ⎠ ⎦Λ ⎣⎝ < ⎪⎭ ⎪⎩

±32875(&,5&8/$,5(¬3/$102<(1

)LJXUH

)LJXUH

- 63 -

- 63 -

[R\ GpVLJQHOHSODQGHV\PpWULHGHODSRXWUHFLUFXODLUHRXYHUWHRXIHUPpH RHVWOHFHQWUH GHO¶DUFGHFHUFOHFRQVWLWXpSDUODOLJQHPR\HQQH


/D SRXWUH HVW VXSSRVpH FKDUJpH GDQV VRQ SODQ HW VRXPLVH j XQ FKDPS GH WHPSpUDWXUH 7 = 7* + $< &HWWH SRXWUH D pWp pWXGLpH GDQV OH FDV LVRWKHUPH DX WRPH ,, SDJHV HW


/HV UpVXOWDWV V¶pWHQGHQW DLVpPHQW HQ UHPSODoDQW

⎛0 ⎞ 0= 1 SDU ⎜⎜ = ⎟⎟ − α $ HW SDU (, (6 (, = ⎝ =⎠


⎛0 ⎞ 0= 1 SDU ⎜⎜ = ⎟⎟ − α $ HW SDU (, (6 (, = ⎝ =⎠

1 + α 7* − 7 (6

1 + α 7* − 7 (6

/HVGpSODFHPHQWVWDQJHQWLHO8HWQRUPDO9GH*ODURWDWLRQ ω GHODVHFWLRQ6VDWLVIRQWDX[ pTXDWLRQV ⎧ G 9 ⎞ ⎛ 1 ⎞ ⎛ 0 + α 7* − 7 ⎟ ⎪ + 9 = 5 ⎜ − α $⎟ − 5 ⎜ ⎝ (, ⎠ ⎝ (6 ⎠ ⎪ Gθ ⎪ G8 ⎪ ⎛ 1 ⎞ =9+5⎜ + α 7* − 7 ⎟ ⎨DYHF G θ (6 ⎝ ⎠ ⎪ ⎪ ⎪HW G ω = 5 ⎛ 0 − α $ ⎞ ⎜ ⎟ ⎪ Gθ ⎝ (, ⎠ ⎩

/HVGpSODFHPHQWVWDQJHQWLHO8HWQRUPDO9GH*ODURWDWLRQ ω GHODVHFWLRQ6VDWLVIRQWDX[ pTXDWLRQV ⎧ G 9 ⎞ ⎛ 1 ⎞ ⎛ 0 + α 7* − 7 ⎟ ⎪ + 9 = 5 ⎜ − α $⎟ − 5 ⎜ ⎝ (, ⎠ ⎝ (6 ⎠ ⎪ Gθ ⎪ G8 ⎪ ⎛ 1 ⎞ =9+5⎜ + α 7* − 7 ⎟ ⎨DYHF G θ (6 ⎝ ⎠ ⎪ ⎪ ⎪HW G ω = 5 ⎛ 0 − α $ ⎞ ⎜ ⎟ ⎪ Gθ ⎝ (, ⎠ ⎩

([HPSOHpOpYDWLRQXQLIRUPHGHWHPSpUDWXUH 7* − 7 GDQVXQDQQHDXFLUFXODLUHILJXUH


Gω = O¶pTXLOLEUH G¶XQ GHPL Gθ DQQHDX GRQQH G¶DXWUH SDUW 1 OD SUHPLqUH pTXDWLRQ GRQQH DORUV OH GpSODFHPHQW UDGLDO9VXLYDQW< 9 = α 5 7* − 7


/D WURLVLqPH pTXDWLRQ GRQQH 0 SXLVTXH $ HW


/HVIRUPXOHV HW GRQQHQWHQILQ Σ = HW ε [ = ε \ = ε ] = α 7 − 7


&HVUpVXOWDWVSHXYHQWrWUHREWHQXVSOXVVLPSOHPHQWSXLVTX¶RQDXQVROLGHKRPRJqQHLVRWURSH OLEUHGHVHGpIRUPHUHWVRXPLVjXQHpOpYDWLRQXQLIRUPHGHWHPSpUDWXUH


([HPSOH JUDGLHQW WUDQYHUVDO $ GH WHPSpUDWXUH GDQV XQ DQQHDX FLUFXODLUH $ FRQVWDQW ILJXUH


/¶pTXLOLEUHG¶XQGHPLDQQHDXGRQQH1


/HVpTXDWLRQV GRQQHQWGHSOXV

0 = α $ HW9 8 (,


0 = α $ HW9 8 (,

/HVpTXDWLRQV HW GRQQHQWHQILQ


σ ; = − α $ ⋅ ( ⋅ < VHXOHFRQWUDLQWHQRQQXOOH


ε ; = HW ε < = ε = = + ν α $ < VHXOHVGpIRUPDWLRQVQRQQXOOHV


- 64 -

- 64 -






⎛0 ⎞ 0= 1 SDU ⎜⎜ = ⎟⎟ − α $ HW SDU (6 (, = ⎝ (, = ⎠


⎛0 ⎞ 0= 1 SDU ⎜⎜ = ⎟⎟ − α $ HW SDU (6 (, = ⎝ (, = ⎠

1 + α 7* − 7 (6

1 + α 7* − 7 (6

/HVGpSODFHPHQWVWDQJHQWLHO8HWQRUPDO9GH*ODURWDWLRQ ω GHODVHFWLRQ6VDWLVIRQWDX[ pTXDWLRQV ⎧ G 9 ⎞ ⎛ 1 ⎞ ⎛ 0 + α 7* − 7 ⎟ ⎪ + 9 = 5 ⎜ − α $⎟ − 5 ⎜ (, (6 ⎝ ⎠ ⎝ ⎠ ⎪ Gθ ⎪ G8 ⎪ ⎛ 1 ⎞ =9+5⎜ + α 7* − 7 ⎟ ⎨DYHF G θ (6 ⎝ ⎠ ⎪ ⎪ ⎪HW G ω = 5 ⎛ 0 − α $ ⎞ ⎜ ⎟ ⎪ Gθ ⎝ (, ⎠ ⎩

/HVGpSODFHPHQWVWDQJHQWLHO8HWQRUPDO9GH*ODURWDWLRQ ω GHODVHFWLRQ6VDWLVIRQWDX[ pTXDWLRQV ⎧ G 9 ⎞ ⎛ 1 ⎞ ⎛ 0 + α 7* − 7 ⎟ ⎪ + 9 = 5 ⎜ − α $⎟ − 5 ⎜ (, (6 ⎝ ⎠ ⎝ ⎠ ⎪ Gθ ⎪ G8 ⎪ ⎛ 1 ⎞ =9+5⎜ + α 7* − 7 ⎟ ⎨DYHF G θ (6 ⎝ ⎠ ⎪ ⎪ ⎪HW G ω = 5 ⎛ 0 − α $ ⎞ ⎜ ⎟ ⎪ Gθ ⎝ (, ⎠ ⎩
















0 = α $ HW9 8 (,


0 = α $ HW9 8 (,







- 64 -

- 64 -

)LJXUH

)LJXUH

±)/(;,213/$1('¶81(32875(5(&7,/,*1(¬3/$102<(1

±)/(;,213/$1('¶81(32875(5(&7,/,*1(¬3/$102<(1

&H SUREOqPH WUqV LPSRUWDQW D pWp pWXGLp DX WRPH ,, FKDSLWUH ,,, GDQV OH FDV LVRWKHUPH ⎛0 ⎞ 0 7 = 7 (OOHVV¶pWHQGHQWDLVpPHQWDX[FDVWKHUPLTXHHQUHPSODoDQW = SDU ⎜⎜ = − α $ ⎟⎟ (, = (, ⎝ = ⎠ ⎛0 ⎞ 0 HW < SDU ⎜⎜ < + α % ⎟⎟ (, (, < ⎝ < ⎠

&H SUREOqPH WUqV LPSRUWDQW D pWp pWXGLp DX WRPH ,, FKDSLWUH ,,, GDQV OH FDV LVRWKHUPH ⎛0 ⎞ 0 7 = 7 (OOHVV¶pWHQGHQWDLVpPHQWDX[FDVWKHUPLTXHHQUHPSODoDQW = SDU ⎜⎜ = − α $ ⎟⎟ (, = (, ⎝ = ⎠ ⎛0 ⎞ 0 HW < SDU ⎜⎜ < + α % ⎟⎟ (, (, < ⎝ < ⎠

(QSDUWLFXOLHUODIOqFKHY[ VDWLVIDLWjODIRUPXOH


G Y 0= G ω= = −α $ = (, = G[ G[

'DQVOHSODQ[R\ RQDXUDLW

G Z 0< Gω = −α %= − < G[ G [ (, <

G Y 0= G ω= = −α $ = (, = G[ G[


([HPSOHSRXWUHLVRVWDWLTXH

G Z 0< Gω = −α %= − < G[ G [ (, <


)LJXUH

- 65 -

)LJXUH

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- 65 -

)LJXUH

±)/(;,213/$1('¶81(32875(5(&7,/,*1(¬3/$102<(1

±)/(;,213/$1('¶81(32875(5(&7,/,*1(¬3/$102<(1

&H SUREOqPH WUqV LPSRUWDQW D pWp pWXGLp DX WRPH ,, FKDSLWUH ,,, GDQV OH FDV LVRWKHUPH ⎛0 ⎞ 0 7 = 7 (OOHVV¶pWHQGHQWDLVpPHQWDX[FDVWKHUPLTXHHQUHPSODoDQW = SDU ⎜⎜ = − α $ ⎟⎟ (, = (, ⎝ = ⎠

&H SUREOqPH WUqV LPSRUWDQW D pWp pWXGLp DX WRPH ,, FKDSLWUH ,,, GDQV OH FDV LVRWKHUPH ⎛0 ⎞ 0 7 = 7 (OOHVV¶pWHQGHQWDLVpPHQWDX[FDVWKHUPLTXHHQUHPSODoDQW = SDU ⎜⎜ = − α $ ⎟⎟ (, = (, ⎝ = ⎠

HW

⎛0 ⎞ 0< SDU ⎜⎜ < + α % ⎟⎟ (, < ⎝ (, < ⎠

HW

(QSDUWLFXOLHUODIOqFKHY[ VDWLVIDLWjODIRUPXOH 'DQVOHSODQ[R\ RQDXUDLW

G Y 0= G ω= = −α $ = (, G[ G[ =

⎛0 ⎞ 0< SDU ⎜⎜ < + α % ⎟⎟ (, < ⎝ (, < ⎠


G Z 0< Gω = −α %= − < (, < G[ G[



G Y 0= G ω= = −α $ = (, G[ G[ =

G Z 0< Gω = −α %= − < (, < G[ G[


)LJXUH

- 65 -

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G Y GY = − α $ = − α $ [ Y = − α $ [ Y/ = − α $ / G[ G [

G Y GY = − α $ = − α $ [ Y = − α $ [ Y/ = − α $ / G[ G [

([HPSOHSRXWUHK\SHUVWDWLTXH


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,OV¶DJLWGXPrPHSUREOqPHTXHGDQVO¶H[HPSOHPDLVO¶H[WUpPLWpHVWVLPSOHPHQWDSSX\pH


&HWWHVWUXFWXUHHVWGRQFH[WpULHXUHPHQWK\SHUVWDWLTXHGHGHJUpVRXVO¶DFWLRQGXJUDGLHQW WUDQVYHUVDO$ODIOqFKHpWDQWHPSrFKpHHQ&FHODFUpHXQHUpDFWLRQ 5 F HWSDUFRQVpTXHQWXQ PRPHQWIOpFKLVVDQW 0 = 0 = = 5 F / − [ jO¶DEVFLVVH[


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G Y 5 F = / − [ − α $ G [ (,


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GY 5 ⎛ [ ⎞⎟ −α $ [ = ω = F ⎜ /[ − ⎜ G[ (, ⎝ ⎟⎠

SXLV

Y [ =


5F (,

⎛ [ [ ⎞ [ ⎜/ − ⎟ −α $ ⎜ ⎟⎠ ⎝

/DOLDLVRQK\SHUVWDWLTXHHQ&LPSRVHY/ G¶RO¶RQWLUH 5F =

GY 5 ⎛ [ ⎞⎟ −α $ [ = ω = F ⎜ /[ − G[ (, ⎝⎜ ⎟⎠

SXLV

Y [ =

5F (,

⎛ [ [ ⎞ [ ⎜/ − ⎟ −α $ ⎜ ⎟⎠ ⎝

/DOLDLVRQK\SHUVWDWLTXHHQ&LPSRVHY/ G¶RO¶RQWLUH

α$(, /

5F =

/HVpTXDWLRQVGHO¶pTXLOLEUHJOREDOGRQQHQWDORUV 5 = − 5F = −

G Y 5 F = / − [ − α $ G [ (,

α$(, /

/HVpTXDWLRQVGHO¶pTXLOLEUHJOREDOGRQQHQWDORUV

α$(, HW 0 H = − / 5 F = − α $ ( , /

5 = − 5F = −

α$(, HW 0 H = − / 5 F = − α $ ( , /

- 66 -

- 66 -



G Y GY = − α $ = − α $ [ Y = − α $ [ Y/ = − α $ / G[ G [

G Y GY = − α $ = − α $ [ Y = − α $ [ Y/ = − α $ / G[ G [



)LJXUH

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G Y 5 F = / − [ − α $ G [ (,


GY 5 ⎛ [ ⎞⎟ −α $ [ = ω = F ⎜ /[ − G[ (, ⎝⎜ ⎟⎠

SXLV

Y [ =

5F (,

⎛ [ [ ⎞ [ ⎜/ − ⎟ −α $ ⎜ ⎟⎠ ⎝

/DOLDLVRQK\SHUVWDWLTXHHQ&LPSRVHY/ G¶RO¶RQWLUH 5F =

G Y 5 F = / − [ − α $ G [ (,


GY 5 ⎛ [ ⎞⎟ −α $ [ = ω = F ⎜ /[ − G[ (, ⎝⎜ ⎟⎠

SXLV

Y [ =

α$(, HW 0 H = − / 5 F = − α $ ( , /

- 66 -

5F (,

⎛ [ [ ⎞ [ ⎜/ − ⎟ −α $ ⎜ ⎟⎠ ⎝

/DOLDLVRQK\SHUVWDWLTXHHQ&LPSRVHY/ G¶RO¶RQWLUH

α$(, /



5F =

α$(, /


α$(, HW 0 H = − / 5 F = − α $ ( , /

- 66 -

±62//,&,7$7,216&20%,1e(6

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- 67 -

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- 67 -



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&RQVLGpURQV O¶RVVDWXUH GHV ILJXUHV D HW E FRQVWLWXpH GH GHX[ SRXWUH SULVPDWLTXHV LGHQWLTXHV VRXGpHV HQWUH HOOHV SHUSHQGLFXODLUHPHQW HQ $ /H PRQWDJH D OLHX GDQV OHV GHX[ FDVVDQVMHXQLVHUUDJHjODWHPSpUDWXUH 7


3RUWRQVjODWHPSpUDWXUHXQLIRUPH 7 ≠ 7


'DQV OH FDV LVRVWDWLTXH D OD VWUXFWXUH OLEUH GH VH GpIRUPHU VXELW OD GLODWDWLRQ XQLIRUPH HW LVRWURSH LPSRVpH SDU O¶pOpYDWLRQ GH WHPSpUDWXUH 7 − 7 HW DXFXQH FRQWUDLQWH QH SUHQG QDLVVDQFH


'DQV OH FDV H[WpULHXUHPHQW K\SHUVWDWLTXH E OD OLDLVRQ ©UHGRQGDQWHª HQ % HPSrFKH OHV GpSODFHPHQWV WUDQVODWLRQV HW URWDWLRQV GH OD VHFWLRQ 6% LQWURGXLVDQW XQH IRUFH GH OLDLVRQ 5 % ; % <% HW XQ FRXSOH 0 % 6XU FKDTXH VHFWLRQ GURLWH GH O¶RVVDWXUH RQ DXUD GRQF XQ HIIRUWQRUPDO1XQHIIRUWWUDQFKDQW 7 = 7< HWXQPRPHQWIOpFKLVVDQW 0 = 0 =


2QWURXYHWRXVOHVFDOFXOVIDLW


(, (, α 7 − 7 ; % = <% = − α 7 − 7 HW 0 % = / /

; % = <% = −

(, (, α 7 − 7 α 7 − 7 HW 0 % = / /

2QHQGpGXLWDLVpPHQWVXUFKDTXHVHFWLRQGURLWHOHVYLVVHXUVHWOHVFRQWUDLQWHV


±&2175$,17(67+(50,48(6'$16/(675(,//,6

±&2175$,17(67+(50,48(6'$16/(675(,//,6

$X WRPH ,, FKDSLWUH ,, QRXV DYRQV GRQQp XQH PpWKRGH JpQpUDOH GH FDOFXO GHV WUHLOOLV GDQVFDVLVRWKHUPH


1RXVDOORQVHQpWHQGUHOHVUpVXOWDWVDXFDVROHPRQWDJHD\DQWpWpUpDOLVpVDQVMHXQLVHUUDJH jODWHPSpUDWXUH 7 RQSRUWHjODWHPSpUDWXUHXQLIRUPH7


/HV GpILQLWLRQV OH FDOFXO GX GHJUp G¶K\SHUVWDWLFLWp OHV pTXDWLRQV G¶pTXLOLEUH GHV Q°XGV UHVWHQWELQHVULQFKDQJpV


3DU FRQWUH GDQV OHV pTXDWLRQV DX[ GpSODFHPHQWV QRGDX[ O¶DOORQJHPHQW δ /LM GH OD EDUUH


⎛ 1/ ⎞ $ L $ M HVWODVRPPHG¶XQHGLODWDWLRQ ⎜ ⎟ GXHjO¶HIIRUWQRUPDO 1LM HWG¶XQHGLODWDWLRQ ⎝ (6 ⎠ LM


WKHUPLTXH /α LM 7 − 7 &HVpTXDWLRQVV¶pFULYHQWDORUVGDQVOHFDVG¶XQWUHLOOLVSODQ


⎛ 1/ ⎞ X M − X L FRV ϕLM + Y M − YL VLQ ϕLM = ⎜ ⎟ + α/ LM 7 − 7 ⎝ (6 ⎠ LM

'DQVOHFDVG¶XQWUHLOOLV'HOOHVV¶pFULYHQW

⎛ 1/ ⎞ X M − X L α LM + Y M − YL βLM + Z M − Z L γ LM = ⎜ ⎟ + α/ LM 7 − 7 ⎝ (6 ⎠ LM


'DQVOHFDVG¶XQWUHLOOLV'HOOHVV¶pFULYHQW ′


′

- 68 -

- 68 -











(, (, α 7 − 7 ; % = <% = − α 7 − 7 HW 0 % = / /

; % = <% = −

(, (, α 7 − 7 α 7 − 7 HW 0 % = / /



±&2175$,17(67+(50,48(6'$16/(675(,//,6

±&2175$,17(67+(50,48(6'$16/(675(,//,6














'DQVOHFDVG¶XQWUHLOOLV'HOOHVV¶pFULYHQW


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'DQVOHFDVG¶XQWUHLOOLV'HOOHVV¶pFULYHQW ′


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′

([HPSOHWUHLOOLVHQIRUPHGHFDUUpDYHFGHX[GLDJRQDOHV/HVEDUUHVODWpUDOHVVRQWHQDFLHU GH FDUDFWpULVWLTXHV / ( 6 α /HV GLDJRQDOHV VRQW HQ DOXPLQLXP GH FDUDFWpULVWLTXHV / (′ 6′ α′


/¶DVVHPEODJHHVWHIIHFWXpVDQVMHXQLVHUUDJHjODWHPSpUDWXUH 7 2QSRUWHjODWHPSpUDWXUH 7 α′ pWDQWGLIIpUHQWGH α RQDXUDGHVHIIRUWVQRUPDX[HWGHVFRQWUDLQWHVGDQVOHVEDUUHV TXHQRXVQRXVSURSRVRQVGHFDOFXOHU


)LJXUH

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/D ILJXUH GRQQH OD JpRPpWULH OHV D[HV HW OHV OLDLVRQV /H WUHLOOLV pWDQW H[WpULHXUHPHQW LVRVWDWLTXHHQO¶DEVHQFHGHFKDUJHPHQWPpFDQLTXHOHVDFWLRQVGHOLDLVRQVVRQWQXOOHV


/¶pTXLOLEUH GHV Q°XGV V¶pFULW 1 + 1′ ODWpUDOHHW 1′ GDQVXQHGLDJRQDOH

= 1 GpVLJQDQW O¶HIIRUW QRUPDO GDQV XQH EDUUH



'¶DXWUH SDUW SXLVTXH X $ = X % = Y % = Y F = δ / OHV pTXDWLRQV DX[ GpSODFHPHQWV QRGDX[ VH UpGXLVHQWj 1/ 1′/ δ/ = + α / 7 − 7 HW δ / = + α′ / 7 − 7 (6 (′6′


/DUpVROXWLRQGRQQH

/DUpVROXWLRQGRQQH

1=

δ / α′ (′ 6′ + α ( 6 α′ − α 7 − 7 = 7 − 7 1′ = − 1 / (′ 6′ + ( 6 + (6 (′6′

%LHQVUVLO¶RQDYDLW α = α′ RQWURXYHUDLW 1 = 1′ = HW

δ/ = α 7 − 7 /

1=

δ / α′ (′ 6′ + α ( 6 α′ − α 7 − 7 = 7 − 7 1′ = − 1 / (′ 6′ + ( 6 + (6 (′6′


δ/ = α 7 − 7 /

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)LJXUH

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/DUpVROXWLRQGRQQH

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1=

δ / α′ (′ 6′ + α ( 6 α′ − α 7 − 7 = 7 − 7 1′ = − 1 / (′ 6′ + ( 6 + (6 (′6′


- 69 -

δ/ = α 7 − 7 /

1=

δ / α′ (′ 6′ + α ( 6 α′ − α 7 − 7 = 7 − 7 1′ = − 1 / (′ 6′ + ( 6 + (6 (′6′


- 69 -

δ/ = α 7 − 7 /

±)/$0%(0(177+(50,48(

±)/$0%(0(177+(50,48(

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'DQVO¶H[HPSOHGXWUHLOOLVFLGHVVXVVXSSRVRQV 7 > 7 FRPPH α′ > α OHVEDUUHVGLDJRQDOHV VRQWFRPSULPpHV


(WDQW DUWLFXOpHV HQ OHXUV GHX[ H[WUpPLWpV HOOHV IODPEHQW VXLYDQW OH SUHPLHU PRGH G¶(XOHU π (′ , ORUVTXHO¶LQWHQVLWpGHO¶HIIRUW 1′ DWWHLQWODYDOHXUFULWLTXHIRQGDPHQWDOH F¶HVWjGLUH / SRXUXQHpOpYDWLRQGHWHPSpUDWXUH


7 − 7 =

⎛ π (′ , ⎞ ⎟ ⎜ + ⎜ / α′ − α ⎝ (6 (′ 6′ ⎟⎠

7 − 7 =

'RQQRQVXQDXWUHH[HPSOHFRQVLGpURQVXQHSRXWUHSULVPDWLTXHGRQWOHSODQPR\HQ[R\ HVWFHOXLGHPRLQGUHUpVLVWDQFHjODIOH[LRQHQFDVWUpHHQVHVGHX[H[WUpPLWpVOHPRQWDJHHVW VXSSRVpUpDOLVpVDQVSUpFRQWUDLQWHVjODWHPSpUDWXUH 7

⎛ π (′ , ⎞ ⎟ ⎜ + ⎜ / α′ − α ⎝ (6 (′ 6′ ⎟⎠


)LJXUH

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8QH pOpYDWLRQ GH WHPSpUDWXUH 7 − 7 LQWURGXLVDQW XQH FRPSUHVVLRQ G¶LQWHQVLWp α (6 7 − 7 OH IODPEHPHQW DXUD OLHX ORUVTXH FHW HIIRUW DWWHLQGUD OD FKDUJH FULWLTXH


π (, G¶DSUqVOHVUpVXOWDWVREWHQXVDXWRPH,,FKDSLWUH9,,GRQFSRXUXQH / pOpYDWLRQGHWHPSpUDWXUH

IRQGDPHQWDOH

7 − 7 =


IRQGDPHQWDOH

π , α 6/

7 − 7 =

π , α 6/

0RQWURQVVXUXQH[HPSOHQXPpULTXHODVHQVLELOLWpG¶XQHWHOOHSRXWUHDXIODPEDJHWKHUPLTXH HQ VXSSRVDQW TX¶LO V¶DJLW G¶XQ WX\DX HQ FXLYUH GH FKDXIIDJH FHQWUDO WUDYHUVDQW GHX[ PXUV SDUDOOqOHVGLVWDQWVGH/ PGDQVOHVTXHOVLOSHXWrWUHFRQVLGpUpFRPPHHQFDVWUp


- 70 -

- 70 -

±)/$0%(0(177+(50,48(

±)/$0%(0(177+(50,48(







7 − 7 =

⎛ π (′ , ⎞ ⎟ ⎜ + ⎜ / α′ − α ⎝ (6 (′ 6′ ⎟⎠

7 − 7 =


⎛ π (′ , ⎞ ⎟ ⎜ + ⎜ / α′ − α ⎝ (6 (′ 6′ ⎟⎠


)LJXUH

)LJXUH




IRQGDPHQWDOH

7 − 7 =

π , α 6/


IRQGDPHQWDOH

7 − 7 =

π , α 6/



- 70 -

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3UHQRQVSRXUGLDPqWUHVGXWX\DXPPLQWpULHXU HWPPH[WpULHXU SRXUOHFXLYUHRQD α = ⋅ − . −


/HPRQWDJHpWDQWUpDOLVp&VDQVMHXQLVHUUDJHO¶pOpYDWLRQGHWHPSpUDWXUHSURYRTXDQWOH IODPEHPHQWVXLYDQWOHSUHPLHUPRGHHVWGRQF π φL + φH 7 − 7 = = GHJUpV α /


1RXVYHQRQVGHYRLUGHX[IODPEHPHQWVWKHUPLTXHVSDUIOH[LRQSODQHG¶XQHSRXWUHRQSHXW DXVVL DYRLU SRXU GHV UDLVRQV WKHUPLTXHV G¶DXWUHV W\SHV GH IODPEHPHQW SDU H[HPSOH OH IODPEHPHQWSDUWRUVLRQRXOHGpYHUVHPHQWODWpUDO


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&+$3,75(,9

&+$3,75(,9

7+(50,48('(63/$48(6

7+(50,48('(63/$48(6

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[R\ GpVLJQDQW OH SODQ PR\HQ G¶XQH SODTXH QRXV DYRQV G¶DERUG GpILQL OHV pOpPHQWV JpRPpWULTXHVVXUOHVTXHOVUDLVRQQHUIHXLOOHWVILEUHVORQJLWXGLQDOHVHWWUDQVYHUVDOHVFRXSXUHV 1RXVDYRQVHQVXLWHGpILQLOHYLVVHXUVXUXQHFRXSXUHpOpPHQWDLUHVHVFLQTFRPSRVDQWHV17 4 8 0 TXH QRXV DYRQV H[SULPpHV HQ IRQFWLRQ GHV FRQWUDLQWHV 1RXV DYRQV GRQQp OHV IRUPXOHVGHFKDQJHPHQWGHEDVHSRXUOHVYLVVHXUVFHTXLQRXVDSHUPLVGHGpILQLUOHWHQVHXU GHV IRUFHV GH PHPEUDQH OH YHFWHXU GHV HIIRUWV WUDQFKDQWV WUDQVYHUVDX[ HW OD PDWULFH GH IOH[LRQWRUVLRQ


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7RXV FHV UpVXOWDWV GHPHXUHQW LQFKDQJpV GDQV OH FDV TXL QRXV LQWpUHVVH PDLQWHQDQW G¶XQ FKDUJHPHQWPpFDQLTXHHWWKHUPLTXH&RPPHGDQVOHFKDSLWUHSUpFpGHQWFRQVDFUpDX[SRXWUHV QRXV DOORQV LFL VLPSOLILHU OH FKDPS GH WHPSpUDWXUH GDQV OD SODTXH HQ OH OLQpDULVDQW VXLYDQW O¶pSDLVVHXUF¶HVWjGLUHHQSUHQDQW


7 3 = 7 3 + $ [ \ ⋅ ]

7 3 = 7 3 + $ [ \ ⋅ ]

$[\ GpVLJQH GRQF OH JUDGLHQW WUDQVYHUVDO GH WHPSpUDWXUH LQGpSHQGDQW GH ] VXSSRVp FRQVWDQW VXU XQH ILEUH QRUPDOH /D YDULDWLRQ 73 − 7 GH WHPSpUDWXUH SDU UDSSRUW j OD WHPSpUDWXUHGHUpIpUHQFH 7 HVWDLQVLODVRPPHGHGHX[WHUPHV


7 3 − 7 LQGpSHQGDQWHGH]HWTXLSURGXLWXQHGpIRUPDWLRQGHODSODTXH©GDQVVRQSODQª


$ [ \ ⋅ ] TXLSURGXLWXQHFRXUEXUHGXIHXLOOHWPR\HQ


1RXV DYRQV GpVLJQp SDU 3 OD SDUWLFXOH FRXUDQWH GH FRWH ] G¶XQH ILEUH QRUPDOH HW SDU 3 OD SDUWLFXOHGHFHQWUDOHGHFHWWHILEUH


&HWWH GpFRPSRVLWLRQ QRXV SHUPHW FRPPH GDQV OH FDV LVRWKHUPH GH FRQVLGpUHU VpSDUpPHQW GHX[SUREOqPHVFHOXLG¶XQHSODTXHFKDUJpH©GDQVVRQSODQªHWFHOXLG¶XQHSODTXHFKDUJpH ©WUDQVYHUVDOHPHQWª


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&+$3,75(,9

&+$3,75(,9

7+(50,48('(63/$48(6

7+(50,48('(63/$48(6









7 3 = 7 3 + $ [ \ ⋅ ]

7 3 = 7 3 + $ [ \ ⋅ ]











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±3/$48(&+$5*(('$166213/$1

±3/$48(&+$5*(('$166213/$1

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±&RQWUDLQWHVSODQHV

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6RXYHQWGHVSODTXHVDLQVLFKDUJpHVVDWLVIRQWULJRXUHXVHPHQWDX[K\SRWKqVHVGHFRQWUDLQWHV SODQHVFDVpWXGLpDXGXFKDSLWUH,,5DSSHORQVTXHGDQVFHFDVRQDQpFHVVDLUHPHQWOD UHODWLRQ G ν GLY I + ( α ⋅ ∆ 7 =




3OXV JpQpUDOHPHQW XQH SODTXH FKDUJpH GDQV VRQ SODQ QH VDWLVIDLW SDV QpFHVVDLUHPHQW DX[ K\SRWKqVHV GH FRQWUDLQWHV SODQHV PDLV QRXV DYRQV PRQWUp DX WRPH , FKDSLWUH 9, TX¶RQFRPPHWWDLWXQHWUqVIDLEOHHUUHXUHQFRQVLGpUDQWFHVK\SRWKqVHVFRPPHVDWLVIDLWHVF¶HVW jGLUHHQFRQVLGpUDQW ± G¶XQHSDUW σ ] τ [] τ \] FRPPHLGHQWLTXHPHQWQXOV


± G¶DXWUHSDUW σ ] σ \ τ [\ FRPPHLQGpSHQGDQWVGH]


&HWWHHUUHXUHVWG¶DXWDQWSOXVIDLEOHTXHODSODTXHHVWSOXVPLQFH


5DSSHORQVGDQVFHFDVOHVUHODWLRQVGH+RRNH'XKDPHO


⎡ σ[ ⎢ [Σ] = ⎢τ[\ ⎢ ⎢⎣

τ [\ σ\

⎤ ⎡ ε[ ⎥ ⎢ ⎥ [E] = ⎢ γ [\ ⎥ ⎢ ⎢⎣ ⎥⎦

γ [\ ε\

⎤ ⎥ ⎥ ⎥ ε ] ⎥⎦

⎧ ⎪ε [ = ( σ [ − νσ \ + α 7 − 7 ⎪ ⎪ε = σ − νσ + α 7 − 7 [ ⎪⎪ \ ( \ ⎨ ⎪ε = ν σ + σ + α 7 − 7 \ ⎪ ] ( [ ⎪ ⎪τ [\ = ( γ [\ ⎪⎩ + ν


⎡ σ[ ⎢ [Σ] = ⎢τ[\ ⎢ ⎢⎣

τ [\ σ\

⎤ ⎡ ε[ ⎥ ⎢ ⎥ [E] = ⎢ γ [\ ⎥ ⎢ ⎢⎣ ⎥⎦

γ [\ ε\

⎤ ⎥ ⎥ ⎥ ε ] ⎥⎦



- 74 -

- 74 -

±3/$48(&+$5*(('$166213/$1

±3/$48(&+$5*(('$166213/$1



±&RQWUDLQWHVSODQHV

±&RQWUDLQWHVSODQHV













⎡ σ[ ⎢ [Σ] = ⎢τ[\ ⎢ ⎢⎣

τ [\ σ\

⎤ ⎡ ε[ ⎥ ⎢ ⎥ [E] = ⎢ γ [\ ⎥ ⎢ ⎢⎣ ⎥⎦

γ [\ ε\

⎤ ⎥ ⎥ ⎥ ε ] ⎥⎦



- 74 -

⎡ σ[ ⎢ [Σ] = ⎢τ[\ ⎢ ⎢⎣

τ [\ σ\

⎤ ⎡ ε[ ⎥ ⎢ ⎥ [E] = ⎢ γ [\ ⎥ ⎢ ⎢⎣ ⎥⎦

γ [\ ε\

⎤ ⎥ ⎥ ⎥ ε ] ⎥⎦



- 74 -

DYHFpJDOHPHQW

ε] = −

+ ν ν ε [ + ε \ + α 7 − 7 − ν − ν

DYHFpJDOHPHQW

ε] = −

+ ν ν ε [ + ε \ + α 7 − 7 − ν − ν

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⎧ ∂ σ [ ∂ τ [\ + = ⎪I [ + ∂[ ∂\ ⎪ ⎪ ∂ τ [\ ∂ σ \ ⎪ (TXLOLEUHORFDO + = ⎨I \ + ∂[ ∂\ ⎪ ⎪ ⎪∆ σ + σ + + ν GLY IG + ( α ⋅ ∆ 7 = [ \ ⎪ ⎩


&HWWH WURLVLqPH pTXDWLRQ HVW O¶H[SUHVVLRQ HQ WHUPHV GH FRQWUDLQWHV HW FRPSWH WHQX GHV GHX[ DXWUHVGHO¶pTXDWLRQGHFRPSDWLELOLWp


∂ γ [\ ∂ ε[ ∂ ε\ + = ∂[ ∂\ ∂ \ ∂ [

∂ γ [\ ∂ ε[ ∂ ε\ + = ∂[ ∂\ ∂ \ ∂ [

5HPDUTXHVLO¶RQXWLOLVHXQHIRQFWLRQG¶$LU\ φ [ \ FHOOHFLVDWLVIDLWO¶pTXDWLRQ ∆ ∆ φ = − ν ∆ 9 − ( α ∆ 7

5HPDUTXHVLO¶RQXWLOLVHXQHIRQFWLRQG¶$LU\ φ [ \ FHOOHFLVDWLVIDLWO¶pTXDWLRQ

5HPDUTXHOHVFRQWUDLQWHVpWDQWFRQVWDQWHVVXLYDQWO¶pSDLVVHXURQD

∆ ∆ φ = − ν ∆ 9 − ( α ∆ 7


1 [ = K ⋅ σ [ 1 \ = K ⋅ σ \ 7[ = 7\ = K ⋅ τ [\

1 [ = K ⋅ σ [ 1 \ = K ⋅ σ \ 7[ = 7\ = K ⋅ τ [\

G G /HVpTXDWLRQV V¶pFULYHQWDORUVDXVVLSXLVTXH S = K ⋅ I IRUFHVXUIDFLTXH


⎧ ∂ 1 [ ∂ 7\ + = ⎪S [ + ∂[ ∂\ ⎪ ⎪ ∂ 7[ ∂ 1 \ ⎪ + = ⎨S \ + ∂[ ∂\ ⎪ ⎪ ⎪∆ 1 + 1 + + ν GLY SG + ( K α ⋅ ∆ 7 = [ \ ⎪ ⎩

⎧ ∂ 1 [ ∂ 7\ + = ⎪S [ + ∂[ ∂\ ⎪ ⎪ ∂ 7[ ∂ 1 \ ⎪ + = ⎨S \ + ∂[ ∂\ ⎪ ⎪ ⎪∆ 1 + 1 + + ν GLY SG + ( K α ⋅ ∆ 7 = [ \ ⎪ ⎩

5HPDUTXH GDQV OH FDV WUqV FRXUDQW R O¶RQ HVW GHQ UpJLPH VWDWLRQQDLUH VDQV IRUFH YROXPLTXHVQLVRXUFHVGHFKDOHXUDXVHLQGXVROLGHRQD I = HW ∆ 7 = FHTXLVLPSOLILHOHV pTXDWLRQV HW


- 75 -

- 75 -

DYHFpJDOHPHQW

ε] = −

+ ν ν ε [ + ε \ + α 7 − 7 − ν − ν

DYHFpJDOHPHQW

ε] = −

+ ν ν ε [ + ε \ + α 7 − 7 − ν − ν







∂ γ [\ ∂ ε[ ∂ ε\ + = ∂[ ∂\ ∂ \ ∂ [

∂ γ [\ ∂ ε[ ∂ ε\ + = ∂[ ∂\ ∂ \ ∂ [


∆ ∆ φ = − ν ∆ 9 − ( α ∆ 7



∆ ∆ φ = − ν ∆ 9 − ( α ∆ 7


1 [ = K ⋅ σ [ 1 \ = K ⋅ σ \ 7[ = 7\ = K ⋅ τ [\

1 [ = K ⋅ σ [ 1 \ = K ⋅ σ \ 7[ = 7\ = K ⋅ τ [\



⎧ ∂ 1 [ ∂ 7\ + = ⎪S [ + ∂[ ∂\ ⎪ ⎪ ∂ 7[ ∂ 1 \ ⎪ + = ⎨S \ + ∂[ ∂\ ⎪ ⎪ ⎪∆ 1 + 1 + + ν GLY SG + ( K α ⋅ ∆ 7 = [ \ ⎪ ⎩

⎧ ∂ 1 [ ∂ 7\ + = ⎪S [ + ∂[ ∂\ ⎪ ⎪ ∂ 7[ ∂ 1 \ ⎪ + = ⎨S \ + ∂[ ∂\ ⎪ ⎪ ⎪∆ 1 + 1 + + ν GLY SG + ( K α ⋅ ∆ 7 = [ \ ⎪ ⎩



- 75 -

- 75 -

([HPSOHFRXURQQHFLUFXODLUHPLQFHVRXPLVHjXQJUDGLHQWUDGLDOGHWHPSpUDWXUH


)LJXUH

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7 = $ /RJ U + %

DYHF

$=

7 = $ /RJ U + %

7 /RJ 5 − 7 /RJ 5 7 − 7 HW % = ⎛ 5 ⎞ ⎛ 5 ⎞ ⎟⎟ ⎟⎟ /RJ ⎜⎜ /RJ ⎜⎜ ⎝ 5 ⎠ ⎝ 5 ⎠

DYHF

$=

7 /RJ 5 − 7 /RJ 5 7 − 7 HW % = ⎛ 5 ⎞ ⎛ 5 ⎞ ⎟⎟ ⎟⎟ /RJ ⎜⎜ /RJ ⎜⎜ ⎝ 5 ⎠ ⎝ 5 ⎠

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6DVROXWLRQJpQpUDOHV¶pFULW X = &U +

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X = &U +

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7 = $ /RJ U + %

DYHF

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7 /RJ 5 − 7 /RJ 5 7 − 7 HW % = ⎛5 ⎞ ⎛5 ⎞ /RJ ⎜⎜ ⎟⎟ /RJ ⎜⎜ ⎟⎟ ⎝ 5 ⎠ ⎝ 5 ⎠

7 = $ /RJ U + %

DYHF

$=

7 /RJ 5 − 7 /RJ 5 7 − 7 HW % = ⎛5 ⎞ ⎛5 ⎞ /RJ ⎜⎜ ⎟⎟ /RJ ⎜⎜ ⎟⎟ ⎝ 5 ⎠ ⎝ 5 ⎠





6DVROXWLRQJpQpUDOHV¶pFULW

6DVROXWLRQJpQpUDOHV¶pFULW X = &U +

' G¶RO¶RQWLUH U

- 76 -

X = &U +

' G¶RO¶RQWLUH U

- 76 -

⎧ GX ' ⎪ε U = G U = & − U ⎪ ⎪ X ' ⎪ ⎨εθ = = & + U U ⎪ ⎪ ⎪ε = ν& + + ν α 7 − 7 ⎪ ] − ν − ν ⎩

⎧ ( ⎪σ U = − ν ⎪ ⎨ ⎪ ( ⎪σθ = − ν ⎩

'⎤ ( α ⎡ ⎢⎣& + ν − − ν U ⎥⎦ − − ν 7 − 7 '⎤ ( α ⎡ ⎢⎣& + ν + − ν U ⎥⎦ − − ν 7 − 7


⎧ ( ⎪σ U = − ν ⎪ ⎨ ⎪ ( ⎪σθ = − ν ⎩

'⎤ ( α ⎡ ⎢⎣& + ν − − ν U ⎥⎦ − − ν 7 − 7 '⎤ ( α ⎡ ⎢⎣& + ν + − ν U ⎥⎦ − − ν 7 − 7

/HV FRQGLWLRQV DX[ OLPLWHV VRQW σ U = SRXU 5 HW 5 2Q HQ GpGXLW OHV FRQVWDQWHV G¶LQWpJUDWLRQ&HW' ⎧ ⎞ ⎛ 7 5 − 7 5 ⎪& = α ⎜⎜ − 7 ⎟⎟ ⎪ ⎠ ⎝ 5 − 5 ⎨ ⎪ + ν 5 5 (7 − 7 ) ⎪' = α − ν 5 − 5 ⎩


±3/$48(&+$5*e(75$169(56$/(0(17

±3/$48(&+$5*e(75$169(56$/(0(17

3DUGpILQLWLRQODSODTXHHVWVRXPLVH


± jXQJUDGLHQWWUDQVYHUVDOGHWHPSpUDWXUH

∂7 = $ [ \ ∂]


∂7 = $ [ \ ∂]

± jGHVIRUFHVQRUPDOHVjODSODTXHVXSSRVpHVDSSOLTXpHVDXIHXLOOHWPR\HQSRQFWXHOOHVRX VXUIDFLTXHVGHGHQVLWp S ] RXHQFRUHOLQpLTXHVXUOHVERUGGHGHQVLWp4


± jGHVFRXSOHVOLQpLTXHVGHIOH[LRQHWWRUVLRQVXUOHVERUGV


/H IHXLOOHW PR\HQ GHYLHQW XQH VXUIDFH FRXUEH /HV HIIRUWV GH PHPEUDQH 1 7 VRQW LGHQWLTXHPHQWQXOV


/HV K\SRWKqVHV TXL QRXV SHUPHWWHQW G¶pODERUHU XQH PpWKRGH GH FDOFXO VRQW FHOOHV GX FDV LVRWKHUPH WRPH , QRXV \ DMRXWRQV O¶K\SRWKqVH GX JUDGLHQW WUDQVYHUVDO GH WHPSpUDWXUH $[\ LQGpSHQGDQWGH]F¶HVWjGLUHFRQVWDQWVXLYDQWO¶pSDLVVHXU


&HWWH PpWKRGH HVW XQH PpWKRGH GHV GpSODFHPHQWV TXL FRQVLVWH GDQV XQ SUHPLHU WHPSV j H[SULPHUWRXVOHVSDUDPqWUHVGXSUREOqPHHQIRQFWLRQGHODIOqFKHZ[\ SULVHSDUOHSRLQW FRXUDQW 3 GXIHXLOOHWPR\HQSRXUpWDEOLUHQVXLWHO¶pTXDWLRQGH/DJUDQJHjODTXHOOHVDWLVIDLW FHWWHIOqFKH(WDEOLVVRQVPDLQWHQDQWFHVIRUPXOHVHQFRRUGRQQpHVFDUWpVLHQQHV


- 77 -

- 77 -



⎧ ( ⎪σ U = − ν ⎪ ⎨ ⎪ ( ⎪σθ = − ν ⎩

'⎤ ( α ⎡ ⎢⎣& + ν − − ν U ⎥⎦ − − ν 7 − 7 '⎤ ( α ⎡ ⎢⎣& + ν + − ν U ⎥⎦ − − ν 7 − 7

⎧ ( ⎪σ U = − ν ⎪ ⎨ ⎪ ( ⎪σθ = − ν ⎩

'⎤ ( α ⎡ ⎢⎣& + ν − − ν U ⎥⎦ − − ν 7 − 7 '⎤ ( α ⎡ ⎢⎣& + ν + − ν U ⎥⎦ − − ν 7 − 7



±3/$48(&+$5*e(75$169(56$/(0(17

±3/$48(&+$5*e(75$169(56$/(0(17



∂7 ± jXQJUDGLHQWWUDQVYHUVDOGHWHPSpUDWXUH = $ [ \ ∂]












- 77 -

- 77 -

∂7 = $ [ \ ∂]

UHpWDSHH[SUHVVLRQGHVWUDQVODWLRQV


⎧ ∂Z ⎪X = − ] ∂ [ ⎪ ∂Z ⎪ 3R3 ⎨Y = −] ∂\ ⎪ ⎪ ⎪ Z = Z [ \ ⎩

HpWDSHH[SUHVVLRQGHVGpIRUPDWLRQV

γ [] HW γ \] VHURQWFDOFXOpVSOXVORLQRQDGHSOXV

⎧ ∂X ∂ Z = −] ⎪ε [ = ∂[ ∂ [ ⎪ ∂Y ∂ Z ⎪⎪ = −] ⎨ε \ = ∂\ ∂ \ ⎪ ⎪ ∂ Z ⎛ ∂X ∂Y ⎞ ⎟⎟ = − ] + ⎪γ [\ = ⎜⎜ ∂[ ∂\ ⎝∂\ ∂[ ⎠ ⎩⎪

+ ν ν ε [ + ε \ + α 7 − 7 − ν − ν

ε] = −

+ ν ν ε [ + ε \ + α 7 − 7 − ν − ν

HpWDSHH[SUHVVLRQGHVFRQWUDLQWHV

$O¶DLGHGHODORLGH+RRNH'XKDPHORQREWLHQW


( (α ⎧ ⎪σ [ = − ν ε [ + ν ε \ − − ν 7 − 7 ⎪ ⎪ ( (α ⎪ ε \ + ν ε [ − 7 − 7 ⎨σ \ = − ν − ν ⎪ ⎪ ⎪τ = ( γ [\ [\ + ν ⎩⎪

( (α ⎧ ⎪σ [ = − ν ε [ + ν ε \ − − ν 7 − 7 ⎪ ⎪ ( (α ⎪ ε \ + ν ε [ − 7 − 7 ⎨σ \ = − ν − ν ⎪ ⎪ ⎪τ = ( γ [\ [\ + ν ⎩⎪

6RLW

6RLW ⎧ ( ⎡ ∂ Z +ν ⎪σ [ = − ⎢ − ν ⎣⎢ ∂ [ ⎪ ⎪ ( ⎡ ∂ Z ⎪ +ν ⎢ ⎨σ \ = − − ν ⎣⎢ ∂ \ ⎪ ⎪ ⎪τ = − ( ∂ Z ⋅ ] ⎪ [\ + ν ∂ [ ∂ \ ⎩

⎤ ∂ Z + α + ν $ ⎥ ⋅ ] ∂\ ⎦⎥ ⎤ ∂ Z + α + ν $ ⎥ ⋅ ] ∂[ ⎦⎥

⎧ ( ⎡ ∂ Z +ν ⎪σ [ = − ⎢ − ν ⎣⎢ ∂ [ ⎪ ⎪ ( ⎡ ∂ Z ⎪ +ν ⎢ ⎨σ \ = − − ν ⎣⎢ ∂ \ ⎪ ⎪ ⎪τ = − ( ∂ Z ⋅ ] ⎪ [\ + ν ∂ [ ∂ \ ⎩

- 78 -



⎧ ∂Z ⎪X = − ] ∂ [ ⎪ ∂Z ⎪ 3R3 ⎨Y = − ] ∂\ ⎪ ⎪ ⎪ Z = Z [ \ ⎩

⎧ ∂X ∂ Z = −] ⎪ε [ = ∂[ ∂ [ ⎪ ⎪⎪ ∂Y ∂ Z = −] ⎨ε \ = ∂\ ∂ \ ⎪ ⎪ ∂ Z ⎛ ∂X ∂Y ⎞ ⎟⎟ = − ] + ⎪γ [\ = ⎜⎜ ∂[ ∂\ ⎝∂\ ∂[ ⎠ ⎪⎩


+ ν ν ε [ + ε \ + α 7 − 7 − ν − ν

ε] = −

+ ν ν ε [ + ε \ + α 7 − 7 − ν − ν





( (α ⎧ ⎪σ [ = − ν ε [ + ν ε \ − − ν 7 − 7 ⎪ ⎪ ( (α ⎪ ε \ + ν ε [ − 7 − 7 ⎨σ \ = −ν − ν ⎪ ⎪ ⎪τ = ( γ ⎪⎩ [\ + ν [\

( (α ⎧ ⎪σ [ = − ν ε [ + ν ε \ − − ν 7 − 7 ⎪ ⎪ ( (α ⎪ ε \ + ν ε [ − 7 − 7 ⎨σ \ = −ν − ν ⎪ ⎪ ⎪τ = ( γ ⎪⎩ [\ + ν [\

6RLW


⎧ ∂X ∂ Z = −] ⎪ε [ = ∂[ ∂ [ ⎪ ⎪⎪ ∂Y ∂ Z = −] ⎨ε \ = ∂\ ∂ \ ⎪ ⎪ ∂ Z ⎛ ∂X ∂Y ⎞ ⎟⎟ = − ] + ⎪γ [\ = ⎜⎜ ∂[ ∂\ ⎝∂\ ∂[ ⎠ ⎪⎩

ε] = −

⎤ ∂ Z + α + ν $ ⎥ ⋅ ] ∂[ ⎦⎥


⎧ ∂Z ⎪X = − ] ∂ [ ⎪ ∂Z ⎪ 3R3 ⎨Y = −] ∂\ ⎪ ⎪ ⎪ Z = Z [ \ ⎩

⎤ ∂ Z + α + ν $ ⎥ ⋅ ] ∂\ ⎦⎥

- 78 -





⎧ ∂X ∂ Z = −] ⎪ε [ = ∂[ ∂ [ ⎪ ∂Y ∂ Z ⎪⎪ = −] ⎨ε \ = ∂\ ∂ \ ⎪ ⎪ ∂ Z ⎛ ∂X ∂Y ⎞ ⎟⎟ = − ] + ⎪γ [\ = ⎜⎜ ∂[ ∂\ ⎝∂\ ∂[ ⎠ ⎩⎪

ε] = −

⎧ ∂Z ⎪X = − ] ∂ [ ⎪ ∂Z ⎪ 3R3 ⎨Y = − ] ∂\ ⎪ ⎪ ⎪ Z = Z [ \ ⎩

6RLW ⎧ ( ⎡ ∂ Z +ν ⎪σ [ = − ⎢ − ν ⎢⎣ ∂ [ ⎪ ⎪ ( ⎡ ∂ Z ⎪ +ν ⎢ ⎨σ \ = − − ν ⎣⎢ ∂ \ ⎪ ⎪ ⎪τ = − ( ∂ Z ⋅ ] [\ ⎪ + ν ∂ [ ∂ \ ⎩

⎤ ∂ Z + α + ν $ ⎥ ⋅ ] ∂\ ⎥⎦ ⎤ ∂ Z + α + ν $ ⎥ ⋅ ] ∂[ ⎦⎥

- 78 -

⎧ ( ⎡ ∂ Z +ν ⎪σ [ = − ⎢ − ν ⎢⎣ ∂ [ ⎪ ⎪ ( ⎡ ∂ Z ⎪ +ν ⎢ ⎨σ \ = − − ν ⎣⎢ ∂ \ ⎪ ⎪ ⎪τ = − ( ∂ Z ⋅ ] [\ ⎪ + ν ∂ [ ∂ \ ⎩

⎤ ∂ Z + α + ν $ ⎥ ⋅ ] ∂\ ⎥⎦ ⎤ ∂ Z + α + ν $ ⎥ ⋅ ] ∂[ ⎦⎥

- 78 -

HpWDSHH[SUHVVLRQGHVYLVVHXUV

'=

2QSRVH


( K − ν

'=

2QSRVH

⎧ ∂ Z ⎪ 8 [ = − 8 \ = ' − ν ∂[ ∂\ ⎪ ⎪ ⎞ ⎛ ∂ Z ∂ Z ⎪ ⎟ ⎜ ⎪0 [ = − ' ⎜ ∂ [ + ν ∂ \ α + ν $ ⎟ ⎠ ⎝ ⎪ ⎪ ⎞ ⎛∂ Z ∂ Z ⎪ +ν α + ν $ ⎟ ⎨0 \ = − ' ⎜⎜ ⎟ ∂[ ⎪ ⎠ ⎝ ∂\ ⎪ ⎪4 = − ' ∂ (∆ Z + α + ν $ ) ⎪ [ ∂[ ⎪ ⎪ ∂ ⎪4 [ = − ' (∆ Z + α + ν $ ) ∂\ ⎪⎩

( K − ν

⎧ ∂ Z ⎪8 [ = − 8 \ = ' − ν ∂[ ∂\ ⎪ ⎪ ⎞ ⎛ ∂ Z ∂ Z ⎪ ⎟ ⎜ ⎪0 [ = − ' ⎜ ∂ [ + ν ∂ \ α + ν $ ⎟ ⎠ ⎝ ⎪ ⎪ ⎞ ⎛∂ Z ∂ Z ⎪ +ν α + ν $ ⎟ ⎨0 \ = − ' ⎜⎜ ⎟ ∂[ ⎪ ⎠ ⎝ ∂\ ⎪ ⎪4 = − ' ∂ (∆ Z + α + ν $ ) ⎪ [ ∂[ ⎪ ⎪ ∂ ⎪4 [ = − ' (∆ Z + α + ν $ ) ∂\ ⎪⎩

HpWDSHpTXDWLRQGH/DJUDQJH


(Q UHPSODoDQW 4 [ HW 4 \ SDU OHXUV H[SUHVVLRQV FLGHVVXV GDQV OD WURLVLqPH pTXDWLRQ


G¶pTXLOLEUHORFDOGHVIRUFHVpTXLOLEUHVXLYDQWODQRUPDOH]RQREWLHQW


∆ ∆ Z =

S] − α + ν ∆ $ '

∆ ∆ Z =

S] − α + ν ∆ $ '

5HPDUTXHRQD ∆ $ = ∆ 7 = (QUpJLPHVWDWLRQQDLUHHQO¶DEVHQFHGHVRXUFHVGHFKDOHXU YROXPLTXHVRQDGRQF ∆ 7 = ] ⋅ ∆ $ HWO¶pTXDWLRQ SRVVqGHDORUVODPrPHH[SUHVVLRQTXH GDQVOHFDVLVRWKHUPH


5HPDUTXH OHV FRQWUDLQWHV DX SRLQW FRXUDQW 3 GH FRWH ] V¶H[SULPHQW HQ IRQFWLRQ GHV K FRPSRVDQWHVGHYLVVHXUVFRPPHGDQVOHFDVLVRWKHUPHDYHF , =


⎧ ⎧ 0[ ] ⎪τ \] = ⎪σ [ = , ⎪ ⎪ ⎪ ⎪ 0\ ⎪ ⎪ ] ⎨τ [] = ⎨σ \ = − , ⎪ ⎪ ⎪ ⎪ ⎪τ = − ⎪σ ≈ ⎪ [\ ⎪ ] ⎩ ⎩

⎞ 4\ ⎛ ⎜ − ] ⎟ ⎜ K ⎝ K ⎟⎠

4[ K

⎞ ⎛ ⎜ − ] ⎟ ⎜ K ⎟⎠ ⎝

8\ 8[ ]=− ] , ,

⎧ ⎧ 0[ ] ⎪τ \] = ⎪σ [ = , ⎪ ⎪ ⎪ ⎪ 0\ ⎪ ⎪ ] ⎨τ [] = ⎨σ \ = − , ⎪ ⎪ ⎪ ⎪ ⎪τ = − ⎪σ ≈ ⎪ [\ ⎪ ] ⎩ ⎩

- 79 -

4[ K

⎞ ⎛ ⎜ − ] ⎟ ⎜ K ⎟⎠ ⎝

8\ 8[ ]=− ] , ,

- 79 -



'=

2QSRVH

⎞ 4\ ⎛ ⎜ − ] ⎟ ⎜ K ⎝ K ⎟⎠

( K − ν

'=

2QSRVH

⎧ ∂ Z ⎪ 8 [ = − 8 \ = ' − ν ∂[ ∂\ ⎪ ⎪ ⎞ ⎛ ∂ Z ∂ Z ⎪ ⎟ ⎜ ⎪0 [ = − ' ⎜ ∂ [ + ν ∂ \ α + ν $ ⎟ ⎠ ⎝ ⎪ ⎪ ⎞ ⎛∂ Z ∂ Z ⎪ +ν α + ν $ ⎟ ⎨0 \ = − ' ⎜⎜ ⎟ ∂[ ⎪ ⎠ ⎝ ∂\ ⎪ ⎪4 = − ' ∂ (∆ Z + α + ν $ ) ⎪ [ ∂[ ⎪ ⎪ ∂ ⎪4 [ = − ' (∆ Z + α + ν $ ) ∂ \ ⎪⎩

( K − ν

⎧ ∂ Z ⎪8 [ = − 8 \ = ' − ν ∂[ ∂\ ⎪ ⎪ ⎞ ⎛ ∂ Z ∂ Z ⎪ ⎟ ⎜ ⎪0 [ = − ' ⎜ ∂ [ + ν ∂ \ α + ν $ ⎟ ⎠ ⎝ ⎪ ⎪ ⎞ ⎛∂ Z ∂ Z ⎪ +ν α + ν $ ⎟ ⎨0 \ = − ' ⎜⎜ ⎟ ∂[ ⎪ ⎠ ⎝ ∂\ ⎪ ⎪4 = − ' ∂ (∆ Z + α + ν $ ) ⎪ [ ∂[ ⎪ ⎪ ∂ ⎪4 [ = − ' (∆ Z + α + ν $ ) ∂ \ ⎪⎩







∆ ∆ Z =

S] − α + ν ∆ $ '

∆ ∆ Z =

S] − α + ν ∆ $ '





⎧ ⎧ 0[ ] ⎪τ \] = ⎪σ [ = , ⎪ ⎪ ⎪ ⎪ 0\ ⎪ ⎪ ] ⎨τ [] = ⎨σ \ = − , ⎪ ⎪ ⎪ ⎪ ⎪τ = − ⎪σ ≈ ⎪ [\ ⎪ ] ⎩ ⎩

- 79 -

⎞ 4\ ⎛ ⎜ − ] ⎟ K ⎜⎝ K ⎟⎠

4[ K

⎞ ⎛ ⎜ − ] ⎟ ⎜ K ⎟⎠ ⎝

8\ 8[ ]=− ] , ,

⎧ ⎧ 0[ ] ⎪τ \] = ⎪σ [ = , ⎪ ⎪ ⎪ ⎪ 0\ ⎪ ⎪ ] ⎨τ [] = ⎨σ \ = − , ⎪ ⎪ ⎪ ⎪ ⎪τ = − ⎪σ ≈ ⎪ [\ ⎪ ] ⎩ ⎩

- 79 -

⎞ 4\ ⎛ ⎜ − ] ⎟ K ⎜⎝ K ⎟⎠

4[ K

⎞ ⎛ ⎜ − ] ⎟ ⎜ K ⎟⎠ ⎝

8\ 8[ ]=− ] , ,

3RXUOHVGpIRUPDWLRQVRQDOHVIRUPXOHV


⎧ ⎞ ⎛ 0[ + ν 0\ + ν ⎧ τ \] γ \] = + α $ ⎟⎟ ] ⎪ε [ = ⎜⎜ ⎪ ( (, ⎠ ⎝ ⎪ ⎪ ⎪ ⎪ ⎞ ⎛ − 0 \ + ν 0 [ + ν ⎪ ⎪ τ [] + α $ ⎟⎟ ] ⎨γ [] = ⎨ε \ = ⎜⎜ ( (, ⎠ ⎝ ⎪ ⎪ ⎪ ⎪ ⎪ε = ⎛⎜ ν 0 \ − 0 [ + α $ ⎞⎟ ] ⎪γ = + ν τ [\ ⎟ ⎪ ] ⎜ ⎪⎩ [\ ( (, ⎝ ⎠ ⎩

⎧ ⎞ ⎛ 0[ + ν 0\ + ν ⎧ τ \] γ \] = + α $ ⎟⎟ ] ⎪ε [ = ⎜⎜ ⎪ ( (, ⎠ ⎝ ⎪ ⎪ ⎪ ⎪ ⎞ ⎛ − 0 \ + ν 0 [ + ν ⎪ ⎪ τ [] + α $ ⎟⎟ ] ⎨γ [] = ⎨ε \ = ⎜⎜ ( (, ⎠ ⎝ ⎪ ⎪ ⎪ ⎪ ⎪ε = ⎛⎜ ν 0 \ − 0 [ + α $ ⎞⎟ ] ⎪γ = + ν τ [\ ⎟ ⎪ ] ⎜ ⎪⎩ [\ ( (, ⎝ ⎠ ⎩

5HPDUTXHQRXVDYRQVGRQQpGDQVOHFDVLVRWKHUPHWRPH, OHVIRUPXOHVGHVSODTXHVHQ FRRUGRQQpHVFXUYLOLJQHVRUWKRJRQDOHV(OOHVSHXYHQWrWUHFRPSOpWpHVDLVpPHQWSDUOHVWHUPHV GH WHPSpUDWXUH FRPPH FHOD D pWp IDLW SRXU OHV IRUPXOHV j FLGHVVXV 1RXV QH OHV UHSUHQGURQVSDVLFL


([HPSOH SODTXH UHFWDQJXODLUH HQFDVWUpH VXU VHV TXDWUH ERUGV HW VRXPLVHV j XQ JUDGLHQW GH WHPSpUDWXUHWUDQVYHUVDOXQLIRUPH


)LJXUH

)LJXUH

&DUDFWpULVWLTXHVGHODSODTXH


± JpRPpWULTXHVF{WpVDHWEpSDLVVHXUK


± PpFDQLTXHVHWWKHUPLTXHV( ν α


K K ± FKDUJHPHQW SXUHPHQW WKHUPLTXH OHV IDFHV ] = − HW ] = VRQW PDLQWHQXV DX[ WHPSpUDWXUHV 7 HW 7 UHVSHFWLYHPHQWDYHF 7 + 7 = 7 HW 7 − 7 = $ ⋅ K


/DWHPSpUDWXUHDXSRLQWFRXUDQW3HVWGRQF


7 = 7 + $ ⋅ ]

7 = 7 + $ ⋅ ]

- 80 -

- 80 -



⎧ ⎞ ⎛ 0[ + ν 0\ + ν ⎧ + α $ ⎟⎟ ] ⎪ε [ = ⎜⎜ ⎪γ \] = ( τ \] (, ⎠ ⎝ ⎪ ⎪ ⎪ ⎪ − + ν 0 0 ⎞ ⎛ + ν ⎪ ⎪ \ [ τ [] + α $ ⎟⎟ ] ⎨γ [] = ⎨ε \ = ⎜⎜ ( (, ⎠ ⎝ ⎪ ⎪ ⎪ ⎪ ⎪ε = ⎛⎜ ν 0 \ − 0 [ + α $ ⎞⎟ ] ⎪γ = + ν τ [\ ⎟ ⎪ ] ⎜ ⎪⎩ [\ ( (, ⎝ ⎠ ⎩

⎧ ⎞ ⎛ 0[ + ν 0\ + ν ⎧ + α $ ⎟⎟ ] ⎪ε [ = ⎜⎜ ⎪γ \] = ( τ \] (, ⎠ ⎝ ⎪ ⎪ ⎪ ⎪ − + ν 0 0 ⎞ ⎛ + ν ⎪ ⎪ \ [ τ [] + α $ ⎟⎟ ] ⎨γ [] = ⎨ε \ = ⎜⎜ ( (, ⎠ ⎝ ⎪ ⎪ ⎪ ⎪ ⎪ε = ⎛⎜ ν 0 \ − 0 [ + α $ ⎞⎟ ] ⎪γ = + ν τ [\ ⎟ ⎪ ] ⎜ ⎪⎩ [\ ( (, ⎝ ⎠ ⎩





)LJXUH

)LJXUH











7 = 7 + $ ⋅ ]

7 = 7 + $ ⋅ ]

- 80 -

- 80 -

/D VROXWLRQ TXHOTXH SHX SDUDGR[DOH j VXLYDQWHV ⎧X = ⎡ ⎪⎪ ⎢ 3R3 ⎨ Y = [E] ⎢ ⎢ ⎪ ⎢⎣ ⎪⎩Z = DYHF

ε] =

SUHPLqUH YXH HVW FRQVWLWXpH SDU OHV IRQFWLRQV ⎤ ⎡σ [ ⎥ ⎢ ⎥ [Σ] ⎢ ⎥ ⎢ ⎢⎣ ε ] ⎥⎦

σ\

⎤ ⎥ ⎥ ⎥ ⎥⎦

+ ν (α α $ ⋅ ] σ [ = σ \ = − $⋅] − ν − ν

/D VROXWLRQ TXHOTXH SHX SDUDGR[DOH j VXLYDQWHV ⎧X = ⎡ ⎪⎪ ⎢ 3R3 ⎨ Y = [E] ⎢ ⎢ ⎪ ⎢⎣ ⎪⎩Z = DYHF

ε] =


σ\

⎤ ⎥ ⎥ ⎥ ⎥⎦

+ ν (α α $ ⋅ ] σ [ = σ \ = − $⋅] − ν − ν

8 [ = = 8 \ 0 [ = −0 \ = −' + ν α $ 4 [ = = 4 \

8 [ = = 8 \ 0 [ = −0 \ = −' + ν α $ 4 [ = = 4 \

/D SODTXH UHVWH GRQF SODQH Z OH JUDGLHQW GH WHPSpUDWXUH WHQG j FRXUEHU OH IHXLOOHW PR\HQPDLVFHWWHFRXUEXUHHVWHQWLqUHPHQWFRQWUpHSDUOHVPRPHQWVG¶HQFDVWUHPHQW


±)/(;,21$;,6<0e75,48('(6',648(6(7&285211(6

±)/(;,21$;,6<0e75,48('(6',648(6(7&285211(6

/DSODTXHOHFKDUJHPHQWOHVOLDLVRQVDGPHWWHQWR]FRPPHD[HGHUpYROXWLRQ/HFKDUJHPHQW HVWWUDQVYHUVDOPpFDQLTXHHWWKHUPLTXH,OV¶DJLWGRQFG¶XQFDVSDUWLFXOLHUGHFHOXLWUDLWpDX (QFRRUGRQQpHVF\OLQGULTXHV U θ ] OHVIRUPXOHV j V¶pFULYHQW


⎧ ∂Z Y = Z = Z U W ⎪X = − ] ∂U ⎪ ⎨ ⎪ ] ∂Z ∂ Z εθ = − γU θ = ⎪ε U = −] U ∂U ∂ U ⎩


⎧ ( ⎪σU = − − ν ⎪ ⎪ ( ⎪ ⎨σθ = − − ν ⎪ ⎪ ⎪τ = ⎪ Uθ ⎩

⎞ ⎛ ∂ Z ν ∂ Z ]⎜ + + α + ν $ ⎟ ⎟ ⎜ ∂U U ∂U ⎠ ⎝ ⎞ ⎛ ∂Z ∂ Z ]⎜ +ν + α + ν $ ⎟ ⎟ ⎜ U ∂U ∂U ⎠ ⎝

ε] =

⎞ ⎛ ∂ Z ν ∂ Z ]⎜ + + α + ν $ ⎟ ⎟ ⎜ ∂U U ∂U ⎠ ⎝ ⎞ ⎛ ∂Z ∂ Z ]⎜ +ν + α + ν $ ⎟ ⎟ ⎜ U ∂U ∂U ⎠ ⎝

⎧ ⎪8 U = = 8 θ ⎪ ⎪ ⎞ ⎛ ∂ Z ν ∂ Z ⎪ + α + ν $ ⎟ ⎨0 U = −' ⎜⎜ + ⎟ U ∂U ⎪ ⎠ ⎝ ∂U ⎪ ⎪0 = ' ⎛⎜ ∂ Z + ν ∂ Z + α + ν $ ⎞⎟ ⎟ ⎪ θ ⎜ U ∂U ∂ U ⎠ ⎝ ⎩


- 81 -

- 81 -

/D VROXWLRQ TXHOTXH SHX SDUDGR[DOH j VXLYDQWHV ⎧X = ⎡ ⎪⎪ ⎢ 3R3 ⎨ Y = [E] ⎢ ⎢ ⎪ ⎢⎣ ⎩⎪Z = DYHF

⎧ ( ⎪σU = − − ν ⎪ ⎪ ( ⎪ ⎨σθ = − − ν ⎪ ⎪ ⎪τ = ⎪ Uθ ⎩


σ\

⎤ ⎥ ⎥ ⎥ ⎥⎦

+ ν (α α $ ⋅ ] σ [ = σ \ = − $⋅] − ν − ν

/D VROXWLRQ TXHOTXH SHX SDUDGR[DOH j VXLYDQWHV ⎧X = ⎡ ⎪⎪ ⎢ 3R3 ⎨ Y = [E] ⎢ ⎢ ⎪ ⎢⎣ ⎩⎪Z = DYHF

ε] =


σ\

⎤ ⎥ ⎥ ⎥ ⎥⎦

+ ν (α α $ ⋅ ] σ [ = σ \ = − $⋅] − ν − ν

8 [ = = 8 \ 0 [ = −0 \ = −' + ν α $ 4 [ = = 4 \

8 [ = = 8 \ 0 [ = −0 \ = −' + ν α $ 4 [ = = 4 \



±)/(;,21$;,6<0e75,48('(6',648(6(7&285211(6

±)/(;,21$;,6<0e75,48('(6',648(6(7&285211(6





⎧ ( ⎪σU = − − ν ⎪ ⎪ ( ⎪ ⎨σθ = − − ν ⎪ ⎪ ⎪τ = ⎪ Uθ ⎩

⎞ ⎛ ∂ Z ν ∂ Z ]⎜ + + α + ν $ ⎟ ⎟ ⎜ ∂U U U ∂ ⎠ ⎝ ⎞ ⎛ ∂Z ∂ Z ]⎜ +ν + α + ν $ ⎟ ⎟ ⎜ U ∂U ∂U ⎠ ⎝

⎧ ( ⎪σU = − − ν ⎪ ⎪ ( ⎪ ⎨σθ = − − ν ⎪ ⎪ ⎪τ = ⎪ Uθ ⎩

⎞ ⎛ ∂ Z ν ∂ Z ]⎜ + + α + ν $ ⎟ ⎟ ⎜ ∂U U U ∂ ⎠ ⎝ ⎞ ⎛ ∂Z ∂ Z ]⎜ +ν + α + ν $ ⎟ ⎟ ⎜ U ∂U ∂U ⎠ ⎝



- 81 -

- 81 -

∂ ⎧ ⎪⎪4 U = − ' ∂ U ∆ Z + α + ν $ ⎨ ⎪4θ = ⎪⎩ ∂ ⎛ ∂ ⎞ ⎜U ⎟ /¶RSpUDWHXUODSODFLHQV¶pFULWLFL ∆= U ∂ U ⎜⎝ ∂ U ⎟⎠ 5HPDUTXHHQUpJLPHVWDWLRQQDLUHRQD

∂ ⎧ ⎪⎪4 U = − ' ∂ U ∆ Z + α + ν $ ⎨ ⎪4θ = ⎪⎩ ∂ ⎛ ∂ ⎞ ⎜U ⎟ /¶RSpUDWHXUODSODFLHQV¶pFULWLFL ∆= U ∂ U ⎜⎝ ∂ U ⎟⎠

∂ G = ∂U GU

0U ⎧ ⎧τ θ ] = ⎪σ U = , ] ⎪ ⎪ ⎪⎪ 0θ 4U ⎪ ] ⎨τ U] = ⎨σθ = − , K ⎪ ⎪ ⎪ ⎪ ⎪σ ] = ⎩⎪τ U θ = ⎩


0U ⎧ ⎧τ θ ] = ⎪σ U = , ] ⎪ ⎪ ⎪⎪ 0θ 4U ⎪ ] ⎨τ U] = ⎨σθ = − , K ⎪ ⎪ ⎪ ⎪ ⎪σ ] = ⎩⎪τ U θ = ⎩

⎞ ⎛ ⎜ − ] ⎟ ⎜ K ⎟⎠ ⎝

⎧ ⎛ 0 U + ν 0θ ⎞ ⎧γ θ ] = + α $⎟ ] ⎪ε U = ⎜ (, ⎪ ⎝ ⎠ ⎪ ⎪ ⎪ ⎪ + ν 4 U ⎛⎜ ] ⎞ ⎪ ⎛ 0θ + ν 0U ⎞ + α $ ⎟ ] ⎨γ U] = − ⎟ ⎨εθ = ⎜ − (, ( K ⎜⎝ K ⎟⎠ ⎝ ⎠ ⎪ ⎪ ⎪ ⎪ ⎪ε = ⎛ ν 0 θ − 0 U + α $ ⎞ ] ⎪γ = ⎜ ⎟ ⎩ Uθ ⎪⎩ ] ⎝ (, ⎠

([HPSOHGLVTXHFLUFXODLUHVRXVJUDGLHQWWUDQVYHUVDOXQLIRUPHGHWHPSpUDWXUH $ =

7 + 7 = 7

∂ G = ∂U GU

⎞ ⎛ ⎜ − ] ⎟ ⎜ K ⎟⎠ ⎝

⎧ ⎛ 0 U + ν 0θ ⎞ ⎧γ θ ] = + α $⎟ ] ⎪ε U = ⎜ (, ⎪ ⎝ ⎠ ⎪ ⎪ ⎪ ⎪ + ν 4 U ⎛⎜ ] ⎞ ⎪ ⎛ 0θ + ν 0U ⎞ + α $ ⎟ ] ⎨γ U] = − ⎟ ⎨εθ = ⎜ − (, ( K ⎜⎝ K ⎟⎠ ⎝ ⎠ ⎪ ⎪ ⎪ ⎪ ⎪ε = ⎛ ν 0 θ − 0 U + α $ ⎞ ] ⎪γ = ⎜ ⎟ ⎩ Uθ ⎪⎩ ] ⎝ (, ⎠

7 − 7 K


7 + 7 = 7

7 − 7 K

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)LJXUH

- 82 -

- 82 -

∂ ⎧ ⎪⎪4 U = − ' ∂ U ∆ Z + α + ν $ ⎨ ⎪4θ = ⎩⎪

∂ ⎧ ⎪⎪4 U = − ' ∂ U ∆ Z + α + ν $ ⎨ ⎪4θ = ⎩⎪

/¶RSpUDWHXUODSODFLHQV¶pFULWLFL

∆=


∂ ⎛ ∂ ⎞ ⎜U ⎟ U ∂ U ⎜⎝ ∂ U ⎟⎠

/¶RSpUDWHXUODSODFLHQV¶pFULWLFL

∂ G = ∂U GU

0U ⎧ ⎧τ θ ] = ⎪σ U = , ] ⎪ ⎪ ⎪⎪ 0θ 4U ⎪ ] ⎨τ U] = ⎨σθ = − , K ⎪ ⎪ ⎪ ⎪ ⎪⎩τ U θ = ⎪σ ] = ⎩


⎞ ⎛ ⎜ − ] ⎟ ⎜ K ⎟⎠ ⎝


∂ ⎛ ∂ ⎞ ⎜U ⎟ U ∂ U ⎜⎝ ∂ U ⎟⎠

∂ G = ∂U GU

0U ⎧ ⎧τ θ ] = ⎪σ U = , ] ⎪ ⎪ ⎪⎪ 0θ 4U ⎪ ] ⎨τ U] = ⎨σθ = − , K ⎪ ⎪ ⎪ ⎪ ⎪⎩τ U θ = ⎪σ ] = ⎩

⎧ ⎛ 0 U + ν 0θ ⎞ ⎧γ θ ] = + α $⎟ ] ⎪ε U = ⎜ (, ⎪ ⎝ ⎠ ⎪ ⎪ ⎪ ⎪ + ν 4 U ⎛⎜ ] ⎞ ⎪ ⎛ 0θ + ν 0U ⎞ + α $ ⎟ ] ⎨γ U] = − ⎟ ⎨εθ = ⎜ − ⎜ (, ( K ⎝ K ⎟⎠ ⎝ ⎠ ⎪ ⎪ ⎪ ⎪ ⎪ε = ⎛ ν 0 θ − 0 U + α $ ⎞ ] ⎪γ = ⎟ ⎩ Uθ ⎪⎩ ] ⎜⎝ (, ⎠

7 + 7 = 7

∆=

⎞ ⎛ ⎜ − ] ⎟ ⎜ K ⎟⎠ ⎝

⎧ ⎛ 0 U + ν 0θ ⎞ ⎧γ θ ] = + α $⎟ ] ⎪ε U = ⎜ (, ⎪ ⎝ ⎠ ⎪ ⎪ ⎪ ⎪ + ν 4 U ⎛⎜ ] ⎞ ⎪ ⎛ 0θ + ν 0U ⎞ + α $ ⎟ ] ⎨γ U] = − ⎟ ⎨εθ = ⎜ − ⎜ (, ( K ⎝ K ⎟⎠ ⎝ ⎠ ⎪ ⎪ ⎪ ⎪ ⎪ε = ⎛ ν 0 θ − 0 U + α $ ⎞ ] ⎪γ = ⎟ ⎩ Uθ ⎪⎩ ] ⎜⎝ (, ⎠

7 − 7 K


7 + 7 = 7

7 − 7 K

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- 82 -

- 82 -

6LO¶RQLVROHSDUODSHQVpHXQGLVTXHGHUD\RQUHWTXHO¶RQpFULWVRQpTXLOLEUHVXLYDQW]RQ WURXYH G 4 U = VRLW ∆ Z + α + ν $ = GU


/¶LQWpJUDWLRQ GH FHWWH pTXDWLRQ GRQQH FRPSWH WHQX GH OD FRQGLWLRQ Z =


Z = −α + ν $ ⋅ & ⋅ U

Z = −α + ν $ ⋅ & ⋅ U

/D FRQVWDQWH G¶LQWpJUDWLRQ & VH GpWHUPLQH j O¶DLGH GH OD FRQGLWLRQ 0 U = VXU OH ERUG OLEUH U D RQWURXYH


⎞ ⎛ G Z ν G Z ⎜ = VRLW & = + + α + ν $ ⎟ ⎟ ⎜ G U + ν U G U ⎠ U =D ⎝


Z U = − α $ U

'¶RODIOqFKH

Z U = − α $ U

'¶RODIOqFKH

2QHQGpGXLWWRXVOHVSDUDPqWUHVGXSUREOqPHHQSDUWLFXOLHU ε U = εθ = α $ ⋅ ] σ U = σθ = 0 U = 0 θ = /HFKDPSGHWHPSpUDWXUHpWDQWOLQpDLUHHWOHGLVTXHOLEUHGHVHGpIRUPHULOHVW QRUPDOG¶DYRLUWURXYpGHVFRQWUDLQWHVHWYLVVHXUVSDUWRXWQXOV


±)/(;,21&
±)/(;,21&

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- 83 -

- 83 -





Z = −α + ν $ ⋅ & ⋅ U

Z = −α + ν $ ⋅ & ⋅ U





'¶RODIOqFKH

Z U = − α $ U

'¶RODIOqFKH

Z U = − α $ U



±)/(;,21&
±)/(;,21&

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- 83 -

- 83 -

/HVIRUPXOHV j V¶pFULYHQWDORUV


⎧ ∂Z Y = Z = Z [ ⎪X = − ] ∂[ ⎪ ⎨ ⎪ ∂ Z ε \ = γ [\ = ⎪ε [ = − ] ∂ [ ⎩

⎧ ( ⎪σ [ = − − ν ⎪ ⎪ ( ⎪ ⎨σ \ = − − ν ⎪ ⎪ ⎪τ = ⎪ [\ ⎩

⎧ ⎪8 [ = = 8 \ ⎪ ⎪ ⎞ ⎛ ∂ Z ⎪ ⎜ 0 = − ' + α + ν $ [ ⎟ [ ⎪ ⎟ ⎜ ∂ [ ⎠ ⎝ ⎪ ⎪ ⎞ ⎛ ∂ Z ⎪ + α + ν $ [ ⎟ ⎨0 \ = ' ⎜⎜ ν ⎟ ⎪ ⎠ ⎝ ∂[ ⎪ ⎪4 = − ' ⎛⎜ ∂ Z + α + ν ∂ $ ⎞⎟ ⎪ [ ⎜ ∂ [ ∂ [ ⎟⎠ ⎝ ⎪ ⎪ ⎪4 [ = ⎪⎩

HWO¶pTXDWLRQGH/DJUDQJH

⎧ ∂Z Y = Z = Z [ ⎪X = − ] ∂[ ⎪ ⎨ ⎪ ∂ Z ε \ = γ [\ = ⎪ε [ = − ] ∂ [ ⎩

⎧ ( ⎪σ [ = − − ν ⎪ ⎪ ( ⎪ ⎨σ \ = − − ν ⎪ ⎪ ⎪τ = ⎪ [\ ⎩

⎧ ⎪8 [ = = 8 \ ⎪ ⎪ ⎞ ⎛ ∂ Z ⎪ ⎜ 0 = − ' + α + ν $ [ ⎟ [ ⎪ ⎟ ⎜ ∂ [ ⎠ ⎝ ⎪ ⎪ ⎞ ⎛ ∂ Z ⎪ + α + ν $ [ ⎟ ⎨0 \ = ' ⎜⎜ ν ⎟ ⎪ ⎠ ⎝ ∂[ ⎪ ⎪4 = − ' ⎛⎜ ∂ Z + α + ν ∂ $ ⎞⎟ ⎪ [ ⎜ ∂ [ ∂ [ ⎟⎠ ⎝ ⎪ ⎪ ⎪4 [ = ⎪⎩

⎤ ⎡ ∂ Z ⎢ + α + ν $ [ ⎥ ⋅ ] ⎦⎥ ⎣⎢ ∂ [ ⎤ ⎡ ∂ Z + α + ν $ [ ⎥ ⋅ ] ⎢ν ⎥⎦ ⎢⎣ ∂ [

∂ Z S] ∂ $ = − α + ν ' ∂ [ ∂ [

5HPDUTXH HQ UpJLPH VWDWLRQQDLUH RQ D


∂ G SXLVTXH OHV SDUDPqWUHV QH GpSHQGHQW = ∂[ G[

⎤ ⎡ ∂ Z ⎢ + α + ν $ [ ⎥ ⋅ ] ⎦⎥ ⎣⎢ ∂ [ ⎤ ⎡ ∂ Z + α + ν $ [ ⎥ ⋅ ] ⎢ν ⎥⎦ ⎢⎣ ∂ [

∂ Z S] ∂ $ = − α + ν ' ∂ [ ∂ [



TXHGH[HWSDVGXWHPSV

TXHGH[HWSDVGXWHPSV

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- 84 -

- 84 -



⎧ ∂Z Y = Z = Z [ ⎪X = − ] ∂[ ⎪ ⎨ ⎪ ∂ Z ε \ = γ [\ = ⎪ε [ = − ] ∂ [ ⎩

⎧ ( ⎪σ [ = − − ν ⎪ ⎪ ( ⎪ ⎨σ \ = − − ν ⎪ ⎪ ⎪τ = ⎪ [\ ⎩

⎧ ⎪8 [ = = 8 \ ⎪ ⎪ ⎞ ⎛ ∂ Z ⎪ ⎟ ⎜ ⎪0 [ = − ' ⎜ ∂ [ + α + ν $ [ ⎟ ⎠ ⎝ ⎪ ⎪ ⎞ ⎛ ∂ Z ⎪ + α + ν $ [ ⎟ ⎨0 \ = ' ⎜⎜ ν ⎟ ⎪ ⎠ ⎝ ∂[ ⎪ ⎪4 = − ' ⎛⎜ ∂ Z + α + ν ∂ $ ⎞⎟ ⎪ [ ⎜ ∂ [ ∂ [ ⎟⎠ ⎝ ⎪ ⎪ ⎪4 [ = ⎪⎩


⎧ ∂Z Y = Z = Z [ ⎪X = − ] ∂[ ⎪ ⎨ ⎪ ∂ Z ε \ = γ [\ = ⎪ε [ = − ] ∂ [ ⎩

⎧ ( ⎪σ [ = − − ν ⎪ ⎪ ( ⎪ ⎨σ \ = − − ν ⎪ ⎪ ⎪τ = ⎪ [\ ⎩

⎧ ⎪8 [ = = 8 \ ⎪ ⎪ ⎞ ⎛ ∂ Z ⎪ ⎟ ⎜ ⎪0 [ = − ' ⎜ ∂ [ + α + ν $ [ ⎟ ⎠ ⎝ ⎪ ⎪ ⎞ ⎛ ∂ Z ⎪ + α + ν $ [ ⎟ ⎨0 \ = ' ⎜⎜ ν ⎟ ⎪ ⎠ ⎝ ∂[ ⎪ ⎪4 = − ' ⎛⎜ ∂ Z + α + ν ∂ $ ⎞⎟ ⎪ [ ⎜ ∂ [ ∂ [ ⎟⎠ ⎝ ⎪ ⎪ ⎪4 [ = ⎪⎩

⎤ ⎡ ∂ Z ⎢ + α + ν $ [ ⎥ ⋅ ] ⎦⎥ ⎣⎢ ∂ [ ⎤ ⎡ ∂ Z + α + ν $ [ ⎥ ⋅ ] ⎢ν ⎥⎦ ⎢⎣ ∂ [

∂ Z S] ∂ $ = − α + ν ' ∂[ ∂ [




⎤ ⎡ ∂ Z ⎢ + α + ν $ [ ⎥ ⋅ ] ⎦⎥ ⎣⎢ ∂ [ ⎤ ⎡ ∂ Z + α + ν $ [ ⎥ ⋅ ] ⎢ν ⎥⎦ ⎢⎣ ∂ [

∂ Z S] ∂ $ = − α + ν ' ∂[ ∂ [



TXHGH[HWSDVGXWHPSV

TXHGH[HWSDVGXWHPSV





- 84 -

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⎛ ∂ Z ⎞ 0[ = − ' ⎜ + α + ν $ ⎟ = −5 $ / − [ ⎟ ⎜ ∂ [ ⎝ ⎠

⎛ ∂ Z ⎞ 0[ = − ' ⎜ + α + ν $ ⎟ = −5 $ / − [ ⎟ ⎜ ∂ [ ⎝ ⎠

G¶R

∂ Z 5 $ = / − [ − α + ν $ ' ∂ [

/¶LQWpJUDWLRQGRQQHFRPSWHWHQXGHVFRQGLWLRQVG¶HQFDVWUHPHQWHQ[

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[ ⎞⎟ ∂ Z 5 $ ⎛⎜ /[ − − α + ν $ ⋅ [ = ' ⎝⎜ ⎠⎟ ∂[ Z [ =

5$ '

5$ =

[ ⎞⎟ ∂ Z 5 $ ⎛⎜ /[ − − α + ν $ ⋅ [ = ' ⎝⎜ ⎠⎟ ∂[

⎛ [ [ ⎞ [ ⎜/ − ⎟ − α + ν $ ⋅ ⎟ ⎜ ⎠ ⎝

/DFRQGLWLRQZ/ SHUPHWGHOHYHUO¶K\SHUVWDWLFLWpRQWURXYH

Z [ =

5$ '

⎛ [ [ ⎞ [ ⎜/ − ⎟ − α + ν $ ⋅ ⎟ ⎜ ⎠ ⎝

/DFRQGLWLRQZ/ SHUPHWGHOHYHUO¶K\SHUVWDWLFLWpRQWURXYH

α ' + ν $ α ( K $ = / − ν /

/HVpTXDWLRQVGHO¶pTXLOLEUHJOREDO

∂ Z 5 $ = / − [ − α + ν $ ' ∂ [

5$ =

α ' + ν $ α ( K $ = / − ν /


5 $ + 5 = HW MH − 5 $ ⋅ / =

5 $ + 5 = HW MH − 5 $ ⋅ / =

GRQQHQW HQVXLWH OD UpDFWLRQ OLQpLTXH HQ [ 5 = −5 $ HW OH PRPHQW G¶HQFDVWUHPHQW OLQpLTXH MH = 5 $ ⋅ / /DIOqFKHZ[ DILQDOHPHQWSRXUH[SUHVVLRQ


Z [ =

⎛ [ [ ⎞ α + ν $ / ⎜ − ⎟ ⎜/ / ⎟⎠ ⎝

Z [ =

⎛ [ [ ⎞ α + ν $ / ⎜ − ⎟ ⎜/ / ⎟⎠ ⎝

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±)/$0%(0(177+(50,48(

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- 85 -

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⎛ ∂ Z ⎞ 0[ = − ' ⎜ + α + ν $ ⎟ = −5 $ / − [ ⎟ ⎜ ∂ [ ⎝ ⎠

⎛ ∂ Z ⎞ 0[ = − ' ⎜ + α + ν $ ⎟ = −5 $ / − [ ⎟ ⎜ ∂ [ ⎝ ⎠

G¶R

∂ Z 5 $ = / − [ − α + ν $ ' ∂ [

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5$ '

⎛ [ [ ⎞ [ ⎜/ − ⎟ − α + ν $ ⋅ ⎜ ⎠⎟ ⎝

/DFRQGLWLRQZ/ SHUPHWGHOHYHUO¶K\SHUVWDWLFLWpRQWURXYH 5$ =

α ' + ν $ α ( K $ = / − ν /


G¶R

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5$ '

⎛ [ [ ⎞ [ ⎜/ − ⎟ − α + ν $ ⋅ ⎜ ⎠⎟ ⎝

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α ' + ν $ α ( K $ = / − ν /


5 $ + 5 = HW MH − 5 $ ⋅ / =

5 $ + 5 = HW MH − 5 $ ⋅ / =



Z [ =

⎛ [ [ ⎞ α + ν $ / ⎜ − ⎟ ⎜/ / ⎟⎠ ⎝

Z [ =

⎛ [ [ ⎞ α + ν $ / ⎜ − ⎟ ⎜/ / ⎟⎠ ⎝



±)/$0%(0(177+(50,48(

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)=

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7 − 7 =

)=

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)=

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SRXU

7 − 7 =

' U

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- 86 -

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)=

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)=

X 5 (K − ν 5

( K α 7 − 7 − ν

&HWWHIRUFHDWWHLQWVDYDOHXUFULWLTXH )&5 = + ν SRXU

7 − 7 =

)=

2QHQGpGXLW

' U

( K α 7 − 7 − ν

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SRXU

- 86 -

)=

X 5 (K − ν 5

7 − 7 =

' U

' − ν ⎛ K ⎞ = ⎜ ⎟ α ⎝5⎠ ( K α 5

- 86 -

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- 87 -

- 87 -

&+$3,75(9

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7+(50,48('(6&248(6

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UHK\SRWKqVH

UHK\SRWKqVH









- 87 -

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+ &HWWHWURLVLqPHWUDQVODWLRQHVWQpJOLJHDEOHGHYDQWOHVGHX[DXWUHV


&HWWH SUHPLqUH K\SRWKqVHV HVW GRQF LQFKDQJpH SDU UDSSRUW DX FDV LVRWKHUPH HW D OHV GHX[ PrPHVFRQVpTXHQFHVSULQFLSDOHV


UHFRQVpTXHQFH

UHFRQVpTXHQFH

⎧X = X + = ω G ⎪⎪ X ⎨X = X − = ω ⎪ ⎪⎩Z = Z

HFRQVpTXHQFH

HFRQVpTXHQFH ⎧ ∂Z X + ⎪ ω = β ∂ α 5 1 ⎪ ⎪ ∂Z X ⎪ + ⎨ ω = − β ∂ α 5 1 ⎪ ⎪ ⎪ ω = ⎛⎜ ∂ X − ∂ X + X + X ⎞⎟ ⎪ ] ⎜β ∂α β ∂α 5J 5J ⎟⎠ ⎝ ⎩

⎧X = X + = ω G ⎪⎪ X ⎨X = X − = ω ⎪ ⎪⎩Z = Z

⎧ ∂Z X + ⎪ ω = β ∂ α 5 1 ⎪ ⎪ ∂Z X ⎪ + ⎨ ω = − β ∂ α 5 1 ⎪ ⎪ ⎪ ω = ⎛⎜ ∂ X − ∂ X + X + X ⎞⎟ ⎪ ] ⎜β ∂α β ∂α 5J 5J ⎟⎠ ⎝ ⎩

HK\SRWKqVH

HK\SRWKqVH

+ σ ] HVWQpJOLJHDEOHGHYDQW σ HW σ FRPPHGDQVOHFDVLVRWKHUPH


&RQVpTXHQFHRQHQGpGXLWHQYHUWXGHODORLGH+RRNH'XKDPHO


⎧ ⎪ε = ( σ − ν σ + α 7 − 7 ⎪ ⎪ε = σ − ν σ + α 7 − 7 ⎪⎪ ( ⎨ ⎪ε = − ν σ + σ + α 7 − 7 = − ν ε + ε + + ν α 7 − 7 ⎪ ] ( − ν − ν ⎪ ⎪γ = + ν τ HWLQYHUVHPHQW ( ⎩⎪

(α ( ⎧ ⎪σ = − ν ε + ν ε − − ν 7 − 7 ⎪ (α ( ⎪ ε + ν ε − 7 − 7 ⎨σ = −ν − ν ⎪ ⎪ ( γ ⎪τ = + ν ⎩

⎧ ⎪ε = ( σ − ν σ + α 7 − 7 ⎪ ⎪ε = σ − ν σ + α 7 − 7 ⎪⎪ ( ⎨ ⎪ε = − ν σ + σ + α 7 − 7 = − ν ε + ε + + ν α 7 − 7 ⎪ ] ( − ν − ν ⎪ ⎪γ = + ν τ HWLQYHUVHPHQW ( ⎩⎪


- 88 -

- 88 -





UHFRQVpTXHQFH

UHFRQVpTXHQFH

⎧X = X + = ω G ⎪⎪ X ⎨X = X − = ω ⎪ ⎪⎩Z = Z

HFRQVpTXHQFH

HFRQVpTXHQFH ⎧ ∂Z X + ⎪ ω = β ∂ α 5 1 ⎪ ⎪ ∂Z X ⎪ + ⎨ ω = − β ∂ α 5 1 ⎪ ⎪ ⎪ ω = ⎛⎜ ∂ X − ∂ X + X + X ⎞⎟ ⎪ ] ⎜β ∂α β ∂α 5J 5J ⎟⎠ ⎝ ⎩

⎧X = X + = ω G ⎪⎪ X ⎨X = X − = ω ⎪ ⎪⎩Z = Z

⎧ ∂Z X + ⎪ ω = β ∂ α 5 1 ⎪ ⎪ ∂Z X ⎪ + ⎨ ω = − β ∂ α 5 1 ⎪ ⎪ ⎪ ω = ⎛⎜ ∂ X − ∂ X + X + X ⎞⎟ ⎪ ] ⎜β ∂α β ∂α 5J 5J ⎟⎠ ⎝ ⎩

HK\SRWKqVH

HK\SRWKqVH





⎧ ⎪ε = ( σ − ν σ + α 7 − 7 ⎪ ⎪ε = σ − ν σ + α 7 − 7 ⎪⎪ ( ⎨ ⎪ε = − ν σ + σ + α 7 − 7 = − ν ε + ε + + ν α 7 − 7 ⎪ ] ( − ν − ν ⎪ ⎪γ = + ν τ HWLQYHUVHPHQW ⎪⎩ (


- 88 -

⎧ ⎪ε = ( σ − ν σ + α 7 − 7 ⎪ ⎪ε = σ − ν σ + α 7 − 7 ⎪⎪ ( ⎨ ⎪ε = − ν σ + σ + α 7 − 7 = − ν ε + ε + + ν α 7 − 7 ⎪ ] ( − ν − ν ⎪ ⎪γ = + ν τ HWLQYHUVHPHQW ⎪⎩ (


- 88 -

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HK\SRWKqVH

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= ⎛ = ⎞ ⎟⎟ =+ + ⎜⎜ = 5 5 1 1 ⎝ ⎠ − 51

= ⎛ = ⎞ ⎟⎟ =+ + ⎜⎜ = 5 5 1 1 ⎝ ⎠ − 51

HWSRXUODWHPSpUDWXUH 7 3 = 7 + 7′ = + 7′′ = DYHF7 = 7 3


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±7+e25,(48$'5$7,48(

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/HVIRUPXOHV GRQQHQWOHVFRPSRVDQWHVGHODURWDWLRQHQOHVSRUWDQWGDQVOHVIRUPXOHV RQREWLHQW ⎧ ⎛ ∂Z X ⎞ + ⎟⎟ ] ⎪X = X − ⎜⎜ β ∂ α 5 1 ⎠ ⎝ ⎪ ⎪ ⎛ G ∂Z X ⎞ ⎪ X = 3R3 ⎨X = X − ⎜⎜ + ⎟⎟ ] ⎝ β ∂ α 5 1 ⎠ ⎪ ⎪ ⎪Z = Z α α W ⎪⎩

/HVIRUPXOHV GRQQHQWOHVFRPSRVDQWHVGHODURWDWLRQHQOHVSRUWDQWGDQVOHVIRUPXOHV RQREWLHQW ⎧ ⎛ ∂Z X ⎞ + ⎟⎟ ] ⎪X = X − ⎜⎜ β ∂ α 5 1 ⎠ ⎝ ⎪ ⎪ ⎛ G ∂Z X ⎞ ⎪ X = 3R3 ⎨X = X − ⎜⎜ + ⎟⎟ ] ⎝ β ∂ α 5 1 ⎠ ⎪ ⎪ ⎪Z = Z α α W ⎪⎩



6L O¶RQ UHPSODFH X X HW Z SDU OHXUV H[SUHVVLRQV FLGHVVXV GDQV OHV IRUPXOHV GX WRPH,FKDSLWUH9,,,RQREWLHQW ⎧ε = ε + ] ε′ + ] ε′′ ⎪ ⎪ ⎨ε = ε + ] ε′ + ] ε′′ ⎪ ′ + ] γ ′′ ⎪⎩γ = γ + ] γ DYHF ⎧ ∂ X X Z − − ⎪ε = 5J 5 β ∂ α 1 ⎪ ⎪ ∂ X X Z ⎪ + − ⎨ ε = 5J 5 β ∂ α 1 ⎪ ⎪ ⎞ ⎛ ⎪γ = ⎜ ∂ X + ∂ X + X − X ⎟ ⎜ ⎝ β ∂ α β ∂ α 5J 5J ⎟⎠ ⎪⎩

6L O¶RQ UHPSODFH X X HW Z SDU OHXUV H[SUHVVLRQV FLGHVVXV GDQV OHV IRUPXOHV GX WRPH,FKDSLWUH9,,,RQREWLHQW ⎧ε = ε + ] ε′ + ] ε′′ ⎪ ⎪ ⎨ε = ε + ] ε′ + ] ε′′ ⎪ ′ + ] γ ′′ ⎪⎩γ = γ + ] γ DYHF ⎧ ∂ X X Z − − ⎪ε = 5J 5 β ∂ α 1 ⎪ ⎪ ∂ X X Z ⎪ + − ⎨ ε = 5J 5 β ∂ α 1 ⎪ ⎪ ⎞ ⎛ ⎪γ = ⎜ ∂ X + ∂ X + X − X ⎟ ⎜ ⎝ β ∂ α β ∂ α 5J 5J ⎟⎠ ⎪⎩

- 89 -

- 89 -

HK\SRWKqVH

HK\SRWKqVH





−

= 51

=+

= ⎛ = ⎞ ⎟ +⎜ 5 1 ⎜⎝ 5 1 ⎟⎠

−

= 51

=+

= ⎛ = ⎞ ⎟ +⎜ 5 1 ⎜⎝ 5 1 ⎟⎠





±7+e25,(48$'5$7,48(

±7+e25,(48$'5$7,48(



/HVIRUPXOHV GRQQHQWOHVFRPSRVDQWHVGHODURWDWLRQHQOHVSRUWDQWGDQVOHVIRUPXOHV RQREWLHQW ⎧ ⎛ ∂Z X ⎞ + ⎟⎟ ] ⎪X = X − ⎜⎜ ⎝ β ∂ α 5 1 ⎠ ⎪ ⎪ ⎛ ∂Z G X ⎞ ⎪ X = 3R3 ⎨X = X − ⎜⎜ + ⎟⎟ ] ⎝ β ∂ α 5 1 ⎠ ⎪ ⎪ ⎪Z = Z α α W ⎩⎪

/HVIRUPXOHV GRQQHQWOHVFRPSRVDQWHVGHODURWDWLRQHQOHVSRUWDQWGDQVOHVIRUPXOHV RQREWLHQW ⎧ ⎛ ∂Z X ⎞ + ⎟⎟ ] ⎪X = X − ⎜⎜ ⎝ β ∂ α 5 1 ⎠ ⎪ ⎪ ⎛ ∂Z G X ⎞ ⎪ X = 3R3 ⎨X = X − ⎜⎜ + ⎟⎟ ] ⎝ β ∂ α 5 1 ⎠ ⎪ ⎪ ⎪Z = Z α α W ⎩⎪



6L O¶RQ UHPSODFH X X HW Z SDU OHXUV H[SUHVVLRQV FLGHVVXV GDQV OHV IRUPXOHV GX WRPH,FKDSLWUH9,,,RQREWLHQW ⎧ε = ε + ] ε′ + ] ε′′ ⎪ ⎪ ⎨ε = ε + ] ε′ + ] ε′′ ⎪ ′ + ] γ ′′ ⎪⎩γ = γ + ] γ DYHF ⎧ ∂ X X Z − − ⎪ε = 5J 5 β ∂ α 1 ⎪ ⎪ ∂ X X Z ⎪ + − ⎨ ε = β ∂ α 5J 5 1 ⎪ ⎪ ⎞ ⎛ ⎪γ = ⎜ ∂ X + ∂ X + X − X ⎟ ⎟ ⎜ 5J 5J β ∂ α β ∂ α ⎪⎩ ⎠ ⎝

6L O¶RQ UHPSODFH X X HW Z SDU OHXUV H[SUHVVLRQV FLGHVVXV GDQV OHV IRUPXOHV GX WRPH,FKDSLWUH9,,,RQREWLHQW ⎧ε = ε + ] ε′ + ] ε′′ ⎪ ⎪ ⎨ε = ε + ] ε′ + ] ε′′ ⎪ ′ + ] γ ′′ ⎪⎩γ = γ + ] γ DYHF ⎧ ∂ X X Z − − ⎪ε = 5J 5 β ∂ α 1 ⎪ ⎪ ∂ X X Z ⎪ + − ⎨ ε = β ∂ α 5J 5 1 ⎪ ⎪ ⎞ ⎛ ⎪γ = ⎜ ∂ X + ∂ X + X − X ⎟ ⎟ ⎜ 5J 5J β ∂ α β ∂ α ⎪⎩ ⎠ ⎝

- 89 -

- 89 -

⎧ ∂Z X ⎛ ⎞ X ∂ ⎛ ⎞ Z ∂ ⎛ ∂Z ⎞ ⎜⎜ ⎟⎟ − ⎜⎜ ⎟⎟ − − ⎜⎜ ⎟⎟ + − ⎪ε′ = 5J ⎝ 5 1 5 1 ⎠ β ∂ α ⎝ 5 1 ⎠ 5 1 β ∂ α ⎝ β ∂ α ⎠ β 5J ∂ α ⎪ ⎪ ⎪ ⎪ ⎪ ∂Z X ⎛ ⎞ X ∂ ⎛ ⎞ Z ∂ ⎛ ∂Z ⎞ ⎟⎟ − ⎜⎜ ⎟⎟ − − ⎜⎜ ⎟⎟ − − ⎪ε′ = ⎜⎜ 5J ⎝ 5 1 5 1 ⎠ β ∂ α ⎝ 5 1 ⎠ 5 1 β ∂ α ⎝ β ∂ α ⎠ β 5J ∂ α ⎪⎪ ⎨ ⎪ ⎪ ∂ X ∂ ⎛ X ⎞ ∂ X ∂ ⎛ X ⎞ ⎪ γ′ = ⎜⎜ ⎟⎟ + ⎜ ⎟ − − ⎪ β 5 β ∂ α ⎝ 5 1 ⎠ β 5 1 ∂ α β ∂ α ⎜⎝ 5 1 ⎟⎠ 1 ∂ α ⎪ ⎪ ∂Z ∂ Z ∂Z ⎪ − + − − − ⎪ β 5J ∂ α β 5J ∂ α β β ∂ α ∂ α ⎪⎩

⎧ ∂Z X ⎛ ⎞ X ∂ ⎛ ⎞ Z ∂ ⎛ ∂Z ⎞ ⎜⎜ ⎟⎟ − ⎜⎜ ⎟⎟ − − ⎜⎜ ⎟⎟ + − ⎪ε′ = 5J ⎝ 5 1 5 1 ⎠ β ∂ α ⎝ 5 1 ⎠ 5 1 β ∂ α ⎝ β ∂ α ⎠ β 5J ∂ α ⎪ ⎪ ⎪ ⎪ ⎪ ∂Z X ⎛ ⎞ X ∂ ⎛ ⎞ Z ∂ ⎛ ∂Z ⎞ ⎟⎟ − ⎜⎜ ⎟⎟ − − ⎜⎜ ⎟⎟ − − ⎪ε′ = ⎜⎜ 5J ⎝ 5 1 5 1 ⎠ β ∂ α ⎝ 5 1 ⎠ 5 1 β ∂ α ⎝ β ∂ α ⎠ β 5J ∂ α ⎪⎪ ⎨ ⎪ ⎪ ∂ X ∂ ⎛ X ⎞ ∂ X ∂ ⎛ X ⎞ ⎪ γ′ = ⎜⎜ ⎟⎟ + ⎜ ⎟ − − ⎪ β 5 β ∂ α ⎝ 5 1 ⎠ β 5 1 ∂ α β ∂ α ⎜⎝ 5 1 ⎟⎠ 1 ∂ α ⎪ ⎪ ∂Z ∂ Z ∂Z ⎪ − + − − − ⎪ β 5J ∂ α β 5J ∂ α β β ∂ α ∂ α ⎪⎩

⎧ ⎛ X ∂ ⎛ ⎞ X ⎞ Z ⎜⎜ ⎟⎟ + ⎜⎜ ⎟⎟ − − ⎪ε′′ = 5 1 β ∂ α ⎝ 5 1 ⎠ 5J 5 1 ⎝ 5 1 5 1 ⎠ 5 1 ⎪ ⎪ ⎪ ∂ ⎛ ∂Z ⎞ ∂Z ⎪ ⎜ ⎟+ − ⎪ 5 1 β ∂ α ⎜⎝ β ∂ α ⎟⎠ 5 1 5J β ∂ α ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎛ X ∂ ⎛ ⎞ X ⎞ Z ⎜⎜ ⎟⎟ + ⎜⎜ ⎟⎟ − − ⎪ε′′ = 5 1 β ∂ α ⎝ 5 1 ⎠ 5J 5 1 ⎝ 5 1 5 1 ⎠ 5 1 ⎪ ⎪ ⎪ ∂ ⎛ ∂Z ⎞ ∂Z ⎪ ⎜⎜ ⎟⎟ + − ⎨ 5 5J 5 β ∂ α β ∂ α β 1 ⎝ ⎠ 1 ∂ α ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎡ ∂ X ∂ ⎛ X ⎞ ∂ X ′′ = ⎢ ⎜⎜ ⎟⎟ + − ⎪γ β ∂α β ∂α 5 5 β ∂ α ⎪ 1 ⎝ 1 ⎠ 5 1 ⎣⎢ 5 1 ⎪ ⎪ ∂ ⎛ X ⎞ ⎛ ⎞ ∂Z ⎪ ⎜⎜ ⎟⎟ − ⎜⎜ ⎟ + − ⎪ 5 1 β ∂ α ⎝ 5 1 ⎠ 5J ⎝ 5 1 5 1 ⎟⎠ β ∂ α ⎪ ⎪ ⎪ ⎛ ⎞ ∂Z ⎛ ⎞ ∂ Z ⎤ ⎪ ⎜⎜ ⎟⎟ ⎟⎟ + + − ⎜⎜ + ⎥ 5J ⎝ 5 1 5 1 ⎠ β ∂ α ⎝ 5 1 5 1 ⎠ β β ∂ α ∂ α ⎦⎥ ⎪⎩

⎧ ⎛ X ∂ ⎛ ⎞ X ⎞ Z ⎜⎜ ⎟⎟ + ⎜⎜ ⎟⎟ − − ⎪ε′′ = 5 1 β ∂ α ⎝ 5 1 ⎠ 5J 5 1 ⎝ 5 1 5 1 ⎠ 5 1 ⎪ ⎪ ⎪ ∂ ⎛ ∂Z ⎞ ∂Z ⎪ ⎜ ⎟+ − ⎪ 5 1 β ∂ α ⎜⎝ β ∂ α ⎟⎠ 5 1 5J β ∂ α ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎛ X ∂ ⎛ ⎞ X ⎞ Z ⎜⎜ ⎟⎟ + ⎜⎜ ⎟⎟ − − ⎪ε′′ = 5 1 β ∂ α ⎝ 5 1 ⎠ 5J 5 1 ⎝ 5 1 5 1 ⎠ 5 1 ⎪ ⎪ ⎪ ∂ ⎛ ∂Z ⎞ ∂Z ⎪ ⎜⎜ ⎟⎟ + − ⎨ 5 5J 5 β ∂ α β ∂ α β 1 ⎝ ⎠ 1 ∂ α ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎡ ∂ X ∂ ⎛ X ⎞ ∂ X ′′ = ⎢ ⎜⎜ ⎟⎟ + − ⎪γ β ∂α β ∂α 5 5 β ∂ α ⎪ 1 ⎝ 1 ⎠ 5 1 ⎣⎢ 5 1 ⎪ ⎪ ∂ ⎛ X ⎞ ⎛ ⎞ ∂Z ⎪ ⎜⎜ ⎟⎟ − ⎜⎜ ⎟ + − ⎪ 5 1 β ∂ α ⎝ 5 1 ⎠ 5J ⎝ 5 1 5 1 ⎟⎠ β ∂ α ⎪ ⎪ ⎪ ⎛ ⎞ ∂Z ⎛ ⎞ ∂ Z ⎤ ⎪ ⎜⎜ ⎟⎟ ⎟⎟ + + − ⎜⎜ + ⎥ 5J ⎝ 5 1 5 1 ⎠ β ∂ α ⎝ 5 1 5 1 ⎠ β β ∂ α ∂ α ⎦⎥ ⎪⎩

- 90 -

- 90 -

⎧ ∂Z X ⎛ ⎞ X ∂ ⎛ ⎞ Z ∂ ⎛ ∂Z ⎞ ⎜⎜ ⎟⎟ − ⎜⎜ ⎟⎟ − − ⎜⎜ ⎟⎟ + − ⎪ε′ = β ∂ α β ∂ α β ∂ α β ∂ 5J 5 5 5 5J ⎝ 1 1 ⎠ ⎝ 1 ⎠ 5 1 ⎝ ⎠ α ⎪ ⎪ ⎪ ⎪ ⎪ ∂Z X ⎛ ⎞ X ∂ ⎛ ⎞ Z ∂ ⎛ ∂Z ⎞ ⎟⎟ − ⎜⎜ ⎟⎟ − − ⎜⎜ ⎟⎟ − − ⎪ε′ = ⎜⎜ β ∂ α β ∂ α β ∂ α β ∂ 5J 5 5 5 5J ⎝ 1 1 ⎠ ⎝ 1 ⎠ 5 1 ⎝ ⎠ α ⎪⎪ ⎨ ⎪ ⎪ ∂ X ∂ ⎛ X ⎞ ∂ X ∂ ⎛ X ⎞ ⎪ γ′ = ⎜⎜ ⎟⎟ + ⎜ ⎟ − − ⎪ β 5 β ∂ α ⎝ 5 1 ⎠ β 5 1 ∂ α β ∂ α ⎜⎝ 5 1 ⎟⎠ 1 ∂ α ⎪ ⎪ ∂Z ∂ Z ∂Z ⎪ − +− −− ⎪ β 5J ∂ α β 5J ∂ α β β ∂ α ∂ α ⎪⎩

⎧ ∂Z X ⎛ ⎞ X ∂ ⎛ ⎞ Z ∂ ⎛ ∂Z ⎞ ⎜⎜ ⎟⎟ − ⎜⎜ ⎟⎟ − − ⎜⎜ ⎟⎟ + − ⎪ε′ = β ∂ α β ∂ α β ∂ α β ∂ 5J 5 5 5 5J ⎝ 1 1 ⎠ ⎝ 1 ⎠ 5 1 ⎝ ⎠ α ⎪ ⎪ ⎪ ⎪ ⎪ ∂Z X ⎛ ⎞ X ∂ ⎛ ⎞ Z ∂ ⎛ ∂Z ⎞ ⎟⎟ − ⎜⎜ ⎟⎟ − − ⎜⎜ ⎟⎟ − − ⎪ε′ = ⎜⎜ β ∂ α β ∂ α β ∂ α β ∂ 5J 5 5 5 5J ⎝ 1 1 ⎠ ⎝ 1 ⎠ 5 1 ⎝ ⎠ α ⎪⎪ ⎨ ⎪ ⎪ ∂ X ∂ ⎛ X ⎞ ∂ X ∂ ⎛ X ⎞ ⎪ γ′ = ⎜⎜ ⎟⎟ + ⎜ ⎟ − − ⎪ β 5 β ∂ α ⎝ 5 1 ⎠ β 5 1 ∂ α β ∂ α ⎜⎝ 5 1 ⎟⎠ 1 ∂ α ⎪ ⎪ ∂Z ∂ Z ∂Z ⎪ − +− −− ⎪ β 5J ∂ α β 5J ∂ α β β ∂ α ∂ α ⎪⎩

⎧ ⎛ X ∂ ⎛ ⎞ X ⎞ Z ⎜⎜ ⎟⎟ + ⎜⎜ ⎟⎟ − − ⎪ε′′ = 5 1 β ∂ α ⎝ 5 1 ⎠ 5J 5 1 ⎝ 5 1 5 1 ⎠ 5 1 ⎪ ⎪ ⎪ ∂ ⎛ ∂Z ⎞ ∂Z ⎪ ⎜ ⎟+ − ⎪ 5 1 β ∂ α ⎜⎝ β ∂ α ⎟⎠ 5 1 5J β ∂ α ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎛ X ∂ ⎛ ⎞ X ⎞ Z ⎜⎜ ⎟⎟ + ⎜⎜ ⎟⎟ − − ⎪ε′′ = 5 1 β ∂ α ⎝ 5 1 ⎠ 5J 5 1 ⎝ 5 1 5 1 ⎠ 5 1 ⎪ ⎪ ⎪ ∂ ⎛ ∂Z ⎞ ∂Z ⎪ ⎜⎜ ⎟⎟ + − ⎨ 5 5J 5 ∂ α β ∂ α β β 1 ⎝ ⎠ 1 ∂ α ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎡ ∂ X ∂ ⎛ X ⎞ ∂ X ′′ = ⎢ ⎜ ⎟+ − ⎪γ ⎢⎣ 5 1 β ∂ α 5 1 β ∂ α ⎜⎝ 5 1 ⎟⎠ 5 1 β ∂ α ⎪ ⎪ ⎪ ∂ ⎛ X ⎞ ⎛ ⎞ ∂Z ⎪ ⎜⎜ ⎟⎟ − ⎜⎜ ⎟⎟ + − ⎪ 5 1 β ∂ α ⎝ 5 1 ⎠ 5J ⎝ 5 1 5 1 ⎠ β ∂ α ⎪ ⎪ ⎪ ⎛ ⎞ ∂Z ⎛ ⎞ ∂ Z ⎤ ⎪ ⎜⎜ ⎟⎟ ⎟⎟ + + − ⎜⎜ + ⎥ 5J ⎝ 5 1 5 1 ⎠ β ∂ α ⎝ 5 1 5 1 ⎠ β β ∂ α ∂ α ⎦⎥ ⎪⎩

⎧ ⎛ X ∂ ⎛ ⎞ X ⎞ Z ⎜⎜ ⎟⎟ + ⎜⎜ ⎟⎟ − − ⎪ε′′ = 5 1 β ∂ α ⎝ 5 1 ⎠ 5J 5 1 ⎝ 5 1 5 1 ⎠ 5 1 ⎪ ⎪ ⎪ ∂ ⎛ ∂Z ⎞ ∂Z ⎪ ⎜ ⎟+ − ⎪ 5 1 β ∂ α ⎜⎝ β ∂ α ⎟⎠ 5 1 5J β ∂ α ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎛ X ∂ ⎛ ⎞ X ⎞ Z ⎜⎜ ⎟⎟ + ⎜⎜ ⎟⎟ − − ⎪ε′′ = 5 1 β ∂ α ⎝ 5 1 ⎠ 5J 5 1 ⎝ 5 1 5 1 ⎠ 5 1 ⎪ ⎪ ⎪ ∂ ⎛ ∂Z ⎞ ∂Z ⎪ ⎜⎜ ⎟⎟ + − ⎨ 5 5J 5 ∂ α β ∂ α β β 1 ⎝ ⎠ 1 ∂ α ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎡ ∂ X ∂ ⎛ X ⎞ ∂ X ′′ = ⎢ ⎜ ⎟+ − ⎪γ ⎢⎣ 5 1 β ∂ α 5 1 β ∂ α ⎜⎝ 5 1 ⎟⎠ 5 1 β ∂ α ⎪ ⎪ ⎪ ∂ ⎛ X ⎞ ⎛ ⎞ ∂Z ⎪ ⎜⎜ ⎟⎟ − ⎜⎜ ⎟⎟ + − ⎪ 5 1 β ∂ α ⎝ 5 1 ⎠ 5J ⎝ 5 1 5 1 ⎠ β ∂ α ⎪ ⎪ ⎪ ⎛ ⎞ ∂Z ⎛ ⎞ ∂ Z ⎤ ⎪ ⎜⎜ ⎟⎟ ⎟⎟ + + − ⎜⎜ + ⎥ 5J ⎝ 5 1 5 1 ⎠ β ∂ α ⎝ 5 1 5 1 ⎠ β β ∂ α ∂ α ⎦⎥ ⎪⎩

- 90 -

- 90 -



/HVIRUPXOHV SHUPHWWHQWG¶pFULUHDYHF 7 − 7 = 7 − 7 + 7′ = + 7′′ =


⎧σ = σ + ] σ′ + ] σ′′ ⎪ ⎪ ⎨σ = σ + ] σ′ + ] σ′′ ⎪ ′ + ] τ ′′ ⎪⎩τ = τ + ] τ

DYHF

⎧σ = σ + ] σ′ + ] σ′′ ⎪ ⎪ ⎨σ = σ + ] σ′ + ] σ′′ ⎪ ′ + ] τ ′′ ⎪⎩τ = τ + ] τ

( (α ⎧ ⎪σ = − ν ε + ν ε − − ν 7 − 7 ⎪ ⎪ ( (α ⎪ ε + ν ε − 7 − 7 ⎨ σ = −ν − ν ⎪ ⎪ ⎪τ = ( γ ⎪⎩ + ν

( (α ⎧ ′ ⎪σ = − ν ε′ + ν ε′ − − ν 7 ′ ⎪ ⎪ ( (α ⎪ ε′ + ν ε′ − 7′ ⎨σ′ = − ν − ν ⎪ ⎪ ⎪ τ′ = ( γ′ ⎪⎩ + ν

( (α ⎧ ′′ ⎪σ = − ν ε′′ + ν ε′′ − − ν 7 ′′ ⎪ ⎪ ( (α ⎪ ε′′ + ν ε′′ − 7 ′′ ⎨σ′′ = −ν − ν ⎪ ⎪ ⎪ τ′′ = ( γ′′ ⎪⎩ + ν

DYHF


( (α ⎧ ′ ⎪σ = − ν ε′ + ν ε′ − − ν 7 ′ ⎪ ⎪ ( (α ⎪ ε′ + ν ε′ − 7′ ⎨σ′ = − ν − ν ⎪ ⎪ ⎪ τ′ = ( γ′ ⎪⎩ + ν

( (α ⎧ ′′ ⎪σ = − ν ε′′ + ν ε′′ − − ν 7 ′′ ⎪ ⎪ ( (α ⎪ ε′′ + ν ε′′ − 7 ′′ ⎨σ′′ = −ν − ν ⎪ ⎪ ⎪ τ′′ = ( γ′′ ⎪⎩ + ν

HpWDSHH[SUHVVLRQGHVYLVVHXUV HW


(Q XWLOLVDQW OHV IRUPXOHV pWDEOLHV DX WRPH , GDQV OH FDV LVRWKHUPH HW HQ HIIHFWXDQW OHV LQWpJUDWLRQV VXLYDQW O¶pSDLVVHXU DSUqV DYRLU UHPSODFp OHV FRQWUDLQWHV SDU OHXUV H[SUHVVLRQV K RQREWLHQWOHVH[SUHVVLRQVGHVFRPSRVDQWHVGHVYLVVHXUVDYHF , =


- 91 -

- 91 -





⎧σ = σ + ] σ′ + ] σ′′ ⎪ ⎪ ⎨σ = σ + ] σ′ + ] σ′′ ⎪ ′ + ] τ ′′ ⎪⎩τ = τ + ] τ

DYHF

⎧σ = σ + ] σ′ + ] σ′′ ⎪ ⎪ ⎨σ = σ + ] σ′ + ] σ′′ ⎪ ′ + ] τ ′′ ⎪⎩τ = τ + ] τ


( (α ⎧ ′ ⎪σ = − ν ε′ + ν ε′ − − ν 7 ′ ⎪ ⎪ ( (α ⎪ ε′ + ν ε′ − 7′ ⎨σ′ = −ν − ν ⎪ ⎪ ⎪ τ′ = ( γ′ ⎪⎩ + ν

( (α ⎧ ′′ ⎪σ = − ν ε′′ + ν ε′′ − − ν 7 ′′ ⎪ ⎪ (α ( ⎪ ε′′ + ν ε′′ − 7 ′′ ⎨σ′′ = − ν − ν ⎪ ⎪ ⎪ τ′′ = ( γ′′ + ν ⎩⎪

DYHF



( (α ⎧ ′′ ⎪σ = − ν ε′′ + ν ε′′ − − ν 7 ′′ ⎪ ⎪ (α ( ⎪ ε′′ + ν ε′′ − 7 ′′ ⎨σ′′ = − ν − ν ⎪ ⎪ ⎪ τ′′ = ( γ′′ + ν ⎩⎪





- 91 -

- 91 -

⎧ ⎧ ⎛ ⎛ σ′ ⎞ σ′ ⎞ ⎟⎟ ⎟ ⎪ 1 = K σ + , ⎜⎜ σ′′ − ⎪ 1 = K σ + , ⎜⎜ σ′′ − 5 1 ⎠ 5 1 ⎟⎠ ⎝ ⎝ ⎪ ⎪ ⎪ ⎪ ′ ⎞ ′ ⎞ ⎪7 = K τ + , ⎛⎜ τ′′ − τ ⎪7 = K τ + , ⎛⎜ τ′′ − τ ⎟ ⎜ 5 ⎟ ⎜ 5 ⎟⎟ ⎪ ⎪ 1 ⎠ 1 ⎠ ⎝ ⎝ ⎪ ⎪ ⎨ ⎨ ⎞ ⎞ ⎛ τ ⎛ τ ′′ ⎪ ⎪ ′ ⎟⎟ ′ ⎟⎟ ⎪8 = , ⎜⎜ 5 − τ ⎪8 = −, ⎜⎜ 5 − τ 1 1 ⎠ ⎠ ⎝ ⎝ ⎪ ⎪ ⎪ ⎪ ⎪0 = , ⎛⎜ − σ + σ′ ⎞⎟ ⎪0 = −, ⎛⎜ − σ + σ′ ⎞⎟ ⎟ ⎟ ⎜ ⎜ 5 ⎪⎩ ⎪⎩ 1 ⎠ ⎠ ⎝ 5 1 ⎝

⎧ ⎧ ⎛ ⎛ σ′ ⎞ σ′ ⎞ ⎟⎟ ⎟ ⎪ 1 = K σ + , ⎜⎜ σ′′ − ⎪ 1 = K σ + , ⎜⎜ σ′′ − 5 1 ⎠ 5 1 ⎟⎠ ⎝ ⎝ ⎪ ⎪ ⎪ ⎪ ′ ⎞ ′ ⎞ ⎪7 = K τ + , ⎛⎜ τ′′ − τ ⎪7 = K τ + , ⎛⎜ τ′′ − τ ⎟ ⎜ 5 ⎟ ⎜ 5 ⎟⎟ ⎪ ⎪ 1 ⎠ 1 ⎠ ⎝ ⎝ ⎪ ⎪ ⎨ ⎨ ⎞ ⎞ ⎛ τ ⎛ τ ′′ ⎪ ⎪ ′ ⎟⎟ ′ ⎟⎟ ⎪8 = , ⎜⎜ 5 − τ ⎪8 = −, ⎜⎜ 5 − τ 1 1 ⎠ ⎠ ⎝ ⎝ ⎪ ⎪ ⎪ ⎪ ⎪0 = , ⎛⎜ − σ + σ′ ⎞⎟ ⎪0 = −, ⎛⎜ − σ + σ′ ⎞⎟ ⎟ ⎟ ⎜ ⎜ 5 ⎪⎩ ⎪⎩ 1 ⎠ ⎠ ⎝ 5 1 ⎝

2Q SHXW DORUV HQ GpGXLUH 4 HW 4 HIIRUWV WUDQFKDQWV WUDQVYHUVDX[ j SDUWLU GHV GHX[ SUHPLqUHV pTXDWLRQV GH O¶pTXLOLEUH ORFDO GHV PRPHQWV OD WURLVLqPH DXWRXU GH ] pWDQW LGHQWLTXHPHQWYpULILpH


HpWDSHpTXDWLRQVDX[GpSODFHPHQWVGH 3


2QOHVREWLHQWHQUHPSODoDQWGDQVOHVWURLVpTXDWLRQVO¶pTXLOLEUHGHVIRUFHVOHVFRPSRVDQWHV GHVYLVVHXUV HW SDUOHXUVH[SUHVVLRQVHQIRQFWLRQGH X X Z REWHQXHVJUkFHDX[pWDSHV SUpFpGHQWHV


&H VRQW GHV pTXDWLRQV DX[ GpULYpHV SDUWLHOOHV TXDWULqPHV SDU UDSSRUW j α α HW W pYHQWXHOOHPHQW


±7+e25,(/,1e$,5(

±7+e25,(/,1e$,5(

(OOHVVHGpGXLWGHODSUpFpGHQWHHQSUHQDQWGHVGpYHORSSHPHQWVOLPLWpVDXSUHPLHUGHJUpHQ] VXLYDQWO¶pSDLVVHXU


UHpWDSHH[SUHVVLRQGHVGpSODFHPHQWV X X Z GH3


⎧ ⎛ ∂Z X ⎞ + ⎟⎟ ] ⎪X = X − ⎜⎜ ⎝ β ∂ α 5 1 ⎠ ⎪ ⎪ ⎛ ∂Z X ⎞ G⎪ X ⎨X = X − ⎜⎜ + ⎟⎟ ] ⎝ β ∂ α 5 1 ⎠ ⎪ ⎪ ⎪Z = Z ⎪⎩



⎧ε = ε + ] ε′ ⎪⎪ ⎨ε = ε + ] ε′ ⎪ ′ ⎪⎩γ = γ + ] γ

⎧ ⎛ ∂Z X ⎞ + ⎟⎟ ] ⎪X = X − ⎜⎜ ⎝ β ∂ α 5 1 ⎠ ⎪ ⎪ ⎛ ∂Z X ⎞ G⎪ X ⎨X = X − ⎜⎜ + ⎟⎟ ] ⎝ β ∂ α 5 1 ⎠ ⎪ ⎪ ⎪Z = Z ⎪⎩

⎧ε = ε + ] ε′ ⎪⎪ ⎨ε = ε + ] ε′ ⎪ ′ ⎪⎩γ = γ + ] γ

- 92 -

- 92 -

⎧ ⎧ ⎛ ⎛ σ′ ⎞ σ′ ⎞ ⎟⎟ ⎟⎟ ⎪ 1 = K σ + , ⎜⎜ σ′′ − ⎪ 1 = K σ + , ⎜⎜ σ′′ − 5 5 1 ⎠ 1 ⎠ ⎝ ⎝ ⎪ ⎪ ⎪ ⎪ ′ ⎞ ′ ⎞ ⎪7 = K τ + , ⎛⎜ τ′′ − τ ⎪7 = K τ + , ⎛⎜ τ′′ − τ ⎜ 5 ⎟⎟ ⎜ 5 ⎟⎟ ⎪ ⎪ 1 ⎠ 1 ⎠ ⎝ ⎝ ⎪ ⎪ ⎨ ⎨ ⎞ ⎞ ⎛ τ ⎛ τ ′′ ⎪ ⎪ ′ ⎟⎟ ′ ⎟⎟ ⎪8 = , ⎜⎜ 5 − τ ⎪8 = −, ⎜⎜ 5 − τ ⎠ ⎠ ⎝ 1 ⎝ 1 ⎪ ⎪ ⎪ ⎪ ⎪0 = , ⎛⎜ − σ + σ′ ⎞⎟ ⎪0 = −, ⎛⎜ − σ + σ′ ⎞⎟ ⎟ ⎟ ⎜ 5 ⎜ 5 ⎪⎩ ⎪⎩ 1 1 ⎠ ⎠ ⎝ ⎝

⎧ ⎧ ⎛ ⎛ σ′ ⎞ σ′ ⎞ ⎟⎟ ⎟⎟ ⎪ 1 = K σ + , ⎜⎜ σ′′ − ⎪ 1 = K σ + , ⎜⎜ σ′′ − 5 5 1 ⎠ 1 ⎠ ⎝ ⎝ ⎪ ⎪ ⎪ ⎪ ′ ⎞ ′ ⎞ ⎪7 = K τ + , ⎛⎜ τ′′ − τ ⎪7 = K τ + , ⎛⎜ τ′′ − τ ⎜ 5 ⎟⎟ ⎜ 5 ⎟⎟ ⎪ ⎪ 1 ⎠ 1 ⎠ ⎝ ⎝ ⎪ ⎪ ⎨ ⎨ ⎞ ⎞ ⎛ τ ⎛ τ ′′ ⎪ ⎪ ′ ⎟⎟ ′ ⎟⎟ ⎪8 = , ⎜⎜ 5 − τ ⎪8 = −, ⎜⎜ 5 − τ ⎠ ⎠ ⎝ 1 ⎝ 1 ⎪ ⎪ ⎪ ⎪ ⎪0 = , ⎛⎜ − σ + σ′ ⎞⎟ ⎪0 = −, ⎛⎜ − σ + σ′ ⎞⎟ ⎟ ⎟ ⎜ 5 ⎜ 5 ⎪⎩ ⎪⎩ 1 1 ⎠ ⎠ ⎝ ⎝









±7+e25,(/,1e$,5(

±7+e25,(/,1e$,5(





⎧ ⎛ ∂Z X ⎞ + ⎟⎟ ] ⎪X = X − ⎜⎜ β ∂ α 5 1 ⎠ ⎝ ⎪ ⎛ ∂Z X ⎞ G ⎪⎪ X ⎨X = X − ⎜⎜ + ⎟⎟ ] ⎝ β ∂ α 5 1 ⎠ ⎪ ⎪ ⎪Z = Z ⎪⎩


⎧ ⎛ ∂Z X ⎞ + ⎟⎟ ] ⎪X = X − ⎜⎜ β ∂ α 5 1 ⎠ ⎝ ⎪ ⎛ ∂Z X ⎞ G ⎪⎪ X ⎨X = X − ⎜⎜ + ⎟⎟ ] ⎝ β ∂ α 5 1 ⎠ ⎪ ⎪ ⎪Z = Z ⎪⎩


⎧ε = ε + ] ε′ ⎪⎪ ⎨ε = ε + ] ε′ ⎪ ′ ⎪⎩γ = γ + ] γ

- 92 -

⎧ε = ε + ] ε′ ⎪⎪ ⎨ε = ε + ] ε′ ⎪ ′ ⎪⎩γ = γ + ] γ

- 92 -

± ε ε γ VRQWGRQQpVSDUOHVIRUPXOHV


′ OHVRQWSDUOHVIRUPXOHV ± ε′ ε′ γ




⎧σ = σ + ] σ′ ⎪⎪ ⎨σ = σ + ] σ′ ⎪ ′ ⎪⎩τ = τ + ] τ

DYHF

⎧σ = σ + ] σ′ ⎪⎪ ⎨σ = σ + ] σ′ ⎪ ′ ⎪⎩τ = τ + ] τ



DYHF





'HVIRUPXOHV RQGpGXLWHQDQQXODQWOHVWHUPHVGXVHFRQGRUGUH


⎧ ⎧ , , σ′ σ′ ⎪ 1 = K σ − ⎪ 1 = K σ − 5 5 1 1 ⎪ ⎪ ⎪ ⎪ ⎪7 = K τ − , τ′ ⎪7 = K τ − , τ′ ⎪ ⎪ 5 1 5 1 ⎪ ⎪ ⎨ ⎨ ⎞ ⎞ ⎛ τ ⎛ τ ⎪ ⎪ ′ ⎟⎟ ′ ⎟⎟ ⎪8 = , ⎜⎜ 5 − τ ⎪8 = , ⎜⎜ 5 − τ 1 1 ⎠ ⎠ ⎝ ⎝ ⎪ ⎪ ⎪ ⎪ ⎪0 = , ⎛⎜ − σ + σ′ ⎞⎟ ⎪0 = , ⎛⎜ − σ + σ′ ⎞⎟ ⎟ ⎟ ⎜ 5 ⎜ ⎪⎩ ⎪⎩ 1 ⎠ ⎠ ⎝ ⎝ 5 1

⎧ ⎧ , , σ′ σ′ ⎪ 1 = K σ − ⎪ 1 = K σ − 5 5 1 1 ⎪ ⎪ ⎪ ⎪ ⎪7 = K τ − , τ′ ⎪7 = K τ − , τ′ ⎪ ⎪ 5 1 5 1 ⎪ ⎪ ⎨ ⎨ ⎞ ⎞ ⎛ τ ⎛ τ ⎪ ⎪ ′ ⎟⎟ ′ ⎟⎟ ⎪8 = , ⎜⎜ 5 − τ ⎪8 = , ⎜⎜ 5 − τ 1 1 ⎠ ⎠ ⎝ ⎝ ⎪ ⎪ ⎪ ⎪ ⎪0 = , ⎛⎜ − σ + σ′ ⎞⎟ ⎪0 = , ⎛⎜ − σ + σ′ ⎞⎟ ⎟ ⎟ ⎜ 5 ⎜ ⎪⎩ ⎪⎩ 1 ⎠ ⎠ ⎝ ⎝ 5 1

- 93 -

- 93 -







⎧σ = σ + ] σ′ ⎪⎪ ⎨σ = σ + ] σ′ ⎪ ′ ⎩⎪τ = τ + ] τ

DYHF

⎧σ = σ + ] σ′ ⎪⎪ ⎨σ = σ + ] σ′ ⎪ ′ ⎩⎪τ = τ + ] τ

( (α ⎧ ⎪σ = − ν ε + ν ε − − ν 7 − 7 ⎪ ⎪ ( (α ⎪ ε + ν ε − 7 − 7 ⎨ σ = − ν − ν ⎪ ⎪ ⎪τ = ( γ + ν ⎩⎪


DYHF

( (α ⎧ ⎪σ = − ν ε + ν ε − − ν 7 − 7 ⎪ ⎪ ( (α ⎪ ε + ν ε − 7 − 7 ⎨ σ = − ν − ν ⎪ ⎪ ⎪τ = ( γ + ν ⎩⎪






⎧ ⎧ , , σ′ σ′ ⎪ 1 = K σ − ⎪ 1 = K σ − 5 5 1 1 ⎪ ⎪ ⎪ ⎪ ⎪7 = K τ − , τ′ ⎪7 = K τ − , τ′ ⎪ ⎪ 5 1 5 1 ⎪ ⎪ ⎨ ⎨ ⎞ ⎞ ⎛ τ ⎛ τ ⎪ ⎪ ′ ⎟⎟ ′ ⎟⎟ ⎪8 = , ⎜⎜ 5 − τ ⎪8 = , ⎜⎜ 5 − τ ⎠ ⎠ ⎝ 1 ⎝ 1 ⎪ ⎪ ⎪ ⎪ ⎪0 = , ⎛⎜ − σ + σ′ ⎞⎟ ⎪0 = , ⎛⎜ − σ + σ′ ⎞⎟ ⎟ ⎟ ⎜ 5 ⎜ 5 ⎪⎩ ⎪⎩ 1 1 ⎠ ⎠ ⎝ ⎝ - 93 -

⎧ ⎧ , , σ′ σ′ ⎪ 1 = K σ − ⎪ 1 = K σ − 5 5 1 1 ⎪ ⎪ ⎪ ⎪ ⎪7 = K τ − , τ′ ⎪7 = K τ − , τ′ ⎪ ⎪ 5 1 5 1 ⎪ ⎪ ⎨ ⎨ ⎞ ⎞ ⎛ τ ⎛ τ ⎪ ⎪ ′ ⎟⎟ ′ ⎟⎟ ⎪8 = , ⎜⎜ 5 − τ ⎪8 = , ⎜⎜ 5 − τ ⎠ ⎠ ⎝ 1 ⎝ 1 ⎪ ⎪ ⎪ ⎪ ⎪0 = , ⎛⎜ − σ + σ′ ⎞⎟ ⎪0 = , ⎛⎜ − σ + σ′ ⎞⎟ ⎟ ⎟ ⎜ 5 ⎜ 5 ⎪⎩ ⎪⎩ 1 1 ⎠ ⎠ ⎝ ⎝ - 93 -

2QHQGpGXLWDORUVOHVH[SUHVVLRQVGHVHIIRUWVWUDQFKDQWVWUDQVYHUVDX[ 4 HW 4 DXPR\HQGHV GHX[ SUHPLqUHV pTXDWLRQV GH O¶pTXLOLEUH ORFDO GHV PRPHQWV WRPH , FKDSLWUH 9,,, IRUPXOHV




&RPPH GDQV OD WKpRULH TXDGUDWLTXH RQ OHV REWLHQW HQ UHPSODoDQW GDQV OHV WURLV pTXDWLRQV G¶pTXLOLEUHORFDOGHVIRUFHVWRPH,FKDSLWUH9,,,IRUPXOHV OHVFRPSRVDQWHVGHVYLVVHXUV HW SDUOHXUVH[SUHVVLRQVHQIRQFWLRQGH X X Z REWHQXHVJUkFHDX[pWDSHVSUpFpGHQWHV




5HPDUTXHGHVIRUPXOHV RQWLUHHQQpJOLJHDQW K GHYDQW 5 1


σ =

σ ′ =

K

⎛ 0 ⎞ ⎛ 0 ⎞ ⎛ 8 ⎞ ⎛ 8 ⎞ ⎜⎜ 1 + ⎟⎟ σ = ⎜⎜ 1 − ⎟⎟ τ = ⎜⎜ 7 − ⎟⎟ = ⎜⎜ 7 + ⎟⎟ 5 K 5 K 5 K 5 1 ⎠ 1 ⎠ 1 ⎠ 1 ⎠ ⎝ ⎝ ⎝ ⎝

1 0 1 0 7 8 7 8 + ′ = σ′ = − τ − = + K 5 1 , K 5 1 , K 5 1 , K 5 1 ,

'¶R

σ ′ =

K

⎛ 0 ⎞ ⎛ 0 ⎞ ⎛ 8 ⎞ ⎛ 8 ⎞ ⎜⎜ 1 + ⎟⎟ σ = ⎜⎜ 1 − ⎟⎟ τ = ⎜⎜ 7 − ⎟⎟ = ⎜⎜ 7 + ⎟⎟ 5 K 5 K 5 K 5 1 ⎠ 1 ⎠ 1 ⎠ 1 ⎠ ⎝ ⎝ ⎝ ⎝

1 0 1 0 7 8 7 8 + ′ = σ′ = − τ − = + K 5 1 , K 5 1 , K 5 1 , K 5 1 ,

'¶R ⎧ ⎛ 1 ⎛ 0 ⎞ 0 ⎞ ⎟⎟ + ] ⎜⎜ + ⎟⎟ ⎪σ = ⎜⎜ 1 + K⎝ 5 1 ⎠ , ⎠ ⎝ K 5 1 ⎪ ⎪ ⎛ 1 ⎛ 0 ⎞ 0 ⎞ ⎪ ⎟⎟ + ] ⎜⎜ − ⎟⎟ ⎨σ = ⎜⎜ 1 − K 5 K 5 , ⎠ 1 ⎠ 1 ⎝ ⎝ ⎪ ⎪ ⎪ τ = ⎛⎜ 7 − 8 ⎞⎟ + ⎛⎜ 7 − 8 ⎞⎟ ] = ⎛⎜ 7 + 8 ⎞⎟ + ⎛⎜ 7 + 8 ⎞⎟ ] ⎪ K ⎜ 5 ⎟ ⎜ K 5 , ⎟⎠ 5 1 ⎟⎠ ⎜⎝ K 5 1 , ⎟⎠ K ⎜⎝ 1 ⎠ ⎝ 1 ⎝ ⎩

σ =

&RPPHGDQVOHFDVGHVSODTXHVRQDSRXUOHVFLVVLRQVWUDQVYHUVHV τ] =

4 ⎜⎛ ] ⎞ 4 ⎜⎛ ] ⎞ − ⎟ HW τ] = − ⎟ K ⎜⎝ K ⎜⎝ K ⎠⎟ K ⎟⎠

⎧ ⎛ 1 ⎛ 0 ⎞ 0 ⎞ ⎟⎟ + ] ⎜⎜ + ⎟⎟ ⎪σ = ⎜⎜ 1 + K⎝ 5 1 ⎠ , ⎠ ⎝ K 5 1 ⎪ ⎪ ⎛ 1 ⎛ 0 ⎞ 0 ⎞ ⎪ ⎟⎟ + ] ⎜⎜ − ⎟⎟ ⎨σ = ⎜⎜ 1 − K 5 K 5 , ⎠ 1 ⎠ 1 ⎝ ⎝ ⎪ ⎪ ⎪ τ = ⎛⎜ 7 − 8 ⎞⎟ + ⎛⎜ 7 − 8 ⎞⎟ ] = ⎛⎜ 7 + 8 ⎞⎟ + ⎛⎜ 7 + 8 ⎞⎟ ] ⎪ K ⎜ 5 ⎟ ⎜ K 5 , ⎟⎠ 5 1 ⎟⎠ ⎜⎝ K 5 1 , ⎟⎠ K ⎜⎝ 1 ⎠ ⎝ 1 ⎝ ⎩

&RPPHGDQVOHFDVGHVSODTXHVRQDSRXUOHVFLVVLRQVWUDQVYHUVHV

τ] =

4 ⎜⎛ ] ⎞ 4 ⎜⎛ ] ⎞ − ⎟ HW τ] = − ⎟ K ⎜⎝ K ⎜⎝ K ⎠⎟ K ⎟⎠

±7+e25,('(60(0%5$1(6

±7+e25,('(60(0%5$1(6

'DQV FH FDV OH ORQJ GH WRXWH ILEUH WUDQVYHUVH OHV GpSODFHPHQWV OHV GpIRUPDWLRQV OHV FRQWUDLQWHVHWODWHPSpUDWXUHVRQWVXSSRVpVFRQVWDQWV


/D WKpRULH GHV PHPEUDQHV VH GpGXLW GH OD WKpRULH OLQpDLUH FLGHVVXV HQ DQQXODQW WRXV OHV IDFWHXUVGH]FRHIILFLHQWVHQ′


- 94 -

- 94 -











σ =

σ ′ =

K

⎛ 0 ⎞ ⎛ 0 ⎞ ⎛ 8 ⎞ ⎛ 8 ⎞ ⎜⎜ 1 + ⎟⎟ σ = ⎜⎜ 1 − ⎟⎟ τ = ⎜⎜ 7 − ⎟⎟ = ⎜⎜ 7 + ⎟⎟ 5 K 5 K 5 K 5 1 ⎠ 1 ⎠ 1 ⎠ 1 ⎠ ⎝ ⎝ ⎝ ⎝

1 0 1 0 7 8 7 8 + ′ = σ′ = − τ − = + K 5 1 , K 5 1 , K 5 1 , K 5 1 ,

'¶R

σ ′ =

K

⎛ 0 ⎞ ⎛ 0 ⎞ ⎛ 8 ⎞ ⎛ 8 ⎞ ⎜⎜ 1 + ⎟⎟ σ = ⎜⎜ 1 − ⎟⎟ τ = ⎜⎜ 7 − ⎟⎟ = ⎜⎜ 7 + ⎟⎟ 5 K 5 K 5 K 5 1 ⎠ 1 ⎠ 1 ⎠ 1 ⎠ ⎝ ⎝ ⎝ ⎝

1 0 1 0 7 8 7 8 + ′ = σ′ = − τ − = + K 5 1 , K 5 1 , K 5 1 , K 5 1 ,

'¶R ⎧ ⎛ 1 ⎛ 0 ⎞ 0 ⎞ ⎟⎟ + ] ⎜⎜ + ⎟⎟ ⎪σ = ⎜⎜ 1 + K⎝ 5 1 ⎠ , ⎠ ⎝ K 5 1 ⎪ ⎪ ⎛ 1 ⎛ 0 ⎞ 0 ⎞ ⎪ ⎟⎟ + ] ⎜⎜ − ⎟⎟ ⎨σ = ⎜⎜ 1 − K⎝ 5 1 ⎠ , ⎠ ⎝ K 5 1 ⎪ ⎪ ⎪ τ = ⎛⎜ 7 − 8 ⎞⎟ + ⎛⎜ 7 − 8 ⎞⎟ ] = ⎛⎜ 7 + 8 ⎞⎟ + ⎛⎜ 7 + 8 ⎞⎟ ] ⎪ K ⎜ 5 ⎟ ⎜ K 5 , ⎟⎠ 5 1 ⎟⎠ ⎜⎝ K 5 1 , ⎟⎠ K ⎜⎝ 1 ⎠ ⎝ 1 ⎝ ⎩


σ =

τ] =

4 ⎜⎛ ] ⎞ 4 ⎜⎛ ] ⎞ − ⎟ HW τ] = − ⎟ K ⎜⎝ K ⎜⎝ K ⎠⎟ K ⎟⎠

⎧ ⎛ 1 ⎛ 0 ⎞ 0 ⎞ ⎟⎟ + ] ⎜⎜ + ⎟⎟ ⎪σ = ⎜⎜ 1 + K⎝ 5 1 ⎠ , ⎠ ⎝ K 5 1 ⎪ ⎪ ⎛ 1 ⎛ 0 ⎞ 0 ⎞ ⎪ ⎟⎟ + ] ⎜⎜ − ⎟⎟ ⎨σ = ⎜⎜ 1 − K⎝ 5 1 ⎠ , ⎠ ⎝ K 5 1 ⎪ ⎪ ⎪ τ = ⎛⎜ 7 − 8 ⎞⎟ + ⎛⎜ 7 − 8 ⎞⎟ ] = ⎛⎜ 7 + 8 ⎞⎟ + ⎛⎜ 7 + 8 ⎞⎟ ] ⎪ K ⎜ 5 ⎟ ⎜ K 5 , ⎟⎠ 5 1 ⎟⎠ ⎜⎝ K 5 1 , ⎟⎠ K ⎜⎝ 1 ⎠ ⎝ 1 ⎝ ⎩


τ] =

4 ⎜⎛ ] ⎞ 4 ⎜⎛ ] ⎞ − ⎟ HW τ] = − ⎟ K ⎜⎝ K ⎜⎝ K ⎠⎟ K ⎟⎠

±7+e25,('(60(0%5$1(6

±7+e25,('(60(0%5$1(6





- 94 -

- 94 -

UHpWDSHGpSODFHPHQWV

UHpWDSHGpSODFHPHQWV

⎧X = X α α W G ⎪⎪ X ⎨X = X α α W ⎪ ⎪⎩Z = Z α α W

HpWDSHGpIRUPDWLRQV

⎧X = X α α W G ⎪⎪ X ⎨X = X α α W ⎪ ⎪⎩Z = Z α α W

HpWDSHGpIRUPDWLRQV

ε = ε α α W

ε = ε α α W

γ = γ α α W

ε = ε α α W

ε = ε α α W

&HVWURLVIRQFWLRQVVRQWGRQQpHVSDUOHVIRUPXOHV


HpWDSHFRQWUDLQWHV

HpWDSHFRQWUDLQWHV

( (α ⎧ ⎪σ = X α α W = − ν ε + ν ε − − ν 7 − 7 ⎪ ⎪ ( (α ⎪ ε + ν ε − 7 − 7 ⎨σ = X α α W = − ν − ν ⎪ ⎪ ⎪τ = τ α α W = ( γ + ν ⎩⎪

HpWDSHYLVVHXUV

⎧ 1 = K σ ⎪⎪ ⎨1 = K σ ⎪ ⎪⎩7 = 7 = K τ

⎧4 = 6 = ⎪⎪ ⎨0 = 0 = ⎪ ⎪⎩8 = 8 =

( (α ⎧ ⎪σ = X α α W = − ν ε + ν ε − − ν 7 − 7 ⎪ ⎪ ( (α ⎪ ε + ν ε − 7 − 7 ⎨σ = X α α W = − ν − ν ⎪ ⎪ ⎪τ = τ α α W = ( γ + ν ⎩⎪

HpWDSHYLVVHXUV

γ = γ α α W

⎧ 1 = K σ ⎪⎪ ⎨1 = K σ ⎪ ⎪⎩7 = 7 = K τ

⎧4 = 6 = ⎪⎪ ⎨0 = 0 = ⎪ ⎪⎩8 = 8 =

2QQ¶DTXHGHVHIIRUWVGHPHPEUDQHELHQHQWHQGX


HpWDSHpTXDWLRQVGXSUREOqPH


/HV pTXDWLRQV G¶pTXLOLEUH ORFDO GHV PRPHQWV VRQW LGHQWLTXHPHQW YpULILpHV OHV pTXDWLRQV G¶pTXLOLEUHGHVIRUFHVV¶pFULYHQW


⎧ ∂ 1 ∂ 7 1 − 1 7 + 7 + + − = ⎪S + 5J 5J β β ∂ α ∂ α ⎪ ⎪ ∂ 1 ∂ 7 1 − 1 7 + 7 ⎪ + + − = ⎨S + 5J 5J β ∂ α β ∂ α ⎪ ⎪ ⎪S + 1 + 1 = ⎪ ] 5 5 1 1 ⎩

⎧ ∂ 1 ∂ 7 1 − 1 7 + 7 + + − = ⎪S + 5J 5J β β ∂ α ∂ α ⎪ ⎪ ∂ 1 ∂ 7 1 − 1 7 + 7 ⎪ + + − = ⎨S + 5J 5J β ∂ α β ∂ α ⎪ ⎪ ⎪S + 1 + 1 = ⎪ ] 5 5 1 1 ⎩

- 95 -

UHpWDSHGpSODFHPHQWV

- 95 -

UHpWDSHGpSODFHPHQWV

⎧X = X α α W G ⎪⎪ X ⎨X = X α α W ⎪ ⎩⎪Z = Z α α W

HpWDSHGpIRUPDWLRQV

⎧X = X α α W G ⎪⎪ X ⎨X = X α α W ⎪ ⎩⎪Z = Z α α W

ε = ε α α W

ε = ε α α W

γ = γ α α W

ε = ε α α W

ε = ε α α W


HpWDSHFRQWUDLQWHV

HpWDSHFRQWUDLQWHV

( (α ⎧ ⎪σ = X α α W = − ν ε + ν ε − − ν 7 − 7 ⎪ ⎪ ( (α ⎪ ε + ν ε − 7 − 7 ⎨σ = X α α W = −ν − ν ⎪ ⎪ ⎪τ = τ α α W = ( γ ⎪⎩ + ν

HpWDSHYLVVHXUV

HpWDSHGpIRUPDWLRQV


⎧ 1 = K σ ⎪⎪ ⎨1 = K σ ⎪ ⎩⎪7 = 7 = K τ

⎧4 = 6 = ⎪⎪ ⎨0 = 0 = ⎪ ⎩⎪8 = 8 =

( (α ⎧ ⎪σ = X α α W = − ν ε + ν ε − − ν 7 − 7 ⎪ ⎪ ( (α ⎪ ε + ν ε − 7 − 7 ⎨σ = X α α W = −ν − ν ⎪ ⎪ ⎪τ = τ α α W = ( γ ⎪⎩ + ν

HpWDSHYLVVHXUV

γ = γ α α W

⎧ 1 = K σ ⎪⎪ ⎨1 = K σ ⎪ ⎩⎪7 = 7 = K τ

⎧4 = 6 = ⎪⎪ ⎨0 = 0 = ⎪ ⎩⎪8 = 8 =







⎧ ∂ 1 ∂ 7 1 − 1 7 + 7 + + − = ⎪S + 5J 5J β ∂ α β ∂ α ⎪ ⎪ ∂ 1 ∂ 7 1 − 1 7 + 7 ⎪ + + − = ⎨S + 5J 5J β β ∂ α ∂ α ⎪ ⎪ ⎪S + 1 + 1 = ⎪ ] 5 5 1 1 ⎩

⎧ ∂ 1 ∂ 7 1 − 1 7 + 7 + + − = ⎪S + 5J 5J β ∂ α β ∂ α ⎪ ⎪ ∂ 1 ∂ 7 1 − 1 7 + 7 ⎪ + + − = ⎨S + 5J 5J β β ∂ α ∂ α ⎪ ⎪ ⎪S + 1 + 1 = ⎪ ] 5 5 1 1 ⎩

- 95 -

- 95 -

2QDDORUVOHFKRL[HQWUHGHX[PpWKRGHV ± XQH PpWKRGH GHV IRUFHV TXL FRQVLVWH j LQWpJUHU OH V\VWqPH GH WURLV pTXDWLRQV j WURLV IRQFWLRQVLQFRQQXHV 1 1 7 = 7


± XQHPpWKRGHGHVGpSODFHPHQWVTXLFRQVLVWHjLQWpJUHUDXVVLOHV\VWqPH PDLVDSUqV\ DYRLUUHPSODFp 1 1 HW 7 7 SDUOHXUVH[SUHVVLRQVHQIRQFWLRQGH X X Z


±(;(03/('(/$63+Ê5(&5(86(

±(;(03/('(/$63+Ê5(&5(86(

2Q DSSHOOH 5 HW 5 OHV UD\RQV LQWpULHXU HW H[WpULHXU DYHF 5 − 5 = K pSDLVVHXU /HV IDFHVLQWpULHXUHHWH[WpULHXUHVRQWPDLQWHQXHVDX[WHPSpUDWXUHV 7 HW 7 UHVSHFWLYHPHQW$X FKDSLWUH,QRXVDYRQVGpWHUPLQpOHFKDPSGHWHPSpUDWXUHGDQVFHWWHFRTXHVSKpULTXHRQD WURXYp $ 5 5 7 5 −7 5 7 = % − DYHF $ = 7 − 7 HW % = U 5 − 5 5 − 5


/DFRTXHpWDQWPLQFHRQD


U = 5 + ] 5 =

$=

%=

GRQF

5 + 5

5 + 5

U = 5 + ] 5 =

5 7 − 7 K

$=

5 7 − 7 K

7 + 7 5 7 + 7 7 − 7 7 −7 + ] − ] + 7 − 7 HW 7 = K 5K K

%=

7 + 7 5 7 + 7 7 − 7 7 −7 + ] − ] + 7 − 7 HW 7 = K 5K K

GRQF

− 7 + 7 7 −7 7 + 7 7 ′ = 7 ′′ = 7 = 7 3 = 5K K

7 = 7 3 =

− 7 + 7 7 −7 7 + 7 7 ′ = 7 ′′ = 5K K

)LJXUH

)LJXUH

- 96 -

- 96 -





±(;(03/('(/$63+Ê5(&5(86(

±(;(03/('(/$63+Ê5(&5(86(





U = 5 + ] 5 =

$=

%=

GRQF

5 + 5

5 + 5

U = 5 + ] 5 =

5 7 − 7 K

$=

5 7 − 7 K

7 + 7 5 7 + 7 7 − 7 7 −7 + ] − ] + 7 − 7 HW 7 = K 5K K

%=

7 + 7 5 7 + 7 7 − 7 7 −7 + ] − ] + 7 − 7 HW 7 = K 5K K

GRQF

− 7 + 7 7 −7 7 + 7 7 ′ = 7 ′′ = 7 = 7 3 = 5K K

7 = 7 3 =

− 7 + 7 7 −7 7 + 7 7 ′ = 7 ′′ = 5K K

)LJXUH

)LJXUH

- 96 -

- 96 -

/HSUREOqPHHVWFRQVLGpUpHQUpJLPHVWDWLRQQDLUHLOSRVVqGHGHSOXVODV\PpWULHVSKpULTXH 3DUODWKpRULHTXDGUDWLTXHRQWURXYH


± OHVGpSODFHPHQWVHQ3


⎧ ⎪X = ⎪ G ⎪ X = 3R3 ⎨X = ⎪ ⎛ 7 + 7 ⎞ ⎪ ⎪Z = α 5 ⎜⎝ − 7 ⎟⎠ ⎩

± OHVGpIRUPDWLRQVHQ3

⎧ ⎪X = ⎪ G ⎪ X = 3R3 ⎨X = ⎪ ⎛ 7 + 7 ⎞ ⎪ ⎪Z = α 5 ⎜⎝ − 7 ⎟⎠ ⎩


] ] ⎞ ⎛ 7 + 7 ⎞⎛ ε = ε = εF = α ⎜ − 7 ⎟ ⎜ − + ⎟ ⎜ ⎝ ⎠ ⎝ 5 5 ⎠⎟

] ] ⎞ ⎛ 7 + 7 ⎞⎛ ε = ε = εF = α ⎜ − 7 ⎟ ⎜ − + ⎟ ⎜ ⎝ ⎠ ⎝ 5 5 ⎠⎟

± OHVFRQWUDLQWHVHQ3


σ = σ = σF =

(α − ν

5 ] ⎞⎟ ⎡ 7 + 7 ⎤ ⎜⎛ ] − 7 + 7 − 7 ⎥⎜ − ⎢⎣ K 5 ⎠⎟ ⎦⎝5

σ = σ = σF =

± OHVYLVVHXUVHQ 3

(α − ν

5 ] ⎞⎟ ⎡ 7 + 7 ⎤ ⎜⎛ ] − 7 + 7 − 7 ⎥⎜ − ⎢⎣ K 5 ⎠⎟ ⎦⎝5

± OHVYLVVHXUVHQ 3

1 = 1 = 7 = 7 = 4 = 4 = = 8 = 8 0 = −0 = −

1 = 1 = 7 = 7 = 4 = 4 = = 8 = 8

' + ν α ⎡ 7 + 7 5 ⎤ ⎢⎣ − 7 − K 7 − 7 ⎥⎦ 5

0 = −0 = −

' + ν α ⎡ 7 + 7 5 ⎤ ⎢⎣ − 7 − K 7 − 7 ⎥⎦ 5

(Q VXSSULPDQW OHV WHUPHV HQ ] RQ REWLHQW OHV UpVXOWDWV GRQQpV SDU OD WKpRULH OLQpDLUH HQ VXSSULPDQWDXVVLOHVWHUPHVHQ]RQREWLHQWOHVUpVXOWDWVGRQQpVSDUODWKpRULHGHVPHPEUDQHV LQDGDSWpHLFLSXLVTX¶RQDXQJUDGLHQWWUDQVYHUVDOGHWHPSpUDWXUH


9R\RQV VXU XQH DSSOLFDWLRQ QXPpULTXH OHV GLIIpUHQFHV HQWUH OD WKpRULH OLQpDLUH HW FHOOH GH WKHUPRpODVWLFLWp 3UHQRQV 7 = ° & 7 = ° & 7 = ° & 5 = P K = FP


( = *3D α = ⋅ − . − HW ν = /HVFRQWUDLQWHV σL ILEUHLQWHUQH HW σH ILEUH H[WHUQH RQWSRXUYDOHXUVHQ03D


7KpRULHOLQpDLUH

7KpRULH'

7KpRULHOLQpDLUH

7KpRULH'

σL

σL

σH

±

±

σH

±

±

/HVpFDUWVVRQWGHO¶RUGUHGHSRXUXQHFRTXHDVVH]pSDLVVHHWXQJUDGLHQWGHWHPSpUDWXUH LPSRUWDQW


- 97 -

- 97 -





⎧ ⎪X = ⎪ G ⎪ X = 3R3 ⎨X = ⎪ ⎛ 7 + 7 ⎞ ⎪ ⎪Z = α 5 ⎜⎝ − 7 ⎟⎠ ⎩


⎧ ⎪X = ⎪ G ⎪ X = 3R3 ⎨X = ⎪ ⎛ 7 + 7 ⎞ ⎪ ⎪Z = α 5 ⎜⎝ − 7 ⎟⎠ ⎩


] ] ⎞ ⎛ 7 + 7 ⎞⎛ ε = ε = εF = α ⎜ − 7 ⎟ ⎜ − + ⎟ ⎜ ⎝ ⎠ ⎝ 5 5 ⎠⎟

] ] ⎞ ⎛ 7 + 7 ⎞⎛ ε = ε = εF = α ⎜ − 7 ⎟ ⎜ − + ⎟ ⎝ ⎠ ⎝⎜ 5 5 ⎠⎟



σ = σ = σF =

( α ⎡ 7 + 7 5 ] ⎞⎟ ⎤ ⎜⎛ ] − 7 + 7 − 7 ⎥⎜ − ⎢ − ν ⎣ K 5 ⎠⎟ ⎦⎝5

σ = σ = σF =

± OHVYLVVHXUVHQ 3

( α ⎡ 7 + 7 5 ] ⎞⎟ ⎤ ⎜⎛ ] − 7 + 7 − 7 ⎥⎜ − ⎢ − ν ⎣ K 5 ⎠⎟ ⎦⎝5

± OHVYLVVHXUVHQ 3

1 = 1 = 7 = 7 = 4 = 4 = = 8 = 8 0 = −0 = −

' + ν α 5

1 = 1 = 7 = 7 = 4 = 4 = = 8 = 8

5 ⎡ 7 + 7 ⎤ ⎢⎣ − 7 − K 7 − 7 ⎥⎦

0 = −0 = −

' + ν α 5

5 ⎡ 7 + 7 ⎤ ⎢⎣ − 7 − K 7 − 7 ⎥⎦







7KpRULHOLQpDLUH

7KpRULH'

7KpRULHOLQpDLUH

7KpRULH'

σL

σL

σH

±

±

σH

±

±



- 97 -

- 97 -

±352%/Ê0(6$;,6<0e75,48(60e5,',(16

±352%/Ê0(6$;,6<0e75,48(60e5,',(16

]′] HVWVXSSRVpD[HGHUpYROXWLRQSRXUODFRTXHOHFKDUJHPHQWOHVOLDLVRQVHWOHFKDPSGH WHPSpUDWXUH OHV IRUFHV VRQW GH SOXV ©PpULGLHQQHVª F¶HVW j GLUH VLWXpHV GDQV GHV SODQV PpULGLHQV




UHpWDSHGpSODFHPHQWV

UHpWDSHGpSODFHPHQWV

HpWDSHGpIRUPDWLRQV

HpWDSHGpIRUPDWLRQV

/HVFRHIILFLHQWVQRQQXOVRQWSRXUH[SUHVVLRQ


⎧ ⎧ ∂ Z X ⎪X = ⎪ ω = ∂ V + 5 ⎪ ⎪ ⎪ ⎛ ∂ Z X ⎞ G ⎪ ⎟⎟ + X = 3R3 ⎨X = X − ] ⎜⎜ ⎨ ω = ⎝ ∂ V 5 ⎠ ⎪ ⎪ ⎪ ⎪ω = ⎪ ] ⎪Z = Z V W ⎩ ⎩

⎧ ⎪ε = − X FRV γ + Z VLQ γ U ⎪ ⎪ ⎪ε = ∂ X − Z ⎪ ∂V 5 ⎪ ⎪ ⎛ ⎞ ⎪ε′ = X FRV γ ⎜ + VLQ γ ⎟ − Z VLQ γ ⎜ ⎟ U 5 U ⎪⎪ ⎝ ⎠ U ⎨ ⎪ G ⎛ ⎞ Z ∂ Z ⎪ε′ = − X G V ⎜⎜ 5 ⎟⎟ − − ∂ V ⎝ ⎠ 5 ⎪ ⎪ ⎪ε′′ = − X VLQ γ FRV γ ⎛⎜ + VLQ γ ⎞⎟ + Z VLQ γ − VLQ γ FRV γ ∂ Z ⎜ ⎟ ⎪ ∂ V U U ⎠ U U ⎝ 5 ⎪ ⎪ X G ⎛ ⎞ Z ∂ Z ⎜⎜ ⎟⎟ − − ⎪ε′′ = − 5 G V ⎝ 5 ⎠ 5 5 ∂ V ⎪⎩


⎧ ⎪ε = − X FRV γ + Z VLQ γ U ⎪ ⎪ ⎪ε = ∂ X − Z ⎪ ∂V 5 ⎪ ⎪ ⎛ ⎞ ⎪ε′ = X FRV γ ⎜ + VLQ γ ⎟ − Z VLQ γ ⎜ ⎟ U 5 U ⎪⎪ ⎝ ⎠ U ⎨ ⎪ G ⎛ ⎞ Z ∂ Z ⎪ε′ = − X G V ⎜⎜ 5 ⎟⎟ − − ∂ V ⎝ ⎠ 5 ⎪ ⎪ ⎪ε′′ = − X VLQ γ FRV γ ⎛⎜ + VLQ γ ⎞⎟ + Z VLQ γ − VLQ γ FRV γ ∂ Z ⎜ ⎟ ⎪ ∂ V U U ⎠ U U ⎝ 5 ⎪ ⎪ X G ⎛ ⎞ Z ∂ Z ⎜⎜ ⎟⎟ − − ⎪ε′′ = − 5 G V ⎝ 5 ⎠ 5 5 ∂ V ⎪⎩

HpWDSHFRQWUDLQWHV

HpWDSHFRQWUDLQWHV

2QHQGpGXLWOHVFRQWUDLQWHVQRQQXOOHV


( (α ⎧ ⎪ σ = − ν ε + ν ε − − ν 7 − 7 = σ + ] σ′ + = σ′′ ⎪ ⎨ ( (α ⎪ ε + ν ε − 7 − 7 = σ + ] σ′ + = σ′′ ⎪σ = − ν − ν ⎩


- 98 -

- 98 -

±352%/Ê0(6$;,6<0e75,48(60e5,',(16

±352%/Ê0(6$;,6<0e75,48(60e5,',(16





UHpWDSHGpSODFHPHQWV

UHpWDSHGpSODFHPHQWV

HpWDSHGpIRUPDWLRQV

HpWDSHGpIRUPDWLRQV




⎧ ⎪ε = − X FRV γ + Z VLQ γ U ⎪ ⎪ ⎪ε = ∂ X − Z ⎪ ∂V 5 ⎪ ⎪ ⎛ ⎞ ⎪ε′ = X FRV γ ⎜ + VLQ γ ⎟ − Z VLQ γ U ⎜⎝ 5 U ⎟⎠ U ⎪⎪ ⎨ ⎪ G ⎛ ⎞ Z ∂ Z ⎪ε′ = − X G V ⎜⎜ 5 ⎟⎟ − − ∂ V ⎝ ⎠ 5 ⎪ ⎪ ⎪ε′′ = − X VLQ γ FRV γ ⎛⎜ + VLQ γ ⎞⎟ + Z VLQ γ − VLQ γ FRV γ ∂ Z ⎜ ⎟ ⎪ ∂ V U U ⎠ U U ⎝ 5 ⎪ ⎪ X G ⎛ ⎞ Z ∂ Z ⎜⎜ ⎟⎟ − − ⎪ε′′ = − 5 G V ⎝ 5 ⎠ 5 5 ∂ V ⎩⎪


⎧ ⎪ε = − X FRV γ + Z VLQ γ U ⎪ ⎪ ⎪ε = ∂ X − Z ⎪ ∂V 5 ⎪ ⎪ ⎛ ⎞ ⎪ε′ = X FRV γ ⎜ + VLQ γ ⎟ − Z VLQ γ U ⎜⎝ 5 U ⎟⎠ U ⎪⎪ ⎨ ⎪ G ⎛ ⎞ Z ∂ Z ⎪ε′ = − X G V ⎜⎜ 5 ⎟⎟ − − ∂ V ⎝ ⎠ 5 ⎪ ⎪ ⎪ε′′ = − X VLQ γ FRV γ ⎛⎜ + VLQ γ ⎞⎟ + Z VLQ γ − VLQ γ FRV γ ∂ Z ⎜ ⎟ ⎪ ∂ V U U ⎠ U U ⎝ 5 ⎪ ⎪ X G ⎛ ⎞ Z ∂ Z ⎜⎜ ⎟⎟ − − ⎪ε′′ = − 5 G V ⎝ 5 ⎠ 5 5 ∂ V ⎩⎪

HpWDSHFRQWUDLQWHV

HpWDSHFRQWUDLQWHV





- 98 -

- 98 -

7 − 7 = 7 − 7 + ] 7 ′ V W + ] 7 ′′ V W

DYHF

7 − 7 = 7 − 7 + ] 7 ′ V W + ] 7 ′′ V W

DYHF

HpWDSHYLVVHXUV

HpWDSHYLVVHXUV

/HVFRPSRVDQWHVQRQQXOOHVRQWSRXUH[SUHVVLRQV


⎧ ⎛ σ′ ⎞ VLQ γ ⎞ ⎛ ⎟⎟ 1 = K σ + , ⎜ σ′′ + σ′ ⎟ ⎪ 1 = K σ + , ⎜⎜ σ′′ − U ⎠ 5 ⎝ ⎠ ⎝ ⎪ ⎪ ⎛ σ ⎞ ⎪ ⎛ VLQ γ ⎞ 0 = − , ⎜ σ + σ′ ⎟⎟ + σ′ ⎟ ⎨0 = , ⎜⎜ − 5 U ⎝ ⎠ ⎝ ⎠ ⎪ ⎪ ⎪G¶ R4 = − ∂ 0 − FRV γ 0 + 0 ⎪ U ∂ V ⎩

⎧ ⎛ σ′ ⎞ VLQ γ ⎞ ⎛ ⎟⎟ 1 = K σ + , ⎜ σ′′ + σ′ ⎟ ⎪ 1 = K σ + , ⎜⎜ σ′′ − U ⎠ 5 ⎝ ⎠ ⎝ ⎪ ⎪ ⎛ σ ⎞ ⎪ ⎛ VLQ γ ⎞ 0 = − , ⎜ σ + σ′ ⎟⎟ + σ′ ⎟ ⎨0 = , ⎜⎜ − 5 U ⎝ ⎠ ⎝ ⎠ ⎪ ⎪ ⎪G¶ R4 = − ∂ 0 − FRV γ 0 + 0 ⎪ U ∂ V ⎩

HpWDSHpTXDWLRQVDX[GpSODFHPHQWV


/HVHVHWHVpTXDWLRQVG¶pTXLOLEUHGHVIRUFHVV¶pFULYHQWLFL


∂ 1 FRV γ 4 ⎧ ⎪S + ∂ V + 1 − 1 U − 5 = ⎪ ⎨ ∂ 4 VLQ γ 1 FRV γ ⎪ ⎪S ] + ∂ V − 1 U + 5 − 4 U = ⎩


(Q\UHPSODoDQW 1 1 HW 4 SDUOHXUVH[SUHVVLRQVHQIRQFWLRQGH X HWZRQREWLHQWOHV pTXDWLRQVDX[GpSODFHPHQWV


±7KpRULHOLQpDLUH

±7KpRULHOLQpDLUH

(OOHVHGpGXLWGHODWKpRULHTXDGUDWLTXHFLGHVVXVSDUVXSSUHVVLRQGHVWHUPHVHQ ] RQSUHQG DLQVL ε′′ = ε′′ = = σ′′ = σ′′ HW 7 ′′ =




(OOHVHGpGXLWGHODWKpRULHTXDGUDWLTXHHQDQQXODQWOHVFRHIILFLHQWVGH ] HWFHX[GH]GRQF HQ SUHQDQW ε′ = ε′′ = σ′ = σ′′ = ε′ = ε′′ = σ′ = σ′′ = 7 ′ = 7 ′′ = (OOH Q¶HVW GRQF SDVDGDSWpHDX[FRTXHVSUpVHQWDQWXQJUDGLHQWWUDQVYHUVDOGHWHPSpUDWXUH


/HVpTXDWLRQVG¶pTXLOLEUHGHYLHQQHQWHQSDUWLFXOLHUSXLVTXH 0 = 0 = = 4 = 4


∂ 1 FRV γ ⎧ ⎪S + ∂ V + 1 − 1 U = ⎪ ⎨ VLQ γ 1 ⎪ ⎪S ] − 1 U + 5 = ⎩

∂ 1 FRV γ ⎧ ⎪S + ∂ V + 1 − 1 U = ⎪ ⎨ VLQ γ 1 ⎪ ⎪S ] − 1 U + 5 = ⎩

- 99 -

- 99 -

7 − 7 = 7 − 7 + ] 7 ′ V W + ] 7 ′′ V W

DYHF

7 − 7 = 7 − 7 + ] 7 ′ V W + ] 7 ′′ V W

DYHF

HpWDSHYLVVHXUV

HpWDSHYLVVHXUV



⎧ ⎛ σ′ ⎞ VLQ γ ⎞ ⎛ ⎟⎟ 1 = K σ + , ⎜ σ′′ + σ′ ⎟ ⎪ 1 = K σ + , ⎜⎜ σ′′ − U ⎠ 5 ⎠ ⎝ ⎝ ⎪ ⎪ ⎛ σ ⎞ ⎪ ⎛ VLQ γ ⎞ 0 = − , ⎜ σ + σ′ ⎟⎟ + σ′ ⎟ ⎨0 = , ⎜⎜ − 5 U ⎝ ⎠ ⎝ ⎠ ⎪ ⎪ ⎪G¶ R4 = − ∂ 0 − FRV γ 0 + 0 ⎪ U ∂ V ⎩

⎧ ⎛ σ′ ⎞ VLQ γ ⎞ ⎛ ⎟⎟ 1 = K σ + , ⎜ σ′′ + σ′ ⎟ ⎪ 1 = K σ + , ⎜⎜ σ′′ − U ⎠ 5 ⎠ ⎝ ⎝ ⎪ ⎪ ⎛ σ ⎞ ⎪ ⎛ VLQ γ ⎞ 0 = − , ⎜ σ + σ′ ⎟⎟ + σ′ ⎟ ⎨0 = , ⎜⎜ − 5 U ⎝ ⎠ ⎝ ⎠ ⎪ ⎪ ⎪G¶ R4 = − ∂ 0 − FRV γ 0 + 0 ⎪ U ∂ V ⎩









±7KpRULHOLQpDLUH

±7KpRULHOLQpDLUH









∂ 1 FRV γ ⎧ ⎪S + ∂ V + 1 − 1 U = ⎪ ⎨ VLQ γ 1 ⎪ ⎪S ] − 1 U + 5 = ⎩

- 99 -

∂ 1 FRV γ ⎧ ⎪S + ∂ V + 1 − 1 U = ⎪ ⎨ VLQ γ 1 ⎪ ⎪S ] − 1 U + 5 = ⎩

- 99 -



6RLWDOHUD\RQPR\HQGXF\OLQGUH ]′] O¶D[HGHUpYROXWLRQODIRUFHVXUIDFLTXHHVWVXSSRVpH QRUPDOH GH PHVXUH DOJpEULTXH S = S ] OD WHPSpUDWXUH HVW GH OD IRUPH [[ 'RQQRQV OHV IRUPXOHVGHODWKpRULHOLQpDLUHOHVFDUDFWpULVWLTXHVJpRPpWULTXHVGXF\OLQGUHFLUFXODLUHD\DQW pWpGRQQpHVDXWRPH,FKDSLWUH9,,SDJH


⎧ ⎧ ∂Z ⎪ ω = ∂ ] ⎪X = ⎪ ⎪ G ∂Z ⎪ ⎪ ω ⎨ ω = X = 3R3 ⎨X = X − = ∂] ⎪ ⎪ ⎪ω = ⎪ Z = Z ] W ⎪ ] ⎪ ⎩ ⎩


/HVpTXDWLRQVG¶pTXLOLEUH V¶pFULYHQWSXLVTXH γ =

1 = & WH S +

π = V = ] + & WH U = D S = 5

∂ 4 1 ∂ 0 − = HW 4 = − ∂] D ∂]

2QREWLHQWDORUVOHVFRHIILFLHQWVGHODWKpRULHOLQpDLUH ε = −


1 = & WH S +

∂ 4 1 ∂ 0 − = HW 4 = − ∂] D ∂]

2QREWLHQWDORUVOHVFRHIILFLHQWVGHODWKpRULHOLQpDLUH

∂X Z ∂ Z Z ε′ = − ε = − ε′ = − D ∂] D ∂]

SXLV

π = V = ] + & WH U = D S = 5

ε = −

∂X Z ∂ Z Z ε′ = − ε = − ε′ = − D ∂] D ∂]

SXLV

σ =

−( ⎛Z ∂ X ⎞ ( α ⎜ +ν ⎟− 7 − 7 ⎜ D ∂ ] ⎟⎠ − ν − ν ⎝

σ =

−( ⎛Z ∂ X ⎞ ( α ⎜ +ν ⎟− 7 − 7 ⎜ D ∂ ] ⎟⎠ − ν − ν ⎝

σ′ =

− ( ⎜⎛ Z ∂ Z ⎞⎟ ( α +ν − 7′ ⎜ ∂ ] ⎟⎠ − ν − ν ⎝ D

σ′ =

− ( ⎜⎛ Z ∂ Z ⎞⎟ ( α +ν − 7′ ⎜ ∂ ] ⎟⎠ − ν − ν ⎝ D

σ =

⎛ ∂ X Z⎞ (α ⎜⎜ + ν ⎟⎟ − 7 − 7 ∂ ] D ⎠ − ν − ν ⎝

σ =

⎛ ∂ X Z⎞ (α ⎜⎜ + ν ⎟⎟ − 7 − 7 ∂ ] D ⎠ − ν − ν ⎝

σ′ =

Z⎞ (α − ( ⎜⎛ ∂ Z 7′ +ν ⎟− ⎜ − ν ⎝ ∂ ] D ⎟⎠ − ν

σ′ =

Z⎞ (α − ( ⎜⎛ ∂ Z 7′ +ν ⎟− ⎜ − ν ⎝ ∂ ] D ⎟⎠ − ν

(

HWHQILQHQQpJOLJHDQW K GHYDQW D

1 =

(K − ν

(


⎛Z ∂ X ⎞ ( α K ⎜⎜ + ν ⎟− 7 − 7 D ∂ ] ⎟⎠ − ν ⎝

1 =

(K − ν

⎛Z ∂ X ⎞ ( α K ⎜⎜ + ν ⎟− 7 − 7 D ∂ ] ⎟⎠ − ν ⎝

- 100 -

- 100 -








1 = & WH S +

π = V = ] + & WH U = D S = 5

∂ 4 1 ∂ 0 − = HW 4 = − ∂] D ∂]

2QREWLHQWDORUVOHVFRHIILFLHQWVGHODWKpRULHOLQpDLUH ε = −


1 = & WH S +

∂ 4 1 ∂ 0 − = HW 4 = − ∂] D ∂]

2QREWLHQWDORUVOHVFRHIILFLHQWVGHODWKpRULHOLQpDLUH

∂X Z ∂ Z Z ε′ = − ε = − ε′ = − D ∂] D ∂]

SXLV

π = V = ] + & WH U = D S = 5

ε = −

∂X Z ∂ Z Z ε′ = − ε = − ε′ = − D ∂] D ∂]

SXLV

σ =

−( ⎛Z ∂ X ⎞ ( α ⎜ +ν ⎟− 7 − 7 ⎜ D ∂ ] ⎟⎠ − ν − ν ⎝

σ =

−( ⎛Z ∂ X ⎞ ( α ⎜ +ν ⎟− 7 − 7 ⎜ D ∂ ] ⎟⎠ − ν − ν ⎝

σ′ =

− ( ⎜⎛ Z ∂ Z ⎞⎟ ( α + ν − 7′ ∂ ] ⎟⎠ − ν − ν ⎜⎝ D

σ′ =

− ( ⎜⎛ Z ∂ Z ⎞⎟ ( α + ν − 7′ ∂ ] ⎟⎠ − ν − ν ⎜⎝ D

σ =

( ⎛ ∂ X Z⎞ (α ⎜⎜ + ν ⎟⎟ − 7 − 7 D ⎠ − ν − ν ⎝ ∂ ]

σ =

( ⎛ ∂ X Z⎞ (α ⎜⎜ + ν ⎟⎟ − 7 − 7 D ⎠ − ν − ν ⎝ ∂ ]

σ′ =

Z⎞ (α − ( ⎜⎛ ∂ Z 7′ +ν ⎟− ⎜ − ν ⎝ ∂ ] D ⎟⎠ − ν

σ′ =

Z⎞ (α − ( ⎜⎛ ∂ Z 7′ +ν ⎟− ⎜ − ν ⎝ ∂ ] D ⎟⎠ − ν


1 =

(K − ν


⎛Z ∂ X ⎞ ( α K ⎜⎜ + ν ⎟− 7 − 7 ∂ ] ⎟⎠ − ν ⎝D

- 100 -

1 =

(K − ν

⎛Z ∂ X ⎞ ( α K ⎜⎜ + ν ⎟− 7 − 7 ∂ ] ⎟⎠ − ν ⎝D

- 100 -

1 =

(K − ν

⎛ ∂ X Z⎞ (αK ⎜⎜ + ν ⎟⎟ − 7 − 7 ∂ ] D ⎠ − ν ⎝

(K − ν

⎛ ∂ X Z⎞ (αK ⎜⎜ + ν ⎟⎟ − 7 − 7 ∂ ] D ⎠ − ν ⎝

1 =

⎤ ⎡Z ∂ Z 0 = −' ⎢ + ν + α + ν 7 ′⎥ ∂] ⎦⎥ ⎣⎢ D

⎤ ⎡Z ∂ Z 0 = −' ⎢ + ν + α + ν 7 ′⎥ ∂] ⎦⎥ ⎣⎢ D

⎡ ∂ Z ∂ X ⎛ 7 − 7 ⎞⎤ 0 = −' ⎢ − + α + ν ⎜⎜ + 7 ′ ⎟⎟⎥ D ∂] ⎢⎣ ∂ ] ⎝ D ⎠⎥⎦

⎡ ∂ Z ∂ X ⎛ 7 − 7 ⎞⎤ 0 = −' ⎢ − + α + ν ⎜⎜ + 7 ′ ⎟⎟⎥ D ∂] ⎢⎣ ∂ ] ⎝ D ⎠⎥⎦

4 =

4 =

∂ 0 ( K DYHF ' = ∂] − ν

∂ 0 ( K DYHF ' = ∂] − ν

/HVSDUDPqWUHV γ τ 7 7 8 8 4 VRQWLFLLGHQWLTXHPHQWQXOV


/HVpTXDWLRQVG¶pTXLOLEUHpFULWHVHQWHUPHVGHGpSODFHPHQWVGRQQHQWILQDOHPHQW


0 =

DYHF N =

HW

S] 1 ∂ Z ∂ 7′ ( α K N Z ] W + 7 − 7 + = − ν − α + ν ' D' 'D ∂ ] ∂ ]

( K − ν = 'D K D

0 =

DYHF N =

∂ X − ν Z = 1 − ν + α + ν 7 − 7 ∂] (K D

HW

S] 1 ∂ Z ∂ 7′ ( α K N Z ] W + 7 − 7 + = − ν − α + ν ' D' 'D ∂ ] ∂ ]

( K − ν = 'D K D

∂ X − ν Z = 1 − ν + α + ν 7 − 7 ∂] (K D

±([HPSOH

±([HPSOH

&\OLQGUHVHPLLQILQLHQFDVWUpVXLYDQWVDVHFWLRQ] VRXPLVjXQHpOpYDWLRQXQLIRUPHGH WHPSpUDWXUH 7 − 7 O¶HQFDVWUHPHQWDpWpUpDOLVpVDQVMHXQLVHUUDJHjODWHPSpUDWXUH 7


2Q D LFL 1 = S = HW 7 ′ = /¶pTXDWLRQ GH OD IOqFKH Z V¶pFULW GRQF HQ UpJLPH VWDWLRQQDLUH


∂ Z ∂]

+ N Z =

∂ Z

(αK 7 − 7 'D

∂ ]

- 101 -

1 =

(K − ν

+ N Z =

(αK 7 − 7 'D

- 101 -

⎛ ∂ X Z⎞ (αK ⎜⎜ + ν ⎟⎟ − 7 − 7 ∂ ] D ⎠ − ν ⎝

(K − ν

⎛ ∂ X Z⎞ (αK ⎜⎜ + ν ⎟⎟ − 7 − 7 ∂ ] D ⎠ − ν ⎝

1 =

⎤ ⎡Z ∂ Z 0 = −' ⎢ + ν + α + ν 7 ′⎥ ∂] ⎥⎦ ⎢⎣ D

⎤ ⎡Z ∂ Z 0 = −' ⎢ + ν + α + ν 7 ′⎥ ∂] ⎥⎦ ⎢⎣ D

⎡ ∂ Z ∂ X ⎛ 7 − 7 ⎞⎤ 0 = −' ⎢ − + α + ν ⎜⎜ + 7 ′ ⎟⎟⎥ D ∂] ⎢⎣ ∂ ] ⎝ D ⎠⎥⎦

⎡ ∂ Z ∂ X ⎛ 7 − 7 ⎞⎤ 0 = −' ⎢ − + α + ν ⎜⎜ + 7 ′ ⎟⎟⎥ D ∂] ⎢⎣ ∂ ] ⎝ D ⎠⎥⎦

4 =

4 =

∂ 0 ( K DYHF ' = ∂] − ν

∂ 0 ( K DYHF ' = ∂] − ν





0 =

DYHF N =

HW

S 1 ∂ Z ∂ 7′ ( α K + 7 − 7 + N Z ] W = ] − ν − α + ν ' D' 'D ∂ ] ∂]

( K − ν = 'D K D

0 =

DYHF N =

∂ X − ν Z = 1 − ν + α + ν 7 − 7 ∂] (K D

HW

S 1 ∂ Z ∂ 7′ ( α K + 7 − 7 + N Z ] W = ] − ν − α + ν ' D' 'D ∂ ] ∂]

( K − ν = 'D K D

∂ X − ν Z = 1 − ν + α + ν 7 − 7 ∂] (K D

±([HPSOH

±([HPSOH





∂ Z ∂]

+ N Z =

(αK 7 − 7 'D

- 101 -

∂ Z ∂]

+ N Z =

(αK 7 − 7 'D

- 101 -

)LJXUH

)LJXUH

(OOHDSRXUVROXWLRQJpQpUDOH


Z = H − N] $ FRV N ] + % VLQ N ] + H N] $ ′ FRV N ] + % ′ VLQ N ] + α D 7 − 7

± SRXU ] → ∞ [GRLWUHVWHUILQLFHTXLLPSOLTXH $′ = %′ = ± SRXU] RQGRLWDYRLUZ HW


± SRXU ] → ∞ [GRLWUHVWHUILQLFHTXLLPSOLTXH $′ = %′ =

GZ = FHTXLLPSRVH G]

± SRXU] RQGRLWDYRLUZ HW

$ = % = −α D 7 − 7

G¶R

Z ] = α D 7 − 7 − H − N] FRV N ] + VLQ N ]

SXLV

G X = α 7 − 7 + ν H − N] FRV N ] + VLQ N ] G]

[

]

]

[

GZ = FHTXLLPSRVH G]

$ = % = −α D 7 − 7

G¶R

Z ] = α D 7 − 7 − H − N] FRV N ] + VLQ N ]

SXLV


[

]

]

[

2Q HQ GpGXLW DORUV WRXV OHV SDUDPqWUHV GX SUREOqPH URWDWLRQV WUDQVODWLRQV GpIRUPDWLRQV FRQWUDLQWHVYLVVHXUVUpDFWLRQVGHOLDLVRQ


- 102 -

- 102 -

)LJXUH

)LJXUH




± SRXU ] → ∞ [GRLWUHVWHUILQLFHTXLLPSOLTXH $′ = %′ = ± SRXU] RQGRLWDYRLUZ HW

± SRXU ] → ∞ [GRLWUHVWHUILQLFHTXLLPSOLTXH $′ = %′ =

GZ = FHTXLLPSRVH G]

± SRXU] RQGRLWDYRLUZ HW

$ = % = −α D 7 − 7

G¶R

Z ] = α D 7 − 7 − H − N] FRV N ] + VLQ N ]

SXLV


[

]

]

[


GZ = FHTXLLPSRVH G]

$ = % = −α D 7 − 7

G¶R

Z ] = α D 7 − 7 − H − N] FRV N ] + VLQ N ]

SXLV


[

]

]

[



- 102 -

- 102 -

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T = D FRV ωW + ϕ

T = D FRV ωW + ϕ

D GpVLJQH O¶DPSOLWXGH RX pORQJDWLRQ PD[LPDOH ω OD SXOVDWLRQ HQ UDGLDQV SDU VHFRQGHV π ω 7= OD SpULRGH HQ VHFRQGHV I = = OD IUpTXHQFH HQ KHUW] ϕ OD SKDVH HQ ω π 7 UDGLDQV


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T = D FRV ωW + ϕ

T = D FRV ωW + ϕ



- 105 -

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5HPDUTXH2QSUHQGUDWRXMRXUVDHW ω SRVLWLIV&HFLQHUHVWUHLQWSDVODJpQpUDOLWpGHQRWUH pWXGHFRPPHOHPRQWUHO¶H[HPSOHVXLYDQWVXSSRVRQVTXHO¶RQDLW


T W = − FRV −W + RQSHXWDXVVLpFULUH


T W = FRV W − − π H[SUHVVLRQRDHW ω VRQWELHQSRVLWLIV


5HPDUTXH2QSHXWGRQQHUjO¶pTXDWLRQKRUDLUH O¶H[SUHVVLRQ T W = D VLQ ωW + ϕ′ DYHF ϕ′ = ϕ +

5HPDUTXH2QSHXWGRQQHUjO¶pTXDWLRQKRUDLUH O¶H[SUHVVLRQ

π

T W = D VLQ ωW + ϕ′ DYHF ϕ′ = ϕ +

π

5HPDUTXH8QHH[SUHVVLRQGHODIRUPH T = $ FRV ωW + % VLQ ωW SHXWWRXMRXUVrWUHUDPHQpH jODIRUPH DYHF $ = D FRV ϕ HW % = −D VLQ ϕ




6XSSRVRQVTXHO¶RQDLW


T = D FRV ωW + ϕ + D FRV ωW + ϕ

2QSHXWWRXMRXUVWURXYHUDHW ϕ WHOVTXH



T = D FRV ωW + ϕ

T = D FRV ωW + ϕ

DYHF

D VLQ ϕ + D VLQ ϕ = D VLQ ϕ

DYHF


HW

D FRV ϕ + D FRV ϕ = D FRV ϕ

HW


TW HVWGRQFDXVVLXQHIRQFWLRQKDUPRQLTXHGHSXOVDWLRQ ω




6XSSRVRQVODIRQFWLRQTW QRQVLQXVRwGDOHPDLVSpULRGLTXH$SSHORQV7ODSpULRGHHWSRVRQV π ω= 7


2QVDLWDORUVTXHPR\HQQDQWTXHOTXHVK\SRWKqVHVPDWKpPDWLTXHVSHXUHVWULFWLYHVTW SHXW rWUHFRQVLGpUpFRPPHODVRPPHG¶XQHVpULHGH)RXULHUHWV¶pFULUH


T W =

$ + $ FRV ωW + % VLQ ωW + $ FRV ωW + % VLQ ωW

+ !!!! + $ Q FRV Q ωW + %Q VLQ Q ωW + !!!!

T W =


+ !!!! + $ Q FRV Q ωW + %Q VLQ Q ωW + !!!!

- 106 -

- 106 -







5HPDUTXH2QSHXWGRQQHUjO¶pTXDWLRQKRUDLUH O¶H[SUHVVLRQ T W = D VLQ ωW + ϕ′ DYHF ϕ′ = ϕ +

5HPDUTXH2QSHXWGRQQHUjO¶pTXDWLRQKRUDLUH O¶H[SUHVVLRQ

π

T W = D VLQ ωW + ϕ′ DYHF ϕ′ = ϕ +

π











T = D FRV ωW + ϕ

T = D FRV ωW + ϕ

DYHF


DYHF


HW


HW










T W =


+ !!!! + $ Q FRV Q ωW + %Q VLQ Q ωW + !!!!

- 106 -

T W =


+ !!!! + $ Q FRV Q ωW + %Q VLQ Q ωW + !!!!

- 106 -

/HVFRHIILFLHQWVGH)RXULHUVHFDOFXOHQWjO¶DLGHGHVIRUPXOHV


7 ⎧ ⎪$ Q = ω T W ⋅ FRV Q ωW ⋅ GW ⎪ π ⎪ ⎨ 7 ⎪ ⎪%Q = ω T W ⋅ VLQ Q ωW ⋅ GW π ⎪ ⎩


∫

∫

∫

∫

5HPDUTXH2QSHXWWRXMRXUVWURXYHU D Q HW ϕQ DYHF $ Q = D Q FRV ϕQ HW %Q = −D Q VLQ ϕQ WHOVTXH T W =

D + D FRV ωW + ϕ + D FRV ωW + ϕ + !! + D Q FRV Q ωW + ϕQ + !!



5HPDUTXH/HWHUPH D Q FRV Q ωW + ϕQ = $ Q FRV Q ωW + %Q VLQ Q ωW VHQRPPHKDUPRQLTXH GH UDQJ Q GH OD IRQFWLRQ TW VRQ DPSOLWXGH HVW D Q VD SXOVDWLRQ Q ω VD SpULRGH π 7 Qω = VDIUpTXHQFH I Q = 7Q = = QI HWVDSKDVH ϕQ Qω Q π


5HPDUTXH/HVVXLWHV {$ Q } {%Q } {D Q }FRQVWLWXHQWOHVVSHFWUHVGHODIRQFWLRQSpULRGLTXH TW 2QSHXWOHVUHSUpVHQWHUJUDSKLTXHPHQWFRPPHOHPRQWUHODILJXUH SRXUODVXLWH {$ Q }


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T W = D [FRV ωW + ϕ + FRV ω W + ϕ ]VRLW


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∫

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∫

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±%DWWHPHQWV

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- 107 -

- 107 -

T W = D FRV

ω + ω W + ϕ + ϕ ω − ω W + ϕ − ϕ FRV

T W = D FRV

'DQV FH SURGXLW GH GHX[ IRQFWLRQV KDUPRQLTXHV OD SUHPLqUH GLWH RQGH PR\HQQH D SRXU ω + ω π HW SRXU SpULRGH 7PR\ = SXOVDWLRQ ωPR\ = OD VHFRQGH GLWH RQGH GH ω + ω ω − ω π PRGXODWLRQDSRXUSXOVDWLRQ ωPRG = HWSRXUSpULRGH 7PRG = HQVXSSRVDQW ω − ω ω > ω /DFRXUEHUHSUpVHQWDWLYHGHTW HVWGRQQpHSDUODILJXUH2QD 7PRG >> 7PR\


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ω + ω W + ϕ + ϕ ω − ω W + ϕ − ϕ FRV

ω HVWXQQRPEUHUDWLRQQHO ω

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0 T + & T + . T = ) W

0 T + & T + . T = ) W

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/DVWUXFWXUHHVWGLWHOLEUHVL)W HOOHHVWGLWHH[FLWpHGDQVOHFDVFRQWUDLUH


- 108 -

T W = D FRV

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ω + ω W + ϕ + ϕ ω − ω W + ϕ − ϕ FRV

T W = D FRV



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ω + ω W + ϕ + ϕ ω − ω W + ϕ − ϕ FRV


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0 T + & T + . T = ) W

0 T + & T + . T = ) W







- 108 -

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/DVWUXFWXUHHVWGLWHFRQVHUYDWULFHVL& HOOHHVWGLWHGLVVLSDWLYHVL&!


8QHVWUXFWXUHSHXWWRXMRXUVrWUHFODVVpHYLVjYLVGHVRQFRPSRUWHPHQWG\QDPLTXHGDQVO¶XQH GHVTXDWUHFDVHVGXWDEOHDXVXLYDQW


6WUXFWXUH

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0 T + . T =

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. 0

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T W = $ FRV ωW + % VLQ ωW = D FRV ωW + ϕ


0 HW OD IUpTXHQFH I = GH FHW RVFLOODWHXU KDUPRQLTXH VRQW GRQF 7 . LQGpSHQGDQWHV GHV FRQGLWLRQV LQLWLDOHV GH SRVLWLRQ HW YLWHVVH T HWT DLQVL TXH GH O¶DPSOLWXGH/HPRXYHPHQWVLQXVRwGDOHVWGLWOLEUHRXQDWXUHO GHPrPHTXHODSXOVDWLRQ ω ODSpULRGH7HWODIUpTXHQFHI

. 0



/D SpULRGH 7 = π

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π 2QUHPDUTXHTXHODYLWHVVH T = D ω FRV ωW + ϕ HWO¶DFFpOpUDWLRQ T = D ω FRV ωW + ϕ + π VRQWDXVVLVLQXVRwGDOHVGHSXOVDWLRQ ω T HVWHQTXDGUDWXUHDYDQFHVXUT T HQRSSRVLWLRQGH SKDVHVXUT


6LO¶RQDjO¶LQVWDQWLQLWLDOW T = T HW T = T RQHQGpGXLWOHVFRQVWDQWHVG¶LQWpJUDWLRQ


$ = T % =

T T T T D = T FRV ϕ = VLQ ϕ = − ωD D ω ω

$ = T % =

T T T T D = T FRV ϕ = VLQ ϕ = − ωD D ω ω

HUH[HPSOH

HUH[HPSOH

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6WUXFWXUH

FRQVHUYDWLYH

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FRQVHUYDWLYH

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0 T + . T =


. 0



0 T + . T =



. 0



/D SpULRGH 7 = π

/D SpULRGH 7 = π





$ = T % =

T T T T D = T FRV ϕ = VLQ ϕ = − ω D D ω ω

$ = T % =

T T T T D = T FRV ϕ = VLQ ϕ = − ω D D ω ω

HUH[HPSOH

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'DQV OH VHFRQG FDV HQ SUHQDQW O¶RULJLQH GHV DEVFLVVHV DX SRLQW GH O¶pTXLOLEUH VWDWLTXH OH SULQFLSHIRQGDPHQWDOV¶pFULW 0J − . [ + G = 0 [ VRLW

0 [ + . [ =


0 [ + . [ =

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&¶HVW OH FDV GX SHQGXOH VROLGH LQGpIRUPDEOH PRELOH DXWRXU G¶XQ D[H KRUL]RQWDO R] GDQV OH FKDPSGHODSHVDQWHXUILJXUH


2QD

2QD

0=PDVVHGXVROLGH

0=PDVVHGXVROLGH

*=FHQWUHGHJUDYLWp

*=FHQWUHGHJUDYLWp

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,=PRPHQWG¶LQHUWLHSDUUDSSRUWjO¶D[HR]


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2[ 2* = θPRGXOH π

2[ 2* = θPRGXOH π

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- 111 -

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0 [ + . [ =

0 [ + . [ =



HH[HPSOHOHSHQGXOH

HH[HPSOHOHSHQGXOH





2QD

2QD

0=PDVVHGXVROLGH

0=PDVVHGXVROLGH

*=FHQWUHGHJUDYLWp

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2*= D ≠

2*= D ≠





2[ 2* = θPRGXOH π

2[ 2* = θPRGXOH π

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/H SULQFLSH IRQGDPHQWDO GH OD G\QDPLTXH SRXU XQ VROLGH HQ URWDWLRQ DXWRXU G¶XQ D[H R] V¶pFULWLFL , ⋅ θ + 0J D ⋅ VLQ θ =


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θ + 0J D θ = ,

'¶R

ω=

0J D 0J D 7 = π , ,

'¶R

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ω=

J D HW 7 = π D J

P O θ HWO¶pQHUJLHSRWHQWLHOOHHPPDJDVLQpHGDQVOHUHVVRUW

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RXHQFRUH

θ + ω θ = DYHF ω = N P O


'¶R

'¶R ODSpULRGH 7 = π O

J D HW 7 = π D J


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ω=

9 = N θ G¶ROHODJUDQJLHQ/ 7±9/¶pTXDWLRQGH/DJUDQJHV¶pFULWLFL

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0J D 0J D 7 = π , ,

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ω=



θ + 0J D θ = ,

P HWODIUpTXHQFH I = 7 N

ODSpULRGH 7 = π O


- 112 -

- 112 -





θ + 0J D θ = ,

'¶R

ω=

0J D 0J D 7 = π , ,


ω=

J D HW 7 = π D J




G ∂/ ∂/ − = VRLW P O θ + N θ = GW ∂θ ∂θ

RXHQFRUH

'¶R

ω=



ω=

J D HW 7 = π D J




G ∂/ ∂/ − = VRLW P O θ + N θ = GW ∂θ ∂θ


'¶R ODSpULRGH 7 = π O


- 112 -

0J D 0J D 7 = π , ,

RXHQFRUH

'¶R

θ + 0J D θ = ,

ODSpULRGH 7 = π O


- 112 -

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0 T + & T + . T = DYHF0&.SRVLWLIV

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0 [ + & [ + . [ =

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3RVRQV ω =

. SXOVDWLRQQDWXUHOOH 0

3RVRQV ω =

&F = . 0 FRHIILFLHQWG¶DPRUWLVVHPHQWFULWLTXH λ=


&F = . 0 FRHIILFLHQWG¶DPRUWLVVHPHQWFULWLTXH

& FRHIILFLHQWG¶DPRUWLVVHPHQWUpGXLWRXHQFRUHWDX[G¶DPRUWLVVHPHQW &F

λ=


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0 [ + & [ + . [ =

0 [ + & [ + . [ =



0 V + & V + . =

0 V + & V + . =



3RVRQV ω =


3RVRQV ω =










- 113 -

- 113 -

⎧ − λωW $ FKUW + % VK UW ⎪T W = H ⎪ ⎪ − λωW FK UW + ϕ ⎨T W = D H ⎪ $ − % − UW ⎞ ⎪ − λωW ⎛ $ + % UW H ⎟ H + ⎜ ⎪T W = H ⎝ ⎠ ⎩


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⎧⎪$ = T ⎨ ⎪⎩% = T + ω T


⎧⎪$ = T ⎨ ⎪⎩% = T + ω T

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- 114 -

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⎧⎪$ = T ⎨ ⎪⎩% = T + ω T


⎧⎪$ = T ⎨ ⎪⎩% = T + ω T







- 114 -

- 114 -

⎧⎪T W = H −λωW $ FRV ΩW + % VLQ ΩW ⎨ ⎪⎩T W = D H − λωW FRV ΩW + ϕ


⎧ ⎧ ⎪$ = T ⎪D FRV ϕ = T RX ⎨ DYHF ⎨ T + λ ω T T + λ ω T ⎪% = ⎪D VLQ ϕ = Ω Ω ⎩ ⎩


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'pILQLVVRQVTXHOTXHVSDUDPqWUHVDVVRFLpVjXQWHOPRXYHPHQWRVFLOODWRLUHDPRUWL 7 π = 7 Ω

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7 π = 7 Ω

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⎛ T ⎞ δ = /RJ ⎜⎜ L ⎟⎟ ⎝ T L + ⎠

⎛ T ⎞ δ = /RJ ⎜⎜ L ⎟⎟ ⎝ T L + ⎠

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δ=λωW=

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− λ

δ=λωW=

- 115 -

− λ

- 115 -


πλ






)LJXUH

'pILQLVVRQVTXHOTXHVSDUDPqWUHVDVVRFLpVjXQWHOPRXYHPHQWRVFLOODWRLUHDPRUWL 3VHXGRSpULRGHO¶D[HGHVWHPSVHVWFRXSpjGHVLQWHUYDOOHVVXFFHVVLIVpJDX[j HVWODSVHXGRSpULRGH

)LJXUH

'pILQLVVRQVTXHOTXHVSDUDPqWUHVDVVRFLpVjXQWHOPRXYHPHQWRVFLOODWRLUHDPRUWL 7 π = 7 Ω

3VHXGRSpULRGHO¶D[HGHVWHPSVHVWFRXSpjGHVLQWHUYDOOHVVXFFHVVLIVpJDX[j HVWODSVHXGRSpULRGH

7 π = 7 Ω



⎛ T ⎞ δ = /RJ ⎜⎜ L ⎟⎟ ⎝ T L + ⎠

⎛ T ⎞ δ = /RJ ⎜⎜ L ⎟⎟ ⎝ T L + ⎠



δ=λωW=

πλ

- 115 -

− λ

δ=λωW=

πλ

- 115 -

− λ

&RQVWDQWHGHWHPSVRXWHPSVGHUHOD[DWLRQ F¶HVWOHSDUDPqWUH τ =

0 WHPSVDX = λω &


0 WHPSVDX = λω &

ERXWGXTXHOO¶DPSOLWXGHHVWGLYLVpHSDUH


'LVVLSDWLRQ UHODWLYH G¶pQHUJLH F¶HVW OH UDSSRUW GH O¶pQHUJLH GLVVLSpH HQWUH OHV LQVWDQWV W HW W7jO¶pQHUJLHPpFDQLTXHDXWHPSVW2QWURXYH


⎛ − πλ ( W − ( W + 7 = − H[S ⎜ ⎜ ( W ⎝ − λ

⎞ ⎟ ⎟ ⎠

⎛ − πλ ( W − ( W + 7 = − H[S ⎜ ⎜ ( W ⎝ − λ

⎞ ⎟ ⎟ ⎠

TXDQWLWp LQGpSHQGDQWH GX WHPSV W /¶pQHUJLH PpFDQLTXH GH OD VWUXFWXUH HVW OD VRPPH GH O¶pQHUJLHFLQpWLTXH 0 T HWGHO¶pQHUJLHSRWHQWLHOOH . T


±6758&785(/,1e$,5(&216(59$7,9((;&,7e(

±6758&785(/,1e$,5(&216(59$7,9((;&,7e(

/DVWUXFWXUHREpLWjO¶pTXDWLRQGLIIpUHQWLHOOH


0 T + . T = ) W

0 T + . T = ) W

6DVROXWLRQJpQpUDOHV¶REWLHQWHQDMRXWDQWjODVROXWLRQJpQpUDOHGHO¶pTXDWLRQKRPRJqQHVDQV VHFRQGPHPEUH XQHVROXWLRQSDUWLFXOLqUHTXHOFRQTXHGHO¶pTXDWLRQFRPSOqWH


1RXVGLVWLQJXHURQVOHVWURLVFDVG¶H[FLWDWLRQKDUPRQLTXHSpULRGLTXHQRQSpULRGLTXH




2Q SHXW SUHQGUH GDQV FH FDV ) W = ) ⋅ VLQ ω W R ) HW ω VRQW GRQQpV /D VROXWLRQ JpQpUDOHV¶pFULWGDQVOHFDVRODSXOVDWLRQG¶H[FLWDWLRQ ω GLIIqUHGHODSXOVDWLRQQDWXUHOOH ω


) . T W = D VLQ ωW + ϕ + ω − ω

) . T W = D VLQ ωW + ϕ + ω − ω

DHW ϕ VRQWGpWHUPLQpVSDUOHVFRQGLWLRQVLQLWLDOHVWDQGLVTXH ψ O¶HVWSDUOHVUHODWLRQV VLQ ψ = WJ ψ = FRV ψ =

ω − ω ω − ω


GRQFRX±

ω ) HVW OD GpIOH[LRQ VWDWLTXH GX UHVVRUW VRXV O¶DFWLRQ GH ) − . ω

−

HVW OH IDFWHXU

ω − ω ω − ω

GRQFRX±


−

HVW OH IDFWHXU

G¶DPSOLILFDWLRQ G\QDPLTXH TXL SHXW rWUH WUqV JUDQG VL ω HW ω VRQW YRLVLQV FDV GH OD UpVRQDQFH


- 116 -

- 116 -


0 WHPSVDX = λω &


0 WHPSVDX = λω &





⎛ − πλ ( W − ( W + 7 = − H[S ⎜ ⎜ ( W ⎝ − λ

⎞ ⎟ ⎟ ⎠

⎛ − πλ ( W − ( W + 7 = − H[S ⎜ ⎜ ( W ⎝ − λ

⎞ ⎟ ⎟ ⎠



±6758&785(/,1e$,5(&216(59$7,9((;&,7e(

±6758&785(/,1e$,5(&216(59$7,9((;&,7e(



0 T + . T = ) W

0 T + . T = ) W









) . T W = D VLQ ωW + ϕ + ω − ω

) . T W = D VLQ ωW + ϕ + ω − ω


ω − ω ω − ω

−

HVW OH IDFWHXU


GRQFRX±


ω − ω ω − ω

GRQFRX±


−

HVW OH IDFWHXU



- 116 -

- 116 -

/H PRXYHPHQW Q¶HVW GRQF SDV KDUPRQLTXH SXLVTXH TW HVW OD VRPPH GH GHX[ IRQFWLRQV KDUPRQLTXHVGHSXOVDWLRQVGLIIpUHQWHV ω IRUFpH HW ω QDWXUHOOH


/DILJXUHGRQQHOHIDFWHXUG¶DPSOLILFDWLRQG\QDPLTXH)$' HWODSKDVH ψ HQIRQFWLRQGX ω UDSSRUW ω


)LJXUH

)LJXUH

± 3RXU ≤ ω < ω RQD ψ = ODUpSRQVHIRUFpHHVWHQSKDVHDYHFO¶H[FLWDWLRQOH)$'HVW YRLVLQGHSRXU ω SHWLWHWLOGHYLHQWWUqVJUDQGORUVTXH ω WHQGYHUV ω UpVRQDQFH

± 3RXU ≤ ω < ω RQD ψ = ODUpSRQVHIRUFpHHVWHQSKDVHDYHFO¶H[FLWDWLRQOH)$'HVW YRLVLQGHSRXU ω SHWLWHWLOGHYLHQWWUqVJUDQGORUVTXH ω WHQGYHUV ω UpVRQDQFH

± 3RXU ω > ω RQD ψ = π ODUpSRQVHIRUFpHHVWHQRSSRVLWLRQGHSKDVHDYHFO¶H[FLWDWLRQ




OH)$'WHQGYHUV]pURORUVTXH ω GHYLHQWWUqVJUDQG

π ODSpULRGHGH)W FHWWHIRQFWLRQVHGpYHORSSHHQVpULHGH)RXULHUFRPPHQRXV ω O¶DYRQVUDSSHOpSOXVKDXW2QDDLQVL

6RLW 7 =

) W =

) +

∞

∑


6RLW 7 =

)Q VLQ Q ω W + ϕQ

) W =

Q =

/DVROXWLRQSDUWLFXOLqUHGHO¶pTXDWLRQFRPSOqWH HVWODVRPPHGHVVROXWLRQVSDUWLFXOLqUHV FRUUHVSRQGDQWjFKDTXHKDUPRQLHGH)W 2QREWLHQWDLQVLODVROXWLRQJpQpUDOH ) T W = D VLQ ωW + ϕ + +

. . YLEUDWLRQQDWXUHOOH


∞

∑

)Q Q

Q =

ω

) +

∞

∑ ) VLQ Q ω W + ϕ Q

Q

Q =


. .

VLQ Q ω W + ψ Q

− ω

YLEUDWLRQQDWXUHOOH

⎛ π⎞ ⎟ YLEUDWLRQIRUFpH⎜⎜ SpULRGH7 = ω ⎟⎠ ⎝

∞

∑

)Q

VLQ Q ω W + ψ Q

Q ω

Q =

− ω


- 117 -

- 117 -





)LJXUH

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± 3RXU ≤ ω < ω RQD ψ = ODUpSRQVHIRUFpHHVWHQSKDVHDYHFO¶H[FLWDWLRQOH)$'HVW

± 3RXU ≤ ω < ω RQD ψ = ODUpSRQVHIRUFpHHVWHQSKDVHDYHFO¶H[FLWDWLRQOH)$'HVW





YRLVLQGHSRXU ω SHWLWHWLOGHYLHQWWUqVJUDQGORUVTXH ω WHQGYHUV ω UpVRQDQFH OH)$'WHQGYHUV]pURORUVTXH ω GHYLHQWWUqVJUDQG


6RLW 7 =

) W =

) +

∞

∑

)Q VLQ Q ω W + ϕQ



6RLW 7 =

) W =

Q =



YRLVLQGHSRXU ω SHWLWHWLOGHYLHQWWUqVJUDQGORUVTXH ω WHQGYHUV ω UpVRQDQFH

∞

∑ Q =

)Q −

Q

ω

VLQ Q ω W + ψ Q

ω


- 117 -

) +

∞

∑ ) VLQ Q ω W + ϕ Q

Q

Q =



∞

∑ Q =

)Q

−

Q ω

VLQ Q ω W + ψ Q

ω


- 117 -

DYHF HW

ψ Q = ϕQ ψ Q = ϕQ + π

VL VL

ω > Q ω ω < Q ω

DYHF HW


VL VL

ω > Q ω ω < Q ω

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&H SUREOqPH HVW WUqV YDVWH HW FRPSRUWH XQH LQILQLWp GH FDV 1RXV WUDLWHURQV GHX[ H[HPSOHV LPSRUWDQWV


([HPSOHUpSRQVHjXQpFKHORQ


)LJXUH

)LJXUH

± $ W < RQD ) = 0 T + . T = G¶R . T T = T FRV ωW + VLQ ωW DYHF ω = ω 0


± $ W > RQD ) = ) 0 T + . T = ) G¶R T ) T = T FRV ωW + VLQ ωW + − FRV ωW ω .


± $ W = LO\DELHQHQWHQGXFRQWLQXLWpSRXUTW HW T W HWXQVDXWGHGLVFRQWLQXLWppJDOj ) SRXU T W 0


([HPSOHUpSRQVHjXQFUpQHDX


YLEUDWLRQQDWXUHOOH

UpSRQVHIRUFpH

YLEUDWLRQQDWXUHOOH

)LJXUH


VL VL

)LJXUH

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DYHF HW

UpSRQVHIRUFpH

- 118 -

ω > Q ω ω < Q ω

DYHF HW


VL VL

ω > Q ω ω < Q ω









)LJXUH

)LJXUH









YLEUDWLRQQDWXUHOOH

)LJXUH

- 118 -

UpSRQVHIRUFpH

YLEUDWLRQQDWXUHOOH

)LJXUH

- 118 -

UpSRQVHIRUFpH

± SRXUWRQD ) = HW 0 T + . T = G¶R T = T FRV ωW +

± SRXUWRQD ) = HW 0 T + . T = G¶R

. T VLQ ωW DYHF ω = ω 0

T = T FRV ωW +

± SRXU < W < τ RQD ) = ) 0 T + . T = ) G¶R T = T FRV ωW +


± SRXU < W < τ RQD ) = ) 0 T + . T = ) G¶R

T ) VLQ ωW + − FRV ωW ω .

T = T FRV ωW +


± SRXU τ < W RQ D ) = HW 0 T + . T = G¶R HQ pFULYDQW OD FRQWLQXLWp GH T HW T SRXU W = τ


) ) ⎛ ⎞ ⎛ T ⎞ T W = ⎜ T − − FRV ωτ ⎟ FRV ωW + ⎜ + VLQ ωτ ⎟ VLQ ωW . ⎝ ⎠ ⎝ω . ⎠


&DV SDUWLFXOLHU OH FDV OLPLWH R τ → ) → ∞ OH SURGXLW ) τ D\DQW XQH YDOHXU ILQLH GRQQpH,VHQRPPHLPSXOVLRQ2QDGDQVFHFDV


± SRXU W < T W = T FRV ωW +

T VLQ ωW ω


T , VLQ ωW ± SRXU W > T W = T FRV ωW + VLQ ωW + ω ω 0

YLEUDWLRQQDWXUHOOH

T , VLQ ωW ± SRXU W > T W = T FRV ωW + VLQ ωW + ω ω 0

YLEUDWLRQQDWXUHOOH

UpSRQVH LPSXOVLRQQHOOH

'DQVFHFDVLO\DFRQWLQXLWpGHTW HW T W SRXU W = HWXQVDXWGHGLVFRQWLQXLWppJDOj

SRXU T W

T VLQ ωW ω

, 0



SRXU T W

, 0

6758&785(/,1e$,5(',66,3$7,9((;&,7e(

6758&785(/,1e$,5(',66,3$7,9((;&,7e(

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6RQ DPSOLWXGH GpFURvW HW WHQG YHUV ]pUR DX ERXW G¶XQ WHPSV SOXV RX PRLQV FRXUW VXLYDQW O¶LPSRUWDQFHGHO¶DPRUWLVVHPHQWRQO¶DSSHOOHUpSRQVHWUDQVLWRLUH


/D VROXWLRQ SDUWLFXOLqUH GH O¶pTXDWLRQ FRPSOqWH QRWpH T∗ VH QRPPH UpSRQVH IRUFpH RX SHUPDQHQWHRXVWDWLRQQDLUH F¶HVWHOOHTXHQRXVDOORQVpWXGLHUPDLQWHQDQW


- 119 -

- 119 -

± SRXUWRQD ) = HW 0 T + . T = G¶R T = T FRV ωW +

± SRXUWRQD ) = HW 0 T + . T = G¶R


T = T FRV ωW +

± SRXU < W < τ RQD ) = ) 0 T + . T = ) G¶R T = T FRV ωW +


± SRXU < W < τ RQD ) = ) 0 T + . T = ) G¶R


T = T FRV ωW +









T VLQ ωW ω


T , VLQ ωW ± SRXU W > T W = T FRV ωW + VLQ ωW + ω 0 ω

YLEUDWLRQQDWXUHOOH

T , VLQ ωW ± SRXU W > T W = T FRV ωW + VLQ ωW + ω 0 ω

YLEUDWLRQQDWXUHOOH



SRXU T W

T VLQ ωW ω

, 0



SRXU T W

, 0

6758&785(/,1e$,5(',66,3$7,9((;&,7e(

6758&785(/,1e$,5(',66,3$7,9((;&,7e(







- 119 -

- 119 -



/¶pTXDWLRQGXPRXYHPHQWSHXWV¶pFULUH


0 T + & T + . T = ) VLQ ω W

HQSUHQDQWpJDOHj]pURODSKDVH ϕ GHODIRUFHH[FLWDWULFHFHTXLQHUHVWUHLQWSDVODJpQpUDOLWp GHO¶pWXGH2QDGDQVFHFDVSRXUODUpSRQVHIRUFpH

DYHF

) VLQ ω W + ψ .

∗

T =

⎧ ⎪ ⎪ FRV ψ = ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ VLQ ψ = ⎪ ⎪ ⎪⎩

⎛ ω ⎞ ⎜ − ⎟ + ⎛⎜ λ ω ⎞⎟ ⎜ ω ⎟ ω⎠ ⎝ ⎝ ⎠

⎛ω ⎞ − ⎜ ⎟ ⎝ ω ⎠

⎞ ⎛ ⎜ − ω ⎟ ⎜ ω ⎟⎠ ⎝

ω ⎞ ⎛ + ⎜ λ ⎟ ω ⎠ ⎝

ω ⎞ ⎛ + ⎜ λ ⎟ ω ⎠ ⎝

DYHF

5HPDUTXHODUpSRQVHIRUFpHHVWKDUPRQLTXHHWVDIUpTXHQFHHVWFHOOHGHO¶H[FLWDWLRQ 5HPDUTXHO¶DPSOLWXGHGHODUpSRQVHIRUFpHHVWpJDOHjO¶DPSOLWXGHVWDWLTXH

−

ω SRXU ω GLIIpUHQWHVYDOHXUVGHO¶DPRUWLVVHPHQW λ /HPD[LPXPGX)$'HVWG¶DXWDQWSOXVJUDQGTXH O¶DPRUWLVVHPHQW HVW IDLEOH LO GHYLHQW LQILQL GDQV OH FDV QRQ UpHO G¶XQ DPRUWLVVHPHQW ULJRXUHXVHPHQWQXO

LQIpULHXURXpJDOjO¶XQLWpTXHOOHTXHVRLWODYDOHXUGXUDSSRUW

ω ω

) VLQ ω W + ψ .

∗

T =

⎧ ⎪ ⎪ FRV ψ = ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ VLQ ψ = ⎪ ⎪ ⎪⎩

⎛ ω ⎞ ⎜ − ⎟ + ⎛⎜ λ ω ⎞⎟ ⎜ ω ⎟ ω⎠ ⎝ ⎝ ⎠

⎛ω ⎞ − ⎜ ⎟ ⎝ ω ⎠

⎞ ⎛ ⎜ − ω ⎟ ⎜ ω ⎟⎠ ⎝

OH)$' HVW

) VLQ ω W + ψ .

⎛ ω ⎞ ⎜ − ⎟ + ⎛⎜ λ ω ⎞⎟ ⎜ ω ⎟ ω⎠ ⎝ ⎝ ⎠

⎛ω ⎞ − ⎜ ⎟ ⎝ ω ⎠

) PXOWLSOLpH . −

ω ω

OH)$' HVW

ω ⎞ ⎛ + ⎜ λ ⎟ ω ⎠ ⎝

DYHF

5HPDUTXHO¶DPSOLWXGHGHODUpSRQVHIRUFpHHVWpJDOHjO¶DPSOLWXGHVWDWLTXH

) PXOWLSOLpH .

⎤ ⎡⎛ ω ⎞ ω ⎞ ⎛ SDUOH)$'IDFWHXUG¶DPSOLILFDWLRQG\QDPLTXH pJDOj ⎢⎜ − ⎟ + ⎜ λ ⎟ ⎥ ⎢⎜⎝ ω ⎟⎠ ω⎠ ⎥ ⎝ ⎣ ⎦

−


/DILJXUHGRQQHOHVYDOHXUVGHFHIDFWHXUHQIRQFWLRQGXUDSSRUWGHVIUpTXHQFHV

/DSHQWHDXSRLQW RULJLQHFRPPXQHGHVFRXUEHVHVWQXOOH3RXU λ ≥ LQIpULHXURXpJDOjO¶XQLWpTXHOOHTXHVRLWODYDOHXUGXUDSSRUW

ω ω

OH)$' HVW

0 T + & T + . T = ) VLQ ω W


ω ⎞ ⎛ + ⎜ λ ⎟ ω ⎠ ⎝

5HPDUTXHODUpSRQVHIRUFpHHVWKDUPRQLTXHHWVDIUpTXHQFHHVWFHOOHGHO¶H[FLWDWLRQ

- 120 -

ω −λ ω ⎞ ⎛ ⎜ − ω ⎟ ⎜ ω ⎟⎠ ⎝

- 120 -


DYHF

ω ⎞ ⎛ + ⎜ λ ⎟ ω ⎠ ⎝

LQIpULHXURXpJDOjO¶XQLWpTXHOOHTXHVRLWODYDOHXUGXUDSSRUW

0 T + & T + . T = ) VLQ ω W

⎞ ⎛ ⎜ − ω ⎟ ⎜ ω ⎟⎠ ⎝

/DSHQWHDXSRLQW RULJLQHFRPPXQHGHVFRXUEHVHVWQXOOH3RXU λ ≥


⎧ ⎪ ⎪ FRV ψ = ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ VLQ ψ = ⎪ ⎪ ⎪⎩



T =

ω ⎞ ⎛ + ⎜ λ ⎟ ω ⎠ ⎝



5HPDUTXHO¶DPSOLWXGHGHODUpSRQVHIRUFpHHVWpJDOHjO¶DPSOLWXGHVWDWLTXH


∗

ω −λ ω

- 120 -

⎞ ⎛ ⎜ − ω ⎟ ⎜ ω ⎟⎠ ⎝

⎤ ⎡⎛ ω ⎞⎟ ω ⎞ ⎛ ⎢ ⎜ SDUOH)$'IDFWHXUG¶DPSOLILFDWLRQG\QDPLTXH pJDOj − + ⎜ λ ⎟ ⎥ ⎢⎜⎝ ω ⎟⎠ ω⎠ ⎥ ⎝ ⎣ ⎦


/DSHQWHDXSRLQW RULJLQHFRPPXQHGHVFRXUEHVHVWQXOOH3RXU λ ≥

5HPDUTXHODUpSRQVHIRUFpHHVWKDUPRQLTXHHWVDIUpTXHQFHHVWFHOOHGHO¶H[FLWDWLRQ

) PXOWLSOLpH .

⎤ ⎡⎛ ω ⎞⎟ ω ⎞ ⎛ ⎢ ⎜ SDUOH)$'IDFWHXUG¶DPSOLILFDWLRQG\QDPLTXH pJDOj − + ⎜ λ ⎟ ⎥ ⎢⎜⎝ ω ⎟⎠ ω⎠ ⎥ ⎝ ⎣ ⎦

0 T + & T + . T = ) VLQ ω W


ω −λ ω ⎞ ⎛ ⎜ − ω ⎟ ⎜ ω ⎟⎠ ⎝

∗

T =

⎧ ⎪ ⎪ FRV ψ = ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ VLQ ψ = ⎪ ⎪ ⎪⎩

) VLQ ω W + ψ .

⎛ ω ⎞ ⎜ − ⎟ + ⎛⎜ λ ω ⎞⎟ ⎜ ω ⎟ ω⎠ ⎝ ⎝ ⎠

⎛ω ⎞ − ⎜ ⎟ ⎝ ω ⎠

⎞ ⎛ ⎜ − ω ⎟ ⎜ ω ⎟⎠ ⎝

ω ⎞ ⎛ + ⎜ λ ⎟ ω ⎠ ⎝

ω −λ ω ⎞ ⎛ ⎜ − ω ⎟ ⎜ ω ⎟⎠ ⎝

ω ⎞ ⎛ + ⎜ λ ⎟ ω ⎠ ⎝

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) PXOWLSOLpH .

⎤ ⎡⎛ ω ⎞ ω ⎞ ⎛ SDUOH)$'IDFWHXUG¶DPSOLILFDWLRQG\QDPLTXH pJDOj ⎢⎜ − ⎟ + ⎜ λ ⎟ ⎥ ⎢⎜⎝ ω ⎟⎠ ω⎠ ⎥ ⎝ ⎣ ⎦

−



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- 120 -

ω ω

OH)$' HVW

)DFWHXUG¶$PSOLILFDWLRQ'\QDPLTXH


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j O¶H[FLWDWLRQ HVW WRXMRXUV FRPSULVH HQWUH − π HW T∗ HVW HQUHWDUG VXU )W /D ILJXUH ω ⎛ π⎞ GRQQHOHVYDOHXUVGH −ψ HQIRQFWLRQGH SRXUGLIIpUHQWHVYDOHXUVGH λ OHSRLQW ⎜ ⎟ ω ⎝ ⎠

HVW FRPPXQ j WRXWHV OHV FRXUEHV F¶HVW j GLUH TXH TXHO TXH VRLW λ T∗ W HVW HQ TXDGUDWXUH ω UHWDUGVXU)W VL ω = ω RQQRWHUDODUDSLGHYDULDWLRQGH ψ DXYRLVLQDJHGH = ω


- 121 -

- 121 -



)LJXUH

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j O¶H[FLWDWLRQ HVW WRXMRXUV FRPSULVH HQWUH − π HW T∗ HVW HQ UHWDUG VXU )W /D ILJXUH ω ⎛ π⎞ GRQQHOHVYDOHXUVGH −ψ HQIRQFWLRQGH SRXUGLIIpUHQWHVYDOHXUVGH λ OHSRLQW ⎜ ⎟ ω ⎝ ⎠

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)$' PD[ =

λ − λ

SRXU

ω = − λ ω

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G ∂ (F − ∂ (F = 4 L G W ∂ TL ∂ TL

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⎛ P / ⎟⎞ ⎛ ⎞ ω = ⎜ N − P / ω ⎟ − ⎜⎜ N − ⎟ ⎝ ⎠ ⎝ ⎠ N N HW ω = ω = P/ P/


⎡ ⎤ ⎧ ⎫⎪ ⎧⎪⎫⎪ ⎥ ⎬ HW {3 }= ⎨ ⎬ G¶R [3 ] = ⎢ ⎪⎩− ⎪⎭ ⎪⎩⎪⎭ ⎢⎣− ⎥⎦

⎡ ⎤ ⎧ ⎫⎪ ⎧⎪⎫⎪ ⎥ ⎬ HW {3 }= ⎨ ⎬ G¶R [3 ] = ⎢ ⎪⎩− ⎪⎭ ⎪⎩⎪⎭ ⎢⎣− ⎥⎦

{3}= ⎪⎨

{3}= ⎪⎨

/HVPRGHVSURSUHVFRUUHVSRQGDQWVVRQWGRQQpVSDU


⎧⎪θ ⎫⎪ ⎧⎪ ⎫⎪ ⎧⎪θ ⎫⎪ ⎧⎪⎫⎪ ⎨ ⎬ = ⎨ ⎬ FRV ω W + ϕ HW ⎨ ⎬ = ⎨ ⎬ FRV ω W + ϕ ⎪⎩θ ⎪⎭ ⎪⎩− ⎪⎭ ⎪⎩θ ⎪⎭ ⎪⎩⎪⎭ VRLW θ = − θ θ = HW θ = θ = − θ


)LJXUH

)LJXUH

- 151 -

- 151 -








[ θ

]

(F =

P/

(S =

N θ + θ θ + θ

− θ θ + θ

[

]


[0 ] = P /

GHUDFLQHV

P / θ − θ θ + θ

(S =

N θ + θ θ + θ

[0 ] = P /

[

]

⎡ − ⎤ ⎡ ⎤ ⎢ ⎥ HW [. ] = N ⎢ ⎥ ⎣⎢− ⎦⎥ ⎣⎢ ⎥⎦


GHUDFLQHV


⎛ P / ⎟⎞ ⎛ ⎞ ω = ⎜ N − P / ω ⎟ − ⎜⎜ N − ⎟ ⎝ ⎠ ⎝ ⎠ N N HW ω = ω = P/ P/


⎡ ⎤ ⎧ ⎫⎪ ⎧⎪⎫⎪ ⎥ ⎬ HW {3 }= ⎨ ⎬ G¶R [3 ] = ⎢ ⎪⎩− ⎪⎭ ⎪⎩⎪⎭ ⎣⎢− ⎦⎥

⎡ ⎤ ⎧ ⎫⎪ ⎧⎪⎫⎪ ⎥ ⎬ HW {3 }= ⎨ ⎬ G¶R [3 ] = ⎢ ⎪⎩− ⎪⎭ ⎪⎩⎪⎭ ⎣⎢− ⎦⎥

{3}= ⎪⎨

{3}= ⎪⎨




- 151 -

]

(F =


⎛ P / ⎟⎞ ⎛ ⎞ ω = ⎜ N − P / ω ⎟ − ⎜⎜ N − ⎟ ⎝ ⎠ ⎝ ⎠ N N HW ω = ω = P/ P/

)LJXUH

[

⎡ − ⎤ ⎡ ⎤ ⎢ ⎥ HW [. ] = N ⎢ ⎥ ⎣⎢− ⎦⎥ ⎣⎢ ⎥⎦




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- 151 -

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⎧⎪θ ⎫⎪ ⎧⎪ ⎫⎪ ⎨ ⎬ = D ⎨ ⎬ FRV ω W + ϕ + D ⎪⎩− ⎪⎭ ⎪⎩θ ⎪⎭

⎧⎪θ ⎫⎪ ⎧⎪ ⎫⎪ ⎨ ⎬ = D ⎨ ⎬ FRV ω W + ϕ + D ⎪⎩− ⎪⎭ ⎪⎩θ ⎪⎭

⎧⎪⎫⎪ ⎨ ⎬ FRV ω W + ϕ ⎪⎩⎪⎭

⎧⎪⎫⎪ ⎨ ⎬ FRV ω W + ϕ ⎪⎩⎪⎭

D D ϕ ϕ VRQWGHVFRQVWDQWHVG¶LQWpJUDWLRQ


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[0]{T}+ [&]{T }+ [. ]{T}= {}

RXHQSUpPXOWLSOLDQWSDU [0 ]−


{T}+ [0]− [&]{T }+ [0 ]− [. ]{T}= {}

[0]{T}+ [&]{T }+ [. ]{T}= {}

{T}+ [0]− [&]{T }+ [0 ]− [. ]{T}= {}



&¶HVWOHFDVVLPSOHDVVH]UpSDQGXROHVWURLVPDWULFHV [0 ] [&] [. ]VRQWGLDJRQDOLVDEOHV SDUOHPrPHFKDQJHPHQWGHEDVHGHPDWULFH [3]2QDDORUVOHVIRUPXOHV


{T}= [3]{T} {T }= [3]{T } {T}= [3]{T}HW

{T}= [3]{T} {T }= [3]{T } {T}= [3]{T}HW

⎤ ⎡ & ⎥ ⎢ [&] [3] = ⎢ & 2 ⎥ ⎢ % ⎥ ⎢⎣ 2 & Q ⎥⎦

[& ]= [ 3 W ]

/D IRUPH TXDGUDWLTXH 5 =

W {T } [&]{T } pWDQW SRVLWLYH PDLV SDV QpFHVVDLUHPHQW GpILQLH

SRVLWLYHRQD &L ≥

/¶pTXDWLRQ pTXLYDXWDORUVjQpTXDWLRQVGpFRXSOpHVHWGRQFLQWpJUDEOHVVpSDUpPHQW PL TL + &L T L + N L TL = L = ! Q

⎤ ⎡ & ⎥ ⎢ [&] [3] = ⎢ & 2 ⎥ ⎢ % ⎥ ⎢⎣ 2 & Q ⎥⎦

[& ]= [ 3 W ]



SRVLWLYHRQD &L ≥

/¶pTXDWLRQ pTXLYDXWDORUVjQpTXDWLRQVGpFRXSOpHVHWGRQFLQWpJUDEOHVVpSDUpPHQW

PL TL + &L T L + N L TL = L = ! Q

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'DQV OD EDVH QRUPDOH EDVH SURSUH FRPPXQH GH [0 ]− [&] HW [0 ]− [. ] OD VWUXFWXUH j Q ''/ SURSRVpH SHXW rWUH FRQVLGpUpH FRPPH OD MX[WDSRVLWLRQ GH Q VWUXFWXUHV j XQ ''/ LQGpSHQGDQWHVGpFRXSOpHVWUDLWDEOHVVpSDUpPHQW


- 152 -

- 152 -



⎧⎪θ ⎫⎪ ⎧⎪ ⎫⎪ ⎨ ⎬ = D ⎨ ⎬ FRV ω W + ϕ + D ⎪⎩− ⎪⎭ ⎪⎩θ ⎪⎭

⎧⎪θ ⎫⎪ ⎧⎪ ⎫⎪ ⎨ ⎬ = D ⎨ ⎬ FRV ω W + ϕ + D ⎪⎩− ⎪⎭ ⎪⎩θ ⎪⎭

⎧⎪⎫⎪ ⎨ ⎬ FRV ω W + ϕ ⎪⎩⎪⎭

⎧⎪⎫⎪ ⎨ ⎬ FRV ω W + ϕ ⎪⎩⎪⎭



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[0]{T}+ [&]{T }+ [. ]{T}= {}


[0]{T}+ [&]{T }+ [. ]{T}= {}


{T}+ [0]− [&]{T }+ [0 ]− [. ]{T}= {}

{T}+ [0]− [&]{T }+ [0 ]− [. ]{T}= {}





{T}= [3]{T} {T }= [3]{T } {T}= [3]{T}HW

{T}= [3]{T} {T }= [3]{T } {T}= [3]{T}HW

⎤ ⎡ & ⎥ ⎢ [&] [3] = ⎢ & 2 ⎥ ⎢ % ⎥ ⎢⎣ 2 & Q ⎥⎦

[& ]= [ 3 ]



SRVLWLYHRQD &L ≥

W


PL TL + &L T L + N L TL = L = ! Q

⎤ ⎡ & ⎥ ⎢ [&] [3] = ⎢ & 2 ⎥ ⎢ % ⎥ ⎢⎣ 2 & Q ⎥⎦

[& ]= [ 3 ]



SRVLWLYHRQD &L ≥

W


PL TL + &L T L + N L TL = L = ! Q





- 152 -

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{T}= {3}T W + {3 }T W + !! + {3Q }TQ W

([HPSOHUHSUHQRQVOHHH[HPSOHGXDYHF . = . = . HW 0 = 0 = 0 /¶pTXDWLRQ V¶pFULWLFL ⎡0 2 ⎤ ⎧⎪ [ ⎫⎪ ⎡ & − &⎤ ⎧⎪ [ ⎫⎪ ⎡. 2⎤ ⎧⎪ [ ⎫⎪ ⎧⎪⎫⎪ ⎢ ⎥ ⎨ ⎬+ ⎢ ⎥ ⎨ ⎬+ ⎢ ⎥ ⎨ ⎬ = ⎨ ⎬ ⎢⎣ 2 0 ⎥⎦ ⎪⎩[ ⎪⎭ ⎢⎣− & &⎥⎦ ⎪⎩[ ⎪⎭ ⎢⎣ 2 . ⎥⎦ ⎪⎩[ ⎪⎭ ⎪⎩⎪⎭ /HVGHX[SXOVDWLRQVQDWXUHOOHVVRQWpJDOHV ω = ω =

. /HVYHFWHXUVSURSUHVFRPPXQV 0


{T}= {3}T W + {3 }T W + !! + {3Q }TQ W



DX[WURLVPDWULFHVVRQW ⎡ ⎤ ⎧⎫ ⎧ ⎫ {3}= ⎪⎨ ⎪⎬ HW {3 }= ⎪⎨ ⎪⎬ G¶RODPDWULFHGHSDVVDJH [3] = ⎢ ⎥ ⎪⎩⎪⎭ ⎪⎩− ⎪⎭ ⎢⎣ − ⎥⎦


/HVpTXDWLRQV V¶pFULYHQWLFL 0 T + . T = HW 0 T + & T + . T =


6XSSRVRQVXQDPRUWLVVHPHQWIDLEOH & < .0


2QDDORUV

T W = D FRV ω W + ϕ

2QDDORUV


(W

T W = D H −λωW FRV Ω W + ϕ

(W


DYHF

ω=

. λ = 0

& Ω = ω − λ .0

DYHF

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ω=

. λ = 0

& Ω = ω − λ .0


⎧⎪ [ = D FRV ωW + ϕ + D H −λωW FRV Ω W + ϕ ⎨ ⎪⎩[ = D FRV ωW + ϕ − D H − λωW FRV Ω W + ϕ


/H SUHPLHU PRGH SURSUH GH YLEUDWLRQ FRUUHVSRQG j D = OHV GHX[ FKDULRWV RQW GHV PRXYHPHQWVKDUPRQLTXHVLGHQWLTXHVFRPPHV¶LOVFRQVWLWXDLHQWXQVHXOVROLGH/¶DPRUWLVVHXU HVW LQRSpUDQW /H VHFRQG PRGH SURSUH FRUUHVSRQG j D = OHV GHX[ FKDULRWV RQW GHV PRXYHPHQWVV\PpWULTXHV [ = − [ ∀W


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2Q QH SHXW SDV WURXYHU XQ FKDQJHPHQW GH EDVH TXL GLDJRQDOLVH OHV WURLV PDWULFHV [0 ] [&] [. ]


- 153 -

- 153 -



{T}= {3}T W + {3 }T W + !! + {3Q }TQ W



{T}= {3}T W + {3 }T W + !! + {3Q }TQ W









2QDDORUV


2QDDORUV


(W


(W


DYHF

ω=

. λ = 0

& Ω = ω − λ .0


DYHF

ω=

. λ = 0

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±&DVJpQpUDO

±&DVJpQpUDO



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{T}= ∑ $ α {T α W }

{T}= ∑ $ α {T α W }

Q

Q

α =

α =

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LQLWLDOHV T L HW T L

LQLWLDOHV T L HW T L

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⎧{T }⎫ ⎡ [2] [,] ⎤ ⎧{T}⎫ ⎥⎪ ⎪ ⎪ ⎪ ⎢ {;}= ⎨ ⎬ = ⎢ ⎥ ⎨ ⎬ = [$ ]{;} ⎪{T}⎪ ⎢ ⎪ ⎪ − − ⎩ ⎭ ⎣ − [0 ] [. ] − [0 ] [&]⎥⎦ ⎩{T }⎭

⎧{T }⎫ ⎡ [2] [,] ⎤ ⎧{T}⎫ ⎥⎪ ⎪ ⎪ ⎪ ⎢ {;}= ⎨ ⎬ = ⎢ ⎥ ⎨ ⎬ = [$ ]{;} ⎪{T}⎪ ⎢ ⎪ ⎪ − − ⎩ ⎭ ⎣ − [0 ] [. ] − [0 ] [&]⎥⎦ ⎩{T }⎭

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{;}= [3]{;}

{; }= [3]{; } [$ ]= [3]− [$][3]

{}

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{;}= [3]{;}

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[0]{T}+ [. ]{T}= {) W }

{}


[0]{T}+ [. ]{T}= {) W }

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Q

{T}= ∑ $ α {T α W }

Q

{T}= ∑ $ α {T α W }

α =

α =

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⎧{T }⎫ ⎡ [2] [,] ⎤ ⎧{T}⎫ ⎥⎪ ⎪ ⎪ ⎪ ⎢ {;}= ⎨ ⎬ = ⎢ ⎥ ⎨ ⎬ = [$ ]{;} ⎪{T}⎪ ⎢ ⎪ ⎪ − − ⎩ ⎭ ⎣ − [0 ] [. ] − [0 ] [&]⎥⎦ ⎩{T }⎭

⎧{T }⎫ ⎡ [2] [,] ⎤ ⎧{T}⎫ ⎥⎪ ⎪ ⎪ ⎪ ⎢ {;}= ⎨ ⎬ = ⎢ ⎥ ⎨ ⎬ = [$ ]{;} ⎪{T}⎪ ⎢ ⎪ ⎪ − − ⎩ ⎭ ⎣ − [0 ] [. ] − [0 ] [&]⎥⎦ ⎩{T }⎭


{;}= [3]{;}

{; }= [3]{; } [$ ]= [3]− [$][3]

{}


{;}= [3]{;}

{; }= [3]{; } [$ ]= [3]− [$][3]


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[0]{T}+ [. ]{T}= {) W }

{}


[0]{T}+ [. ]{T}= {) W }



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{ }

[0]{T}+ [. ]{T}= [3 W ]{)}= {)}

RX

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PL TL + N L TL = )L W

{ }

[0]{T}+ [. ]{T}= [3 W ]{)}= {)}

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RX


TL + ωL TL = )L W PL

RX TL + ωL TL = )L W PL

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{ }

RX

[0]{T}+ [. ]{T}= [3 W ]{)}= {)}

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−

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[0]{T}+ [. ]{T}= [3 W ]{)}= {)}

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RX


TL + ωL TL = )L W PL

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[0 ]{T}+ [& ]{T }+ [. ]{T}= {)}= [3 W ]{)}




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⎧ ⎧X [ W ⎪ω[ = ⎪ ⎪ ⎪ ⎪⎪ 3R3 = 8 [ W ⎨Y [ W ⎨ω\ = − ⎪ ⎪ ⎪ ⎪ ⎪⎩Z [ W ⎪ω] = − ⎩

⎧ ⎧ ∂X ⎪γ \] = ⎪ε [ = ∂ [ ⎪ ⎪ ⎪ ⎪ ε = ⎨γ ][ = ⎨ \ ⎪ ⎪ ⎪ ⎪ ⎪γ [\ = ⎪ε ] = ⎩ ⎩

∂Z ∂[ ∂Y ∂[

∂Z ∂X HWH = GLY 8 = ε [ = ∂[ ∂[ ∂Y ∂[

⎧ ⎧X [ W ⎪ω[ = ⎪ ⎪ ⎪ ⎪⎪ 3R3 = 8 [ W ⎨Y [ W ⎨ω\ = − ⎪ ⎪ ⎪ ⎪ ⎪⎩Z [ W ⎪ω] = − ⎩

⎧ ⎧ ∂X ⎪γ \] = ⎪ε [ = ∂ [ ⎪ ⎪ ⎪ ⎪ ε = ⎨γ ][ = ⎨ \ ⎪ ⎪ ⎪ ⎪ ⎪γ [\ = ⎪ε ] = ⎩ ⎩

∂Z ∂[ ∂Y ∂[

∂Z ∂X HWH = GLY 8 = ε [ = ∂[ ∂[ ∂Y ∂[

- 162 -

- 162 -



⎛& ⎞ λ + µ − ν = DYHF ⎜⎜ / ⎟⎟ = & µ − ν ⎝ 7⎠

⎛& ⎞ λ + µ − ν = DYHF ⎜⎜ / ⎟⎟ = & µ − ν ⎝ 7⎠









±21'(6e/$67,48(63/$1(6

±21'(6e/$67,48(63/$1(6



⎧ ⎧X [ W ⎪ω[ = ⎪ ⎪ ⎪⎪ ⎪ 3R3 = 8 [ W ⎨Y [ W ⎨ω\ = − ⎪ ⎪ ⎪ ⎪ ⎪⎩Z [ W ⎪ω] = − ⎩

⎧ ⎧ ∂X ⎪γ \] = ⎪ε [ = ∂ [ ⎪ ⎪ ⎪ ⎪ ⎨ε \ = ⎨γ ][ = ⎪ ⎪ ⎪ ⎪ ⎪γ [\ = ⎪ε ] = ⎩ ⎩ - 162 -

∂Z ∂[ ∂Y ∂[

∂Z ∂X HWH = GLY 8 = ε [ = ∂[ ∂[ ∂Y ∂[

⎧ ⎧X [ W ⎪ω[ = ⎪ ⎪ ⎪⎪ ⎪ 3R3 = 8 [ W ⎨Y [ W ⎨ω\ = − ⎪ ⎪ ⎪ ⎪ ⎪⎩Z [ W ⎪ω] = − ⎩

⎧ ⎧ ∂X ⎪γ \] = ⎪ε [ = ∂ [ ⎪ ⎪ ⎪ ⎪ ⎨ε \ = ⎨γ ][ = ⎪ ⎪ ⎪ ⎪ ⎪γ [\ = ⎪ε ] = ⎩ ⎩ - 162 -

∂Z ∂[ ∂Y ∂[

∂Z ∂X HWH = GLY 8 = ε [ = ∂[ ∂[ ∂Y ∂[

G /HVVXUIDFHVG¶RQGHVVRQWGHVSODQVRUWKRJRQDX[jR[/DSDUWLHLUURWDWLRQQHOOHHVW 8 / = X L G G G G G L XQLWDLUHGHR[ ODSDUWLHVROpQRwGDOHHVW 8 7 = Y M + Z N M HW N XQLWDLUHVGHR\HWR]


([HPSOH XQ PLOLHX pODVWLTXH RFFXSH WRXW O¶HVSDFH FRPSULV HQWUH OHV SODQV [ = HW [ = K > OH SODQ OLPLWH [ = HVW IL[H HW JDOLOpHQ OH SODQ OLPLWH [ = K HVW OLEUH


YHFWHXUFRQWUDLQWH & [ QXO


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λ + µ ( − ν ∂ X ∂ X DYHF & = = = ρ ρ + ν − ν ∂[ & ∂W

3RXUODUpVRXGUHFKHUFKRQVG¶DERUGGHVVROXWLRQVSDUWLFXOLqUHVGHODIRUPH X = I [ ⋅ J W

F

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I ′′ J = I J

λ + µ ( − ν ∂ X ∂ X DYHF & = = = ρ ρ + ν − ν ∂[ & ∂W


F

2QGRLWDYRLU

I ′′ J = I J

/DYDOHXUFRPPXQHGHFHVGHX[UDSSRUWVHVWGRQFXQHFRQVWDQWHQpFHVVDLUHPHQWQpJDWLYHHQ UDLVRQGXFDUDFWqUHRVFLOODWRLUHGXPRXYHPHQWFRQVWDQWHTXHQRXVQRWRQV − ω


2QDDORUVOHVpTXDWLRQVGLIIpUHQWLHOOHVGpFRXSOpHV


I ′′ [ + F − ω I [ = HW J W + ω ⋅ J W =

I ′′ [ + F − ω I [ = HW J W + ω ⋅ J W =

/HVVROXWLRQVJpQpUDOHVV¶pFULYHQW I [ = α FRV

/HVVROXWLRQVJpQpUDOHVV¶pFULYHQW

ω[ ω[ + β VLQ HW J [ = $ FRV ω W + % VLQ ω W F F

I [ = α FRV


([SULPRQVOHVFRQGLWLRQVDX[OLPLWHV


± HQ [ = X = ∀ W ⇒ I = ⇒ α =

± HQ [ = X = ∀ W ⇒ I = ⇒ α =

- 163 -

- 163 -







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)LJXUH



λ + µ ( − ν ∂ X ∂ X DYHF & = = = ρ ρ + ν − ν ∂[ & ∂W


F

2QGRLWDYRLU

I ′′ J = I J

λ + µ ( − ν ∂ X ∂ X DYHF & = = = ρ ρ + ν − ν ∂[ & ∂W


F

2QGRLWDYRLU

I ′′ J = I J





I ′′ [ + F − ω I [ = HW J W + ω ⋅ J W = /HVVROXWLRQVJpQpUDOHVV¶pFULYHQW I [ = α FRV

I ′′ [ + F − ω I [ = HW J W + ω ⋅ J W = /HVVROXWLRQVJpQpUDOHVV¶pFULYHQW


I [ = α FRV




± HQ [ = X = ∀ W ⇒ I = ⇒ α =

± HQ [ = X = ∀ W ⇒ I = ⇒ α =

- 163 -

- 163 -

⎛ ∂X ⎞ βω ωK ⎟⎟ = ∀ W ⇒ I ′K = ⇒ FRV = ± HQ [ = K σ [ = ∀ W ⇒ ε [ = ⎜⎜ F F ∂ [ ⎝ ⎠K


⎛ βω ⎞ ⎛ ωK ⎞ (OLPLQRQVODVROXWLRQ ⎜ = ⎟ TXLFRUUHVSRQGDXUHSRVRQGRLWGRQFDYRLU FRV ⎜ ⎟ = F ⎝ ⎠ ⎝ F ⎠ Q + F HW ω HVWXQQRPEUHGHODVXLWH ωQ = π Q ∈ 1 K

⎛ βω ⎞ ⎛ ωK ⎞ (OLPLQRQVODVROXWLRQ ⎜ = ⎟ TXLFRUUHVSRQGDXUHSRVRQGRLWGRQFDYRLU FRV ⎜ ⎟ = F ⎝ ⎠ ⎝ F ⎠ Q + F HW ω HVWXQQRPEUHGHODVXLWH ωQ = π Q ∈ 1 K

2QSHXWSUHQGUH β = HWpFULUHODVROXWLRQpOpPHQWDLUHGHUDQJQGLWHKDUPRQLTXHGHUDQJQ


Q + F W Q + F W ⎞ ⎛ Q + [ ⎞ ⎛ X Q [ W = VLQ ⎜ π ⎟ ⎜ $ Q FRV π + %Q VLQ π ⎟ K⎠⎝ K K⎠ ⎝


&HVKDUPRQLTXHVFRQVWLWXHQWOHVPRGHVSURSUHVGHYLEUDWLRQGXPLOLHXVROLGHOHV ωQ HQVRQW ω π OHVSpULRGHVSURSUHVOHV Q OHVIUpTXHQFHVSURSUHV OHVSXOVDWLRQVSURSUHVOHV 7Q = π ωQ


6XSSRVRQV TXH O¶RVFLOODWLRQ DLW OLHX VXLYDQW OH PRGH SURSUH QXPpUR Q GRQF TXH X = X Q = I Q [ ⋅ J Q W


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/HV SODQV PDWpULHOV SDUDOOqOHV j \R] GRQW OHV DEVFLVVHV [ P =

P K DQQXOHQW Q +

Q + [ π UHVWHQWIL[HVLOVFRQVWLWXHQWOHVQ°XGVGHODYLEUDWLRQ/¶RQGH {X Q } K HVWDLQVLXQHRQGHVWDWLRQQDLUH I Q [ = VLQ

S + K UHQGHQW I Q [ Q + H[WUpPDOHVLSSDLU±VLSLPSDLU VRQWFHX[TXLYLEUHQWDYHFO¶DPSOLWXGHPD[LPDOHLOV FRQVWLWXHQW OHV YHQWUHV GH O¶RQGH VWDWLRQQDLUH /HV Q°XGV VRQW UpJXOLqUHPHQW HVSDFpV GH ⎛ K ⎞ ⎜ ⎟ LOHQHVWGHPrPHGHVYHQWUHV ⎝ Q + ⎠

/HV SODQV PDWpULHOV SDUDOOqOHV j \R] GRQW OHV DEVFLVVHV [ S =

/D©SpULRGHVSDWLDOHª λ Q =

UHPDUTXHTXHOHUDSSRUW

K SpULRGHGHODIRQFWLRQ I Q [ HVWODORQJXHXUG¶RQGHRQ Q +

λQ HVWpJDOj&FpOpULWpGHO¶RQGHHWQHGpSHQGSDVGHQ 7Q


P K DQQXOHQW Q +








- 164 -

- 164 -



⎛ βω ⎞ ⎛ ωK ⎞ (OLPLQRQVODVROXWLRQ ⎜ = ⎟ TXLFRUUHVSRQGDXUHSRVRQGRLWGRQFDYRLU FRV ⎜ ⎟ = ⎝ F ⎠ ⎝ F ⎠ Q + F HW ω HVWXQQRPEUHGHODVXLWH ωQ = π Q ∈ 1 K

⎛ βω ⎞ ⎛ ωK ⎞ (OLPLQRQVODVROXWLRQ ⎜ = ⎟ TXLFRUUHVSRQGDXUHSRVRQGRLWGRQFDYRLU FRV ⎜ ⎟ = ⎝ F ⎠ ⎝ F ⎠ Q + F HW ω HVWXQQRPEUHGHODVXLWH ωQ = π Q ∈ 1 K












P K DQQXOHQW Q +








- 164 -


P K DQQXOHQW Q +








- 164 -

/DVROXWLRQJpQpUDOHVHSUpVHQWHFRPPHODVRPPHG¶XQHGRXEOHVpULHGH)RXULHUSDUUDSSRUW j[HWW ∞

X [ W =

∑

∞

X Q [ W =

Q =

∑

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IQ [ ⋅ J Q W

X [ W =

Q =

∑

∞

X Q [ W =

Q =

∑ I [ ⋅ J W Q

Q

Q =

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⎛ ∂ X ψ [ = ⎜⎜ ⎝ ∂ W

⎛ ∂ X ψ [ = ⎜⎜ ⎝ ∂ W

⎞ ⎟⎟ ⎠W= ∞ ⎧ ⎛ Q + [ ⎞ ⎪X [ R = π ⎟ = ϕ [ $ Q VLQ ⎜ K⎠ ⎝ ⎪ = Q ⎪ ⎨ ∞ ⎪⎛ ⎞ Q + F ⎛ Q + [ ⎞ ⎪⎜ ∂ X ⎟ π VLQ ⎜ π ⎟ = ψ [ K⎠ K ⎪⎜⎝ ∂ W ⎟⎠ W = ⎝ Q = ⎩

∞ ⎧ ⎛ Q + [ ⎞ ⎪X [ R = π ⎟ = ϕ [ $ Q VLQ ⎜ K⎠ ⎝ ⎪ = Q ⎪ ⎨ ∞ ⎪⎛ ⎞ Q + F ⎛ Q + [ ⎞ ⎪⎜ ∂ X ⎟ π VLQ ⎜ π ⎟ = ψ [ K⎠ K ⎪⎜⎝ ∂ W ⎟⎠ W = ⎝ Q = ⎩

∑

∑

∑

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⎞ ⎟⎟ ⎠W=

∑


K ⎧ ⎪$ Q = K ϕ [ ⋅ VLQ ⎛⎜ Q + π [ ⎞⎟ ⋅ G [ ⎪ K⎠ ⎝ ⎪ ⎨ K ⎪ ⎛ Q + [ ⎞ ⎪% Q = ψ [ ⋅ VLQ ⎜ π ⎟⋅G[ Q + π & K⎠ ⎪ ⎝ ⎩


3DUH[HPSOHVLj W = RQDEDQGRQQHOHVROLGHVDQVYLWHVVHLQLWLDOH ψ = DYHFOHFKDPSGH X [ GpSODFHPHQWVWDWLTXH ϕ [ = X RQWURXYH %Q = HW $ Q = − Q K Q + π


/HV UpVXOWDWV TXH QRXV YHQRQV G¶REWHQLU SRXU O¶RQGH ORQJLWXGLQDOH {X [ W } VH WUDQVSRVHQW LPPpGLDWHPHQW DX[ RQGHV WUDQVYHUVDOHV {Y [ W } HW {Z [ W } HQ SUHQDQW FRPPH FpOpULWp &7 DXOLHXGH & /


±21'(6&
±21'(6&
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- 165 -

- 165 -

/DVROXWLRQJpQpUDOHVHSUpVHQWHFRPPHODVRPPHG¶XQHGRXEOHVpULHGH)RXULHUSDUUDSSRUW j[HWW

/DVROXWLRQJpQpUDOHVHSUpVHQWHFRPPHODVRPPHG¶XQHGRXEOHVpULHGH)RXULHUSDUUDSSRUW j[HWW

∫

∫

∫

∞

X [ W =

∑

∞

X Q [ W =

Q =

∑

∫

∞

IQ [ ⋅ J Q W

X [ W =

Q =

∑

∞

X Q [ W =

Q =

∑ I [ ⋅ J W Q

Q

Q =



⎛ ∂ X ψ [ = ⎜⎜ ⎝ ∂ W

⎛ ∂ X ψ [ = ⎜⎜ ⎝ ∂ W

⎞ ⎟⎟ ⎠W= ∞ ⎧ ⎛ Q + [ ⎞ ⎪X [ R = π ⎟ = ϕ [ $ Q VLQ ⎜ K⎠ ⎝ ⎪ = Q ⎪ ⎨ ∞ ⎪⎛ ⎞ Q + F ⎛ Q + [ ⎞ ⎪⎜ ∂ X ⎟ π VLQ ⎜ π ⎟ = ψ [ ⎜ ⎟ K⎠ K ⎪⎝ ∂ W ⎠ W = ⎝ Q = ⎩

∑

∑


⎞ ⎟⎟ ⎠W= ∞ ⎧ ⎛ Q + [ ⎞ ⎪X [ R = π ⎟ = ϕ [ $ Q VLQ ⎜ K⎠ ⎝ ⎪ = Q ⎪ ⎨ ∞ ⎪⎛ ⎞ Q + F ⎛ Q + [ ⎞ ⎪⎜ ∂ X ⎟ π VLQ ⎜ π ⎟ = ψ [ ⎜ ⎟ K⎠ K ⎪⎝ ∂ W ⎠ W = ⎝ Q = ⎩

∑

∑








±21'(6&
±21'(6&


- 165 -

- 165 -

∫

∫

∫

∫

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⎧ ⎧ ∂X ⎪γ θ] = ⎪ε U = ∂ U ⎪ ⎪ ⎪ X ⎪ ⎨γ ]U = ⎨εθ = U ⎪ ⎪ ⎪ ⎪ ⎪ε ] = ⎪γ Uθ = ⎩ ⎩

'¶R ⎧ ⎪ωU = ⎪ ⎪ ∂Z ∂Z ⎨ωθ = − ∂U ∂U ⎪ ⎪ ⎛∂Y Y⎞ ⎛∂Y ⎜⎜ + − ⎟⎟ ⎪ω] = ⎜⎜ ⎝ ∂U ∂ U U ⎝ ⎠ ⎩

H = GLY 8 = ε U + εθ + ε ] =

HW

Y⎞ ⎟ U ⎟⎠

∂X X + ∂U U


⎧ ⎪ωU = ⎪ ⎪ ∂Z ∂Z ⎨ωθ = − ∂U ∂U ⎪ ⎪ ⎛∂Y Y⎞ ⎛∂Y ⎜⎜ + − ⎟⎟ ⎪ω] = ⎜⎜ ⎝ ∂U ∂ U U ⎝ ⎠ ⎩

H = GLY 8 = ε U + εθ + ε ] =

HW

Y⎞ ⎟ U ⎟⎠

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⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

∂ X ∂ U

+

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∂ ∂U

⎛∂X X ⎞ ∂ X ⎜⎜ + ⎟⎟ = ⎝ ∂ U U ⎠ &/ ∂ W

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RXELHQ RXHQFRUH

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∂ ∂U

⎛∂Y ⎜⎜ + ⎝ ∂U

∂ ∂U

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RXELHQ RXHQFRUH

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

∂ Y ∂ Y Y ∂ Y + − = ⋅ ∂ U U ∂ U U &7 ∂ W Y ⎞ ∂ Y ⎟= U ⎟⎠ &7 ∂ W

∂ ∂U

⎛∂Y ⎜⎜ + ⎝ ∂U

∂ ∂U

⎛ ∂ ⎞ ∂ Y ⎜⎜ U Y ⎟⎟ = ⎝ U ∂U ⎠ &7 ∂ W

- 166 -

'¶R


RXHQFRUH

- 166 -

'¶R ⎧ ⎪ωU = ⎪ ⎪ ∂Z ∂Z ⎨ωθ = − ∂U ∂U ⎪ ⎪ ⎛∂Y Y⎞ ⎛∂Y ⎜⎜ + − ⎟⎟ ⎪ω] = ⎜⎜ ⎝ ∂U ⎝ ∂U U ⎠ ⎩

H = GLY 8 = ε U + εθ + ε ] =

HW

RXELHQ

Y⎞ ⎟ U ⎟⎠

∂X X + ∂U U


⎧ ⎪ωU = ⎪ ⎪ ∂Z ∂Z ⎨ωθ = − ∂U ∂U ⎪ ⎪ ⎛∂Y Y⎞ ⎛∂Y ⎜⎜ + − ⎟⎟ ⎪ω] = ⎜⎜ ⎝ ∂U ⎝ ∂U U ⎠ ⎩

H = GLY 8 = ε U + εθ + ε ] =

HW

Y⎞ ⎟ U ⎟⎠

∂X X + ∂U U







⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

∂ X ∂ U

+

∂X X ∂ X − = ⋅ U ∂U U &/ ∂ W

∂ ∂U

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∂ ∂U

⎞ ⎛ ∂ ∂ X ⎜⎜ U X ⎟⎟ = ⎠ &/ ∂ W ⎝ U ∂U

RXELHQ RXHQFRUH

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

∂ X ∂ U

+

∂X X ∂ X − = ⋅ U ∂U U &/ ∂ W

∂ ∂U

⎛∂X X ⎞ ∂ X ⎜⎜ + ⎟⎟ = ⎝ ∂ U U ⎠ &/ ∂ W

∂ ∂U

⎞ ⎛ ∂ ∂ X ⎜⎜ U X ⎟⎟ = ⎠ &/ ∂ W ⎝ U ∂U

RXELHQ RXHQFRUH





⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

∂ Y ∂ Y Y ∂ Y + − = ⋅ ∂ U U ∂ U U &7 ∂ W Y ⎞ ∂ Y ⎟= U ⎟⎠ &7 ∂ W

∂ ∂U

⎛∂Y ⎜⎜ + ⎝ ∂U

∂ ∂U

⎛ ∂ ⎞ ∂ Y ⎜⎜ U ∂ U U Y ⎟⎟ = ⎝ ⎠ &7 ∂ W

- 166 -

RXELHQ RXHQFRUH

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

∂ Y ∂ Y Y ∂ Y + − = ⋅ ∂ U U ∂ U U &7 ∂ W Y ⎞ ∂ Y ⎟= U ⎟⎠ &7 ∂ W

∂ ∂U

⎛∂Y ⎜⎜ + ⎝ ∂U

∂ ∂U

⎛ ∂ ⎞ ∂ Y ⎜⎜ U ∂ U U Y ⎟⎟ = ⎝ ⎠ &7 ∂ W

- 166 -

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G ±/HVRQGHVF\OLQGULTXHVD[LDOHV {Z} G /HV RQGHV F\OLQGULTXHV D[LDOHV {Z} SRXU OHVTXHOOHV X = = Y FH VRQW GHV RQGHV WUDQVYHUVDOHV GH JOLVVHPHQW OHV F\OLQGUHV PDWpULHOV G¶D[H R] YLEUHQW HQ WUDQVODWLRQ SDUDOOqOHPHQWjR]VDQVVHGpIRUPHU/¶pTXDWLRQG¶RQGHV¶pFULW ∂ Z ∂ Z ∂ Z + = ⋅ U ∂ U &7 ∂ W ∂U

G ±/HVRQGHVF\OLQGULTXHVD[LDOHV {Z} G /HV RQGHV F\OLQGULTXHV D[LDOHV {Z} SRXU OHVTXHOOHV X = = Y FH VRQW GHV RQGHV WUDQVYHUVDOHV GH JOLVVHPHQW OHV F\OLQGUHV PDWpULHOV G¶D[H R] YLEUHQW HQ WUDQVODWLRQ SDUDOOqOHPHQWjR]VDQVVHGpIRUPHU/¶pTXDWLRQG¶RQGHV¶pFULW ∂ Z ∂ Z ∂ Z + = ⋅ U ∂ U &7 ∂ W ∂U

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2QDDORUV U ϕ θ GpVLJQDQWOHVFRRUGRQQpHVVSKpULTXHVGH 3 GDQV {R [\]}


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⎧ X U W UDGLDO ⎪⎪ G 3R3 = X U W ⎨ VXLYDQWϕ G ⎪ VXLYDQWθ ⎪⎩

⎧ X U W UDGLDO ⎪⎪ G 3R3 = X U W ⎨ VXLYDQWϕ G ⎪ VXLYDQWθ ⎪⎩

⎧ ∂X ⎪ε U = ∂ U ⎧γ ϕθ = ⎪ ⎪ X ⎪ ⎪ ⎨ γ θ U = ⎨εθ = U ⎪ ⎪ ⎪⎩ γ Uϕ = ⎪ε = X ⎪ ϕ U ⎩


- 167 -

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G ±/HVRQGHVF\OLQGULTXHVD[LDOHV {Z} G /HV RQGHV F\OLQGULTXHV D[LDOHV {Z} SRXU OHVTXHOOHV X = = Y FH VRQW GHV RQGHV WUDQVYHUVDOHV GH JOLVVHPHQW OHV F\OLQGUHV PDWpULHOV G¶D[H R] YLEUHQW HQ WUDQVODWLRQ SDUDOOqOHPHQWjR]VDQVVHGpIRUPHU/¶pTXDWLRQG¶RQGHV¶pFULW

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∂ Z ∂ Z ∂ Z + = ⋅ U ∂ U &7 ∂ W ∂U

∂ Z ∂ Z ∂ Z + = ⋅ U ∂ U &7 ∂ W ∂U

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- 167 -

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G /¶RQGHVSKpULTXH {X}HVWGRQFXQHRQGHORQJLWXGLQDOHGHGLODWDWLRQ /HVVXUIDFHVG¶RQGHV VRQWGHVVSKqUHVGHFHQWUHDXFRXUVGHODYLEUDWLRQOHVVSKqUHVPDWpULHOOHVGHFHQWUHQHVH GpSODFHQW SDV HOOHV ©UHVSLUHQWª F¶HVW j GLUH VH GLODWHQW HW VH FRQWUDFWHQW DOWHUQDWLYHPHQW /¶pTXDWLRQG¶RQGHSHXWV¶pFULUHVRXVOHVWURLVIRUPHVpTXLYDOHQWHVVXLYDQWHV

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩


∂ X ∂ X X ∂ X + − = ∂ U U ∂ U U &/ ∂ W ∂ ∂U

⎛∂X X ⎞ ∂ X ⎜⎜ + ⎟⎟ = U ⎠ &/ ∂ W ⎝ ∂U

∂ ⎡ ∂ ∂ X ⎤ = X U ⎥ ∂ U ⎢⎣ U ∂ U ⎦ &/ ∂ W

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

∂ X ∂ X X ∂ X + − = ∂ U U ∂ U U &/ ∂ W ∂ ∂U

⎛∂X X ⎞ ∂ X ⎜⎜ + ⎟⎟ = U ⎠ &/ ∂ W ⎝ ∂U

∂ ⎡ ∂ ∂ X ⎤ = X U ⎥ ∂ U ⎢⎣ U ∂ U ⎦ &/ ∂ W

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∂ X ∂ X X ∂ X + − = ∂ U U ∂ U U & / ∂ W

FDV

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∂ Z ∂ Z ∂ Z + = ∂ U U ∂ U &7 ∂ W

FDV


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FDV

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J + ω J =

J + ω J =

- 168 -

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⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

∂ X ∂ X X ∂ X + − = ∂ U U ∂ U U &/ ∂ W ∂ ∂U

⎛∂X X ⎞ ∂ X ⎜⎜ + ⎟⎟ = U ⎠ &/ ∂ W ⎝ ∂U

∂ ⎡ ∂ ∂ X ⎤ = X U ⎢ ⎥ ∂ U ⎣ U ∂ U ⎦ &/ ∂ W

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

∂ X ∂ X X ∂ X + − = ∂ U U ∂ U U &/ ∂ W ∂ ∂U

⎛∂X X ⎞ ∂ X ⎜⎜ + ⎟⎟ = U ⎠ &/ ∂ W ⎝ ∂U

∂ ⎡ ∂ ∂ X ⎤ = X U ⎢ ⎥ ∂ U ⎣ U ∂ U ⎦ &/ ∂ W

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∂ X ∂ X X ∂ X + − = ∂ U U ∂ U U & / ∂ W

FDV


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∂ Z ∂ Z ∂ Z + = ∂ U U ∂ U &7 ∂ W

FDV

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± SRXUO¶RQGHF\OLQGULTXHGHYLEUDWLRQD[LDOH

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∂ X ∂ X X ∂ X + − = ∂ U U ∂ U U & / ∂ W

FDV



J + ω J =

J + ω J =

- 168 -

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/DIRQFWLRQIU HVWDORUVVROXWLRQG¶XQHpTXDWLRQGLIIpUHQWLHOOHGXVHFRQGRUGUHOLQpDLUHVDQV VHFRQGPHPEUHGXW\SHGH%HVVHOPRGLILpH ⎧ ⎛ ω ⎞ ⎪I ′′U + I ′U + ⎜⎜ − ⎟⎟ I U = U U ⎠ ⎪ ⎝& ⎪ ω ⎪ ⎨I ′′U + I ′U + ⋅ I U = U F ⎪ ⎪ ⎪I ′′U + I ′U + ⎛⎜ ω − ⎞⎟ I U = ⎪ ⎜ F U ⎟ U ⎠ ⎝ ⎩

FDVHW FDV FDV

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I ′′ [ +

⎛ λ ⎞ I ′ [ + ⎜ − ⎟ I [ = DYHF λ ∈ 5 ⎟ ⎜ [ ⎝ [ ⎠

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⎛[⎞ -λ [ = ⎜ ⎟ ⎝⎠

λ

I ′′ [ +

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Q ⎡ ⎤ − Q ⎛ [ ⎞ + !!⎥ + !! + ⎢ ⎜ ⎟ Q ⎝ ⎠ Γ λ + Q + ⎢⎣ Γ λ + ⎥⎦

GDQVODTXHOOH Γ GpVLJQHODIRQFWLRQHXOHULHQQH

⎛[⎞ -λ [ = ⎜ ⎟ ⎝⎠

λ

Q ⎡ ⎤ − Q ⎛ [ ⎞ + !!⎥ + !! + ⎢ ⎜ ⎟ Q ⎝ ⎠ Γ λ + Q + ⎢⎣ Γ λ + ⎥⎦

Γ [ =

∫X


∞

∞ [ − − X

H

GX

Γ [ =

∫X

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I [ = & - λ [ + & - − λ [

I [ = & - λ [ + & - − λ [

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1 λ [ = OLPε →

- λ + ε [ − − λ - − λ − ε [ πε

1 λ [ = OLPε →

- λ + ε [ − − λ - − λ − ε [ πε

- 169 -

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⎧ ⎛ ω ⎞ ⎪I ′′U + I ′U + ⎜⎜ − ⎟⎟ I U = U U ⎠ ⎪ ⎝& ⎪ ω ⎪ ⎨I ′′U + I ′U + ⋅ I U = U F ⎪ ⎪ ⎪I ′′U + I ′U + ⎛⎜ ω − ⎞⎟ I U = ⎪ ⎜F U U ⎟⎠ ⎝ ⎩

⎧ ⎛ ω ⎞ ⎪I ′′U + I ′U + ⎜⎜ − ⎟⎟ I U = U U ⎠ ⎪ ⎝& ⎪ ω ⎪ ⎨I ′′U + I ′U + ⋅ I U = U F ⎪ ⎪ ⎪I ′′U + I ′U + ⎛⎜ ω − ⎞⎟ I U = ⎪ ⎜F U U ⎟⎠ ⎝ ⎩

FDVHW FDV FDV

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I ′′ [ +

⎛ λ ⎞ I ′ [ + ⎜ − ⎟ I [ = DYHF λ ∈ 5 ⎟ ⎜ [ ⎝ [ ⎠


I ′′ [ +

⎛ λ ⎞ I ′ [ + ⎜ − ⎟ I [ = DYHF λ ∈ 5 ⎟ ⎜ [ ⎝ [ ⎠


λ Q ⎤ − Q ⎛ [ ⎞ ⎛[⎞ ⎡ + !!⎥ -λ [ = ⎜ ⎟ ⎢ + !! + ⎜ ⎟ Q ⎝ ⎠ Γ λ + Q + ⎝ ⎠ ⎣⎢ Γ λ + ⎦⎥


λ Q ⎤ − Q ⎛ [ ⎞ ⎛[⎞ ⎡ + !!⎥ -λ [ = ⎜ ⎟ ⎢ + !! + ⎜ ⎟ Q ⎝ ⎠ Γ λ + Q + ⎝ ⎠ ⎣⎢ Γ λ + ⎦⎥

Γ [ =

∫


∞

∞

X [ − H − X G X

Γ [ =

∫X

[ − − X

H

GX





I [ = & - λ [ + & - − λ [

I [ = & - λ [ + & - − λ [





1 λ [ = OLPε →

λ

- λ + ε [ − − - − λ − ε [ πε

- 169 -

1 λ [ = OLPε →

- λ + ε [ − − λ - − λ − ε [ πε

- 169 -

HVWLQGpSHQGDQWHGH - λ [ HWHVWVROXWLRQGH RQO¶DSSHOOHIRQFWLRQGH%HVVHOGHVHFRQGH HVSqFHG¶RUGUH λ /DVROXWLRQJpQpUDOHGH SHXWDORUVV¶pFULUH I [ = & - λ [ + & 1 λ [

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3RVDQW U =

F[ HOOHVGHYLHQQHQW ω

I ′′ [ +

3RVDQW U =

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GDQV OH FDV VSKpULTXH (Q SRVDQW I [ = [ −α K [ RQREWLHQWO¶pTXDWLRQGH%HVVHO

DYHF α = GDQV OHV FDV F\OLQGULTXHV HW α =

K′′ [ +

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I ′′ [ +

⎛ α − λ ⎞ α + ⎟ I U = I ′ [ + ⎜ + ⎟ ⎜ [ [ ⎝ ⎠



⎛ λ ⎞ K′ [ + ⎜ − ⎟ K [ = ⎜ [ ⎟ [ ⎝ ⎠

K′′ [ +

/HVIRQFWLRQVGH%HVVHOVDWLVIRQWDX[IRUPXOHVGHGpULYDWLRQVXLYDQWHV


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G 1λ λ G -λ λ = 1 λ − 1 λ + = - λ − - λ + HW G[ [ G[ [

- 170 -

- 170 -

HVWLQGpSHQGDQWHGH - λ [ HWHVWVROXWLRQGH RQO¶DSSHOOHIRQFWLRQGH%HVVHOGHVHFRQGH HVSqFHG¶RUGUH λ /DVROXWLRQJpQpUDOHGH SHXWDORUVV¶pFULUH

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I [ = & - λ [ + & 1 λ [


I [ = & - λ [ + & 1 λ [






3RVDQW U =


I ′′ [ +

3RVDQW U =

⎛ α − λ ⎞ α + ⎟ I U = I ′ [ + ⎜ + ⎜ [ [ ⎟⎠ ⎝



K′′ [ +

G 1λ λ G -λ λ = 1 λ − 1 λ + = - λ − - λ + HW G[ [ G[ [

- 170 -

I ′′ [ +

⎛ α − λ ⎞ α + ⎟ I U = I ′ [ + ⎜ + ⎜ [ [ ⎟⎠ ⎝



⎛ λ ⎞ K′ [ + ⎜ − ⎟ K [ = ⎜ [ ⎟ [ ⎝ ⎠

K′′ [ +



⎛ λ ⎞ K′ [ + ⎜ − ⎟ K [ = ⎜ [ ⎟ [ ⎝ ⎠


G 1λ λ G -λ λ = 1 λ − 1 λ + = - λ − - λ + HW G[ [ G[ [

- 170 -

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Σ = Σ( + ΣY

2QDDLQVL

Σ = Σ( + ΣY









)LJXUH

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σ = ( ε + (′

∂ε ∂W

σ = ( ε + (′

∂ε ∂W

F¶HVWjGLUHSDUFRQVpTXHQW σ = σ ( + σ Y DYHF


∂ε FRQWUDLQWH YLVTXHXVH TXL IDLW DSSDUDvWUH OH ∂W FRHIILFLHQWGHYLVFRVLWpORQJLWXGLQDOH (′ FDUDFWpULVWLTXHGXPDWpULDX


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σ ( = ( ε FRQWUDLQWH pODVWLTXH HW σ Y = (′

τ [θ = τ =

τ = * γ + *′

∂γ ∂W

F¶HVWjGLUH


τ [θ = τ =

τ = * γ + *′

∂γ ∂W

F¶HVWjGLUH

τ( = * γ FRQWUDLQWHpODVWLTXH HW τ Y = *′

∂γ FRQWUDLQWHYLVTXHXVH ∂W



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- 172 -

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σ = ( ε + (′

∂ε ∂W

σ = ( ε + (′

∂ε ∂W












τ [θ = τ =

τ = * γ + *′

∂γ ∂W

F¶HVWjGLUH


τ [θ = τ =

τ = * γ + *′

∂γ ∂W

F¶HVWjGLUH







- 172 -

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⎛ ∂H ∂ E⎞ ⎟ Σ = (λ H , + µ E) + ⎜⎜ λ′ , + µ′

⎝ ∂W ∂ W ⎟⎠

Σ( ΣY


⎛ ∂H ∂ E⎞ ⎟ Σ = (λ H , + µ E) + ⎜⎜ λ′ , + µ′

⎝ ∂W ∂ W ⎟⎠

Σ( ΣY

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⎛ ∂ 8/ ⎞ ⎟ = ρ ∂ 8/ λ + µ ∆ 8 / + λ′ + µ′ ∆ ⎜⎜ ⎟ ∂ W ⎝ ∂W ⎠

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- 173 -

- 173 -


⎛ ∂H ∂ E⎞ ⎟ Σ = (λ H , + µ E) + ⎜⎜ λ′ , + µ′

⎝ ∂W ∂ W ⎟⎠

Σ( ΣY


⎛ ∂H ∂ E⎞ ⎟ Σ = (λ H , + µ E) + ⎜⎜ λ′ , + µ′

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Σ( ΣY













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⎛ ∂ 8/ ⎞ ⎟ = ρ ∂ 8/ λ + µ ∆ 8 / + λ′ + µ′ ∆ ⎜⎜ ⎟ ∂ W ⎝ ∂W ⎠

- 173 -

⎛ ∂ 8/ ⎞ ⎟ = ρ ∂ 8/ λ + µ ∆ 8 / + λ′ + µ′ ∆ ⎜⎜ ⎟ ∂ W ⎝ ∂W ⎠

- 173 -

± O¶RQGHWUDQVYHUVDOHTXLREpLWjO¶pTXDWLRQ ⎛ ∂ 87 ⎞ ⎟ = ρ ∂ 87 µ ∆ 8 7 + µ′ ∆ ⎜⎜ ⎟ ∂ W ⎝ ∂W ⎠


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HW & =

&′&5 =

J + Nω ⋅ J + ω ⋅ J = HQSRVDQW

HW

λ′ + µ′ FRHIILFLHQWG¶DPRUWLVVHPHQW ρ

& =

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⎞ ⎟⎟ FRHIILFLHQWG¶DPRUWLVVHPHQWUpGXLW ⎠




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$PRUWLVVHPHQWIRUWN!

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HW & =

&′&5 =




⎛ &′ N = ⎜⎜ ⎝ &′&5


HW & =

&′&5 =




⎛ &′ N = ⎜⎜ ⎝ &′&5




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J W = H −ωW $ + %W

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⎛ Q + [ ⎞ 8 Q [ W = VLQ ⎜ π ⎟ ⋅ J Q W K⎠ ⎝

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∑

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∞

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∞

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∑8

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⎧⎪J W = H − NωW $ FRV ΩW + % VLQ ΩW RXELHQ [[ ⎨ ⎪⎩J W = D H − NωW FRV ΩW + ϕ Q + F π K

⎛ Q + [ ⎞ 8 Q [ W = VLQ ⎜ π ⎟ ⋅ J Q W K⎠ ⎝


∑

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∞

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∞

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8 Q [ W

Q =

∑8

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W

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σ[ ⋅ G \ ⋅ G ] ⋅

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⎡⎛ ∂ γ ⎞ ⎛ ∂ γ ⎞ ⎛ ∂ γ ⎞ \] ⎟ + ⎜ ][ ⎟ + ⎜ [\ ⎟ ⎥ + µ′ ⎢⎜⎜ ⎢⎝ ∂ W ⎟⎠ ⎜⎝ ∂ W ⎟⎠ ⎜⎝ ∂ W ⎟⎠ ⎥ ⎣ ⎦

⎤


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λ′ + µ′ ≥ µ′ ≥

λ′ + µ′ ≥ µ′ ≥

- 176 -

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σ[ ⋅ G \ ⋅ G ] ⋅

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⎛ ⎛ ∂ E⎞ ∂ E⎞ ⎟ 3 = 3( + 3Y DYHF 3( = WU ⎜⎜ Σ ( ⋅ ⎟⎟ HW 3Y = WU ⎜⎜ Σ Y ⋅ ∂ W ⎟⎠ ∂W ⎠ ⎝ ⎝

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⎡⎛ ∂ γ ⎞ ⎛ ∂ γ ⎞ ⎛ ∂ γ ⎞ ⎤ \] ⎟ + ⎜ ][ ⎟ + ⎜ [\ ⎟ ⎥ + µ′ ⎢⎜⎜ ⎢⎝ ∂ W ⎟⎠ ⎜⎝ ∂ W ⎟⎠ ⎜⎝ ∂ W ⎟⎠ ⎥ ⎣ ⎦


⎡⎛ ∂ γ ⎞ ⎛ ∂ γ ⎞ ⎛ ∂ γ ⎞ ⎤ \] ⎟ + ⎜ ][ ⎟ + ⎜ [\ ⎟ ⎥ + µ′ ⎢⎜⎜ ⎢⎝ ∂ W ⎟⎠ ⎜⎝ ∂ W ⎟⎠ ⎜⎝ ∂ W ⎟⎠ ⎥ ⎣ ⎦







λ′ + µ′ ≥ µ′ ≥

λ′ + µ′ ≥ µ′ ≥

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(OLPLQDQW1HQWUHOHVGHX[RQREWLHQWO¶pTXDWLRQGHPRXYHPHQW ( ∂ X ∂ X = DYHF & = ρ ∂[ & ∂W

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$X GX FKDSLWUH SUpFpGHQW QRXV DYRQV UpVROX FHWWH pTXDWLRQ GH G¶$OHPEHUW SRXU O¶RQGH SODQH QRXV JDUGHURQV OHV PrPHV FRQGLWLRQV DX[ OLPLWHV FRPPH OH PRQWUH OD ILJXUH H[WUpPLWp [ = HQFDVWUpHH[WUpPLWp [ = / OLEUH2QDDORUV ∞

X [ W =

∑

VLQ

Q =

Q + [ ⎛ Q + FW Q + FW ⎞ π ⎜ $ Q FRV π + %Q VLQ π ⎟ /⎝ / /⎠

/HVQ°XGVGHO¶KDUPRQLTXHGHUDQJQRQWSRXUDEVFLVVHVR

/DIUpTXHQFHHVW I Q =

( ∂ X ∂ X = DYHF & = ρ ∂[ & ∂W


X [ W =

∑ VLQ Q =

/ / / « Q + Q + Q +



Q + & /¶RQGHHVWVWDWLRQQDLUH /


/ / / « Q + Q + Q +


/HVFRHIILFLHQWVGH)RXULHU $ Q HW %Q VHFDOFXOHQWjSDUWLUGHVFRQGLWLRQVLQLWLDOHVGHSRVLWLRQV HW YLWHVVHV RX FRQGLWLRQV GH &DXFK\ F¶HVW j GLUH j SDUWLU GHV IRQFWLRQV GRQQpHV ⎛∂X ⎞ ⎟⎟ ϕ [ = X [ R HW ψ [ = ⎜⎜ ⎝ ∂ W ⎠ W =

/HVFRHIILFLHQWVGH)RXULHU $ Q HW %Q VHFDOFXOHQWjSDUWLUGHVFRQGLWLRQVLQLWLDOHVGHSRVLWLRQV HW YLWHVVHV RX FRQGLWLRQV GH &DXFK\ F¶HVW j GLUH j SDUWLU GHV IRQFWLRQV GRQQpHV ⎛∂X ⎞ ⎟⎟ ϕ [ = X [ R HW ψ [ = ⎜⎜ ⎝ ∂ W ⎠ W =

/ ⎧ ⎪$ Q = ϕ [ VLQ ⎛⎜ Q + π [ ⎞⎟ ⋅ G[ ⎪ / /⎠ ⎝ ⎪ ⎨ / ⎪ ⎛ Q + [ ⎞ ψ [ VLQ ⎜ π ⎟ ⋅ G[ ⎪% Q = Q + π & /⎠ ⎝ ⎩⎪

/ ⎧ ⎪$ Q = ϕ [ VLQ ⎛⎜ Q + π [ ⎞⎟ ⋅ G[ ⎪ / /⎠ ⎝ ⎪ ⎨ / ⎪ ⎛ Q + [ ⎞ ψ [ VLQ ⎜ π ⎟ ⋅ G[ ⎪% Q = Q + π & /⎠ ⎝ ⎩⎪

- 178 -

- 178 -



⎧ ∂1 ∂ X G[ = ρ 6 G[ ⎪− 1 + 1 + ∂[ ∂W ⎪ ⎨ ⎪ ∂X ⎪ 1 = 6 ⋅ σ [ = ( 6 ⋅ ε [ = (6 ∂ [ ⎩

⎧ ∂1 ∂ X G[ = ρ 6 G[ ⎪− 1 + 1 + ∂[ ∂W ⎪ ⎨ ⎪ ∂X ⎪ 1 = 6 ⋅ σ [ = ( 6 ⋅ ε [ = (6 ∂ [ ⎩

∫

∫

∫

∫

(OLPLQDQW1HQWUHOHVGHX[RQREWLHQWO¶pTXDWLRQGHPRXYHPHQW ( ∂ X ∂ X = DYHF & = ρ ∂[ & ∂W

(OLPLQDQW1HQWUHOHVGHX[RQREWLHQWO¶pTXDWLRQGHPRXYHPHQW


X [ W =

∑ Q =

VLQ




( ∂ X ∂ X = DYHF & = ρ ∂[ & ∂W


/ / / « Q + Q + Q +


X [ W =

∑ VLQ Q =




/ / / « Q + Q + Q +


/HVFRHIILFLHQWVGH)RXULHU $ Q HW %Q VHFDOFXOHQWjSDUWLUGHVFRQGLWLRQVLQLWLDOHVGHSRVLWLRQV HW YLWHVVHV RX FRQGLWLRQV GH &DXFK\ F¶HVW j GLUH j SDUWLU GHV IRQFWLRQV GRQQpHV ⎛∂X ⎞ ϕ [ = X [ R HW ψ [ = ⎜⎜ ⎟⎟ ⎝ ∂ W ⎠ W =

/HVFRHIILFLHQWVGH)RXULHU $ Q HW %Q VHFDOFXOHQWjSDUWLUGHVFRQGLWLRQVLQLWLDOHVGHSRVLWLRQV HW YLWHVVHV RX FRQGLWLRQV GH &DXFK\ F¶HVW j GLUH j SDUWLU GHV IRQFWLRQV GRQQpHV ⎛∂X ⎞ ϕ [ = X [ R HW ψ [ = ⎜⎜ ⎟⎟ ⎝ ∂ W ⎠ W =

/ ⎧ ⎪$ Q = ϕ [ VLQ ⎛⎜ Q + π [ ⎞⎟ ⋅ G[ ⎪ / /⎠ ⎝ ⎪ ⎨ / ⎪ ⎛ Q + [ ⎞ ψ [ VLQ ⎜ π ⎟ ⋅ G[ ⎪% Q = Q + π & /⎠ ⎝ ⎪⎩

/ ⎧ ⎪$ Q = ϕ [ VLQ ⎛⎜ Q + π [ ⎞⎟ ⋅ G[ ⎪ / /⎠ ⎝ ⎪ ⎨ / ⎪ ⎛ Q + [ ⎞ ψ [ VLQ ⎜ π ⎟ ⋅ G[ ⎪% Q = Q + π & /⎠ ⎝ ⎪⎩

- 178 -

- 178 -

∫

∫

∫

∫



'¶DSUqVODIRUPXOH GXFKDSLWUHSUpFpGHQWODORLGHFRPSRUWHPHQWYLVFRpODVWLTXHOLQpDLUH V¶pFULW ∂X ∂ ⎛ ∂X ⎞ ⎟ 1 = (6 ⋅ + (′6 ⋅ ⎜⎜ ∂[ ∂ W ⎝ ∂ [ ⎟⎠


HWO¶pTXDWLRQGXPRXYHPHQW GHYLHQW


(

⎛ ∂ X ⎞ ∂ X ∂ X + (′ ⎜ ⎟ = ρ ⎟ ⎜ ∂W ∂[ ⎝∂[ ⎠

,FL QRXV SRVHURQV X = I [ ⋅ J W & =

⎛ &′ FULWLTXH N = ⎜⎜ ⎝ &′&5

( (′ &′ = &′ &5 = DPRUWLVVHPHQW ρ ρ ω&

⎞ ⎟⎟ DPRUWLVVHPHQWUpGXLW ⎠

(

⎛ ∂ X ⎞ ∂ X ∂ X + (′ ⎜ ⎟ = ρ ⎟ ⎜ ∂W ∂[ ⎝∂[ ⎠


⎛ &′ FULWLTXH N = ⎜⎜ ⎝ &′&5



/DVXLWH {ωQ }GHVSXOVDWLRQVSURSUHVQRQDPRUWLHVHVWGRQQpHSDUOHVFRQGLWLRQVDX[OLPLWHV 3RXUFKDTXHKDUPRQLTXHWURLVFDVVRQWjGLVWLQJXHU


$PRUWLVVHPHQWIRUW N Q >


(QSRVDQW UQ = ωQ N Q − RQDOHVWURLVIRUPHVpTXLYDOHQWHV


⎧ − N Q ωQ W ($ Q FK UQ W + %Q VKUQ W ) RXELHQ ⎪ J Q W = H ⎪ ⎪ −N ω W RXHQFRUH ⎨ J Q W = D Q H Q Q FK (UQ W + ϕQ ) ⎪ ⎪ $ − %Q − UQ W ⎞ − N Q ωQ W ⎛ $ Q + %Q UQ W H + Q H ⎜ ⎟ ⎪ J Q W = H ⎝ ⎠ ⎩


$PRUWLVVHPHQWFULWLTXH N Q =


2QDDORUV

J Q W = H

− ωQ W

2QDDORUV

($ Q + %Q W )

J Q W = H −ωQ W ($ Q + %Q W )

$PRUWLVVHPHQWIDLEOH N Q <


(QSRVDQW Ω Q = ωQ − N Q RQDOHVGHX[IRUPHVpTXLYDOHQWHV


⎧ J Q W = H − N Q ωQ W ($ Q FRV Ω Q W + %Q VLQ Ω Q W ) RX ⎪ ⎨ ⎪ J W = D H − N Q ωQ W FRV (Ω W + ϕ ) Q Q Q ⎩ Q


- 179 -

- 179 -







(

⎛ ∂ X ⎞ ∂ X ⎟=ρ ∂ X ⎜ ′ ( + ⎜ ⎟ ∂ W ∂ [ ⎝∂[ ⎠


⎛ &′ FULWLTXH N = ⎜⎜ ⎝ &′&5



(

⎛ ∂ X ⎞ ∂ X ⎟=ρ ∂ X ⎜ ′ ( + ⎜ ⎟ ∂ W ∂ [ ⎝∂[ ⎠


⎛ &′ FULWLTXH N = ⎜⎜ ⎝ &′&5













2QDDORUV

J Q W = H

− ωQ W

($ Q + %Q W )

2QDDORUV

J Q W = H −ωQ W ($ Q + %Q W )







- 179 -

- 179 -

2QUHPDUTXHTXHO¶DPRUWLVVHPHQWFULWLTXHHWO¶DPRUWLVVHPHQWUpGXLWVRQWIRQFWLRQGXUDQJQ GHO¶KDUPRQLTXHFRQVLGpUpSXLVTXHO¶RQD

(&′&5 )Q =

( HW N Q = ρ ωQ


(′ ωQ (

(&′&5 )Q =

( HW N Q = ρ ωQ

(′ ωQ (

*DUGRQV OHV FRQGLWLRQV DX[ OLPLWHV GX FDV SUpFpGHQW QRQ DPRUWL RQ D DORUV Q + & ωQ = π HW N Q FURvWFRPPH Q $SDUWLUG¶XQFHUWDLQUDQJOHVPRGHVSURSUHVVRQW / GRQFIRUWHPHQWDPRUWLV N Q >


±9,%5$7,216'(7256,21'¶8132875(35,60$7,48(

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)LJXUH

)LJXUH

/HPRXYHPHQWGHODVHFWLRQGURLWHFRXUDQWH6G¶DEVFLVVH[HVWXQHURWDWLRQG¶DQJOH θ [ W W

/HPRXYHPHQWGHODVHFWLRQGURLWHFRXUDQWH6G¶DEVFLVVH[HVWXQHURWDWLRQG¶DQJOH θ [ W

DXWRXU GH O¶D[H GH WRUVLRQ [′[ 0 [ W GpVLJQDQW OH PRPHQW GH WRUVLRQ VXU 6 O¶pTXLOLEUH GHV PRPHQWV HW OD ORL GH FRPSRUWHPHQW V¶pFULYHQW SRXU XQH WUDQFKH pOpPHQWDLUH 6 − 6′ G¶pSDLVVHXUG[ ⎧ ∂ 0W ∂ θ W W G[ = , G[ ⎪−0 +0 + ∂[ ⎪ ∂W ⎨ ∂ θ ⎪ 0 W = *⎪⎩ ∂[

DXWRXU GH O¶D[H GH WRUVLRQ [′[ 0 W [ W GpVLJQDQW OH PRPHQW GH WRUVLRQ VXU 6 O¶pTXLOLEUH GHV PRPHQWV HW OD ORL GH FRPSRUWHPHQW V¶pFULYHQW SRXU XQH WUDQFKH pOpPHQWDLUH 6 − 6′ G¶pSDLVVHXUG[ ⎧ ∂ 0W ∂ θ W W G[ = , G[ ⎪−0 +0 + ∂[ ⎪ ∂W ⎨ ∂ θ ⎪ 0 W = *⎪⎩ ∂[

(OLPLQDQW 0 W HQWUHOHVGHX[RQREWLHQWO¶pTXDWLRQGHPRXYHPHQW


*∂ θ ∂ θ = DYHF & = , ∂[ & ∂W

*∂ θ ∂ θ = DYHF & = , ∂[ & ∂W

- 180 -

- 180 -



(&′&5 )Q =

( HW N Q = ρ ωQ

(′ ωQ (

(&′&5 )Q =

( HW N Q = ρ ωQ

(′ ωQ (



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)LJXUH

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/HPRXYHPHQWGHODVHFWLRQGURLWHFRXUDQWH6G¶DEVFLVVH[HVWXQHURWDWLRQG¶DQJOH θ [ W W

/HPRXYHPHQWGHODVHFWLRQGURLWHFRXUDQWH6G¶DEVFLVVH[HVWXQHURWDWLRQG¶DQJOH θ [ W

DXWRXU GH O¶D[H GH WRUVLRQ [′[ 0 [ W GpVLJQDQW OH PRPHQW GH WRUVLRQ VXU 6 O¶pTXLOLEUH GHV PRPHQWV HW OD ORL GH FRPSRUWHPHQW V¶pFULYHQW SRXU XQH WUDQFKH pOpPHQWDLUH 6 − 6′ G¶pSDLVVHXUG[ ⎧ ∂ 0W ∂ θ W W G[ = , G[ ⎪−0 +0 + ∂[ ⎪ ∂W ⎨ ⎪ 0 W = *- ∂ θ ⎪⎩ ∂[

DXWRXU GH O¶D[H GH WRUVLRQ [′[ 0 W [ W GpVLJQDQW OH PRPHQW GH WRUVLRQ VXU 6 O¶pTXLOLEUH GHV PRPHQWV HW OD ORL GH FRPSRUWHPHQW V¶pFULYHQW SRXU XQH WUDQFKH pOpPHQWDLUH 6 − 6′ G¶pSDLVVHXUG[ ⎧ ∂ 0W ∂ θ W W G[ = , G[ ⎪−0 +0 + ∂[ ⎪ ∂W ⎨ ⎪ 0 W = *- ∂ θ ⎪⎩ ∂[



*∂ θ ∂ θ = DYHF & = , ∂[ & ∂W

- 180 -

*∂ θ ∂ θ = DYHF & = , ∂[ & ∂W

- 180 -

&HWWH pTXDWLRQ GH G¶$OHPEHUW HVW IRUPHOOHPHQW LGHQWLTXH j O¶pTXDWLRQ GX SDUDJUDSKH SUpFpGHQW (Q SRVDQW θ [ W = I [ ⋅ J W RQ REWLHQW DYHF OHV FRQGLWLRQV DX[ OLPLWHV GH OD ILJXUH θ = SRXU [ = HW [ = / OLDLVRQVJOLVVLqUHVDX[H[WUpPLWpVTXLHPSrFKHQWWRXWH URWDWLRQODLVVDQWOLEUHOHJDXFKLVVHPHQW ∞

θ [ W =

∑ Q =

&HWWH pTXDWLRQ GH G¶$OHPEHUW HVW IRUPHOOHPHQW LGHQWLTXH j O¶pTXDWLRQ GX SDUDJUDSKH SUpFpGHQW (Q SRVDQW θ [ W = I [ ⋅ J W RQ REWLHQW DYHF OHV FRQGLWLRQV DX[ OLPLWHV GH OD ILJXUH θ = SRXU [ = HW [ = / OLDLVRQVJOLVVLqUHVDX[H[WUpPLWpVTXLHPSrFKHQWWRXWH URWDWLRQODLVVDQWOLEUHOHJDXFKLVVHPHQW ∞

FW ⎞ FW [⎞ ⎛ ⎛ VLQ ⎜ Q π ⎟ ⋅ ⎜ $ Q FRV Q π + %Q VLQ Q π ⎟ /⎠ / /⎠ ⎝ ⎝

θ [ W =

∑ VLQ ⎛⎜⎝ Q π / ⎞⎟⎠ ⋅ ⎛⎜⎝ $ [

Q FRV

Qπ

Q =

/HVFRHIILFLHQWV $ Q HW %Q VHGpWHUPLQHQWjO¶DLGHGHVFRQGLWLRQVLQLWLDOHVGH&DXFK\

FW ⎞ FW + %Q VLQ Q π ⎟ /⎠ /


/ ⎧ ⎪$ Q = ϕ [ ⋅ VLQ Q π [ ⋅ G[ ⎧ ⎪ / / ⎪ϕ [ = θ [ R ⎪ ⎪ DYHF ⎨ ⎨ ⎛ ∂θ⎞ / ⎪ ⎪ ψ [ = ⎜⎜ ⎟⎟ [ ⎪% Q = ⎪ ψ [ ⋅ VLQ Q π ⋅ G[ ⎝ ∂ W ⎠ W = ⎩ QπF / ⎪ ⎩

/ ⎧ ⎪$ Q = ϕ [ ⋅ VLQ Q π [ ⋅ G[ ⎧ ⎪ / / ⎪ϕ [ = θ [ R ⎪ ⎪ DYHF ⎨ ⎨ ⎛ ∂θ⎞ / ⎪ ⎪ ψ [ = ⎜⎜ ⎟⎟ [ ⎪% Q = ⎪ ψ [ ⋅ VLQ Q π ⋅ G[ ⎝ ∂ W ⎠ W = ⎩ QπF / ⎪ ⎩

6LSDUH[HPSOHRQDEDQGRQQHODSRXWUHjO¶LQVWDQW W = VDQVYLWHVVHLQLWLDOH ψ = DSUqV /⎞ ⎛/ ⎞ ⎛ DYRLULPSRVpHQVWDWLTXHXQHURWDWLRQ θ ⎜ R ⎟ = α jODVHFWLRQPpGLDQH ⎜ [ = ⎟ RQDOD ⎠ ⎝ ⎠ ⎝ IRQFWLRQ ϕ [ = θ [ R GRQQpHSDUODILJXUH


∫

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)LJXUH

)LJXUH

2QHQGpGXLW

α α α $ = « $ = $ = $ = − $ = $ = π π π

HW %Q = ∀ Q G¶RODVROXWLRQ

2QHQGpGXLW $ =


α ⎛ [ π FW [ FW VLQ π ⋅ FRV − VLQ π ⋅ FRV π ⎜ / / / / π ⎝ [ FW [ FW ⎞ + VLQ π ⋅ FRV π − VLQ π ⋅ FRV π + !⎟ / / / / ⎠

α ⎛ [ π FW [ FW VLQ π ⋅ FRV − VLQ π ⋅ FRV π ⎜ / / / / π ⎝ [ FW [ FW ⎞ + VLQ π ⋅ FRV π − VLQ π ⋅ FRV π + !⎟ / / / / ⎠

θ [ W =

α α α $ = « $ = $ = $ = $ = − π π π

θ [ W =

- 181 -

- 181 -

&HWWH pTXDWLRQ GH G¶$OHPEHUW HVW IRUPHOOHPHQW LGHQWLTXH j O¶pTXDWLRQ GX SDUDJUDSKH SUpFpGHQW (Q SRVDQW θ [ W = I [ ⋅ J W RQ REWLHQW DYHF OHV FRQGLWLRQV DX[ OLPLWHV GH OD ILJXUH θ = SRXU [ = HW [ = / OLDLVRQVJOLVVLqUHVDX[H[WUpPLWpVTXLHPSrFKHQWWRXWH URWDWLRQODLVVDQWOLEUHOHJDXFKLVVHPHQW

&HWWH pTXDWLRQ GH G¶$OHPEHUW HVW IRUPHOOHPHQW LGHQWLTXH j O¶pTXDWLRQ GX SDUDJUDSKH SUpFpGHQW (Q SRVDQW θ [ W = I [ ⋅ J W RQ REWLHQW DYHF OHV FRQGLWLRQV DX[ OLPLWHV GH OD ILJXUH θ = SRXU [ = HW [ = / OLDLVRQVJOLVVLqUHVDX[H[WUpPLWpVTXLHPSrFKHQWWRXWH URWDWLRQODLVVDQWOLEUHOHJDXFKLVVHPHQW

∞

θ [ W =

∑ Q =

∞

FW ⎞ FW [⎞ ⎛ ⎛ VLQ ⎜ Q π ⎟ ⋅ ⎜ $ Q FRV Q π + %Q VLQ Q π ⎟ /⎠ / /⎠ ⎝ ⎝

θ [ W =

∑ VLQ ⎛⎜⎝ Q π / ⎞⎟⎠ ⋅ ⎛⎜⎝ $ [

Q FRV

Q =


Qπ

FW ⎞ FW + %Q VLQ Q π ⎟ /⎠ /


/ ⎧ ⎪$ Q = ϕ [ ⋅ VLQ Q π [ ⋅ G[ ⎧ ⎪ / / ⎪ϕ [ = θ [ R ⎪ ⎪ DYHF ⎨ ⎨ ⎛ ∂θ⎞ / ⎪ ⎪ ψ [ = ⎜⎜ ⎟⎟ [ ⎪% Q = ⎪ ψ [ ⋅ VLQ Q π ⋅ G[ ⎝ ∂ W ⎠ W = ⎩ Q F / π ⎪ ⎩

/ ⎧ ⎪$ Q = ϕ [ ⋅ VLQ Q π [ ⋅ G[ ⎧ ⎪ / / ⎪ϕ [ = θ [ R ⎪ ⎪ DYHF ⎨ ⎨ ⎛ ∂θ⎞ / ⎪ ⎪ ψ [ = ⎜⎜ ⎟⎟ [ ⎪% Q = ⎪ ψ [ ⋅ VLQ Q π ⋅ G[ ⎝ ∂ W ⎠ W = ⎩ Q F / π ⎪ ⎩



∫

∫

∫

∫

)LJXUH

2QHQGpGXLW

α α α $ = « $ = $ = $ = − $ = $ = π π π


)LJXUH

2QHQGpGXLW $ =


α ⎛ [ π FW [ FW − VLQ π ⋅ FRV π ⎜ VLQ π ⋅ FRV / / / / π ⎝ [ FW [ FW ⎞ + VLQ π ⋅ FRV π − VLQ π ⋅ FRV π + !⎟ / / / / ⎠

θ [ W =

α α α $ = « $ = $ = $ = $ = − π π π

- 181 -

α ⎛ [ π FW [ FW − VLQ π ⋅ FRV π ⎜ VLQ π ⋅ FRV / / / / π ⎝ [ FW [ FW ⎞ + VLQ π ⋅ FRV π − VLQ π ⋅ FRV π + !⎟ / / / / ⎠

θ [ W =

- 181 -

9LEUDWLRQVQDWXUHOOHVDPRUWLHV


'¶DSUqVODIRUPXOH GXFKDSLWUHSUpFpGHQWODORLGXFRPSRUWHPHQWYLVFRpODVWLTXHV¶pFULW LFL


0W = * - ⋅

∂θ ∂ + *′ - ⋅ ∂[ ∂W

⎛ ∂θ ⎞ ⎜⎜ ⎟⎟ ⎝∂[ ⎠

0W = * - ⋅

HWO¶pTXDWLRQ GXPRXYHPHQWGHYLHQW *-⋅

⎛ &′ N = ⎜⎜ ⎝ &′&5

∂ θ ∂ + *′ - ⋅ ∂W ∂[

⎛ ∂ θ ⎞ ⎟=,∂ θ ⎜ ⎜ ⎟ ∂ W ⎝∂[ ⎠

**′ &′ = &′&5 = ⋅ & DPRUWLVVHPHQW FULWLTXH , , ω

∞

I Q [ ⋅ J Q W DYHF I Q [ = VLQ

Q =

(Q SRVDQW θ = I [ ⋅ J W F =

⎛ &′ N = ⎜⎜ ⎝ &′&5

/DVXLWH {ωQ }GHVSXOVDWLRQVSURSUHVQRQDPRUWLHVHVWGRQQpHSDUOHVFRQGLWLRQVDX[OLPLWHV

∑

*-⋅

∂ θ ∂ + *′ - ⋅ ∂W ∂[

⎛ ∂ θ ⎞ ⎟=,∂ θ ⎜ ⎜ ⎟ ∂ W ⎝∂[ ⎠

&HWWHGHUQLqUHHVWIRUPHOOHPHQWLGHQWLTXHj

⎞ ⎟⎟ DPRUWLVVHPHQWUpGXLW RQREWLHQWODVROXWLRQ θ [ W ⎠

⎛&⎞ LFL ωQ = Q π ⎜ ⎟ HW θ [ W = ⎝/⎠

⎛ ∂θ ⎞ ⎜⎜ ⎟⎟ ⎝∂[ ⎠

HWO¶pTXDWLRQ GXPRXYHPHQWGHYLHQW

&HWWHGHUQLqUHHVWIRUPHOOHPHQWLGHQWLTXHj (Q SRVDQW θ = I [ ⋅ J W F =

∂θ ∂ + *′ - ⋅ ∂[ ∂W

Q π[ /



/DVXLWH {ωQ }GHVSXOVDWLRQVSURSUHVQRQDPRUWLHVHVWGRQQpHSDUOHVFRQGLWLRQVDX[OLPLWHV ⎛&⎞ LFL ωQ = Q π ⎜ ⎟ HW θ [ W = ⎝/⎠

∞

∑ I [ ⋅ J W DYHF I [ = VLQ Q

Q =

3RXU J Q W WURLVFDVVRQWjGLVWLQJXHU




J Q W = H − N Q ωQ W ($ Q FK UQ W + %Q VKUQ W )DYHF UQ = ωQ N Q −

Q

Q

Q π[ /

J Q W = H − N Q ωQ W ($ Q FK UQ W + %Q VKUQ W )DYHF UQ = ωQ N Q −



J Q W = H −ωQ W ($ Q + %Q W )

J Q W = H −ωQ W ($ Q + %Q W )



J Q W = H − N Q ωQ W ($ Q FRV Ω Q W + %Q VLQ Ω Q W )DYHF Ω Q = ωQ − N Q


- 182 -

- 182 -





0W = * - ⋅

∂θ ∂ + *′ - ⋅ ∂[ ∂W

⎛ ∂θ ⎞ ⎜⎜ ⎟⎟ ⎝∂[ ⎠

0W = * - ⋅

HWO¶pTXDWLRQ GXPRXYHPHQWGHYLHQW *-⋅

⎛ &′ N = ⎜⎜ ⎝ &′&5

∂ θ ∂ + *′ - ⋅ ∂ W ∂[

⎛ ∂ θ ⎞ ⎟=,∂ θ ⎜ ⎟ ⎜ ∂ W ⎝∂[ ⎠


/DVXLWH {ωQ }GHVSXOVDWLRQVSURSUHVQRQDPRUWLHVHVWGRQQpHSDUOHVFRQGLWLRQVDX[OLPLWHV ∞

∑

I Q [ ⋅ J Q W DYHF I Q [ = VLQ

Q =

*-⋅

∂ θ ∂ + *′ - ⋅ ∂ W ∂[

⎛ ∂ θ ⎞ ⎟=,∂ θ ⎜ ⎟ ⎜ ∂ W ⎝∂[ ⎠

&HWWHGHUQLqUHHVWIRUPHOOHPHQWLGHQWLTXHj


⎛&⎞ LFL ωQ = Q π ⎜ ⎟ HW θ [ W = ⎝/⎠

⎛ ∂θ ⎞ ⎜⎜ ⎟⎟ ⎝∂[ ⎠

HWO¶pTXDWLRQ GXPRXYHPHQWGHYLHQW

&HWWHGHUQLqUHHVWIRUPHOOHPHQWLGHQWLTXHj (Q SRVDQW θ = I [ ⋅ J W F =

∂θ ∂ + *′ - ⋅ ∂[ ∂W

Q π[ /

(Q SRVDQW θ = I [ ⋅ J W F =

⎛ &′ N = ⎜⎜ ⎝ &′&5



/DVXLWH {ωQ }GHVSXOVDWLRQVSURSUHVQRQDPRUWLHVHVWGRQQpHSDUOHVFRQGLWLRQVDX[OLPLWHV ⎛&⎞ LFL ωQ = Q π ⎜ ⎟ HW θ [ W = ⎝/⎠

∞

∑ I [ ⋅ J W DYHF I [ = VLQ Q

Q =





J Q W = H − N Q ωQ W ($ Q FK UQ W + %Q VKUQ W )DYHF UQ = ωQ N Q − $PRUWLVVHPHQWFULWLTXH N Q =

Q

Q

Q π[ /

J Q W = H − N Q ωQ W ($ Q FK UQ W + %Q VKUQ W )DYHF UQ = ωQ N Q − $PRUWLVVHPHQWFULWLTXH N Q =

J Q W = H −ωQ W ($ Q + %Q W ) $PRUWLVVHPHQWIDLEOH N Q <

J Q W = H −ωQ W ($ Q + %Q W ) $PRUWLVVHPHQWIDLEOH N Q <



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1RWDWLRQV Y [ W ∂Y ω= ∂[ (, = (, ] 7 = 7\ [ W

1RWDWLRQV Y [ W ∂Y ω= ∂[ (, = (, ] 7 = 7\ [ W

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0 = 0 ] [ W PRPHQWIOpFKLVVDQW ρ PDVVHYROXPLTXHGXPDWpULDX , = ,] PRPHQWTXDGUDWLTXHGHIOH[LRQ GH 6

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∂7 ∂ Y ∂0 ∂ ω] = ρ 6 HW 7 + =ρ, ∂[ ∂[ ∂W ∂ W

∂7 ∂ Y ∂0 ∂ ω] = ρ 6 HW 7 + =ρ, ∂[ ∂[ ∂W ∂ W

(Q QpJOLJHDQW OD IOqFKH GXH j O¶HIIRUW WUDQFKDQW FRPPH QRXV O¶DYRQV MXVWLILp DX WRPH ,, ∂ Y 0 ∂Y 2Q FKDSLWUH9RQDODORLGHFRPSRUWHPHQWHQIOH[LRQ = HWODUHODWLRQ ω] = (, ∂[ ∂[


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1RWDWLRQV Y [ W ∂Y ω= ∂[ (, = (, ] 7 = 7\ [ W

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∂7 ∂ Y ∂0 ∂ ω] = ρ 6 HW 7 + =ρ, ∂[ ∂[ ∂W ∂ W

∂7 ∂ Y ∂0 ∂ ω] = ρ 6 HW 7 + =ρ, ∂[ ∂[ ∂W ∂ W





- 183 -

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ρ6

∂ Y ∂ Y ∂ Y + (, =ρ, ∂W ∂[ ∂ [ ∂ W

ρ6

∂ Y ∂ Y ∂ Y + (, =ρ, ∂W ∂[ ∂ [ ∂ W

2QSHXWUpVRXGUHDQDO\WLTXHPHQWFHWWHpTXDWLRQ(QIDLWVDXISRXUOHVKDUPRQLTXHVGHUDQJ ⎛ ∂ ω] ⎞⎟ pOHYp OH FRXSOH LQHUWLHO ⎜ − ρ , G [ DSSOLTXp j OD WUDQFKH 6 − 6′ HW G j VRQ ⎜ ∂ W ⎠⎟ ⎝ ⎛ ∂0 ⎞ ⎟ G [ /¶pTXDWLRQ V¶pFULW DFFpOpUDWLRQDQJXODLUHHVWQpJOLJHDEOHGHYDQWOHFRXSOH ⎜⎜ 7 + ∂ [ ⎟⎠ ⎝


DORUV

DORUV

∂ Y ∂ Y ρ 6 + (, = ∂W ∂ [

1RXVODUpVRXGURQVGDQVFHFDVHQFKHUFKDQWGHVVROXWLRQVGHODIRUPH Y [ W = I [ ⋅ J W 2QGRLWDYRLU

ρ6

∂ Y ∂ Y + (, = ∂ W ∂ [


J (, I =− = −ω G¶R J ρ6 I

J (, I =− = −ω G¶R J ρ6 I

J W = $ FRV ωW + % VLQ ωW HW


⎛ ρ6 ω ⎞ ⎛ ρ6 ω ⎞ I [ = α FRV ⎜ [ ⎟ + β VLQ ⎜ [⎟ ⎜ ⎟ ⎜ ⎟ (, (, ⎝ ⎠ ⎝ ⎠

⎛ ρ6 ω ⎞ ⎛ ρ6 ω ⎞ I [ = α FRV ⎜ [ ⎟ + β VLQ ⎜ [⎟ ⎜ ⎟ ⎜ ⎟ (, (, ⎝ ⎠ ⎝ ⎠

⎛ ρ6 ω ⎞ ⎛ ρ6 ω ⎞ + γ FK ⎜ [ ⎟ + δ VK ⎜ [⎟ ⎜ ⎟ ⎜ ⎟ (, (, ⎝ ⎠ ⎝ ⎠

⎛ ρ6 ω ⎞ ⎛ ρ6 ω ⎞ + γ FK ⎜ [ ⎟ + δ VK ⎜ [⎟ ⎜ ⎟ ⎜ ⎟ (, (, ⎝ ⎠ ⎝ ⎠

/DSRXWUHHVWVXSSRVpHVLPSOHPHQWDSSX\pHHQ2HW$FHTXLSHUPHWG¶pFULUHOHVFRQGLWLRQV DX[OLPLWHV ⎧ Y = SRXU [ = ⇒ I R = ⎪ ⎪⎪ Y = SRXU [ = / ⇒ I / = ∀W ⎨ ⎪ 0 = SRXU [ = ⇒ I ′′R = ⎪ ⎪⎩ 0 = SRXU [ = / ⇒ I ′′/ =

/DSRXWUHHVWVXSSRVpHVLPSOHPHQWDSSX\pHHQ2HW$FHTXLSHUPHWG¶pFULUHOHVFRQGLWLRQV DX[OLPLWHV ⎧ Y = SRXU [ = ⇒ I R = ⎪ ⎪⎪ Y = SRXU [ = / ⇒ I / = ∀W ⎨ ⎪ 0 = SRXU [ = ⇒ I ′′R = ⎪ ⎪⎩ 0 = SRXU [ = / ⇒ I ′′/ =

⎛ ρ6 ω ⎞ 2QHQWLUH α = γ = δ = ≠ β HW VLQ ⎜ /⎟ = ⎜ ⎟ (, ⎝ ⎠


/DSXOVDWLRQ ω HVWGRQFO¶XQGHVQRPEUHVGHODVXLWH {ωQ }DYHF ωQ = Q

π (, ⋅ ρ6 /


- 184 -

ρ6

π (, ⋅ ρ6 /

- 184 -

∂ Y ∂ Y ∂ Y + (, =ρ, ∂W ∂[ ∂ [ ∂ W

ρ6

∂ Y ∂ Y ∂ Y + (, =ρ, ∂W ∂[ ∂ [ ∂ W



DORUV

DORUV

ρ6

∂ Y ∂ Y + (, = ∂W ∂ [


ρ6

∂ Y ∂ Y + (, = ∂ W ∂ [


J (, I =− = −ω G¶R J ρ6 I

J (, I =− = −ω G¶R J ρ6 I



⎛ ρ6 ω ⎞ ⎛ ρ6 ω ⎞ I [ = α FRV ⎜ [ ⎟ + β VLQ ⎜ [⎟ ⎜ ⎟ ⎜ ⎟ (, (, ⎝ ⎠ ⎝ ⎠

⎛ ρ6 ω ⎞ ⎛ ρ6 ω ⎞ I [ = α FRV ⎜ [ ⎟ + β VLQ ⎜ [⎟ ⎜ ⎟ ⎜ ⎟ (, (, ⎝ ⎠ ⎝ ⎠

⎛ ρ6 ω ⎞ ⎛ ρ6 ω ⎞ + γ FK ⎜ [ ⎟ + δ VK ⎜ [⎟ ⎜ ⎟ ⎜ ⎟ (, (, ⎝ ⎠ ⎝ ⎠

⎛ ρ6 ω ⎞ ⎛ ρ6 ω ⎞ + γ FK ⎜ [ ⎟ + δ VK ⎜ [⎟ ⎜ ⎟ ⎜ ⎟ (, (, ⎝ ⎠ ⎝ ⎠

/DSRXWUHHVWVXSSRVpHVLPSOHPHQWDSSX\pHHQ2HW$FHTXLSHUPHWG¶pFULUHOHVFRQGLWLRQV DX[OLPLWHV ⎧ Y = SRXU [ = ⇒ I R = ⎪ ⎪⎪ Y = SRXU [ = / ⇒ I / = ∀W ⎨ ⎪ 0 = SRXU [ = ⇒ I ′′R = ⎪ ⎩⎪ 0 = SRXU [ = / ⇒ I ′′/ =

/DSRXWUHHVWVXSSRVpHVLPSOHPHQWDSSX\pHHQ2HW$FHTXLSHUPHWG¶pFULUHOHVFRQGLWLRQV DX[OLPLWHV ⎧ Y = SRXU [ = ⇒ I R = ⎪ ⎪⎪ Y = SRXU [ = / ⇒ I / = ∀W ⎨ ⎪ 0 = SRXU [ = ⇒ I ′′R = ⎪ ⎩⎪ 0 = SRXU [ = / ⇒ I ′′/ =




- 184 -

π (, ⋅ ρ 6 /


- 184 -

π (, ⋅ ρ 6 /

2QSHXWDORUVSUHQGUH β = HWpFULUHODVROXWLRQKDUPRQLTXHGHUDQJQ Y Q [ W = VLQ Q π

2QSHXWDORUVSUHQGUH β = HWpFULUHODVROXWLRQKDUPRQLTXHGHUDQJQ

[ ⋅ $ Q FRV ωQ W + %Q VLQ ωQ W /

Y Q [ W = VLQ Q π



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Y [ W =

∑

∞

Y [ W =

Y Q [ W

Q =

∑ Y [ W Q

Q =

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⎛∂Y⎞ ⎟⎟ 2QD jGLUHjSDUWLUGHVIRQFWLRQV ϕ [ = Y [ R HW ψ [ = ⎜⎜ ⎝ ∂ W ⎠ W =


$Q =

/

/

∫

ϕ [ ⋅ VLQ

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/

∫

ψ [ ⋅ VLQ

Q π[ ⋅ G[ /

$Q =

/

/

∫

ϕ [ ⋅ VLQ

Q π[ ⋅ G[ HW %Q = / ωQ /

/

∫ ψ [ ⋅ VLQ

Q π[ ⋅ G[ /

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ρ6

∂ Y ∂ Y ∂ + + (′ , ⋅ (, ∂W ∂W ∂[

⎛ ∂ Y ⎞ ⎟ ⎜ ⎜ ∂ [ ⎟ ⎝ ⎠

$YHF Y [ W = I [ ⋅ J W RQD J (, I = −ω = (′ ρ6 I J + J (

−

2QDGRQFFRPPHGDQVOHFDVFRQVHUYDWLI

HW

J +

∂ Y ∂ Y ∂ + + (′ , ⋅ (, ∂W ∂W ∂[

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ωQ = Q

ρ6

π /

J (, I = −ω = (′ ρ6 I J + J (


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ωQ = Q

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HW

J +

π /

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(′ ω ⋅ J + ω ⋅ J = RX J + Nω ⋅ J + ω ⋅ J = (

- 185 -

- 185 -

2QSHXWDORUVSUHQGUH β = HWpFULUHODVROXWLRQKDUPRQLTXHGHUDQJQ Y Q [ W = VLQ Q π

2QSHXWDORUVSUHQGUH β = HWpFULUHODVROXWLRQKDUPRQLTXHGHUDQJQ


Y Q [ W = VLQ Q π



/DVROXWLRQJpQpUDOHV¶pFULW ∞

Y [ W =

∑

∞

Y [ W =

Y Q [ W

Q =

∑ Y [ W Q

Q =





$Q = /

/

∫

Q π[ ϕ [ ⋅ VLQ ⋅ G[ HW %Q = / ωQ /

/

∫

Q π[ ψ [ ⋅ VLQ ⋅ G[ /

$Q = /

/

∫

Q π[ ϕ [ ⋅ VLQ ⋅ G[ HW %Q = / ωQ /

/

∫ ψ [ ⋅ VLQ

Q π[ ⋅ G[ /





ρ6

∂ Y ∂ ∂ Y + (, + (′ , ⋅ ∂W ∂W ∂ [

⎛ ∂ Y ⎞ ⎟ ⎜ ⎜ ∂ [ ⎟ ⎝ ⎠

$YHF Y [ W = I [ ⋅ J W RQD

J (, I = −ω = (′ ρ6 I J + J (


HW

J +

π /

ρ6

∂ Y ∂ ∂ Y + (, + (′ , ⋅ ∂W ∂W ∂ [

⎛ ∂ Y ⎞ ⎟ ⎜ ⎜ ∂ [ ⎟ ⎝ ⎠

$YHF Y [ W = I [ ⋅ J W RQD −

ωQ = Q

−

J (, I = −ω = (′ ρ6 I J + J (


π[ (, HW I Q [ = VLQ Q / ρ6

(′ ω ⋅ J + ω ⋅ J = RX J + Nω ⋅ J + ω ⋅ J = (

- 185 -

ωQ = Q HW

J +

π /

π[ (, HW I Q [ = VLQ Q / ρ6

(′ ω ⋅ J + ω ⋅ J = RX J + Nω ⋅ J + ω ⋅ J = (

- 185 -

(′ ρ

3RVRQV

&′ =

&′&5 =

⎛ &′ N = ⎜⎜ ⎝ &′&5

&′ = FRHIILFLHQWG¶DPRUWLVVHPHQW


&′ =

&′&5 =

⎛ &′ N = ⎜⎜ ⎝ &′&5


(′ ρ

3RVRQV




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J Q W = H − N Q ωQ W ($ Q FK UQ W + %Q VK UQ W ) $PRUWLVVHPHQWFULWLTXH N Q =

J Q W = H

− ωQ W

($ Q + %Q W )


J Q W = H −ωQ W ($ Q + %Q W )



(QSRVDQW Ω Q = ωQ − N Q RQD


J Q W = H − N Q ωQ W ($ Q FRV Ω Q W + %Q VLQ Ω Q W )


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- 186 -

- 186 -

(′ ρ

3RVRQV

&′ =

&′&5 =

⎛ &′ N = ⎜⎜ ⎝ &′&5



&′ =

&′&5 =

⎛ &′ N = ⎜⎜ ⎝ &′&5


(′ ρ

3RVRQV











J Q W = H

− ωQ W

($ Q + %Q W )


J Q W = H −ωQ W ($ Q + %Q W )















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**′ = −

, \ VLQ ϕ , ] FRV ϕ -− . 56 56


[

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, \ VLQ ϕ , ] FRV ϕ -− . 56 56


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⎧ ⎛ < FRV ϕ = VLQ ϕ ⎞ + ⎟ GV ⎪ ⎜ − 5 5 ⎠ ⎪⎝ ⎪ ⎛ Gϕ⎞ ⎪ ⎟⎟ GV G 3 ⎨ G< − = ⎜⎜ + ⎝ 7 GV ⎠ ⎪ ⎪ ⎪ G= + < ⎛⎜ + G ϕ ⎞⎟ GV ⎜ 7 GV ⎟ ⎪ ⎝ ⎠ ⎩


⎧ ⎛ ∂ X Y FRV ϕ Z VLQ ϕ ⎞ ∂X ∂X ⎟⎟ GV + − + G< + G= ⎪ ⎜⎜ ∂ V 5 5 < = ∂ ∂ ⎠ ⎪⎝ ⎪ ⎛ G ϕ ⎞⎞ ∂Y ∂Y ⎪ ⎛ ∂ Y X FRV ϕ ⎟⎟ ⎟⎟ GV + G 3R3 ⎨ ⎜⎜ G< + G= + − Z ⎜⎜ + 5 ∂< ∂= ⎝ 7 GV ⎠⎠ ⎪ ⎝ ∂V ⎪ ⎪ ⎛⎜ ∂ Z + X VLQ ϕ + Z ⎛⎜ + G ϕ ⎞⎟ ⎞⎟ GV + ∂ Z G < + ∂ Z G = ⎜ 7 GV ⎟⎟ ⎪ ⎜ ∂V 5 ∂< ∂= ⎝ ⎠⎠ ⎩⎝

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- 188 -

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⎧ ⎛ ∂ X Y FRV ϕ Z VLQ ϕ ⎞ ∂X ∂X ⎟⎟ GV + − + G< + G= ⎪ ⎜⎜ 5 5 ⎠ ∂< ∂= ⎪ ⎝ ∂V ⎪ ⎛ G ϕ ⎞⎞ ∂Y ∂Y ⎪ ⎛ ∂ Y X FRV ϕ ⎟⎟ ⎟⎟ GV + G 3R3 ⎨ ⎜⎜ G< + G= + − Z ⎜⎜ + V 5 7 G V < = ∂ ∂ ∂ ⎝ ⎠⎠ ⎪⎝ ⎪ ⎪ ⎛⎜ ∂ Z + X VLQ ϕ + Z ⎛⎜ + G ϕ ⎞⎟ ⎞⎟ GV + ∂ Z G < + ∂ Z G = ⎜ 7 GV ⎟⎟ ⎪ ⎜ ∂V 5 ∂< ∂= ⎝ ⎠⎠ ⎩⎝

⎧ ⎛ ∂ X Y FRV ϕ Z VLQ ϕ ⎞ ∂X ∂X ⎟⎟ GV + − + G< + G= ⎪ ⎜⎜ 5 5 ⎠ ∂< ∂= ⎪ ⎝ ∂V ⎪ ⎛ G ϕ ⎞⎞ ∂Y ∂Y ⎪ ⎛ ∂ Y X FRV ϕ ⎟⎟ ⎟⎟ GV + G 3R3 ⎨ ⎜⎜ G< + G= + − Z ⎜⎜ + V 5 7 G V < = ∂ ∂ ∂ ⎝ ⎠⎠ ⎪⎝ ⎪ ⎪ ⎛⎜ ∂ Z + X VLQ ϕ + Z ⎛⎜ + G ϕ ⎞⎟ ⎞⎟ GV + ∂ Z G < + ∂ Z G = ⎜ 7 GV ⎟⎟ ⎪ ⎜ ∂V 5 ∂< ∂= ⎝ ⎠⎠ ⎩⎝

- 188 -

- 188 -

G¶R

G¶R

⎧ ∂X ⎞ ∂ X Y FRV ϕ Z VLQ ϕ ⎛ G ϕ ⎞ ⎛ ∂ X ⎟ ⎟⎟ ⎜⎜ = −< − + + ⎜⎜ + ⎪ ∂ ∂ ] ⎟⎠ V 5 5 7 G V < ∂ ⎠⎝ ⎝ ⎪ε = < FRV ϕ = VLQ ϕ ⎪ [ + − ⎪ 5 5 ⎪ ⎪ ⎪ ∂Y ∂Z ⎪ ε] = ⎪ ε< = ∂ < ∂] ⎪ ⎪ ⎪ ⎪ γ = ∂ Z + ∂ Y ⎨ <= ∂ < ∂ ] ⎪ ⎪ ⎪ ∂Z ∂Z⎞ ∂ Z X VLQ ϕ ⎛ G ϕ ⎞ ⎛ ⎟ ⎟ ⎜⎜ Y + = ⎪ −< − + ⎜⎜ + ⎟ ∂ ∂ = ⎟⎠ ∂ V 5 7 G V < ∂X ⎪ ⎠⎝ ⎝ γ = + ⎪ =; ∂ ] < FRV ϕ = VLQ ϕ + − ⎪ 5 5 ⎪ ⎪ ∂Y ∂Y⎞ ∂ Y X FRV ϕ ⎛ G ϕ ⎞ ⎛ ⎪ ⎟ ⎟⎟ ⎜⎜ − Z + ] −< + + ⎜⎜ + ∂< ∂ ] ⎟⎠ ⎪ 5 ∂X ∂V ⎝ 7 GV ⎠ ⎝ γ = + ⎪ ;< ∂ < < FRV ϕ = VLQ ϕ + − ⎪ ⎩ 5 5

⎧ ⎪ ∂Z ∂Y + ⎪ ω; = ∂< ∂] ⎪ ⎪ ⎪ ∂Z ∂Z ⎞ ∂ Z X VLQ ϕ ⎛ G ϕ ⎞ ⎛ ⎪ ⎟ ⎟⎟ ⎜⎜ Y + ] −< − + ⎜⎜ + ⎪⎪ ∂< ∂ ] ⎟⎠ 5 ∂ X ∂V ⎝ 7 GV ⎠ ⎝ ω = − ⎨ < ∂] < FRV ϕ = VLQ ϕ + − ⎪ 5 5 ⎪ ⎪ ∂Y ∂Y⎞ ∂ Y X FRV ϕ ⎛ G ϕ ⎞ ⎛ ⎪ ⎟ ⎟⎟ ⎜⎜ − Z + = −< + + ⎜⎜ + ∂ ∂ ] ⎟⎠ 5 7 G V < ⎪ ∂X ∂V ⎠ ⎝ ⎝ ⎪ ω= = ∂ < + < FRV ϕ = VLQ ϕ + − ⎪ 5 5 ⎪⎩

⎧ ∂X ⎞ ∂ X Y FRV ϕ Z VLQ ϕ ⎛ G ϕ ⎞ ⎛ ∂ X ⎟ ⎟⎟ ⎜⎜ = −< − + + ⎜⎜ + ⎪ ∂ ∂ ] ⎟⎠ V 5 5 7 G V < ∂ ⎠⎝ ⎝ ⎪ε = < FRV ϕ = VLQ ϕ ⎪ [ + − ⎪ 5 5 ⎪ ⎪ ⎪ ∂Y ∂Z ⎪ ε] = ⎪ ε< = ∂ < ∂] ⎪ ⎪ ⎪ ⎪ γ = ∂ Z + ∂ Y ⎨ <= ∂ < ∂ ] ⎪ ⎪ ⎪ ∂Z ∂Z⎞ ∂ Z X VLQ ϕ ⎛ G ϕ ⎞ ⎛ ⎟ ⎟ ⎜⎜ Y + = ⎪ −< − + ⎜⎜ + ⎟ ∂ ∂ = ⎟⎠ ∂ V 5 7 G V < ∂X ⎪ ⎠⎝ ⎝ γ = + ⎪ =; ∂ ] < FRV ϕ = VLQ ϕ + − ⎪ 5 5 ⎪ ⎪ ∂Y ∂Y⎞ ∂ Y X FRV ϕ ⎛ G ϕ ⎞ ⎛ ⎪ ⎟ ⎟⎟ ⎜⎜ − Z + ] −< + + ⎜⎜ + ∂< ∂ ] ⎟⎠ ⎪ 5 ∂X ∂V ⎝ 7 GV ⎠ ⎝ γ = + ⎪ ;< ∂ < < FRV ϕ = VLQ ϕ + − ⎪ ⎩ 5 5

⎧ ⎪ ∂Z ∂Y + ⎪ ω; = ∂< ∂] ⎪ ⎪ ⎪ ∂Z ∂Z ⎞ ∂ Z X VLQ ϕ ⎛ G ϕ ⎞ ⎛ ⎪ ⎟ ⎟⎟ ⎜⎜ Y + ] −< − + ⎜⎜ + ⎪⎪ ∂< ∂ ] ⎟⎠ 5 ∂ X ∂V ⎝ 7 GV ⎠ ⎝ ω = − ⎨ < ∂] < FRV ϕ = VLQ ϕ + − ⎪ 5 5 ⎪ ⎪ ∂Y ∂Y⎞ ∂ Y X FRV ϕ ⎛ G ϕ ⎞ ⎛ ⎪ ⎟ ⎟⎟ ⎜⎜ − Z + = −< + + ⎜⎜ + ∂ ∂ ] ⎟⎠ 5 7 G V < ⎪ ∂X ∂V ⎠ ⎝ ⎝ ⎪ ω= = ∂ < + < FRV ϕ = VLQ ϕ + − ⎪ 5 5 ⎪⎩



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- 189 -

- 189 -

G¶R

G¶R

⎧ ∂X ⎞ ∂ X Y FRV ϕ Z VLQ ϕ ⎛ G ϕ ⎞ ⎛ ∂ X ⎟ ⎟⎟ ⎜⎜ = −< − + + ⎜⎜ + ⎪ 5 5 ∂ ] ⎟⎠ ⎝ 7 GV ⎠ ⎝ ∂ < ⎪ ε = ∂V < FRV ϕ = VLQ ϕ ⎪ [ + − ⎪ 5 5 ⎪ ⎪ ⎪ ∂Y ∂Z ⎪ ε] = ⎪ ε< = ∂ < ∂] ⎪ ⎪ ⎪ ⎪ γ = ∂ Z + ∂ Y ⎨ <= ∂ < ∂ ] ⎪ ⎪ ⎪ ∂Z ∂Z⎞ ∂ Z X VLQ ϕ ⎛ G ϕ ⎞ ⎛ ⎟ ⎟⎟ ⎜⎜ Y + = ⎪ −< − + ⎜⎜ + ∂ ∂ = ⎟⎠ ∂ V 5 7 G V < ∂ X ⎪ ⎠ ⎝ ⎝ ⎪ γ =; = ∂ ] + < FRV ϕ = VLQ ϕ + − ⎪ 5 5 ⎪ ⎪ ∂Y ∂Y⎞ ∂ Y X FRV ϕ ⎛ G ϕ ⎞ ⎛ ⎪ ⎟ ⎟⎟ ⎜⎜ − Z + ] −< + + ⎜⎜ + ∂< ∂ ] ⎟⎠ ⎪ 5 ∂X ∂V ⎝ 7 GV ⎠ ⎝ γ = + ⎪ ;< ∂ < < FRV ϕ = VLQ ϕ + − ⎪ ⎩ 5 5

⎧ ⎪ ∂Z ∂Y + ⎪ ω; = ∂< ∂] ⎪ ⎪ ⎪ ∂Z ∂Z ⎞ ∂ Z X VLQ ϕ ⎛ G ϕ ⎞ ⎛ ⎪ ⎟ ⎟ ⎜⎜ Y + ] −< − + ⎜⎜ + ⎟ ⎪⎪ ∂ ∂ ] ⎟⎠ ∂ V 5 7 G V < ∂X ⎠⎝ ⎝ ω = − ⎨ < ∂] < FRV ϕ = VLQ ϕ + − ⎪ 5 5 ⎪ ⎪ ∂Y ∂Y⎞ ∂ Y X FRV ϕ ⎛ G ϕ ⎞ ⎛ ⎪ ⎟ ⎟⎟ ⎜⎜ − Z + = −< + + ⎜⎜ + ∂< ∂ ] ⎟⎠ 5 ⎪ ∂X ∂V ⎝ 7 GV ⎠ ⎝ ω = + ⎪ = ∂< < FRV ϕ = VLQ ϕ + − ⎪ 5 5 ⎩⎪

⎧ ∂X ⎞ ∂ X Y FRV ϕ Z VLQ ϕ ⎛ G ϕ ⎞ ⎛ ∂ X ⎟ ⎟⎟ ⎜⎜ = −< − + + ⎜⎜ + ⎪ 5 5 ∂ ] ⎟⎠ ⎝ 7 GV ⎠ ⎝ ∂ < ⎪ ε = ∂V < FRV ϕ = VLQ ϕ ⎪ [ + − ⎪ 5 5 ⎪ ⎪ ⎪ ∂Y ∂Z ⎪ ε] = ⎪ ε< = ∂ < ∂] ⎪ ⎪ ⎪ ⎪ γ = ∂ Z + ∂ Y ⎨ <= ∂ < ∂ ] ⎪ ⎪ ⎪ ∂Z ∂Z⎞ ∂ Z X VLQ ϕ ⎛ G ϕ ⎞ ⎛ ⎟ ⎟⎟ ⎜⎜ Y + = ⎪ −< − + ⎜⎜ + ∂ ∂ = ⎟⎠ ∂ V 5 7 G V < ∂ X ⎪ ⎠ ⎝ ⎝ ⎪ γ =; = ∂ ] + < FRV ϕ = VLQ ϕ + − ⎪ 5 5 ⎪ ⎪ ∂Y ∂Y⎞ ∂ Y X FRV ϕ ⎛ G ϕ ⎞ ⎛ ⎪ ⎟ ⎟⎟ ⎜⎜ − Z + ] −< + + ⎜⎜ + ∂< ∂ ] ⎟⎠ ⎪ 5 ∂X ∂V ⎝ 7 GV ⎠ ⎝ γ = + ⎪ ;< ∂ < < FRV ϕ = VLQ ϕ + − ⎪ ⎩ 5 5

⎧ ⎪ ∂Z ∂Y + ⎪ ω; = ∂< ∂] ⎪ ⎪ ⎪ ∂Z ∂Z ⎞ ∂ Z X VLQ ϕ ⎛ G ϕ ⎞ ⎛ ⎪ ⎟ ⎟ ⎜⎜ Y + ] −< − + ⎜⎜ + ⎟ ⎪⎪ ∂ ∂ ] ⎟⎠ ∂ V 5 7 G V < ∂X ⎠⎝ ⎝ ω = − ⎨ < ∂] < FRV ϕ = VLQ ϕ + − ⎪ 5 5 ⎪ ⎪ ∂Y ∂Y⎞ ∂ Y X FRV ϕ ⎛ G ϕ ⎞ ⎛ ⎪ ⎟ ⎟⎟ ⎜⎜ − Z + = −< + + ⎜⎜ + ∂< ∂ ] ⎟⎠ 5 ⎪ ∂X ∂V ⎝ 7 GV ⎠ ⎝ ω = + ⎪ = ∂< < FRV ϕ = VLQ ϕ + − ⎪ 5 5 ⎩⎪





- 189 -

- 189 -

DYHF G S ⋅ GV ⋅ GV

IRUFHGHFKDUJHPHQWDSSOLTXpHjODWUDQFKHGV

DYHF G S ⋅ GV


FRXSOHGHFKDUJHPHQWDSSOLTXpjODWUDQFKHGV

⋅ GV


R

UpVXOWDQWHHWPRPHQWGXYLVVHXUVXU6

R


ρ

PDVVHYROXPLTXHGXPDWpULDX

ρ


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, - .


ΓU ΓH ΓF

DFFpOpUDWLRQUHODWLYHG¶HQWUDvQHPHQWGH&RULROLVGH3

ΓU ΓH ΓF


−ρ

⋅ GV HW

−ρ

⋅ GV

I

SVHXGRYHFWHXUURWDWLRQGH6 jO¶LQVWDQWWGDQV {* ;<=}

−ρ

⋅ GV HW

PRPHQWVHQ*GHVIRUFHVG¶LQHUWLHG¶HQWUDvQHPHQWHWGH&RULROLVDSSOLTXpHV jODWUDQFKHGV

−ρ

⋅ GV

WHQVHXUGHVPRPHQWVTXDGUDWLTXHVGH6 HQ*DYHF

I

⎡, ; ⎢ [I ] = ⎢ ⎢ ⎢⎣

,<

WHQVHXUGHVPRPHQWVTXDGUDWLTXHVGH6 HQ*DYHF ⎡, ; ⎢ [I ] = ⎢ ⎢ ⎢⎣

DYHF

9U YLWHVVHUHODWLYHGDQV {* ;<=}GHODSDUWLFXOH3

,<

⎤ ⎥ ⎥ GDQV {* ;<=} ⎥ , = ⎥⎦

2QUDSSHOOHTXHO¶DFFpOpUDWLRQGH&RULROLVHVWGRQQpHSDUODIRUPXOH

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}

⎧ ⎛ ∂ X ⎛ ∂ Z ∂ Y ⎞⎞ ∂ 1 FRV ϕ VLQ ϕ ⎟⎟ − 7< + 7] = ρ 6 ⎜ + ΓH[ + ⎜⎜ θ < − θ] ⎪ S; + ⎜ ∂W ∂V 5 5 ∂W ∂ W ⎟⎠ ⎟⎠ ⎝ ⎪ ⎝ ⎪ ⎛ ∂ Y ⎛ Gϕ⎞ ⎛ ∂ X ∂ Z ⎞⎞ ∂ 7< FRV ϕ ⎪ ⎟⎟ 7= = ρ 6 ⎜ + ΓH< + ⎜⎜ θ = ⎟⎟ ⎟ + − θ; 1 − ⎜⎜ + ⎨ S< + ⎟ ⎜ ∂ ∂ ∂ V 5 7 G V W W ∂ W ⎝ ⎠ ⎝ ⎠⎠ ⎪ ⎝ ⎪ ⎪ S + ∂ 7] − VLQ ϕ 1 + ⎛⎜ + G ϕ ⎞⎟ 7 = ρ 6 ⎛⎜ ∂ Z + Γ + ⎛⎜ θ ∂ Y − θ ∂ X ⎞⎟ ⎞⎟ H] \ = < ⎜ ⎟ ⎜ ; ∂W ⎪ ⎜ ∂ W ∂V ∂ W ⎟⎠ ⎟⎠ 5 ⎝ 7 GV ⎠ ⎝ ⎝ ⎩


{

}

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⎧ ⎛ ∂ X ⎛ ∂ Z ∂ Y ⎞⎞ ∂ 1 FRV ϕ VLQ ϕ ⎟⎟ − 7< + 7] = ρ 6 ⎜ + ΓH[ + ⎜⎜ θ < − θ] ⎪ S; + ⎜ ∂W ∂V 5 5 ∂W ∂ W ⎟⎠ ⎟⎠ ⎝ ⎪ ⎝ ⎪ ⎛ ∂ Y ⎛ Gϕ⎞ ⎛ ∂ X ∂ Z ⎞⎞ ∂ 7< FRV ϕ ⎪ ⎟⎟ 7= = ρ 6 ⎜ + ΓH< + ⎜⎜ θ = ⎟⎟ ⎟ + − θ; 1 − ⎜⎜ + ⎨ S< + ⎟ ⎜ ∂ ∂ ∂ V 5 7 G V W W ∂ W ⎝ ⎠ ⎝ ⎠⎠ ⎪ ⎝ ⎪ ⎪ S + ∂ 7] − VLQ ϕ 1 + ⎛⎜ + G ϕ ⎞⎟ 7 = ρ 6 ⎛⎜ ∂ Z + Γ + ⎛⎜ θ ∂ Y − θ ∂ X ⎞⎟ ⎞⎟ H] \ = < ⎜ ⎟ ⎜ ; ∂W ⎪ ⎜ ∂ W ∂V ∂ W ⎟⎠ ⎟⎠ 5 ⎝ 7 GV ⎠ ⎝ ⎝ ⎩

- 190 -

- 190 -

DYHF G S ⋅ GV ⋅ GV


DYHF G S ⋅ GV



⋅ GV


R


R


ρ


ρ


, - .


, - .


ΓU ΓH ΓF


ΓU ΓH ΓF


−ρ

⋅ GV HW

−ρ

⋅ GV

I


−ρ

⋅ GV HW


−ρ

⋅ GV

WHQVHXUGHVPRPHQWVTXDGUDWLTXHVGH6 HQ*DYHF

I

⎡, ; ⎢ [I ] = ⎢ ⎢ ⎢⎣

,<

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DYHF


}


⎧ ⎛ ∂ X ⎛ ∂ Z ∂ Y ⎞⎞ ∂ 1 FRV ϕ VLQ ϕ ⎟⎟ − 7< + 7] = ρ 6 ⎜ + ΓH[ + ⎜⎜ θ < − θ] ⎪ S; + ⎜ ∂V 5 5 ∂W ∂ W ⎟⎠ ⎟⎠ ⎝ ⎪ ⎝ ∂W ⎪ ⎛ ∂ Y ⎛ Gϕ⎞ ⎛ ∂ X ∂ Z ⎞⎞ ∂ 7< FRV ϕ ⎪ ⎟⎟ 7= = ρ 6 ⎜ + ΓH< + ⎜⎜ θ = ⎟ ⎟ + − θ; 1 − ⎜⎜ + ⎨ S< + ⎜ ∂W ∂ ∂ W ⎟⎠ ⎟⎠ V 5 7 G V ⎝ ⎠ ⎝ ∂W ⎪ ⎝ ⎪ ⎪ S + ∂ 7] − VLQ ϕ 1 + ⎛⎜ + G ϕ ⎞⎟ 7 = ρ 6 ⎛⎜ ∂ Z + Γ + ⎛⎜ θ ∂ Y − θ ∂ X ⎞⎟ ⎞⎟ H] \ ⎜ 7 GV ⎟ < ⎜ ; ∂W ⎪ = ∂V ⎜ ∂ W ∂ W ⎟⎠ ⎟⎠ 5 ⎝ ⎠ ⎝ ⎝ ⎩

- 190 -

,<

⎤ ⎥ ⎥ GDQV {* ;<=} ⎥ , = ⎥⎦




⎤ ⎥ ⎥ GDQV {* ;<=} ⎥ , = ⎥⎦


{

DYHF




⎤ ⎥ ⎥ GDQV {* ;<=} ⎥ , = ⎥⎦


{


DYHF

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{

}


⎧ ⎛ ∂ X ⎛ ∂ Z ∂ Y ⎞⎞ ∂ 1 FRV ϕ VLQ ϕ ⎟⎟ − 7< + 7] = ρ 6 ⎜ + ΓH[ + ⎜⎜ θ < − θ] ⎪ S; + ⎜ ∂V 5 5 ∂W ∂ W ⎟⎠ ⎟⎠ ⎝ ⎪ ⎝ ∂W ⎪ ⎛ ∂ Y ⎛ Gϕ⎞ ⎛ ∂ X ∂ Z ⎞⎞ ∂ 7< FRV ϕ ⎪ ⎟⎟ 7= = ρ 6 ⎜ + ΓH< + ⎜⎜ θ = ⎟ ⎟ + − θ; 1 − ⎜⎜ + ⎨ S< + ⎜ ∂W ∂ ∂ W ⎟⎠ ⎟⎠ V 5 7 G V ⎝ ⎠ ⎝ ∂W ⎪ ⎝ ⎪ ⎪ S + ∂ 7] − VLQ ϕ 1 + ⎛⎜ + G ϕ ⎞⎟ 7 = ρ 6 ⎛⎜ ∂ Z + Γ + ⎛⎜ θ ∂ Y − θ ∂ X ⎞⎟ ⎞⎟ H] \ ⎜ 7 GV ⎟ < ⎜ ; ∂W ⎪ = ∂V ⎜ ∂ W ∂ W ⎟⎠ ⎟⎠ 5 ⎝ ⎠ ⎝ ⎝ ⎩

- 190 -

⎧ ⎛ ∂ ω\ ⎞ ∂ 0 [ FRV ϕ VLQ ϕ ⎪ P; + − + MH[ + MF[ ⎟ 0< + 0] = ρ ⎜ ,[ ⎜ ⎟ ∂V 5 5 ∂W ⎪ ⎝ ⎠ ⎪ ⎛ ∂ ω[ ⎞ ⎪ ⎛ Gϕ⎞ ∂ 0 < FRV ϕ ⎟⎟ 0 = − 7= = ρ ⎜ , < + + MH< + MF< ⎟ 0 ; − ⎜⎜ + ⎨ P< + ⎜ ⎟ ∂V 5 ∂W ⎝ 7 GV ⎠ ⎪ ⎝ ⎠ ⎪ ⎛ ⎞ ⎪ P + ∂ 0 ] − VLQ ϕ 0 + ⎛⎜ + G ϕ ⎞⎟ 0 + 7 = ρ ⎜ , ∂ ω] + M + M ⎟ H] F] ⎟ ; ⎜ \ ⎟ < ⎜ ] ∂ W ⎪ = ∂V 5 ⎝ 7 GV ⎠ ⎝ ⎠ ⎩

⎧ ⎛ ∂ ω\ ⎞ ∂ 0 [ FRV ϕ VLQ ϕ ⎪ P; + − + MH[ + MF[ ⎟ 0< + 0] = ρ ⎜ ,[ ⎜ ⎟ ∂V 5 5 ∂W ⎪ ⎝ ⎠ ⎪ ⎛ ∂ ω[ ⎞ ⎪ ⎛ Gϕ⎞ ∂ 0 < FRV ϕ ⎟⎟ 0 = − 7= = ρ ⎜ , < + + MH< + MF< ⎟ 0 ; − ⎜⎜ + ⎨ P< + ⎜ ⎟ ∂V 5 ∂W ⎝ 7 GV ⎠ ⎪ ⎝ ⎠ ⎪ ⎛ ⎞ ⎪ P + ∂ 0 ] − VLQ ϕ 0 + ⎛⎜ + G ϕ ⎞⎟ 0 + 7 = ρ ⎜ , ∂ ω] + M + M ⎟ H] F] ⎟ ; ⎜ \ ⎟ < ⎜ ] ∂ W ⎪ = ∂V 5 ⎝ 7 GV ⎠ ⎝ ⎠ ⎩

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= Z = 7] = 0 [ = 0 < = ω[ = ω< = 7

ϕ =

= Z = 7] = 0 [ = 0 < = ω[ = ω< = 7



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- 191 -

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⎧ ⎛ ∂ ω\ ⎞ ∂ 0 [ FRV ϕ VLQ ϕ ⎪ P; + − + MH[ + MF[ ⎟ 0< + 0] = ρ ⎜ ,[ ⎜ ⎟ ∂V 5 5 ∂W ⎪ ⎝ ⎠ ⎪ ⎛ ∂ ω[ ⎞ ⎪ ⎛ Gϕ⎞ ∂ 0 < FRV ϕ ⎟⎟ 0 = − 7= = ρ ⎜ , < + + MH< + MF< ⎟ 0 ; − ⎜⎜ + ⎨ P< + ⎜ ⎟ ∂ V 5 7 G V ∂W ⎝ ⎠ ⎪ ⎝ ⎠ ⎪ ⎪ P + ∂ 0 ] − VLQ ϕ 0 + ⎛⎜ + G ϕ ⎞⎟ 0 + 7 = ρ ⎛⎜ , ∂ ω] + M + M ⎞⎟ H] F] ⎟ ; ⎜ \ ⎟ < ⎜ ] ∂ W ⎪ = ∂V 5 ⎝ 7 GV ⎠ ⎝ ⎠ ⎩

⎧ ⎛ ∂ ω\ ⎞ ∂ 0 [ FRV ϕ VLQ ϕ ⎪ P; + − + MH[ + MF[ ⎟ 0< + 0] = ρ ⎜ ,[ ⎜ ⎟ ∂V 5 5 ∂W ⎪ ⎝ ⎠ ⎪ ⎛ ∂ ω[ ⎞ ⎪ ⎛ Gϕ⎞ ∂ 0 < FRV ϕ ⎟⎟ 0 = − 7= = ρ ⎜ , < + + MH< + MF< ⎟ 0 ; − ⎜⎜ + ⎨ P< + ⎜ ⎟ ∂ V 5 7 G V ∂W ⎝ ⎠ ⎪ ⎝ ⎠ ⎪ ⎪ P + ∂ 0 ] − VLQ ϕ 0 + ⎛⎜ + G ϕ ⎞⎟ 0 + 7 = ρ ⎛⎜ , ∂ ω] + M + M ⎞⎟ H] F] ⎟ ; ⎜ \ ⎟ < ⎜ ] ∂ W ⎪ = ∂V 5 ⎝ 7 GV ⎠ ⎝ ⎠ ⎩



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⎧ X = X + = ⋅ ω\ − < ⋅ ω] ⎪ ⎪ 3R3 ⎨ Y = Y − = ⋅ ω[ ⎪ ⎪⎩ Z = Z + < ⋅ ω[

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⎧ ⎪ ω[ ⎪ ⎪ ⎛ Gϕ⎞ ∂Z VLQ ϕ ⎪ ⎟⎟ Y − ⎨ ω< = X − ⎜⎜ + ∂V 5 ⎝ 7 GV ⎠ ⎪ ⎪ ⎪ ω = FRV ϕ X − ⎛⎜ + G ϕ ⎞⎟ Z + ∂ Y ⎜ 7 GV ⎟ ⎪ ] ∂V 5 ⎝ ⎠ ⎩

⎧ ⎪ ω[ ⎪ ⎪ ⎛ Gϕ⎞ ∂Z VLQ ϕ ⎪ ⎟⎟ Y − ⎨ ω< = X − ⎜⎜ + ∂V 5 ⎝ 7 GV ⎠ ⎪ ⎪ ⎪ ω = FRV ϕ X − ⎛⎜ + G ϕ ⎞⎟ Z + ∂ Y ⎜ 7 GV ⎟ ⎪ ] ∂V 5 ⎝ ⎠ ⎩

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⎧ ⎪ ω[ ⎪ ⎪ ⎛ Gϕ⎞ ∂Z VLQ ϕ ⎪ ⎟⎟ Y − ⎨ ω< = X − ⎜⎜ + ∂V 5 7 G V ⎝ ⎠ ⎪ ⎪ ⎪ ω = FRV ϕ X − ⎛⎜ + G ϕ ⎞⎟ Z + ∂ Y ⎜ 7 GV ⎟ ⎪ ] ∂V 5 ⎝ ⎠ ⎩

⎧ ⎪ ω[ ⎪ ⎪ ⎛ Gϕ⎞ ∂Z VLQ ϕ ⎪ ⎟⎟ Y − ⎨ ω< = X − ⎜⎜ + ∂V 5 7 G V ⎝ ⎠ ⎪ ⎪ ⎪ ω = FRV ϕ X − ⎛⎜ + G ϕ ⎞⎟ Z + ∂ Y ⎜ 7 GV ⎟ ⎪ ] ∂V 5 ⎝ ⎠ ⎩









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⎧ ⎛ ∂Λ ⎞ ⎟ = ∂ X − FRV ϕ Y + VLQ ϕ Z ⎪ ⎜⎜ 5 5 ⎪ ⎝ ∂ V ⎟⎠ ; ∂ V ⎪ ⎪ ⎛ Gϕ ⎞ ∂ Λ ⎪ ⎛⎜ ∂ Λ ⎞⎟ ∂ Y FRV ϕ ⎟⎟ Z = + X − ⎜⎜ + ⎨⎜ ⎟ ∂V ⎪ ⎝ ∂V ⎠ ∂V 5 ⎝ 7 GV ⎠ < ⎪ ⎪ ⎪ ⎛⎜ ∂ Λ ⎞⎟ = ∂ Z − VLQ ϕ X + ⎛⎜ + G ϕ ⎞⎟ Y ⎜ 7 GV ⎟ ⎪ ⎜ ∂V ⎟ 5 ⎝ ⎠ ⎠= ∂ V ⎩⎝ ⎧ ⎛∂ω⎞ ∂ ω[ FRV ϕ VLQ ϕ ⎟⎟ = − ω< + ω= ⎪ ⎜⎜ ∂ ∂ V V 5 5 ⎠; ⎪⎝ ⎪ ⎪ ∂ ω\ FRV ϕ ⎛ Gϕ⎞ ⎪ ⎛∂ω⎞ ⎟⎟ = ⎟⎟ ω= + ω; − ⎜⎜ + ⎨ ⎜⎜ ∂V 5 ⎪ ⎝ ∂ V ⎠< ⎝ 7 GV ⎠ ⎪ ⎪ ⎪ ⎛⎜ ∂ ω ⎞⎟ = ∂ ω] − VLQ ϕ ω + ⎛⎜ + G ϕ ⎞⎟ ω ; ⎜ ⎟ < ⎪ ⎜ ∂V ⎟ ∂V 5 ⎠= ⎝ 7 GV ⎠ ⎩⎝


⎧ ⎛ ∂Λ ⎞ ⎟ = ∂ X − FRV ϕ Y + VLQ ϕ Z ⎪ ⎜⎜ 5 5 ⎪ ⎝ ∂ V ⎟⎠ ; ∂ V ⎪ ⎪ ⎛ Gϕ ⎞ ∂ Λ ⎪ ⎛⎜ ∂ Λ ⎞⎟ ∂ Y FRV ϕ ⎟⎟ Z = + X − ⎜⎜ + ⎨⎜ ⎟ ∂V ⎪ ⎝ ∂V ⎠ ∂V 5 ⎝ 7 GV ⎠ < ⎪ ⎪ ⎪ ⎛⎜ ∂ Λ ⎞⎟ = ∂ Z − VLQ ϕ X + ⎛⎜ + G ϕ ⎞⎟ Y ⎜ 7 GV ⎟ ⎪ ⎜ ∂V ⎟ 5 ⎝ ⎠ ⎠= ∂ V ⎩⎝ ⎧ ⎛∂ω⎞ ∂ ω[ FRV ϕ VLQ ϕ ⎟⎟ = − ω< + ω= ⎪ ⎜⎜ ∂ ∂ V V 5 5 ⎠; ⎪⎝ ⎪ ⎪ ∂ ω\ FRV ϕ ⎛ Gϕ⎞ ⎪ ⎛∂ω⎞ ⎟⎟ = ⎟⎟ ω= + ω; − ⎜⎜ + ⎨ ⎜⎜ ∂V 5 ⎪ ⎝ ∂ V ⎠< ⎝ 7 GV ⎠ ⎪ ⎪ ⎪ ⎛⎜ ∂ ω ⎞⎟ = ∂ ω] − VLQ ϕ ω + ⎛⎜ + G ϕ ⎞⎟ ω ; ⎜ ⎟ < ⎪ ⎜ ∂V ⎟ ∂V 5 ⎠= ⎝ 7 GV ⎠ ⎩⎝

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- 193 -

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⎧ ⎛ ∂Λ ⎞ ⎟ = ∂ X − FRV ϕ Y + VLQ ϕ Z ⎪ ⎜⎜ ⎟ ∂ 5 V 5 ⎪⎝ ⎠; ∂ V ⎪ ⎪ ⎛ Gϕ ⎞ ∂ Λ ⎪ ⎛⎜ ∂ Λ ⎞⎟ ∂ Y FRV ϕ ⎟ Z = + X − ⎜⎜ + ⎨ ∂ V ⎪ ⎜⎝ ∂ V ⎟⎠ ∂V 5 7 G V ⎟⎠ ⎝ < ⎪ ⎪ ⎪ ⎛⎜ ∂ Λ ⎞⎟ = ∂ Z − VLQ ϕ X + ⎛⎜ + G ϕ ⎞⎟ Y ⎜ 7 GV ⎟ ⎪ ⎜ ∂V ⎟ 5 ⎝ ⎠ ⎠= ∂ V ⎩⎝ ⎧ ⎛∂ω⎞ ∂ ω[ FRV ϕ VLQ ϕ ⎟⎟ = − ω< + ω= ⎪ ⎜⎜ ∂V 5 5 ⎪ ⎝ ∂ V ⎠; ⎪ ⎪ ∂ ω\ FRV ϕ ⎛ Gϕ⎞ ⎪ ⎛∂ω⎞ ⎟⎟ = ⎟⎟ ω= + ω; − ⎜⎜ + ⎨ ⎜⎜ ∂V 5 ⎪ ⎝ ∂ V ⎠< ⎝ 7 GV ⎠ ⎪ ⎪ ⎪ ⎛⎜ ∂ ω ⎞⎟ = ∂ ω] − VLQ ϕ ω + ⎛⎜ + G ϕ ⎞⎟ ω ; ⎜ ⎟ < ⎪ ⎜ ∂V ⎟ ∂V 5 ⎠= ⎝ 7 GV ⎠ ⎩⎝


⎧ ⎛ ∂Λ ⎞ ⎟ = ∂ X − FRV ϕ Y + VLQ ϕ Z ⎪ ⎜⎜ ⎟ ∂ 5 V 5 ⎪⎝ ⎠; ∂ V ⎪ ⎪ ⎛ Gϕ ⎞ ∂ Λ ⎪ ⎛⎜ ∂ Λ ⎞⎟ ∂ Y FRV ϕ ⎟ Z = + X − ⎜⎜ + ⎨ ∂ V ⎪ ⎜⎝ ∂ V ⎟⎠ ∂V 5 7 G V ⎟⎠ ⎝ < ⎪ ⎪ ⎪ ⎛⎜ ∂ Λ ⎞⎟ = ∂ Z − VLQ ϕ X + ⎛⎜ + G ϕ ⎞⎟ Y ⎜ 7 GV ⎟ ⎪ ⎜ ∂V ⎟ 5 ⎝ ⎠ ⎠= ∂ V ⎩⎝ ⎧ ⎛∂ω⎞ ∂ ω[ FRV ϕ VLQ ϕ ⎟⎟ = − ω< + ω= ⎪ ⎜⎜ ∂V 5 5 ⎪ ⎝ ∂ V ⎠; ⎪ ⎪ ∂ ω\ FRV ϕ ⎛ Gϕ⎞ ⎪ ⎛∂ω⎞ ⎟⎟ = ⎟⎟ ω= + ω; − ⎜⎜ + ⎨ ⎜⎜ ∂V 5 ⎪ ⎝ ∂ V ⎠< ⎝ 7 GV ⎠ ⎪ ⎪ ⎪ ⎛⎜ ∂ ω ⎞⎟ = ∂ ω] − VLQ ϕ ω + ⎛⎜ + G ϕ ⎞⎟ ω ; ⎜ ⎟ < ⎪ ⎜ ∂V ⎟ ∂V 5 ⎠= ⎝ 7 GV ⎠ ⎩⎝



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⎧ ⎪ ⎪ *R* ⎨ Y [ W ⎪ ⎪⎩

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∂Y ∂W

⎧ ⎪ ⎪ ΓU ⎨ ⎪ ⎪⎩

∂ Y ∂W

⎧ ⎪ ⎪ ΓH ⎨ − θ ⋅ Y ⎪ ⎪⎩

⎧ ⎪ ⎪⎪ ΓF ⎨ ⎪ ⎪ θ ⋅ ∂ Y ⎪⎩ ∂W

⎧ ⎪ G⎪ J ⎨ − J FRV θ ⋅ W ⎪ ⎪⎩ J VLQ θ ⋅ W

⎧ ⎪ ⎪ *R* ⎨ Y [ W ⎪ ⎪⎩

⎧ ⎪ ⎪ 9U ⎨ ⎪ ⎪⎩

∂Y ∂W

⎧ ⎪ ⎪ ΓU ⎨ ⎪ ⎪⎩

∂ Y ∂ W

- 196 -

⎧ ⎪ ⎪ ΓH ⎨ − θ ⋅ Y ⎪ ⎪⎩

⎧ ⎪ ⎪⎪ ΓF ⎨ ⎪ ⎪ θ ⋅ ∂ Y ⎪⎩ ∂W

⎧ ⎪ G⎪ J ⎨ − J FRV θ ⋅ W ⎪ ⎪⎩ J VLQ θ ⋅ W

- 196 -

)LJXUHD

)LJXUHD

)LJXUHE

)LJXUHE

)LJXUHF

)LJXUHF









⎧ ⎪ ⎪ *R* ⎨ Y [ W ⎪ ⎪⎩

⎧ ⎪ ⎪ 9U ⎨ ⎪ ⎪⎩

∂Y ∂W

⎧ ⎪ ⎪ ΓU ⎨ ⎪ ⎪⎩

∂ Y ∂W

⎧ ⎪ ⎪ ΓH ⎨ − θ ⋅ Y ⎪ ⎪⎩

- 196 -

⎧ ⎪ ⎪⎪ ΓF ⎨ ⎪ ⎪ θ ⋅ ∂ Y ⎪⎩ ∂W

⎧ ⎪ G⎪ J ⎨ − J FRV θ ⋅ W ⎪ ⎪⎩ J VLQ θ ⋅ W

⎧ ⎪ ⎪ *R* ⎨ Y [ W ⎪ ⎪⎩

⎧ ⎪ ⎪ 9U ⎨ ⎪ ⎪⎩

∂Y ∂W

⎧ ⎪ ⎪ ΓU ⎨ ⎪ ⎪⎩

∂ Y ∂ W

⎧ ⎪ ⎪ ΓH ⎨ − θ ⋅ Y ⎪ ⎪⎩

- 196 -

⎧ ⎪ ⎪⎪ ΓF ⎨ ⎪ ⎪ θ ⋅ ∂ Y ⎪⎩ ∂W

⎧ ⎪ G⎪ J ⎨ − J FRV θ ⋅ W ⎪ ⎪⎩ J VLQ θ ⋅ W

/HVpTXDWLRQVV¶pFULYHQW


⎧ ∂ 7\ ⎞ ⎛ ∂ Y = ρ 6 ⎜ − θ ⋅ Y + J ⋅ FRV θ W ⎟ °IRUPXOH ⎪ ⎟ ⎜ ⎪ ∂[ ⎠ ⎝ ∂W ⎪ ⎪ ∂ 0] ⎪ ∂ [ + 7\ = °IRUPXOH HQQpJOLJHDQWOHVFRXSOHVLQHUWLHOV ⎪ ⎨ ⎪ ∂Y ⎪ ω] = ∂ [ °IRUPXOH ⎪ ⎪ ⎪ 0 = (, ∂ ω] °IRUPXOH HQO¶ DEVHQFHG¶ DPRUWLVVHPHQW ⎪ ] ∂[ ⎩ 2QHQGpGXLWO¶pTXDWLRQGXPRXYHPHQW ∂ Y + ∂ [ / &

2QHQGpGXLWO¶pTXDWLRQGXPRXYHPHQW

⎛ ∂ Y ⎞ (, J ⎜ − θ ⋅ Y ⎟ = − FRV θ W DYHF & / = ⎟ ⎜ ∂W ρ 6 / & ⎝ ⎠

(Q SRVDQW Y [ W = I [ ⋅ J W HW FRPSWH WHQX GHV FRQGLWLRQV DX[ OLPLWHV I = I ′′ = I / = I ′′/ = RQ WURXYH OD VROXWLRQ JpQpUDOH GH O¶pTXDWLRQ VDQV VHFRQG PHPEUH ∞

Y [ W =

∑

VLQ Q π

Q =

ωQ = Q π

DYHF

[ ($ Q FRV ωQ W + %Q VLQ ωQ W ) /

& −θ /

Y [ W = FRV θ W

∞

∑

D Q VLQ

Q =

DQ =

± SRXUQLPSDLU D Q =

[

(Q SRVDQW Y [ W = I [ ⋅ J W HW FRPSWH WHQX GHV FRQGLWLRQV DX[ OLPLWHV I = I ′′ = I / = I ′′/ = RQ WURXYH OD VROXWLRQ JpQpUDOH GH O¶pTXDWLRQ VDQV VHFRQG PHPEUH ∞

Y [ W =

ωQ = Q π

⎛ Q π & Q π ⎜ θ − ⎜ / ⎝

⎞ ⎟ ⎟ ⎠

DQ =

DQ =

[

]

J − (− )Q ⎛ Q π & ⎞⎟ Q π ⎜ θ − ⎜ / ⎟⎠ ⎝

J ⎛ Q π & ⎞⎟ Q π ⎜ θ − ⎟ ⎜ / ⎝ ⎠


⎛∂ Y ⎞ (, J ⎜ − θ ⋅ Y ⎟ = − FRV θ W DYHF & / = ⎟ ⎜ ∂ W ρ 6 / & ⎝ ⎠

∞

Y [ W =

∑ Q =

ωQ = Q π

[ VLQ Q π ($ Q FRV ωQ W + %Q VLQ ωQ W ) /

& −θ /

Y [ W = FRV θ W

∞

∑

D Q VLQ

Q =

DQ =

[

∞

⎛ Q π & Q π ⎜ θ − ⎜ / ⎝

⎞ ⎟ ⎟ ⎠

- 197 -

∑ VLQ Q π / ($ [

Q FRV ωQ W

+ %Q VLQ ωQ W )

ωQ = Q π

& −θ /

&KHUFKRQVXQHVROXWLRQSDUWLFXOLqUHGHO¶pTXDWLRQFRPSOqWHVRXVODIRUPH Y [ W = FRV θ W

∞

∑D

Q VLQ

Q =

Qπ[ /

2QWURXYHSDUODPpWKRGHFODVVLTXHGHFDOFXOGHVFRHIILFLHQWVGH)RXULHU

]

DQ =

± SRXUQSDLU

Y [ W =

Q =

J − (− )Q ⎛ Q π & ⎞ ⎟ Q π ⎜ θ − ⎜ / ⎟⎠ ⎝

J

⎛ ∂ Y ⎞ (, J ⎜ − θ ⋅ Y ⎟ = − FRV θ W DYHF & / = ⎟ ⎜ ∂ W ρ 6 / & ⎝ ⎠

(Q SRVDQW Y [ W = I [ ⋅ J W HW FRPSWH WHQX GHV FRQGLWLRQV DX[ OLPLWHV I = I ′′ = I / = I ′′/ = RQ WURXYH OD VROXWLRQ JpQpUDOH GH O¶pTXDWLRQ VDQV VHFRQG PHPEUH

Qπ[ /


DQ =

∂ Y + ∂ [ / &

DYHF

&KHUFKRQVXQHVROXWLRQSDUWLFXOLqUHGHO¶pTXDWLRQFRPSOqWHVRXVODIRUPH


Qπ[ /

⎧ ∂ 7\ ⎞ ⎛ ∂ Y = ρ 6 ⎜ − θ ⋅ Y + J ⋅ FRV θ W ⎟ °IRUPXOH ⎪ ⎟ ⎜ ∂W [ ∂ ⎪ ⎠ ⎝ ⎪ ⎪ ∂ 0] ⎪ ∂ [ + 7\ = °IRUPXOH HQQpJOLJHDQWOHVFRXSOHVLQHUWLHOV ⎪ ⎨ ⎪ ∂Y ⎪ ω] = ∂ [ °IRUPXOH ⎪ ⎪ ⎪ 0 = (, ∂ ω] °IRUPXOH HQO¶ DEVHQFHG¶ DPRUWLVVHPHQW ⎪ ] ∂[ ⎩

(Q SRVDQW Y [ W = I [ ⋅ J W HW FRPSWH WHQX GHV FRQGLWLRQV DX[ OLPLWHV I = I ′′ = I / = I ′′/ = RQ WURXYH OD VROXWLRQ JpQpUDOH GH O¶pTXDWLRQ VDQV VHFRQG PHPEUH

± SRXUQSDLU

Q VLQ



DYHF

∑D

- 197 -

⎧ ∂ 7\ ⎞ ⎛ ∂ Y = ρ 6 ⎜ − θ ⋅ Y + J ⋅ FRV θ W ⎟ °IRUPXOH ⎪ ⎟ ⎜ ∂W [ ∂ ⎪ ⎠ ⎝ ⎪ ⎪ ∂ 0] ⎪ ∂ [ + 7\ = °IRUPXOH HQQpJOLJHDQWOHVFRXSOHVLQHUWLHOV ⎪ ⎨ ⎪ ∂Y ⎪ ω] = ∂ [ °IRUPXOH ⎪ ⎪ ⎪ 0 = (, ∂ ω] °IRUPXOH HQO¶ DEVHQFHG¶ DPRUWLVVHPHQW ⎪ ] ∂[ ⎩

+ %Q VLQ ωQ W )




∂ Y + ∂ [ / &

∞

Q =

- 197 -

Q FRV ωQ W

& −θ /

Y [ W = FRV θ W

]

[


± SRXUQSDLU

∑ VLQ Q π / ($ Q =

J − (− )Q ⎛ Q π & ⎞⎟ Q π ⎜ θ − ⎜ / ⎟⎠ ⎝

J

⎛ ∂ Y ⎞ (, J ⎜ − θ ⋅ Y ⎟ = − FRV θ W DYHF & / = ⎟ ⎜ ∂W ρ 6 / & ⎝ ⎠

Qπ[ /


DQ =

∂ Y + ∂ [ / &

DYHF


± SRXUQSDLU

⎧ ∂ 7\ ⎞ ⎛ ∂ Y = ρ 6 ⎜ − θ ⋅ Y + J ⋅ FRV θ W ⎟ °IRUPXOH ⎪ ⎟ ⎜ ⎪ ∂[ ⎠ ⎝ ∂W ⎪ ⎪ ∂ 0] ⎪ ∂ [ + 7\ = °IRUPXOH HQQpJOLJHDQWOHVFRXSOHVLQHUWLHOV ⎪ ⎨ ⎪ ∂Y ⎪ ω] = ∂ [ °IRUPXOH ⎪ ⎪ ⎪ 0 = (, ∂ ω] °IRUPXOH HQO¶ DEVHQFHG¶ DPRUWLVVHPHQW ⎪ ] ∂[ ⎩

DQ =


[

]

J − (− )Q ⎛ Q π & ⎞ ⎟ Q π ⎜ θ − ⎜ / ⎟⎠ ⎝

J ⎛ Q π & ⎞⎟ Q π ⎜ θ − ⎟ ⎜ / ⎝ ⎠

- 197 -

2QUHPDUTXHTXH D Q GHYLHQWLQILQLVL θ =

Q π & /

2QUHPDUTXHTXH D Q GHYLHQWLQILQLVL θ =

Q π & /

2Q D GRQF XQH VXLWH GH YDOHXUV FULWLTXHV SRXU OD YLWHVVH GH URWDWLRQ θ Q π & θ &5 Q = = ωQ & 5 ′ /


/RUVTXH θ VH UDSSURFKH GH θ &5 Q OH QLqPH PRGH HQWUH HQ UpVRQDQFH VRQ DPSOLWXGH DXJPHQWDQWFRQVLGpUDEOHPHQW


±752,6,Ê0($33/,&$7,219,%5$7,2163/$1(6'¶81$11($8

±752,6,Ê0($33/,&$7,219,%5$7,2163/$1(6'¶81$11($8

8QDQQHDXFLUFXODLUHGHVHFWLRQGURLWHFRQVWDQWH6DGPHW[R\ FRPPHSODQGHV\PpWULH/HV D[HV VRQW JDOLOpHQV $ O¶LQVWDQW LQLWLDO O¶DQQHDX HVW DX UHSRV RQ DSSOLTXH DORUV XQH IRUFH UDGLDOHH[FLWDWULFHXQLIRUPpPHQWUpSDUWLH S < = S VLQ ω W RQVHSURSRVHG¶pWXGLHUODUpSRQVH DPRUWLH


)LJXUH

- 198 -

)LJXUH

- 198 -

Q π & 2QUHPDUTXHTXH D Q GHYLHQWLQILQLVL θ = /

Q π & 2QUHPDUTXHTXH D Q GHYLHQWLQILQLVL θ = /





±752,6,Ê0($33/,&$7,219,%5$7,2163/$1(6'¶81$11($8

±752,6,Ê0($33/,&$7,219,%5$7,2163/$1(6'¶81$11($8



)LJXUH

- 198 -

)LJXUH

- 198 -

/HPRXYHPHQWHVWSXUHPHQWUDGLDOHWFRQVLVWHHQXQH©UHVSLUDWLRQªGHO¶DQQHDX


6RQ pTXDWLRQ V¶REWLHQW SDU pOLPLQDWLRQ GH O¶HIIRUW QRUPDO 1W HQWUH O¶pTXDWLRQ G¶pTXLOLEUH VHORQ *< LqPH IRUPXOH HW O¶pTXDWLRQ GH FRPSRUWHPHQW YLVFRpODVWLTXH OLQpDLUH qUHIRUPXOH TXLV¶pFULYHQW


S VLQ ω W +

1 G Y −6 ⎛ GY⎞ ⎜ ( Y + (′ ⎟ = ρ 6 HW 1 = 5 5 ⎜⎝ G W ⎟⎠ GW

2QWURXYH

S VLQ ω W + 2QWURXYH

G Y (′ G Y ( S + + Y= VLQ ω W GW ρ6 GW ρ5 ρ5

G Y (′ G Y ( S + + Y= VLQ ω W GW ρ6 GW ρ5 ρ5 2QSRVHDORUV ω=

( ρ5

2QSRVHDORUV SXOVDWLRQQDWXUHOOHQRQDPRUWLH

(′&5 = 5 ρ ( DPRUWLVVHPHQWFULWLTXH λ=

(′ (′&5

1 G Y −6 ⎛ GY⎞ ⎜ ( Y + (′ ⎟ = ρ 6 HW 1 = 5 5 ⎜⎝ G W ⎟⎠ GW

ω=

( ρ5

SXOVDWLRQQDWXUHOOHQRQDPRUWLH


DPRUWLVVHPHQWUpGXLW

&HODSHUPHWG¶pFULUHO¶pTXDWLRQGXPRXYHPHQW

(′ (′&5

DPRUWLVVHPHQWUpGXLW


G Y GY S + λ ω + ω Y = VLQ ω W G W ρ 6 GW

G Y GY S + λ ω + ω Y = VLQ ω W G W ρ 6 GW

3RXUO¶pTXDWLRQVDQVVHFRQGPHPEUHPRXYHPHQWQDWXUHODPRUWL RQGLVWLQJXHWURLVFDV


HUFDV λ > DPRUWLVVHPHQWIRUW


2QSRVH U = ω λ − HWO¶RQD


Y W = H −λωW $ FK UW + % VK UW LqPHFDV λ = DPRUWLVVHPHQWFULWLTXH


Y W = H −ωW $ + %W

Y W = H −ωW $ + %W

LqPHFDV λ < DPRUWLVVHPHQWIDLEOH


2QSRVH Ω = ω − λ HWO¶RQD


Y W = H −λωW $ FRV ΩW + % VLQ ΩW


- 199 -

- 199 -





S VLQ ω W +

1 G Y −6 ⎛ GY⎞ ⎜⎜ ( Y + (′ ⎟ = ρ 6 HW 1 = 5 5 ⎝ G W ⎟⎠ GW

2QWURXYH

S VLQ ω W + 2QWURXYH

G Y (′ G Y ( S + + Y= VLQ ω W GW ρ 6 GW ρ5 ρ5

G Y (′ G Y ( S + + Y= VLQ ω W GW ρ 6 GW ρ5 ρ5 2QSRVHDORUV ω=

( ρ5

2QSRVHDORUV SXOVDWLRQQDWXUHOOHQRQDPRUWLH


(′ (′&5

1 G Y −6 ⎛ GY⎞ ⎜⎜ ( Y + (′ ⎟ = ρ 6 HW 1 = 5 5 ⎝ G W ⎟⎠ GW

ω=

( ρ5

SXOVDWLRQQDWXUHOOHQRQDPRUWLH


DPRUWLVVHPHQWUpGXLW


G Y GY S + λ ω + ω Y = VLQ ω W GW ρ6 G W

(′ (′&5

DPRUWLVVHPHQWUpGXLW


G Y GY S + λ ω + ω Y = VLQ ω W GW ρ6 G W









Y W = H −ωW $ + %W

Y W = H −ωW $ + %W







- 199 -

- 199 -

2QREWLHQWG¶DXWUHSDUWFRPPHUpSRQVHIRUFpHVROXWLRQSDUWLFXOLqUHGHO¶pTXDWLRQFRPSOqWH

Y∗ W =

DYHF

ρ5 (6

⎧ ⎪ ⎪ FRV ψ = ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ VLQ ψ = ⎪ ⎪ ⎪⎩

VLQ ω W + ψ ω ⎞⎟ ⎟

⎛ ω ⎞ ⎛ ⎜ − + ⎜ λ ⎟ ⎜ ω ω⎠ ⎝ ⎠ ⎝


Y∗ W =

DYHF

⎧ ⎪ ⎪ FRV ψ = ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ VLQ ψ = ⎪ ⎪ ⎪⎩

ω − ω

⎛ ω ⎞ ⎛ ⎟ + ⎜ λ ω ⎞⎟ ⎜ − ⎜ ω ⎟ ⎝ ω⎠ ⎠ ⎝

ω −λ ω

ρ5 (6

⎛ ω ⎞ ⎛ ⎟ + ⎜ λ ω ⎞⎟ ⎜ − ⎜ ω ⎟ ⎝ ω⎠ ⎠ ⎝

VLQ ω W + ψ

⎛ ω ⎞ ⎛ ⎜ − ⎟ + ⎜ λ ω ⎞⎟ ⎜ ω ⎟ ⎝ ω⎠ ⎝ ⎠

−

ω ω

⎛ ω ⎞ ⎛ ⎟ + ⎜ λ ω ⎞⎟ ⎜ − ⎜ ω ⎟ ⎝ ω⎠ ⎠ ⎝

ω −λ ω

⎛ ω ⎞ ⎛ ⎟ + ⎜ λ ω ⎞⎟ ⎜ − ⎜ ω ⎟ ⎝ ω⎠ ⎠ ⎝

±)2508/('(5$
±)2508/('(5$
(OOHV SHUPHWWHQW GH FDOFXOHU GH IDoRQ DSSURFKpH OHV SXOVDWLRQV QDWXUHOOHV QRQ DPRUWLHV ωQ (WXGLRQVHQOHSULQFLSHVXUGHVH[HPSOHV


±)RUPXOHGH5D\OHLJK

±)RUPXOHGH5D\OHLJK

&RQVLGpURQV OD SRXWUH GX SDUDJUDSKH VXU GHX[ DSSXLV HW VXSSRVRQV TX¶HOOH YLEUH HQ IOH[LRQVXLYDQWQLqPHPRGHSURSUHDYHFODFRQGLWLRQLQLWLDOH Y Q [ R =


2QDDORUV Y Q [ W = I Q [ ⋅ VLQ ωQ W


$XQLQVWDQWWVRQpQHUJLHFLQpWLTXHDSRXUH[SUHVVLRQ

$XQLQVWDQWWVRQpQHUJLHFLQpWLTXHDSRXUH[SUHVVLRQ

/

(F =

/

∫

∫

HWVRQpQHUJLHSRWHQWLHOOHpODVWLTXHLQWHUQH : = ( SL (S = SXLVTXH

/

∫

/

⎛∂Y ⎞ ρ 6 ⎜⎜ Q ⎟⎟ G[ = ωQ ⋅ FRV ωQ W ρ 6 (I Q [ ) G[ ∂ W ⎝ ⎠

0 G[ = (,

/

∫

⎛ ∂ YQ (, ⎜ ⎜ ∂ W ⎝

(F =

/

⎛∂Y ⎞ ρ 6 ⎜⎜ Q ⎟⎟ G[ = ωQ ⋅ FRV ωQ W ρ 6 (I Q [ ) G[ ∂ W ⎝ ⎠

∫

∫

HWVRQpQHUJLHSRWHQWLHOOHpODVWLTXHLQWHUQH : = ( SL

⎞ ⎟ G[ = VLQ ωQ W ⎟ ⎠

/

∫

(, (I Q′′ [ ) G[

(S =

∂ Y 0 = ∂ [ (,

SXLVTXH

/

∫

0 G[ = (,

/

∫

⎛ ∂ YQ (, ⎜ ⎜ ∂ W ⎝

⎞ ⎟ G[ = VLQ ωQ W ⎟ ⎠

/

∫ (, (I ′′ [ ) G[ Q

∂ Y 0 = ∂ [ (,

- 200 -

- 200 -



Y∗ W =

DYHF

ρ5 (6

⎧ ⎪ ⎪ FRV ψ = ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ VLQ ψ = ⎪ ⎪ ⎪⎩

VLQ ω W + ψ ω ⎞⎟ ⎟

⎛ ω ⎞ ⎛ ⎜ − + ⎜ λ ⎟ ⎜ ω ω⎠ ⎝ ⎝ ⎠

Y∗ W =

DYHF − ω ⎞⎟ ⎟

⎧ ⎪ ⎪ FRV ψ = ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ VLQ ψ = ⎪ ⎪ ⎪⎩

ω ω

⎛ ω ⎞ ⎛ ⎜ − + ⎜ λ ⎟ ⎜ ω ω⎠ ⎠ ⎝ ⎝

ω −λ ω

ρ5 (6

⎛ ω ⎞ ⎛ ⎟ + ⎜ λ ω ⎞⎟ ⎜ − ⎜ ω ⎟ ⎝ ω⎠ ⎠ ⎝

VLQ ω W + ψ

⎛ ω ⎞ ⎛ ⎜ − ⎟ + ⎜ λ ω ⎞⎟ ⎜ ω ⎟ ⎝ ω⎠ ⎝ ⎠

−

ω ω

⎛ ω ⎞ ⎛ ⎟ + ⎜ λ ω ⎞⎟ ⎜ − ⎜ ω ⎟ ⎝ ω⎠ ⎠ ⎝

ω −λ ω

⎛ ω ⎞ ⎛ ⎟ + ⎜ λ ω ⎞⎟ ⎜ − ⎜ ω ⎟ ⎝ ω⎠ ⎠ ⎝

±)2508/('(5$
±)2508/('(5$


±)RUPXOHGH5D\OHLJK

±)RUPXOHGH5D\OHLJK





$XQLQVWDQWWVRQpQHUJLHFLQpWLTXHDSRXUH[SUHVVLRQ /

$XQLQVWDQWWVRQpQHUJLHFLQpWLTXHDSRXUH[SUHVVLRQ /

/

⎛∂Y ⎞ ρ 6 ⎜⎜ Q ⎟⎟ G[ = ωQ ⋅ FRV ωQ W ρ 6 (I Q [ ) G[ (F = ⎝ ∂W ⎠

∫

∫

HWVRQpQHUJLHSRWHQWLHOOHpODVWLTXHLQWHUQH : = ( SL (S = SXLVTXH

/

∫

0 G[ = (,

/

∫

⎛ ∂ YQ (, ⎜ ⎜ ∂ W ⎝

⎞ ⎟ G[ = VLQ ωQ W ⎟ ⎠

∂ Y 0 = ∂ [ (,

∫

∫

HWVRQpQHUJLHSRWHQWLHOOHpODVWLTXHLQWHUQH : = ( SL /

∫

(, (I Q′′ [ ) G[

(S =

SXLVTXH

- 200 -

/

⎛∂Y ⎞ ρ 6 ⎜⎜ Q ⎟⎟ G[ = ωQ ⋅ FRV ωQ W ρ 6 (I Q [ ) G[ (F = ⎝ ∂W ⎠

/

∫

0 G[ = (,

/

∫

⎛ ∂ YQ (, ⎜ ⎜ ∂ W ⎝

⎞ ⎟ G[ = VLQ ωQ W ⎟ ⎠

∂ Y 0 = ∂ [ (,

- 200 -

/

∫ (, (I ′′ [ ) G[ Q

(QO¶DEVHQFHG¶DPRUWLVVHPHQWO¶pQHUJLHPpFDQLTXH ( P = ( F + ( S HVWFRQVWDQWHpJDORQVVHV YDOHXUVDX[LQVWDQWV W = HW W =

π 2QREWLHQWODIRUPXOHGH5D\OHLJK ωQ /

∫ (, (I ′′ [ ) G[ Q

ωQ =

/

(QO¶DEVHQFHG¶DPRUWLVVHPHQWO¶pQHUJLHPpFDQLTXH ( P = ( F + ( S HVWFRQVWDQWHpJDORQVVHV YDOHXUVDX[LQVWDQWV W = HW W =

/

∫ (, (I ′′ [ ) G[

Q

ωQ =

∫ ρ6 (I [ ) G[ Q

π 2QREWLHQWODIRUPXOHGH5D\OHLJK ωQ

/

∫ ρ6 (I [ ) G[ Q

'DQV O¶H[HPSOH FKRLVL RQ D SX FDOFXOHU H[DFWHPHQW OHV GpIRUPpHV PRGDOHV HW RQ D WURXYp Qπ[ (Q SRUWDQW FHWWH H[SUHVVLRQ GDQV OD IRUPXOH GH 5D\OHLJK RQ REWLHQW I Q [ = %Q VLQ / DORUVOHVYDOHXUVH[DFWHVGH ωQ


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⎛⎛ [ ⎞ [⎞ ⎛[⎞ HQ SUHQDQW I [ = % ⎜ ⎜ ⎟ − ⎜ ⎟ + ⎟ TXL D OD PrPH IRUPH TXH OD IRQFWLRQ H[DFWH ⎜⎝ / ⎠ ⎝ / ⎠ / ⎟⎠ ⎝ [ I [ = % VLQ π HWTXLVDWLVIDLWFRPPHHOOHDX[FRQGLWLRQV /

⎛ ⎛ [ ⎞ [⎞ ⎛[⎞ HQ SUHQDQW I [ = % ⎜ ⎜ ⎟ − ⎜ ⎟ + ⎟ TXL D OD PrPH IRUPH TXH OD IRQFWLRQ H[DFWH ⎜⎝ / ⎠ ⎝ / ⎠ / ⎟⎠ ⎝ [ I [ = % VLQ π HWTXLVDWLVIDLWFRPPHHOOHDX[FRQGLWLRQV /

⎛/⎞ I = I / = I′′ = I′′/ = I / − [ = I [ I⎜ ⎟ = % ⎝⎠

⎛/⎞ I = I / = I′′ = I′′/ = I / − [ = I [ I⎜ ⎟ = % ⎝⎠

2QWURXYHWRXVFDOFXOVIDLWV


ω DS =

(, (, = ρ6 / ρ6 /

ω DS =

/DYDOHXUH[DFWHGpWHUPLQpHDXHVW

ω = π

(, (, = ρ6 / ρ6 /


(, (, = ρ6 / ρ6 /

ω = π

/¶HUUHXUUHODWLYHHVWGRQFpJDOHjSDUH[FqV

(, (, = ρ6 / ρ6 /


- 201 -

- 201 -

(QO¶DEVHQFHG¶DPRUWLVVHPHQWO¶pQHUJLHPpFDQLTXH ( P = ( F + ( S HVWFRQVWDQWHpJDORQVVHV

(QO¶DEVHQFHG¶DPRUWLVVHPHQWO¶pQHUJLHPpFDQLTXH ( P = ( F + ( S HVWFRQVWDQWHpJDORQVVHV

YDOHXUVDX[LQVWDQWV W = HW W =


ωQ =

∫

YDOHXUVDX[LQVWDQWV W = HW W =


∫ (, (I ′′ [ ) G[

(, (I Q′′ [ ) G[

/

Q

ωQ =

∫ ρ6 (I [ ) G[ Q

/

∫ ρ6 (I [ ) G[ Q









⎛/⎞ I = I / = I′′ = I′′/ = I / − [ = I [ I⎜ ⎟ = % ⎝⎠

⎛/⎞ I = I / = I′′ = I′′/ = I / − [ = I [ I⎜ ⎟ = % ⎝⎠



ω DS =

(, (, = ρ6 / ρ6 /


ω = π

ω DS =

(, (, = ρ6 / ρ6 /


(, (, = ρ6 / ρ6 /


- 201 -

ω = π

(, (, = ρ6 / ρ6 /


- 201 -

±0pWKRGHGH5LW]

±0pWKRGHGH5LW]

2QGpPRQWUHTXHOHPLQLPXPDEVROXGXTXRWLHQWGH5D\OHLJK


/

∫

/

5=

/

∫ (, (I ′′ [ ) ⋅ G[

(, (I Q′′ [ ) ⋅ G[

Q

/

5=

∫ ρ6 (I [ ) ⋅ G[ Q

∫ ρ6 (I [ ) ⋅ G[

Q

HVW REWHQX ORUVTXH OD IRQFWLRQ I [ HVW FHOOH GH OD YpULWDEOH GpIRUPpH PRGDOH GX PRGH IRQGDPHQWDO5pJDOHDORUVOHFDUUpGHODSXOVDWLRQSURSUHIRQGDPHQWDOH


6RLW ϕ [ ϕ [ !! ϕP [ PIRQFWLRQVVDWLVIDLVDQWDX[FRQGLWLRQVDX[OLPLWHV2QSHXW DSSUR[LPHU OD YpULWDEOH FRXUEH SDU OD IRQFWLRQ D ϕ [ + D ϕ [ + !! + D P ϕP [ HQ GpWHUPLQDQWOHVFRHIILFLHQWV D L GHIDoRQjUHQGUH5PLQLPDOF¶HVWjGLUHHQUpVROYDQWOHVP pTXDWLRQV ∂5 = L = ! P ∂ DL


5HSUHQRQV O¶H[HPSOH SUpFpGHQW GH OD SRXWUH VXU GHX[ DSSXLV DYHF ϕ [ = ϕ [ =

[ ⎛ [⎞ ⎜ − ⎟ HW / ⎝ /⎠

[ ⎛ [ ⎞ ⎛⎜ [ [ ⎞⎟ IRQFWLRQVTXLVDWLVIRQWDX[FRQGLWLRQV − + ⎜ − ⎟ / ⎝ / ⎠ ⎝⎜ / / ⎠⎟



ϕ = ϕ / = ϕ / − [ = ϕ [

(QSRVDQW λ =

ϕ = ϕ / = ϕ / − [ = ϕ [

I [ = D ϕ [ + D ϕ [

2QDGRQF

I [ = D ϕ [ + D ϕ [

2QDGRQF

D RQWURXYHSRXUTXRWLHQWGH5D\OHLJK D /

5 =

(QSRVDQW λ =


(, λ + λ + ρ6 / λ + λ +

= 6D YDOHXU PLQLPDOH HVW ω DS

5 =

(, REWHQXH SRXU λ = − /¶HUUHXU UHODWLYH ρ6 /

(, λ + λ + ρ6 / λ + λ +


HVWLFLGHSDUH[FqV

HVWLFLGHSDUH[FqV


- 202 -

- 202 -

±0pWKRGHGH5LW]

±0pWKRGHGH5LW]



/

∫ (, (I ′′ [ ) ⋅ G[ Q

/

5=

/

∫ (, (I ′′ [ ) ⋅ G[

Q

/

5=

∫ ρ6 (I [ ) ⋅ G[ Q

[ ⎛ [⎞ ⎜ − ⎟ HW / ⎝ /⎠

∫ ρ6 (I [ ) ⋅ G[

Q






[ ⎛ [⎞ ⎜ − ⎟ HW / ⎝ /⎠




ϕ = ϕ / = ϕ / − [ = ϕ [

I [ = D ϕ [ + D ϕ [

2QDGRQF (QSRVDQW λ =


5 =

ϕ = ϕ / = ϕ / − [ = ϕ [

I [ = D ϕ [ + D ϕ [

2QDGRQF (QSRVDQW λ =


(, λ + λ + ρ6 / λ + λ +



5 =

(, λ + λ + ρ6 / λ + λ +


HVWLFLGHSDUH[FqV

HVWLFLGHSDUH[FqV

- 202 -

[ ⎛ [⎞ ⎜ − ⎟ HW / ⎝ /⎠


- 202 -

&+$3,75(;

&+$3,75(;

'<1$0,48('(63/$48(6

'<1$0,48('(63/$48(6

/HFKDSLWUH,,GXWRPH,pWDLWFRQVDFUpjOD6WDWLTXHGHVSODTXHVOHVSULQFLSDX[UpVXOWDWV TXH QRXV \ DYRQV REWHQXV UHVWHQW YDODEOHV VDQV DXFXQ FKDQJHPHQW HQ '\QDPLTXH G¶DXWUHV GRLYHQWVLPSOHPHQWFRPSOpWpV


1RXVUHSUHQGURQVLFL

1RXVUHSUHQGURQVLFL

± ODGpILQLWLRQGXYLVVHXUVXUXQHFRXSXUHpOpPHQWDLUH


± O¶H[SUHVVLRQGHVFRPSRVDQWHVGXYLVVHXUHQIRQFWLRQGHVFRQWUDLQWHV


± OHGpFRXSODJHGHVHIIRUWVLQWHUQHVHQ


HIIRUWVGHPHPEUDQH17

HIIRUWVGHPHPEUDQH17

HIIRUWVWUDQVYHUVDX[480


/HVpTXDWLRQVG¶pTXLOLEUHpWDEOLHVHQ6WDWLTXHGRLYHQWrWUHFRPSOpWpHVSDUOHVHIIRUWVG¶LQHUWLH HWV¶pFULYHQWHQFDUWpVLHQQHVSRXUXQHILEUHQRUPDOHGHFHQWUH 3 GHVHFWLRQG6


DYHF

DYHF

ΓU

DFFpOpUDWLRQUHODWLYHGHODSDUWLFXOH 3 UHODWLYHj {R [\]}

ΓU


ΓH

DFFpOpUDWLRQG¶HQWUDvQHPHQWGH 3

ΓH


DFFpOpUDWLRQGH&RULROLVGH 3


YLWHVVHGHURWDWLRQLQVWDQWDQpHGXUHSqUH {R [\]}OLpjODSODTXHSDUUDSSRUWj XQJDOLOpHQ

9U G S

YLWHVVHUHODWLYHGH 3 SDUUDSSRUWj {R [\]}

S I GV

YLWHVVHGHURWDWLRQLQVWDQWDQpHGXUHSqUH {R [\]}OLpjODSODTXHSDUUDSSRUWj XQJDOLOpHQ YLWHVVHUHODWLYHGH 3 SDUUDSSRUWj {R [\]}

IRUFHVXUIDFLTXHH[WpULHXUHSHVDQWHXUSUHVVLRQ

9U G S

WHQVHXUFHQWUDOG¶LQHUWLHGHODILEUHQRUPDOH

S I GV



- 203 -

- 203 -

&+$3,75(;

&+$3,75(;

'<1$0,48('(63/$48(6

'<1$0,48('(63/$48(6



1RXVUHSUHQGURQVLFL

1RXVUHSUHQGURQVLFL







HIIRUWVGHPHPEUDQH17

HIIRUWVGHPHPEUDQH17





DYHF

DYHF

ΓU


ΓU


ΓH


ΓH



YLWHVVHGHURWDWLRQLQVWDQWDQpHGXUHSqUH {R [\]}OLpjODSODTXHSDUUDSSRUWj XQJDOLOpHQ

9U G S

YLWHVVHUHODWLYHGH 3 SDUUDSSRUWj {R [\]}

S I GV


YLWHVVHGHURWDWLRQLQVWDQWDQpHGXUHSqUH {R [\]}OLpjODSODTXHSDUUDSSRUWj XQJDOLOpHQ YLWHVVHUHODWLYHGH 3 SDUUDSSRUWj {R [\]}


9U G S


S I GV


- 203 -


- 203 -

⎡ ⎤ ⎥ K ⎢ ⎢ ⎥ GDQVOHUHSqUH {3 [\]} = ⎢ ⎥ ⎢⎣ ⎥⎦

[I ]

[I ]

URWDWLRQGHODILEUHUHODWLYHPHQWj {R [\]}

⎡ ⎤ ⎥ K ⎢ ⎢ ⎥ GDQVOHUHSqUH {3 [\]} = ⎢ ⎥ ⎢⎣ ⎥⎦


−ρ

⋅ GV

PRPHQWHQ 3 GHVIRUFHVG¶LQHUWLHG¶HQWUDvQHPHQWDSSOLTXpHVjODILEUH

−ρ

⋅ GV


−ρ

⋅ GV

PRPHQWHQ 3 GHVIRUFHVG¶LQHUWLHGH&RULROLVDSSOLTXpHVjODILEUH

−ρ

⋅ GV


(QSURMHFWLRQVXUOHVD[HV[\]OHVpTXDWLRQV V¶pFULYHQW


⎧ ⎛ ∂ X ⎛ ∂ Z ∂ Y ⎞⎞ ∂ 1 [ ∂ 7\ ⎟⎟ + = ρ K ⎜ + ΓH[ + ⎜⎜ θ \ − θ] ⎪ S[ + ⎜ ∂[ ∂\ ∂W ∂ W ⎟⎠ ⎟⎠ ⎝ ⎪ ⎝ ∂W ⎪ ⎪ ⎛ ∂ Y ⎛ ∂ X ∂ Z ⎞⎞ ∂ 7[ ∂ 1 \ ⎪ ⎟⎟ + = ρ K ⎜ + ΓH\ + ⎜⎜ θ ] − θ[ ⎨ S\ + ⎜ ∂[ ∂\ ∂ W ⎟⎠ ⎟⎠ ⎝ ∂W ⎪ ⎝ ∂W ⎪ ⎪ ⎪ S + ∂ 4 [ + ∂ 4 \ = ρ K ⎛⎜ ∂ Z + Γ + ⎛⎜ θ ∂ Y − θ ∂ X ⎞⎟ ⎞⎟ H] \ ] ⎜ [ ∂W ⎪ ⎜ ∂ W ∂[ ∂\ ∂ W ⎟⎠ ⎟⎠ ⎝ ⎝ ⎩

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

⎛ K ∂ ω[ ⎞ ∂ 8[ ∂ 0 \ + + 4\ = ρ ⎜ + MH[ + MF[ ⎟ ⎜ ⎟ ∂[ ∂\ ⎝ ∂ W ⎠ ⎛ K ∂ ω\ ⎞ ∂ 0[ ∂ 8\ − 4[ = ρ ⎜ + MH\ + MF\ ⎟ + ⎜ ⎟ ∂\ ∂[ ∂ W ⎝ ⎠

⎧ ⎛ ∂ X ⎛ ∂ Z ∂ Y ⎞⎞ ∂ 1 [ ∂ 7\ ⎟⎟ + = ρ K ⎜ + ΓH[ + ⎜⎜ θ \ − θ] ⎪ S[ + ⎜ ∂[ ∂\ ∂W ∂ W ⎟⎠ ⎟⎠ ⎝ ⎪ ⎝ ∂W ⎪ ⎪ ⎛ ∂ Y ⎛ ∂ X ∂ Z ⎞⎞ ∂ 7[ ∂ 1 \ ⎪ ⎟⎟ + = ρ K ⎜ + ΓH\ + ⎜⎜ θ ] − θ[ ⎨ S\ + ⎜ ∂[ ∂\ ∂ W ⎟⎠ ⎟⎠ ⎝ ∂W ⎪ ⎝ ∂W ⎪ ⎪ ⎪ S + ∂ 4 [ + ∂ 4 \ = ρ K ⎛⎜ ∂ Z + Γ + ⎛⎜ θ ∂ Y − θ ∂ X ⎞⎟ ⎞⎟ H] \ ] ⎜ [ ∂W ⎪ ⎜ ∂ W ∂[ ∂\ ∂ W ⎟⎠ ⎟⎠ ⎝ ⎝ ⎩

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

⎛ K ∂ ω[ ⎞ ∂ 8[ ∂ 0 \ + + 4\ = ρ ⎜ + MH[ + MF[ ⎟ ⎜ ⎟ ∂[ ∂\ ⎝ ∂ W ⎠ ⎛ K ∂ ω\ ⎞ ∂ 0[ ∂ 8\ − 4[ = ρ ⎜ + MH\ + MF\ ⎟ + ⎜ ⎟ ∂\ ∂[ ∂ W ⎝ ⎠

5HPDUTXHV

5HPDUTXHV

± O¶pTXDWLRQ G¶pTXLOLEUH GHV PRPHQWV DXWRXU GH O¶D[H WUDQVYHUVDO ]′] HVW LGHQWLTXHPHQW YpULILpH


± GDQVODPDMRULWpGHVFDVOHPRXYHPHQWDOLHXHQO¶DEVHQFHGHSHVDQWHXURXSOXW{WDXWRXUGH ODFRQILJXUDWLRQG¶pTXLOLEUHVWDWLTXHGDQVOHFKDPSGHSHVDQWHXU


± ORUVTXHOHUHSqUH {R [\]}OLpjODSODTXHHVWJDOLOpHQRQDELHQVU ΓH = ΓF = MH =


MF =

MF =

±9,%5$7,216'¶81(3/$48(©'$166213/$1ª

±9,%5$7,216'¶81(3/$48(©'$166213/$1ª

&¶HVW OH FDV R WRXWHV OHV IRUFHV H[WpULHXUHV GRQQpHV GH OLDLVRQV UpHOOHV G¶LQHUWLH VRQW SDUDOOqOHVDXSODQPR\HQ[R\ HWFRQVWDQWHVVXLYDQWO¶pSDLVVHXU


- 204 -

- 204 -

⎡ ⎤ ⎥ K ⎢ ⎢ ⎥ GDQVOHUHSqUH {3 [\]} = ⎢ ⎥ ⎢⎣ ⎥⎦

[I ]

[I ]


⎡ ⎤ ⎥ K ⎢ ⎢ ⎥ GDQVOHUHSqUH {3 [\]} = ⎢ ⎥ ⎢⎣ ⎥⎦


−ρ

⋅ GV


−ρ

⋅ GV


−ρ

⋅ GV


−ρ

⋅ GV




⎧ ⎛ ∂ X ⎛ ∂ Z ∂ Y ⎞⎞ ∂ 1 [ ∂ 7\ ⎟⎟ ⎟ + = ρ K ⎜ + ΓH[ + ⎜⎜ θ \ − θ] ⎪ S[ + ⎟ ⎜ ∂[ W W ∂\ ∂ ∂ W ∂ ⎝ ⎠⎠ ⎪ ⎝ ⎪ ⎪ ⎛ ∂ Y ⎛ ∂ X ∂ Z ⎞⎞ ∂ 7[ ∂ 1 \ ⎪ ⎟⎟ + = ρ K ⎜ + ΓH\ + ⎜⎜ θ ] − θ[ ⎨ S\ + ⎜ ∂ ∂\ ∂ W ⎟⎠ ⎟⎠ [ ⎝ ∂W ⎪ ⎝ ∂W ⎪ ⎪ ⎪ S + ∂ 4 [ + ∂ 4 \ = ρ K ⎛⎜ ∂ Z + Γ + ⎛⎜ θ ∂ Y − θ ∂ X ⎞⎟ ⎞⎟ H] \ ] ⎜ [ ∂W ⎪ ⎜ ∂ W ∂[ ∂\ ∂ W ⎟⎠ ⎟⎠ ⎝ ⎝ ⎩

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

⎛ K ∂ ω[ ⎞ ∂ 8[ ∂ 0 \ + + 4\ = ρ ⎜ + MH[ + MF[ ⎟ ⎜ ⎟ ∂[ ∂\ ⎝ ∂ W ⎠ ⎛ K ∂ ω\ ⎞ ∂ 0[ ∂ 8\ ⎟ − 4[ = ρ ⎜ + + + M M H\ F\ ⎜ ∂ W ⎟ ∂\ ∂[ ⎝ ⎠

⎧ ⎛ ∂ X ⎛ ∂ Z ∂ Y ⎞⎞ ∂ 1 [ ∂ 7\ ⎟⎟ ⎟ + = ρ K ⎜ + ΓH[ + ⎜⎜ θ \ − θ] ⎪ S[ + ⎟ ⎜ ∂[ W W ∂\ ∂ ∂ W ∂ ⎝ ⎠⎠ ⎪ ⎝ ⎪ ⎪ ⎛ ∂ Y ⎛ ∂ X ∂ Z ⎞⎞ ∂ 7[ ∂ 1 \ ⎪ ⎟⎟ + = ρ K ⎜ + ΓH\ + ⎜⎜ θ ] − θ[ ⎨ S\ + ⎜ ∂ ∂\ ∂ W ⎟⎠ ⎟⎠ [ ⎝ ∂W ⎪ ⎝ ∂W ⎪ ⎪ ⎪ S + ∂ 4 [ + ∂ 4 \ = ρ K ⎛⎜ ∂ Z + Γ + ⎛⎜ θ ∂ Y − θ ∂ X ⎞⎟ ⎞⎟ H] \ ] ⎜ [ ∂W ⎪ ⎜ ∂ W ∂[ ∂\ ∂ W ⎟⎠ ⎟⎠ ⎝ ⎝ ⎩

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

⎛ K ∂ ω[ ⎞ ∂ 8[ ∂ 0 \ + + 4\ = ρ ⎜ + MH[ + MF[ ⎟ ⎜ ⎟ ∂[ ∂\ ⎝ ∂ W ⎠ ⎛ K ∂ ω\ ⎞ ∂ 0[ ∂ 8\ ⎟ − 4[ = ρ ⎜ + + + M M H\ F\ ⎜ ∂ W ⎟ ∂\ ∂[ ⎝ ⎠

5HPDUTXHV

5HPDUTXHV







MF =

MF =

±9,%5$7,216'¶81(3/$48(©'$166213/$1ª

±9,%5$7,216'¶81(3/$48(©'$166213/$1ª



- 204 -

- 204 -

/HVHIIRUWVLQWHUQHVWUDQVYHUVDX[480VRQWDORUVLGHQWLTXHPHQWQXOVVHXOVOHVHIIRUWVGH PHPEUDQH1HW7QHOHVRQWSDV/HIHXLOOHWPR\HQUHVWHSODQ&HFDVDpWppWXGLpHQVWDWLTXH DXWRPH,,FKDSLWUH9,FRQWUDLQWHVSODQHVHWFRQWUDLQWHVTXDVLSODQHV


/HVpTXDWLRQVGXPRXYHPHQWYLVFRpODVWLTXHRQWpWpGRQQpHVDXFKDSLWUH9,,,IRUPXOHV UDSSHORQVOHV


G I + λ + µ JUDG ⋅ GLY 3R3 + µ ∆ 3R3

⎛ ⎞ ⎛ ⎞ ⎛ ∂ 3R3 ⎞ ⎟ + µ′ ∆ ⎜ ∂ 3R3 ⎟ = ρ ⎜ ∂ 3R3 + ΓH + ΓF ⎟ + λ′ + µ′ JUDG ⋅ GLY ⎜⎜ ⎜ ⎟ ⎟ ⎟ ⎜ ⎝ ∂W ⎠ ⎝ ∂W ⎠ ⎝ ∂W ⎠

RXSXLVTXH ∆ = JDUG ⋅ GLY −

G I + λ + µ JUDG ⋅ GLY 3R3 − µ

3R3

⎛ ∂ 3R3 ⎞ ⎟ − µ′ + λ′ + µ′ JUDG ⋅ GLY ⎜⎜ ⎟ ⎝ ∂W ⎠

⎛ ∂ 3R3 ⎞ ⎟ − µ′ + λ′ + µ′ JUDG ⋅ GLY ⎜⎜ ⎟ ⎝ ∂W ⎠

⋅

⋅

⎛ ∂ 3R3 ⎞ ∂ 3R3 =ρ⎜ + ΓH + ΓF ⎟ ⎟ ⎜ ∂W ⎝ ∂W ⎠



⎛ ∂ 3R3 ⎞ ∂ 3R3 =ρ⎜ + ΓH + ΓF ⎟ ⎟ ⎜ ∂W ⎝ ∂W ⎠

G I + λ + µ JUDG ⋅ GLY 3R3 + µ ∆ 3R3


3R3

⎛ ∂ 3R3 ⎞ ⎟ − µ′ + λ′ + µ′ JUDG ⋅ GLY ⎜⎜ ⎟ ⎝ ∂W ⎠

⎛ ∂ 3R3 ⎞ ⎟ − µ′ + λ′ + µ′ JUDG ⋅ GLY ⎜⎜ ⎟ ⎝ ∂W ⎠

⋅

⋅

⎛ ∂ 3R3 ⎞ ∂ 3R3 =ρ⎜ + ΓH + ΓF ⎟ ⎟ ⎜ ∂W ⎝ ∂W ⎠

⎛ ∂ 3R3 ⎞ ∂ 3R3 =ρ⎜ + ΓH + ΓF ⎟ ⎟ ⎜ ∂W ⎝ ∂W ⎠

5HPDUTXHV

5HPDUTXHV

± OHVGpULYDWLRQVSDUUDSSRUWDXWHPSVVHIRQWGDQVOHUHSqUH {R [\]}OLpjODSODTXH


± HQO¶DEVHQFHG¶DPRUWLVVHPHQWLQWHUQHRQD λ′ = R = µ′


G ± OHYHFWHXU I GpVLJQHODIRUFHYROXPLTXHHQ3QpFHVVDLUHPHQWSDUDOOqOHDXSODQPR\HQGHOD SODTXH[R\


± LFL GLY 3R3 = ε [ + ε \ + ε ] = ε [ + ε \ −

− ν ν ε [ + ε \ = ε [ + ε \ − ν − ν

± LFL GLY 3R3 = ε [ + ε \ + ε ] = ε [ + ε \ −

± RQ SURMHWWH OHV pTXDWLRQV GX PRXYHPHQW VXU OHV D[HV R[ HW R\ R OHV FRPSRVDQWHV GH 3R3 VRQWX[\W HWY[\W OHVGHX[pTXDWLRQVDLQVLREWHQXVVRQWLGHQWLTXHVDX[GHX[ SUHPLqUHVpTXDWLRQVG¶pTXLOLEUHGHVIRUFHV GDQVOHVTXHOOHVRQUHPSODFH 1 [ 1 \ 7[

− ν ν ε [ + ε \ = ε [ + ε \ − ν − ν

± RQ SURMHWWH OHV pTXDWLRQV GX PRXYHPHQW VXU OHV D[HV R[ HW R\ R OHV FRPSRVDQWHV GH 3R3 VRQWX[\W HWY[\W OHVGHX[pTXDWLRQVDLQVLREWHQXVVRQWLGHQWLTXHVDX[GHX[ SUHPLqUHVpTXDWLRQVG¶pTXLOLEUHGHVIRUFHV GDQVOHVTXHOOHVRQUHPSODFH 1 [ 1 \ 7[

7\ SDUOHXUVH[SUHVVLRQVHQIRQFWLRQGH X HW Y

7\ SDUOHXUVH[SUHVVLRQVHQIRQFWLRQGH X HW Y

- 205 -

- 205 -





G I + λ + µ JUDG ⋅ GLY 3R3 + µ ∆ 3R3




3R3

⎛ ∂ 3R3 ⎞ ⎟ − µ′ + λ′ + µ′ JUDG ⋅ GLY ⎜⎜ ⎟ ⎝ ∂W ⎠

⎛ ∂ 3R3 ⎞ ⎟ − µ′ + λ′ + µ′ JUDG ⋅ GLY ⎜⎜ ⎟ ⎝ ∂W ⎠

⋅

⋅

⎛ ∂ 3R3 ⎞ ∂ 3R3 =ρ⎜ + ΓH + ΓF ⎟ ⎟ ⎜ ∂W ⎝ ∂W ⎠

⎛ ∂ 3R3 ⎞ ∂ 3R3 =ρ⎜ + ΓH + ΓF ⎟ ⎟ ⎜ ∂W ⎝ ∂W ⎠

G I + λ + µ JUDG ⋅ GLY 3R3 + µ ∆ 3R3




3R3

⎛ ∂ 3R3 ⎞ ⎟ − µ′ + λ′ + µ′ JUDG ⋅ GLY ⎜⎜ ⎟ ⎝ ∂W ⎠

⎛ ∂ 3R3 ⎞ ⎟ − µ′ + λ′ + µ′ JUDG ⋅ GLY ⎜⎜ ⎟ ⎝ ∂W ⎠

⋅

⋅

⎛ ∂ 3R3 ⎞ ∂ 3R3 =ρ⎜ + ΓH + ΓF ⎟ ⎟ ⎜ ∂W ⎝ ∂W ⎠

⎛ ∂ 3R3 ⎞ ∂ 3R3 =ρ⎜ + ΓH + ΓF ⎟ ⎟ ⎜ ∂W ⎝ ∂W ⎠

5HPDUTXHV

5HPDUTXHV







± LFL GLY 3R3 = ε [ + ε \ + ε ] = ε [ + ε \ −

− ν ν ε [ + ε \ = ε [ + ε \ − ν − ν

± RQ SURMHWWH OHV pTXDWLRQV GX PRXYHPHQW VXU OHV D[HV R[ HW R\ R OHV FRPSRVDQWHV GH 3R3 VRQWX[\W HWY[\W OHVGHX[pTXDWLRQVDLQVLREWHQXVVRQWLGHQWLTXHVDX[GHX[ SUHPLqUHVpTXDWLRQVG¶pTXLOLEUHGHVIRUFHV GDQVOHVTXHOOHVRQUHPSODFH 1 [ 1 \ 7[ 7\ SDUOHXUVH[SUHVVLRQVHQIRQFWLRQGH X HW Y

- 205 -

± LFL GLY 3R3 = ε [ + ε \ + ε ] = ε [ + ε \ −

− ν ν ε [ + ε \ = ε [ + ε \ − ν − ν

± RQ SURMHWWH OHV pTXDWLRQV GX PRXYHPHQW VXU OHV D[HV R[ HW R\ R OHV FRPSRVDQWHV GH 3R3 VRQWX[\W HWY[\W OHVGHX[pTXDWLRQVDLQVLREWHQXVVRQWLGHQWLTXHVDX[GHX[ SUHPLqUHVpTXDWLRQVG¶pTXLOLEUHGHVIRUFHV GDQVOHVTXHOOHVRQUHPSODFH 1 [ 1 \ 7[ 7\ SDUOHXUVH[SUHVVLRQVHQIRQFWLRQGH X HW Y

- 205 -

([HPSOH9LEUDWLRQVFLUFRQIpUHQWLHOOHVOLEUHVDPRUWLHVG¶XQHFRXURQQH


)LJXUH

)LJXUH

/DFRXURQQHG¶pSDLVVHXUKGHUD\RQVDLQWpULHXU HWEH[WpULHXU HVWHQFDVWUpHVXLYDQWVRQ ERUGLQWpULHXUVXUXQVROLGHLQGpIRUPDEOHJDOLOpHQ


2Q V¶LQWpUHVVH DX[ PRXYHPHQWV FLUFRQIpUHQWLHOV OH GpSODFHPHQW GH 3 VH UpGXLVDQW j VD FRPSRVDQWHRUWKRUDGLDOHYUW


/DVHFRQGHpTXDWLRQ HQSURMHFWLRQVXUO¶D[HFLUFRQIpUHQWLHOV¶pFULW


⎛ ∂ Y ∂ Y Y ⎞ ∂ Y ∂ ⎛⎜ ∂ Y ∂ Y Y ⎞⎟ = ρ µ⎜ + − ⎟ + µ′ + − ⎜ U ∂ U U ⎠⎟ ∂ W ⎝⎜ ∂ U U ∂ U U ⎠⎟ ∂ W ⎝ ∂U

(QSRVDQW Y U W = I U ⋅ J W HOOHGHYLHQW

[µ J W + µ′ J W ] ⎡⎢I ′′U + I ′U − ⎣

U


⎤ I U ⎥ = ρ I U ⋅ J W U ⎦

VRLWHQFRUHHQVpSDUDQWOHVIRQFWLRQVIU HWJW J = µ′ J + J µ


[µ J W + µ′ J W ] ⎡⎢I ′′U + I ′U − ⎣

U

⎤ I U ⎥ = ρ I U ⋅ J W U ⎦

VRLWHQFRUHHQVpSDUDQWOHVIRQFWLRQVIU HWJW

I ′′ + I ′ − I U U = − ω ρ ⋅I µ

J = µ′ J + J µ

- 206 -

I ′′ + I ′ − I U U = − ω ρ ⋅I µ

- 206 -



)LJXUH

)LJXUH









[µ J W + µ′ J W ] ⎡⎢I ′′U + I ′U − ⎣

U


⎤ I U ⎥ = ρ I U ⋅ J W U ⎦


I ′′ + I ′ − I U U = − ω ρ ⋅I µ

- 206 -


[µ J W + µ′ J W ] ⎡⎢I ′′U + I ′U − ⎣

U

⎤ I U ⎥ = ρ I U ⋅ J W U ⎦


I ′′ + I ′ − I U U = − ω ρ ⋅I µ

- 206 -

3RVRQV & =

µ FpOpULWpGHO¶RQGHQDWXUHOOHQRQDPRUWLH ρ

3RVRQV & =

µ′ FRHIILFLHQWG¶DPRUWLVVHPHQW ρ

&′ =

&′&5 =

⎛ &′ N = ⎜⎜ ⎝ &′&5

&′ =

&′&5 =

⎛ &′ N = ⎜⎜ ⎝ &′&5



2QSHXWDORUVpFULUHOHVGHX[pTXDWLRQVGpFRXSOpHV

µ FpOpULWpGHO¶RQGHQDWXUHOOHQRQDPRUWLH ρ µ′ FRHIILFLHQWG¶DPRUWLVVHPHQW ρ




⎧ J + N ω J + J = ⎪ ⎪ ⎨ ⎞ ⎛ ⎪ I ′′ + I ′ + ⎜ ω − ⎟ I = ⎜ F U ⎟ U ⎪⎩ ⎠ ⎝ /DVHFRQGHGXW\SHGH%HVVHODSRXUVROXWLRQJpQpUDOH


⎛ ωU ⎞ ⎛ ωU ⎞ I U = α - ⎜ ⎟ + β 1 ⎜ ⎟ ⎝ F ⎠ ⎝ F ⎠

⎛ ωU ⎞ ⎛ ωU ⎞ I U = α - ⎜ ⎟ + β 1 ⎜ ⎟ ⎝ F ⎠ ⎝ F ⎠

- HW 1 GpVLJQDQW OHV IRQFWLRQV GH %HVVHO GH SUHPLqUH HW GH VHFRQGH HVSqFH UHVSHFWLYHPHQW G¶RUGUH


/HVFRQGLWLRQVDX[OLPLWHVV¶pFULYHQW


⎛ ωD ⎞ ⎛ ωD ⎞ ±3RXU U = D Y = ∀W ⇒ I D = ⇒ α - ⎜ ⎟ = ⎟ + β 1 ⎜ F ⎝ ⎠ ⎝ F ⎠

⎛ ωD ⎞ ⎛ ωD ⎞ ±3RXU U = D Y = ∀W ⇒ I D = ⇒ α - ⎜ ⎟ = ⎟ + β 1 ⎜ F ⎝ ⎠ ⎝ F ⎠

±3RXU U = E ERUGOLEUH τUθ = ∀W ⇒ γ Uθ = ∀W


&RPPH γ

Uθ

=

⎛ ∂ Y Y ⎜⎜ − U ⎝ ∂ U

⎞ ⎟⎟ FHWWHFRQGLWLRQV¶pFULW ⎠

&RPPH γ

Uθ

=

⎛ ∂ Y Y ⎜⎜ − U ⎝ ∂ U

⎞ ⎟⎟ FHWWHFRQGLWLRQV¶pFULW ⎠

I E = F¶HVWjGLUHFRPSWHWHQXGHVIRUPXOHVGHGpULYDWLRQ GXFKDSLWUH9,,, ⎛ ωE ⎞ ⎛ ωE ⎞ SRXUOHVIRQFWLRQVGH%HVVHO α - ⎜ ⎟ = ⎟ + β 1 ⎜ ⎝ F ⎠ ⎝ F ⎠

I E = F¶HVWjGLUHFRPSWHWHQXGHVIRUPXOHVGHGpULYDWLRQ GXFKDSLWUH9,,, ⎛ ωE ⎞ ⎛ ωE ⎞ SRXUOHVIRQFWLRQVGH%HVVHO α - ⎜ ⎟ = ⎟ + β 1 ⎜ ⎝ F ⎠ ⎝ F ⎠

- 207 -

- 207 -

I ′E −

3RVRQV & =

I ′E −

µ FpOpULWpGHO¶RQGHQDWXUHOOHQRQDPRUWLH ρ

3RVRQV & =

µ′ FRHIILFLHQWG¶DPRUWLVVHPHQW ρ

&′ =

&′&5 =

⎛ &′ N = ⎜⎜ ⎝ &′&5

&′ =

&′&5 =

⎛ &′ N = ⎜⎜ ⎝ &′&5




µ FpOpULWpGHO¶RQGHQDWXUHOOHQRQDPRUWLH ρ µ′ FRHIILFLHQWG¶DPRUWLVVHPHQW ρ






⎛ ωU ⎞ ⎛ ωU ⎞ I U = α - ⎜ ⎟ + β 1 ⎜ ⎟ ⎝ F ⎠ ⎝ F ⎠

⎛ ωU ⎞ ⎛ ωU ⎞ I U = α - ⎜ ⎟ + β 1 ⎜ ⎟ ⎝ F ⎠ ⎝ F ⎠





⎛ ωD ⎞ ⎛ ωD ⎞ ±3RXU U = D Y = ∀W ⇒ I D = ⇒ α - ⎜ ⎟ = ⎟ + β 1 ⎜ ⎝ F ⎠ ⎝ F ⎠

⎛ ωD ⎞ ⎛ ωD ⎞ ±3RXU U = D Y = ∀W ⇒ I D = ⇒ α - ⎜ ⎟ = ⎟ + β 1 ⎜ ⎝ F ⎠ ⎝ F ⎠



&RPPH γ

Uθ

=

⎛ ∂ Y Y ⎞ ⎜⎜ ⎟ FHWWHFRQGLWLRQV¶pFULW − ∂ U U ⎟⎠ ⎝

&RPPH γ

Uθ

=

⎛ ∂ Y Y ⎞ ⎜⎜ ⎟ FHWWHFRQGLWLRQV¶pFULW − ∂ U U ⎟⎠ ⎝

I E = F¶HVWjGLUHFRPSWHWHQXGHVIRUPXOHVGHGpULYDWLRQ GXFKDSLWUH9,,, ⎛ ωE ⎞ ⎛ ωE ⎞ SRXUOHVIRQFWLRQVGH%HVVHO α - ⎜ ⎟ = ⎟ + β 1 ⎜ F ⎝ ⎠ ⎝ F ⎠

I E = F¶HVWjGLUHFRPSWHWHQXGHVIRUPXOHVGHGpULYDWLRQ GXFKDSLWUH9,,, ⎛ ωE ⎞ ⎛ ωE ⎞ SRXUOHVIRQFWLRQVGH%HVVHO α - ⎜ ⎟ = ⎟ + β 1 ⎜ F ⎝ ⎠ ⎝ F ⎠

- 207 -

- 207 -

I ′E −

I ′E −

5DVVHPEORQVFHVGHX[FRQGLWLRQVDX[OLPLWHV


⎧ ⎛ ωD ⎞ ⎛ ωD ⎞ ⎪ α - ⎜ F ⎟ + β 1 ⎜ F ⎟ = ⎝ ⎠ ⎝ ⎠ ⎪ ⎨ ⎪ ⎛ ωE ⎞ ⎛ ωE ⎞ ⎪ α - ⎜ F ⎟ + β 1 ⎜ F ⎟ = ⎠ ⎝ ⎝ ⎠ ⎩

⎧ ⎛ ωD ⎞ ⎛ ωD ⎞ ⎪ α - ⎜ F ⎟ + β 1 ⎜ F ⎟ = ⎝ ⎠ ⎝ ⎠ ⎪ ⎨ ⎪ ⎛ ωE ⎞ ⎛ ωE ⎞ ⎪ α - ⎜ F ⎟ + β 1 ⎜ F ⎟ = ⎠ ⎝ ⎝ ⎠ ⎩

/D VROXWLRQ WULYLDOH α = β = FRUUHVSRQG DX UHSRV HW GRLW rWUH UHMHWpH /D VXLWH {ωQ } GHV SXOVDWLRQV SURSUHV QRQ DPRUWLHV HVW GRQF FRQVWLWXpH SDU OD VXLWH GHV UDFLQHV GH O¶pTXDWLRQ FDUDFWpULVWLTXH


⎛ ωD ⎞ - ⎜ ⎟ ⎝ F ⎠

⎛ ωD ⎞ 1 ⎜ ⎟ ⎝ F ⎠

⎛ ωE ⎞ - ⎜ ⎟ ⎝ F ⎠

⎛ ωE ⎞ 1 ⎜ ⎟ ⎝ F ⎠

=

/¶pTXDWLRQGLIIpUHQWLHOOHHQJW V¶pFULW

⎛ ωD ⎞ - ⎜ ⎟ ⎝ F ⎠

⎛ ωD ⎞ 1 ⎜ ⎟ ⎝ F ⎠

⎛ ωE ⎞ - ⎜ ⎟ ⎝ F ⎠

⎛ ωE ⎞ 1 ⎜ ⎟ ⎝ F ⎠

=


J + N Q ⋅ ωQ ⋅ J + ω ⋅ J =

J + N Q ⋅ ωQ ⋅ J + ω ⋅ J =

3RXUFKDTXHKDUPRQLTXHWURLVFDVGRLYHQWrWUHHQYLVDJpV


HUFDVDPRUWLVVHPHQWIRUW N Q >


(QSRVDQW UQ = ωQ N Q − RQD


J Q W = H − N Q ωQ W ($ Q FK UQ W + %Q VK UQ W ) HFDVDPRUWLVVHPHQWFULWLTXH N Q =


J Q W = H −ωQ W ($ Q + %Q W )

J Q W = H −ωQ W ($ Q + %Q W )

HFDVDPRUWLVVHPHQWIDLEOH N Q <




J Q W = H − N Q ωQ W ($ Q FRV Ω Q W + %Q VLQ Ω Q W ) /DVROXWLRQJpQpUDOHYUW V¶pFULWILQDOHPHQW


∞

Y U W =

∑

∞

I Q U ⋅ J Q W

Y U W =

Q =

∑ I U ⋅ J W Q

Q

Q =

- 208 -

- 208 -



⎧ ⎛ ωD ⎞ ⎛ ωD ⎞ ⎪ α - ⎜ F ⎟ + β 1 ⎜ F ⎟ = ⎝ ⎠ ⎝ ⎠ ⎪ ⎨ ⎪ ⎛ ωE ⎞ ⎛ ωE ⎞ ⎪ α - ⎜ F ⎟ + β 1 ⎜ F ⎟ = ⎝ ⎠ ⎝ ⎠ ⎩

⎧ ⎛ ωD ⎞ ⎛ ωD ⎞ ⎪ α - ⎜ F ⎟ + β 1 ⎜ F ⎟ = ⎝ ⎠ ⎝ ⎠ ⎪ ⎨ ⎪ ⎛ ωE ⎞ ⎛ ωE ⎞ ⎪ α - ⎜ F ⎟ + β 1 ⎜ F ⎟ = ⎝ ⎠ ⎝ ⎠ ⎩



⎛ ωD ⎞ - ⎜ ⎟ ⎝ F ⎠

⎛ ωD ⎞ 1 ⎜ ⎟ ⎝ F ⎠

⎛ ωE ⎞ - ⎜ ⎟ ⎝ F ⎠

⎛ ωE ⎞ 1 ⎜ ⎟ ⎝ F ⎠

=


⎛ ωD ⎞ - ⎜ ⎟ ⎝ F ⎠

⎛ ωD ⎞ 1 ⎜ ⎟ ⎝ F ⎠

⎛ ωE ⎞ - ⎜ ⎟ ⎝ F ⎠

⎛ ωE ⎞ 1 ⎜ ⎟ ⎝ F ⎠

=


J + N Q ⋅ ωQ ⋅ J + ω ⋅ J =

J + N Q ⋅ ωQ ⋅ J + ω ⋅ J =









J Q W = H −ωQ W ($ Q + %Q W )

J Q W = H −ωQ W ($ Q + %Q W )







∞

Y U W =

∑

∞

I Q U ⋅ J Q W

Q =

- 208 -

Y U W =

∑ I U ⋅ J W Q

Q =

- 208 -

Q

2QHQGpGXLWOHWHQVHXUGpIRUPDWLRQ E TXLDSRXUVHXOFRPSRVDQWHQRQQXOOHHQFRRUGRQQpHV F\OLQGULTXHV γ Uθ =

⎛∂Y ⎜ − ⎜⎝ ∂ U

Y⎞ ⎟= U ⎟⎠

∞

∑ Q =

2QHQGpGXLWOHWHQVHXUGpIRUPDWLRQ E TXLDSRXUVHXOFRPSRVDQWHQRQQXOOHHQFRRUGRQQpHV F\OLQGULTXHV

⎡ ωQ ⎛ω U⎞ ⎛ ω U ⎞⎤ J Q W ⋅ ⎢α Q - ⎜ Q ⎟ + β Q 1 ⎜ Q ⎟ ⎥ F F ⎝ ⎠ ⎝ F ⎠⎦ ⎣

SXLVOHWHQVHXUFRQWUDLQWH Σ GRQWODVHXOHFRPSRVDQWHQRQQXOOHHVW τUθ = µ γ Uθ + µ′

γ Uθ =

∂ γ Uθ ∂W

⎛∂Y ⎜ − ⎜⎝ ∂ U

Y⎞ ⎟= U ⎟⎠

∞

∑ Q =

⎡ ωQ ⎛ω U⎞ ⎛ ω U ⎞⎤ J Q W ⋅ ⎢α Q - ⎜ Q ⎟ + β Q 1 ⎜ Q ⎟ ⎥ F F ⎝ ⎠ ⎝ F ⎠⎦ ⎣


∂ γ Uθ ∂W

2QDHQFRUHXQV\VWqPHG¶RQGHVVWDWLRQQDLUHVGHW\SH©WUDQVYHUVDOHVª


±9,%5$7,21675$169(56$/(6'(63/$48(6

±9,%5$7,21675$169(56$/(6'(63/$48(6

3RXU pWDEOLU O¶pTXDWLRQ GX PRXYHPHQW j ODTXHOOH REpLW OD IOqFKH Z[ \ W QRXV SURFpGRQV FRPPHGDQVOHFDVVWDWLTXHHWQRXVDSSXLHURQVVXUOHVGHX[PrPHVK\SRWKqVHV


1RXVH[SULPHURQVG¶DERUGOHVSDUDPqWUHVGXSUREOqPHHQIRQFWLRQGHZ


UHpWDSHGpSODFHPHQWV

UHpWDSHGpSODFHPHQWV ⎧ ∂Z ⎪ X = −] ∂ [ ⎪ ∂Z ⎪ 3R3 ⎨ Y = −] ∂\ ⎪ ⎪ ⎪ Z = Z [ \ ⎩

HpWDSHGpIRUPDWLRQV

⎧ ∂Z ⎪ X = −] ∂ [ ⎪ ∂Z ⎪ 3R3 ⎨ Y = −] ∂\ ⎪ ⎪ ⎪ Z = Z [ \ ⎩

⎧ ∂X ∂ Z = −] ⎪ ε[ = ∂[ ∂ [ ⎪ ⎪ ∂Y ∂ Z ⎪ = −] ⎨ ε\ = ∂\ ∂ \ ⎪ ⎪ ⎪ γ = ⎛⎜ ∂ X + ∂ Y ⎞⎟ = −] ∂ Z [\ ⎪ ∂[ ⋅∂\ ⎜⎝ ∂ \ ∂ [ ⎟⎠ ⎩

2QHQGpGXLW

ε] = −

HpWDSHGpIRUPDWLRQV

⎧ ∂X ∂ Z = −] ⎪ ε[ = ∂[ ∂ [ ⎪ ⎪ ∂Y ∂ Z ⎪ = −] ⎨ ε\ = ∂\ ∂ \ ⎪ ⎪ ⎪ γ = ⎛⎜ ∂ X + ∂ Y ⎞⎟ = −] ∂ Z [\ ⎪ ∂[ ⋅∂\ ⎜⎝ ∂ \ ∂ [ ⎟⎠ ⎩

2QHQGpGXLW

ε] = −

ν ν ε [ + ε \ = ]⋅∆Z − ν − ν

ν ν ε [ + ε \ = ]⋅∆Z − ν − ν

- 209 -

- 209 -



γ Uθ =

⎛∂Y ⎜ − ⎜⎝ ∂ U

Y⎞ ⎟= U ⎟⎠

∞

∑ Q =

⎡ ωQ ⎛ω U⎞ ⎛ ω U ⎞⎤ J Q W ⋅ ⎢α Q - ⎜ Q ⎟ + β Q 1 ⎜ Q ⎟ ⎥ F ⎝ F ⎠ ⎝ F ⎠⎦ ⎣


γ Uθ =

∂ γ Uθ ∂W

⎛∂Y ⎜ − ⎜⎝ ∂ U

Y⎞ ⎟= U ⎟⎠

∞

∑ Q =

⎡ ωQ ⎛ω U⎞ ⎛ ω U ⎞⎤ J Q W ⋅ ⎢α Q - ⎜ Q ⎟ + β Q 1 ⎜ Q ⎟ ⎥ F ⎝ F ⎠ ⎝ F ⎠⎦ ⎣


∂ γ Uθ ∂W



±9,%5$7,21675$169(56$/(6'(63/$48(6

±9,%5$7,21675$169(56$/(6'(63/$48(6





UHpWDSHGpSODFHPHQWV

UHpWDSHGpSODFHPHQWV ⎧ ∂Z ⎪ X = −] ∂ [ ⎪ ∂Z ⎪ 3R3 ⎨ Y = −] ∂\ ⎪ ⎪ ⎪ Z = Z [ \ ⎩

HpWDSHGpIRUPDWLRQV

⎧ ∂Z ⎪ X = −] ∂ [ ⎪ ∂Z ⎪ 3R3 ⎨ Y = −] ∂\ ⎪ ⎪ ⎪ Z = Z [ \ ⎩

⎧ ∂X ∂ Z = −] ⎪ ε[ = ∂[ ∂ [ ⎪ ⎪ ∂Y ∂ Z ⎪ = −] ⎨ ε\ = ∂\ ∂ \ ⎪ ⎪ ⎪ γ = ⎛⎜ ∂ X + ∂ Y ⎞⎟ = −] ∂ Z [\ ⎜ ⎟ ⎪ ∂[ ⋅∂\ ⎝∂\ ∂[ ⎠ ⎩

2QHQGpGXLW

ε] = −

HpWDSHGpIRUPDWLRQV

⎧ ∂X ∂ Z = −] ⎪ ε[ = ∂[ ∂ [ ⎪ ⎪ ∂Y ∂ Z ⎪ = −] ⎨ ε\ = ∂\ ∂ \ ⎪ ⎪ ⎪ γ = ⎛⎜ ∂ X + ∂ Y ⎞⎟ = −] ∂ Z [\ ⎜ ⎟ ⎪ ∂[ ⋅∂\ ⎝∂\ ∂[ ⎠ ⎩

2QHQGpGXLW

ε] = −

ν ν ε [ + ε \ = ]⋅∆Z − ν − ν

- 209 -

ν ν ε [ + ε \ = ]⋅∆Z − ν − ν

- 209 -

HpWDSHFRQWUDLQWHV

HpWDSHFRQWUDLQWHV

/D ORL GX FRPSRUWHPHQW YLVFRpODVWLTXH QRXV GRQQH FRPSWH WHQX GH OD VHFRQGH K\SRWKqVH σ] =


⎧ ( (′ ε [ + ν ε \ + ⎪ σ[ = − ν − ν′ ⎪ ⎪ ( (′ ⎪ ε \ + ν ε [ + ⎨ σ\ = − ν − ν′ ⎪ ⎪ ⎪ τ = ( γ + (′ ∂ γ ⎪ [\ − ν [\ − ν′ ∂ W [\ ⎩

∂ ε \ + ν′ ε [ ∂W

F¶HVWjGLUHHQUHPSODoDQW ε [ ε \ γ [\ SDUOHXUVH[SUHVVLRQV

⎧ ⎡ ( ⎛ ∂ Z (′ ∂ ∂ Z ⎞⎟ ⎜ ⎪ σ[ = − ⎢ + + ν ⎜ ⎟ ∂ \ ⎠ − ν′ ∂ W ⎪ ⎢⎣ − ν ⎝ ∂ [ ⎪ ⎪ ⎡ ( ⎛ ∂ Z ⎪ ∂ Z ⎞⎟ (′ ∂ ⎜ + ν + ⎨ σ\ = − ⎢ ⎜ ⎟ ∂ [ ⎠ − ν′ ∂ W ⎢⎣ − ν ⎝ ∂ \ ⎪ ⎪ ⎪ ⎪ τ = − ⎡ ( ∂ Z + (′ ∂ ∂ Z ⎤ ⋅ ] ⎥ ⎢ [\ ⎪ ⎣⎢ + ν ∂ [ ∂ \ + ν′ ∂ W ∂ [ ∂ \ ⎦⎥ ⎩

⎛ ∂ Z ∂ Z ⎞⎟⎤ ⎜ ′ + ν ⎥⋅] ⎜ ∂ [ ∂ \ ⎟⎠⎥⎦ ⎝ ⎛ ∂ Z ∂ Z ⎞⎟⎤ ⎜ ′ + ν ⎥⋅] ⎜ ∂ \ ∂ [ ⎟⎠⎥⎦ ⎝

∂ ε [ + ν′ ε \ ∂W ∂ ε \ + ν′ ε [ ∂W


HpWDSHYLVVHXUV

⎧ ⎡ ( ⎛ ∂ Z (′ ∂ ∂ Z ⎞⎟ ⎜ ⎪ σ[ = − ⎢ + + ν ⎜ ⎟ ∂ \ ⎠ − ν′ ∂ W ⎪ ⎢⎣ − ν ⎝ ∂ [ ⎪ ⎪ ⎡ ( ⎛ ∂ Z ⎪ ∂ Z ⎞⎟ (′ ∂ ⎜ + ν + ⎨ σ\ = − ⎢ ⎜ ⎟ ∂ [ ⎠ − ν′ ∂ W ⎢⎣ − ν ⎝ ∂ \ ⎪ ⎪ ⎪ ⎪ τ = − ⎡ ( ∂ Z + (′ ∂ ∂ Z ⎤ ⋅ ] ⎥ ⎢ [\ ⎪ ⎣⎢ + ν ∂ [ ∂ \ + ν′ ∂ W ∂ [ ∂ \ ⎦⎥ ⎩

⎛ ∂ Z ∂ Z ⎞⎟⎤ ⎜ ′ + ν ⎥⋅] ⎜ ∂ [ ∂ \ ⎟⎠⎥⎦ ⎝ ⎛ ∂ Z ∂ Z ⎞⎟⎤ ⎜ ′ + ν ⎥⋅] ⎜ ∂ \ ∂ [ ⎟⎠⎥⎦ ⎝

HpWDSHYLVVHXUV

K ⎧ ⎪ ⎪ 8 [ = − 8 \ = − τ [\ ⋅ ] ⋅ G] = ' − ν ∂ Z + '′ − ν′ ∂ ∂ Z ⎪ ∂W ∂[ ∂\ ∂[ ∂\ K ⎪ − ⎪ K ⎪ ⎪⎪ ⎛ ∂ Z ∂ ⎛⎜ ∂ Z ∂ Z ⎞⎟ ∂ Z ⎞⎟ ′ ′ σ [ ⋅ ] ⋅ G] = −' ⎜ + ν − + ν ' ⎨ 0[ = ⎜ ∂ [ ∂ W ⎜⎝ ∂ [ ∂ \ ⎟⎠ ∂ \ ⎟⎠ ⎪ ⎝ K − ⎪ ⎪ K ⎪ ⎛ ∂ Z ∂ Z ⎞⎟ ∂ ⎛⎜ ∂ Z ∂ Z ⎞⎟ ⎪ ⎜ ′ ′ + ν + + ν ' ⎪ 0 \ = − σ \ ⋅ ] ⋅ G] = ' ⎜ ∂ W ⎜⎝ ∂ \ ∂ [ ⎟⎠ ∂ [ ⎟⎠ ⎝ ∂\ ⎪ K − ⎪⎩

∫


∂ ε [ + ν′ ε \ ∂W

∫

K ⎧ ⎪ ⎪ 8 [ = − 8 \ = − τ [\ ⋅ ] ⋅ G] = ' − ν ∂ Z + '′ − ν′ ∂ ∂ Z ⎪ ∂W ∂[ ∂\ ∂[ ∂\ K ⎪ − ⎪ K ⎪ ⎪⎪ ⎛ ∂ Z ∂ ⎛⎜ ∂ Z ∂ Z ⎞⎟ ∂ Z ⎞⎟ ′ ′ σ [ ⋅ ] ⋅ G] = −' ⎜ + ν − + ν ' ⎨ 0[ = ⎜ ∂ [ ∂ W ⎜⎝ ∂ [ ∂ \ ⎟⎠ ∂ \ ⎟⎠ ⎪ ⎝ K − ⎪ ⎪ K ⎪ ⎛ ∂ Z ∂ Z ⎞⎟ ∂ ⎛⎜ ∂ Z ∂ Z ⎞⎟ ⎪ ⎜ ′ ′ + ν + + ν ' ⎪ 0 \ = − σ \ ⋅ ] ⋅ G] = ' ⎜ ∂ W ⎜⎝ ∂ \ ∂ [ ⎟⎠ ∂ [ ⎟⎠ ⎝ ∂\ ⎪ K − ⎪⎩

∫

∫

∫

∫

/HVpTXDWLRQVG¶pTXLOLEUHGHVPRPHQWV SHUPHWWHQWDORUVG¶H[SULPHUOHVHIIRUWVWUDQFKDQWV WUDQVYHUVDX[ 4 [ HW 4 \


- 210 -

- 210 -

HpWDSHFRQWUDLQWHV

HpWDSHFRQWUDLQWHV




∂ ε \ + ν′ ε [ ∂W


⎧ ⎡ ( ⎛ ∂ Z (′ ∂ ∂ Z ⎞⎟ ⎜ ⎪ σ[ = − ⎢ + + ν ⎜ ⎟ ∂ \ ⎠ − ν′ ∂ W ⎪ ⎢⎣ − ν ⎝ ∂ [ ⎪ ⎪ ⎡ ( ⎛ ∂ Z ⎪ ∂ Z ⎞⎟ (′ ∂ ⎜ + ν + ⎨ σ\ = − ⎢ ⎜ ⎟ ∂ [ ⎠ − ν′ ∂ W ⎪ ⎣⎢ − ν ⎝ ∂ \ ⎪ ⎪ ⎪ τ = − ⎡ ( ∂ Z + (′ ∂ ∂ Z ⎤ ⋅ ] ⎥ ⎢ [\ ⎪ ⎣⎢ + ν ∂ [ ∂ \ + ν′ ∂ W ∂ [ ∂ \ ⎦⎥ ⎩

⎛ ∂ Z ∂ Z ⎞⎟⎤ ⎜ ′ + ν ⎥⋅] ⎜ ∂ [ ∂ \ ⎟⎠⎥⎦ ⎝ ⎛ ∂ Z ∂ Z ⎞⎟⎤ ⎜ ′ + ν ⎥⋅] ⎜ ∂ \ ∂ [ ⎟⎠⎦⎥ ⎝

∂ ε [ + ν′ ε \ ∂W ∂ ε \ + ν′ ε [ ∂W


HpWDSHYLVVHXUV

⎧ ⎡ ( ⎛ ∂ Z (′ ∂ ∂ Z ⎞⎟ ⎜ ⎪ σ[ = − ⎢ + + ν ⎜ ⎟ ∂ \ ⎠ − ν′ ∂ W ⎪ ⎢⎣ − ν ⎝ ∂ [ ⎪ ⎪ ⎡ ( ⎛ ∂ Z ⎪ ∂ Z ⎞⎟ (′ ∂ ⎜ + ν + ⎨ σ\ = − ⎢ ⎜ ⎟ ∂ [ ⎠ − ν′ ∂ W ⎪ ⎣⎢ − ν ⎝ ∂ \ ⎪ ⎪ ⎪ τ = − ⎡ ( ∂ Z + (′ ∂ ∂ Z ⎤ ⋅ ] ⎥ ⎢ [\ ⎪ ⎣⎢ + ν ∂ [ ∂ \ + ν′ ∂ W ∂ [ ∂ \ ⎦⎥ ⎩

⎛ ∂ Z ∂ Z ⎞⎟⎤ ⎜ ′ + ν ⎥⋅] ⎜ ∂ [ ∂ \ ⎟⎠⎥⎦ ⎝ ⎛ ∂ Z ∂ Z ⎞⎟⎤ ⎜ ′ + ν ⎥⋅] ⎜ ∂ \ ∂ [ ⎟⎠⎦⎥ ⎝

HpWDSHYLVVHXUV

K ⎧ ⎪ ⎪ 8 [ = − 8 \ = − τ [\ ⋅ ] ⋅ G] = ' − ν ∂ Z + '′ − ν′ ∂ ∂ Z ⎪ ∂W ∂[ ∂\ ∂[ ∂\ K ⎪ − ⎪ K ⎪ ⎪⎪ ⎛ ∂ Z ∂ Z ⎞⎟ ∂ ⎛⎜ ∂ Z ∂ Z ⎞⎟ ′ ′ σ [ ⋅ ] ⋅ G] = −' ⎜ + ν − + ν ' ⎨ 0[ = ⎜ ∂ [ ∂ W ⎜⎝ ∂ [ ∂ \ ⎟⎠ ∂ \ ⎟⎠ ⎪ ⎝ K − ⎪ ⎪ K ⎪ ⎛ ∂ Z ∂ Z ⎞⎟ ∂ ⎛⎜ ∂ Z ∂ Z ⎞⎟ ⎪ = − σ \ ⋅ ] ⋅ G] = ' ⎜ +ν + '′ + ν′ 0 ⎪ \ ⎟ ⎜ ∂ \ ⎜ ∂W ⎝ ∂\ ∂[ ⎠ ∂ [ ⎟⎠ ⎝ ⎪ K − ⎩⎪

∫


∂ ε [ + ν′ ε \ ∂W

∫

K ⎧ ⎪ ⎪ 8 [ = − 8 \ = − τ [\ ⋅ ] ⋅ G] = ' − ν ∂ Z + '′ − ν′ ∂ ∂ Z ⎪ ∂W ∂[ ∂\ ∂[ ∂\ K ⎪ − ⎪ K ⎪ ⎪⎪ ⎛ ∂ Z ∂ Z ⎞⎟ ∂ ⎛⎜ ∂ Z ∂ Z ⎞⎟ ′ ′ σ [ ⋅ ] ⋅ G] = −' ⎜ + ν − + ν ' ⎨ 0[ = ⎜ ∂ [ ∂ W ⎜⎝ ∂ [ ∂ \ ⎟⎠ ∂ \ ⎟⎠ ⎪ ⎝ K − ⎪ ⎪ K ⎪ ⎛ ∂ Z ∂ Z ⎞⎟ ∂ ⎛⎜ ∂ Z ∂ Z ⎞⎟ ⎪ = − σ \ ⋅ ] ⋅ G] = ' ⎜ +ν + '′ + ν′ 0 ⎪ \ ⎟ ⎜ ∂ \ ⎜ ∂W ⎝ ∂\ ∂[ ⎠ ∂ [ ⎟⎠ ⎝ ⎪ K − ⎩⎪

∫

∫

∫

∫



- 210 -

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(TXDWLRQVGXPRXYHPHQW


3RUWRQVFHVH[SUHVVLRQVGH 4 [ HW 4 \ GDQVODWURLVLqPHpTXDWLRQ G¶pTXLOLEUHGHVIRUFHV


RQREWLHQWO¶pTXDWLRQGXPRXYHPHQW


⎛ ∂ Z ⎞ ∂ ⋅ ∆ ∆ Z = S ] − K ρ ⎜ + Γ H ] ⎟ ⎟ ⎜ ∂W ⎝ ∂W ⎠

' ⋅ ∆ ∆ Z + '′ ⋅

⎛ ∂ Z ⎞ ∂ ⋅ ∆ ∆ Z = S ] − K ρ ⎜ + Γ H ] ⎟ ⎟ ⎜ ∂W ⎝ ∂W ⎠

' ⋅ ∆ ∆ Z + '′ ⋅

5HPDUTXHV

5HPDUTXHV

1RXV DYRQV YX DX GX FKDSLWUH 9,,, TXH OHV FRQWUDLQWHV YLVTXHXVHV V¶H[SULPDLHQW HQ IRQFWLRQ GHV YLWHVVHV GH GpIRUPDWLRQ FRPPH OHV FRQWUDLQWHV pODVWLTXHV HQ IRQFWLRQ GHV GpIRUPDWLRQVDXPR\HQGHGHX[FRHIILFLHQWV1RXVDYRQVLQWURGXLWOHVFRHIILFLHQWV (′ HW *′ SXLV λ′ HW µ′ OHVFRHIILFLHQWV ν′ HW '′ VRQWGpILQLVSDUOHVIRUPXOHV


*′ =

(′ (′ K HW '′ = + ν′ − ν′

*′ =

/¶pTXDWLRQ GX PRXYHPHQW TXH QRXV YHQRQV G¶pWDEOLU HVW DVVH] FRPSOH[H FDU WUqV JpQpUDOHHOOHSHUPHWGHWHQLUFRPSWH ± GHO¶DPRUWLVVHPHQWLQWHUQHSDUOHWHUPH '′ ⋅

(′ (′ K HW '′ = + ν′ − ν′

/¶pTXDWLRQ GX PRXYHPHQW TXH QRXV YHQRQV G¶pWDEOLU HVW DVVH] FRPSOH[H FDU WUqV JpQpUDOHHOOHSHUPHWGHWHQLUFRPSWH

∂ ⋅ ∆ ∆ Z ∂W

± GHO¶DPRUWLVVHPHQWLQWHUQHSDUOHWHUPH '′ ⋅

± GHVIRUFHVGHFKDUJHPHQWSDUOHWHUPH S ]

∂ ⋅ ∆ ∆ Z ∂W


± GHV IRUFHV G¶LQHUWLH G¶HQWUDvQHPHQW SDU OHV WHUPHV − K ρ Γ H ] HW &RULROLVSDUOHWHUPH

± GHO¶LQHUWLHGHURWDWLRQGHVILEUHVSDUOHWHUPH ρ


HW GH

K ∂ ∆ Z ∂ W


'DQV OD PDMRULWp GHV SUREOqPHV SOXVLHXUV GH FHV WHUPHV VRQW QXOV (Q SDUWLFXOLHU OHV K ∂ VRQW ∆ Z HW WHUPHV OLpV DX[ LQHUWLHV GH URWDWLRQ GHV ILEUHV ρ ∂ W

HW GH

K ∂ ∆ Z ∂ W

'DQV OD PDMRULWp GHV SUREOqPHV SOXVLHXUV GH FHV WHUPHV VRQW QXOV (Q SDUWLFXOLHU OHV K ∂ VRQW ∆ Z HW WHUPHV OLpV DX[ LQHUWLHV GH URWDWLRQ GHV ILEUHV ρ ∂ W

⎛ ∂ Z ⎞ WRXWjIDLWQpJOLJHDEOHVGHYDQW − K ρ ⎜ + Γ H ] ⎟ OLpDX[LQHUWLHVGHWUDQVODWLRQ ⎟ ⎜ ∂W ⎝ ⎠

⎛ ∂ Z ⎞ WRXWjIDLWQpJOLJHDEOHVGHYDQW − K ρ ⎜ + Γ H ] ⎟ OLpDX[LQHUWLHVGHWUDQVODWLRQ ⎟ ⎜ ∂W ⎝ ⎠

/¶LQWpJUDWLRQGHO¶pTXDWLRQGXPRXYHPHQWGRQQHODIOqFKHZ[\W OHVIRUPXOHV SHUPHWWHQW G¶HQ GpGXLUH OHV DXWUHV SDUDPqWUHV GX SUREOqPH /HV FRQVWDQWHV G¶LQWpJUDWLRQVRQWGpWHUPLQpHVjO¶DLGHGHVFRQGLWLRQVDX[OLPLWHVGHVFRQGLWLRQVGHOLDLVRQ HWGHVFRQGLWLRQVLQLWLDOHV


- 211 -

- 211 -







⎛ ∂ Z ⎞ ∂ ⋅ ∆ ∆ Z = S ] − K ρ ⎜ + Γ H ] ⎟ ⎟ ⎜ ∂W ⎝ ∂W ⎠

' ⋅ ∆ ∆ Z + '′ ⋅

⎛ ∂ Z ⎞ ∂ ⋅ ∆ ∆ Z = S ] − K ρ ⎜ + Γ H ] ⎟ ⎟ ⎜ ∂W ⎝ ∂W ⎠

' ⋅ ∆ ∆ Z + '′ ⋅

5HPDUTXHV

5HPDUTXHV



*′ =

(′ (′ K HW '′ = + ν′ − ν′

*′ =


∂ ⋅ ∆ ∆ Z ∂W



∂ ⋅ ∆ ∆ Z ∂W




(′ (′ K HW '′ = + ν′ − ν′

HW GH

K ∂ ∆ Z ∂ W

'DQV OD PDMRULWp GHV SUREOqPHV SOXVLHXUV GH FHV WHUPHV VRQW QXOV (Q SDUWLFXOLHU OHV K ∂ VRQW ∆ Z HW WHUPHV OLpV DX[ LQHUWLHV GH URWDWLRQ GHV ILEUHV ρ ∂ W ⎛ ∂ Z ⎞ WRXWjIDLWQpJOLJHDEOHVGHYDQW − K ρ ⎜ + Γ H ] ⎟ OLpDX[LQHUWLHVGHWUDQVODWLRQ ⎟ ⎜ ∂W ⎝ ⎠



HW GH

K ∂ ∆ Z ∂ W

'DQV OD PDMRULWp GHV SUREOqPHV SOXVLHXUV GH FHV WHUPHV VRQW QXOV (Q SDUWLFXOLHU OHV K ∂ VRQW ∆ Z HW WHUPHV OLpV DX[ LQHUWLHV GH URWDWLRQ GHV ILEUHV ρ ∂ W ⎛ ∂ Z ⎞ WRXWjIDLWQpJOLJHDEOHVGHYDQW − K ρ ⎜ + Γ H ] ⎟ OLpDX[LQHUWLHVGHWUDQVODWLRQ ⎟ ⎜ ∂W ⎝ ⎠



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±)/(;,21&
±)/(;,21&
&¶HVWOHFDVRODIOqFKHZ[W HVWLQGpSHQGDQWHGHODFRRUGRQQpH\FHFDVDpWppWXGLpHQ 6WDWLTXHDXWRPH,/HVIRUPXOHV j VHVLPSOLILHQWWRXWHVOHVGpULYpHVSDUUDSSRUWj\ pWDQWQXOOHV


7UDLWRQV O¶H[HPSOH GH OD SODTXH UHFWDQJXODLUH HQFDVWUpHOLEUH GH OD ILJXUH GDQV OH FDV GHV YLEUDWLRQV QDWXUHOOHV DPRUWLHV GH IOH[LRQ F\OLQGULTXH HQ UHSqUH JDOLOpHQ {R [\]} HW HQ QpJOLJHDQWOHVLQHUWLHVGHURWDWLRQGHVILEUHV


)LJXUH

)LJXUH

/¶pTXDWLRQGXPRXYHPHQWV¶pFULWLFL

'


∂ Z ∂ ∂ Z ∂ Z + '′ + ρK = ∂W ∂[ ∂[ ∂ W

(OOHSHXWV¶pFULUHVLO¶RQSRVH Z [ W = I [ ⋅ J W

'

J W ' I[ = −ω − = ρ K I [ J W + '′ J W '

HWVHGpFRXSOHDLQVLHQGHX[pTXDWLRQV


'′ ρ K ω I [ HW J W ⋅ ω ⋅ J W + ω ⋅ J W = ' '

I [ =

/DSUHPLqUHDGPHWSRXUVROXWLRQJpQpUDOH

I [ = α ⋅ FRV

∂ Z ∂ ∂ Z ∂ Z + '′ + ρK = ∂W ∂[ ∂[ ∂ W

(OOHSHXWV¶pFULUHVLO¶RQSRVH Z [ W = I [ ⋅ J W

J W ' I[ = −ω − = ρ K I [ J W + '′ J W '

I [ =

'′ ρ K ω I [ HW J W ⋅ ω ⋅ J W + ω ⋅ J W = ' '


ρ K ω ρ K ω ρ K ω ρ K ω [ + β ⋅ VLQ [ + γ ⋅ FK [ + δ ⋅ VK [ ' ' ' '

I [ = α ⋅ FRV


- 212 -

- 212 -

±)/(;,21&
±)/(;,21&




)LJXUH

)LJXUH


'


∂ Z ∂ ∂ Z ∂ Z ′ + ' + ρ K = ∂ W ∂ [ ∂ [ ∂ W

(OOHSHXWV¶pFULUHVLO¶RQSRVH Z [ W = I [ ⋅ J W −

'

∂ Z ∂ ∂ Z ∂ Z ′ + ' + ρ K = ∂ W ∂ [ ∂ [ ∂ W

(OOHSHXWV¶pFULUHVLO¶RQSRVH Z [ W = I [ ⋅ J W −

J W ' I[ = −ω = ρ K I [ J W + '′ J W '


'′ ρ K ω I [ HW J W ⋅ ω ⋅ J W + ω ⋅ J W = ' '


I [ = α ⋅ FRV

J W ' I[ = −ω = ρ K I [ J W + '′ J W '

HWVHGpFRXSOHDLQVLHQGHX[pTXDWLRQV I [ =

I [ =

'′ ρ K ω I [ HW J W ⋅ ω ⋅ J W + ω ⋅ J W = ' '



- 212 -

I [ = α ⋅ FRV


- 212 -



⎧ ⎪ Z R W = ∀W ⎪ ⎪ ⎛∂Z ⎞ ⎟⎟ = ∀W ⎪ ⎜⎜ ⎪ ⎝ ∂ [ ⎠[ = ⎨ ⎪ 0 = SRXU[ = / ∀W ⎪ [ ⎪ ⎪ 4 = SRXU[ = / ∀W ⎪ [ ⎩


⇒ I R = ⇒ I ′R = ⇒ I ′′/ =

⇒ I ′′′/ =

(OOHVGRQQHQW α + γ = β + δ = HW

⇒ I R = ⇒ I ′R = ⇒ I ′′/ =

⇒ I ′′′/ =


⎧⎛ ⎞ ⎛ ⎞ ⎪ ⎜ FRV ρ K ω / + FK ρ K ω / ⎟ ⋅ α + ⎜ VLQ ρ K ω / + VK ρ K ω / ⎟ ⋅ β = ⎟ ⎜ ⎟ ' ' ' ' ⎪⎜ ⎠ ⎝ ⎠ ⎪⎝ ⎨ ⎪⎛ ⎞ ⎛ ⎞ ⎪ ⎜ VLQ ρ K ω / − VK ρ K ω / ⎟ ⋅ α − ⎜ FRV ρ K ω / + FK ρ K ω / ⎟ ⋅ β = ⎟ ⎜ ⎟ ' ' ' ' ⎪⎜ ⎠ ⎝ ⎠ ⎩⎝


,O\DPRXYHPHQWVLO¶RQDG¶DXWUHVVROXWLRQVTXHODVROXWLRQWULYLDOH α = β = γ = δ = F¶HVWj GLUHVLOHGpWHUPLQDQWGXV\VWqPHGHVGHX[GHUQLqUHVpTXDWLRQVHVWQXOFHTXLV¶pFULW


FRV

ρ K ω ρ K ω / ⋅ FK / = − ' '

FRV

/HVVROXWLRQVGHFHWWHpTXDWLRQFDUDFWpULVWLTXHVRQW


' ' ⎛ π⎞ ω = ω = ⎜ ⎟ ρ K / ρ K / ⎝ ⎠

ω =

⎛ Q − π ⎞ ωQ = ⎜ ⎟ ⎝ ⎠

ρ K ω ρ K ω / ⋅ FK / = − ' '

' ρ K /

' ρ K /

$FKDTXHYDOHXU ωQ GHODSXOVDWLRQ ω FRUUHVSRQGXQHIRQFWLRQ I Q [ WHOOHTXH

' ' ⎛ π⎞ ω = ω = ⎜ ⎟ ρ K / ρ K / ⎝ ⎠

ω =

⎛ Q − π ⎞ ωQ = ⎜ ⎟ ⎝ ⎠

' ρ K /

' ρ K /


⎛ ⎛ ρ K ωQ ρ K ωQ ⎞⎟ ρ K ωQ ρ K ωQ ⎞⎟ I Q [ = α Q ⋅ ⎜ FRV [ − FK [ + βQ ⋅ ⎜ FRV [ − VK [ ⎜ ⎟ ⎜ ⎟ ' ' ' ' ⎝ ⎠ ⎝ ⎠


- 213 -

- 213 -





⇒ I R = ⇒ I ′R = ⇒ I ′′/ =

⇒ I ′′′/ =


⇒ I R = ⇒ I ′R = ⇒ I ′′/ =

⇒ I ′′′/ =






FRV

ρ K ω ρ K ω / ⋅ FK / = − ' '

FRV

/HVVROXWLRQVGHFHWWHpTXDWLRQFDUDFWpULVWLTXHVRQW ' ' ⎛ π⎞ ω = ω = ⎜ ⎟ ρK / ρK / ⎝ ⎠

ω =

⎛ Q − π ⎞ ωQ = ⎜ ⎟ ⎝ ⎠

ρ K ω ρ K ω / ⋅ FK / = − ' '


' ρ K /

' ρ K /


' ' ⎛ π⎞ ω = ω = ⎜ ⎟ ρK / ρK / ⎝ ⎠

ω =

⎛ Q − π ⎞ ωQ = ⎜ ⎟ ⎝ ⎠

' ρ K /

' ρ K /




- 213 -

- 213 -

HWO¶RQSHXWSUHQGUHSRXU α Q HW βQ


⎧ ρ K ωQ ρ K ωQ ⎪ α Q = FRV / + FK / ' ' ⎪ ⎪ ⎨ ⎪ ⎪ β = VLQ ρ K ωQ / − VK ρ K ωQ / Q ' ' ⎩⎪

(QFHTXLFRQFHUQHODIRQFWLRQ J Q W HOOHVDWLVIDLWjO¶pTXDWLRQGLIIpUHQWLHOOH JQ +

3RVRQV & =

⎧ ρ K ωQ ρ K ωQ ⎪ α Q = FRV / + FK / ' ' ⎪ ⎪ ⎨ ⎪ ⎪ β = VLQ ρ K ωQ / − VK ρ K ωQ / Q ' ' ⎩⎪

(QFHTXLFRQFHUQHODIRQFWLRQ J Q W HOOHVDWLVIDLWjO¶pTXDWLRQGLIIpUHQWLHOOH

'′ ωQ ⋅ J Q + ωQ ⋅ J Q = RX JQ + N Q ωQ ⋅ J Q + ωQ ⋅ J Q = '

( ρ − ν

& = FpOpULWpGHO¶RQGH

(′ &′ = FRHIILFLHQWG¶DPRUWLVVHPHQW ρ − ν′

&′ =

&′&5 Q =

⎛ &′ ⎞ ⎟⎟ FRHIILFLHQWG¶DPRUWLVVHPHQWUpGXLW N Q = ⎜⎜ ⎝ &′&5 Q ⎠

& FRHIILFLHQWG¶DPRUWLVVHPHQWFULWLTXH ωQ

JQ +

3RVRQV & =


( ρ − ν



&′ =

&′&5 Q =



3RXUFKDTXHKDUPRQLTXHRQGLVWLQJXHGRQFWURLVFDVVXLYDQWOHVYDOHXUVGHO¶DPRUWLVVHPHQW






J Q W = H − N Q ωQ W ($ Q FK UQ W + %Q VK UQ W )


HFDVDPRUWLVVHPHQWFULWLTXH N Q =






J Q W = H −ωQ W ($ Q + %Q W )

J Q W = H −ωQ W ($ Q + %Q W )



- 214 -

- 214 -



⎧ ρ K ωQ ρ K ωQ ⎪ α Q = FRV / + FK / ' ' ⎪ ⎪ ⎨ ⎪ ⎪ β = VLQ ρ K ωQ / − VK ρ K ωQ / ⎪⎩ Q ' '

(QFHTXLFRQFHUQHODIRQFWLRQ J Q W HOOHVDWLVIDLWjO¶pTXDWLRQGLIIpUHQWLHOOH JQ +

3RVRQV & =

⎧ ρ K ωQ ρ K ωQ ⎪ α Q = FRV / + FK / ' ' ⎪ ⎪ ⎨ ⎪ ⎪ β = VLQ ρ K ωQ / − VK ρ K ωQ / ⎪⎩ Q ' '

(QFHTXLFRQFHUQHODIRQFWLRQ J Q W HOOHVDWLVIDLWjO¶pTXDWLRQGLIIpUHQWLHOOH


( ρ − ν



&′ =

&′&5 Q =



JQ +

3RVRQV & =


( ρ − ν



&′ =

&′&5 Q =

















J Q W = H −ωQ W ($ Q + %Q W )

J Q W = H −ωQ W ($ Q + %Q W )



- 214 -

- 214 -

/DVROXWLRQJpQpUDOHV¶pFULWILQDOHPHQW Z [ W =

/DVROXWLRQJpQpUDOHV¶pFULWILQDOHPHQW

∞

∑

I Q [ ⋅ J Q W

Z [ W =

Q =

∞

∑ I [ ⋅ J W Q

Q

Q =

/HVFRQVWDQWHV $ Q HW %Q VHGpWHUPLQHQWDXPR\HQGHVFRQGLWLRQVLQLWLDOHVRXGH&DXFK\


⎛∂Z ⎞ ⎟⎟ ϕ [ = Z [ R HW ψ [ = ⎜⎜ ⎝ ∂ W ⎠W =

⎛∂Z ⎞ ⎟⎟ ϕ [ = Z [ R HW ψ [ = ⎜⎜ ⎝ ∂ W ⎠W =

± 9,%5$7,216$;,6<0e75,48(6'(6',648(6 (7&285211(6&,5&8/$,5(6

± 9,%5$7,216$;,6<0e75,48(6'(6',648(6 (7&285211(6&,5&8/$,5(6

/DIOH[LRQD[LV\PpWULTXHDpWpWUDLWpHHQ6WDWLTXHDXWRPH,


/HVIRUPXOHVGHYLHQQHQWHQ'\QDPLTXHHQFRRUGRQQpHVF\OLQGULTXHV θ ] R ]′] GpVLJQH O¶D[HGHUpYROXWLRQ


⎧ ∂Z ⎪ X = −] ∂ U ⎪ ⎪ 3R3 ⎨ Y = ⎪ ⎪ ⎪ Z = Z U ⎩

⎧ ∂Z ⎪ X = −] ∂ U ⎪ ⎪ 3R3 ⎨ Y = ⎪ ⎪ ⎪ Z = Z U ⎩

GpSODFHPHQWUDGLDO GpSODFHPHQWFLUFRQIpUHQWLHO

GpSODFHPHQWD[LDORXWUDQVYHUVDO

⎧ ∂X ∂ Z = −] ⎪ εU = ∂U ∂ U ⎪ ⎪ X ∂Z ⎪ ⎨ εθ = = − ] U U ∂U ⎪ ⎪ ⎪ γ = ⎪ Uθ ⎩

⎧ ⎡ ( ⎛ ∂ Z ν ∂ Z ⎞ ′ ∂ ⎛ ∂ Z ν′ ∂ Z ⎞⎤ ⎟+ ( ⎜ ⎜ ⎟⎥ ⎪ σU = −] ⎢ + + ⎜ U ∂ U ⎟⎠⎥⎦ U ∂ U ⎟⎠ − ν′ ∂ W ⎜⎝ ∂ U ⎪ ⎢⎣ − ν ⎝ ∂ U ⎪ ⎪ ⎡ ( ⎛ ∂Z ⎪ (′ ∂ ⎛⎜ ∂ Z ∂ Z ⎞⎟⎤ ∂ Z ⎞⎟ ⎜ ′ + + ν + ν ⎥ ⎨ σθ = −] ⎢ ⎜ ∂ [ ⎟⎠⎥⎦ ∂ U ⎟⎠ − ν′ ∂ W ⎜⎝ U ∂ U ⎢⎣ − ν ⎝ U ∂ U ⎪ ⎪ ⎪ ⎪ τ = ⎪ Uθ ⎩

GpSODFHPHQWUDGLDO GpSODFHPHQWFLUFRQIpUHQWLHO GpSODFHPHQWD[LDORXWUDQVYHUVDO


⎧ ⎡ ( ⎛ ∂ Z ν ∂ Z ⎞ ′ ∂ ⎛ ∂ Z ν′ ∂ Z ⎞⎤ ⎟+ ( ⎜ ⎜ ⎟⎥ ⎪ σU = −] ⎢ + + ⎜ U ∂ U ⎟⎠⎥⎦ U ∂ U ⎟⎠ − ν′ ∂ W ⎜⎝ ∂ U ⎪ ⎢⎣ − ν ⎝ ∂ U ⎪ ⎪ ⎡ ( ⎛ ∂Z ⎪ (′ ∂ ⎛⎜ ∂ Z ∂ Z ⎞⎟⎤ ∂ Z ⎞⎟ ⎜ ′ + + ν + ν ⎥ ⎨ σθ = −] ⎢ ⎜ ∂ [ ⎟⎠⎥⎦ ∂ U ⎟⎠ − ν′ ∂ W ⎜⎝ U ∂ U ⎢⎣ − ν ⎝ U ∂ U ⎪ ⎪ ⎪ ⎪ τ = ⎪ Uθ ⎩

- 215 -

/DVROXWLRQJpQpUDOHV¶pFULWILQDOHPHQW Z [ W =

- 215 -


∞

∑

I Q [ ⋅ J Q W

Z [ W =

Q =

∞

∑ I [ ⋅ J W Q

Q

Q =



⎛∂Z ⎞ ⎟⎟ ϕ [ = Z [ R HW ψ [ = ⎜⎜ ⎝ ∂ W ⎠W =

⎛∂Z ⎞ ⎟⎟ ϕ [ = Z [ R HW ψ [ = ⎜⎜ ⎝ ∂ W ⎠W =

± 9,%5$7,216$;,6<0e75,48(6'(6',648(6 (7&285211(6&,5&8/$,5(6

± 9,%5$7,216$;,6<0e75,48(6'(6',648(6 (7&285211(6&,5&8/$,5(6





⎧ ∂Z ⎪ X = −] ∂ U ⎪ ⎪ 3R3 ⎨ Y = ⎪ ⎪ ⎪ Z = Z U ⎩

⎧ ∂Z ⎪ X = −] ∂ U ⎪ ⎪ 3R3 ⎨ Y = ⎪ ⎪ ⎪ Z = Z U ⎩




⎧ ⎡ ( ⎛ ∂ Z ν ∂ Z ⎞ ′ ∂ ⎛ ∂ Z ν′ ∂ Z ⎞⎤ ⎜ ⎟+ ( ⎜ ⎟⎥ ⎪ σU = −] ⎢ + + ⎜ U ∂ U ⎟⎠ − ν′ ∂ W ⎜⎝ ∂ U U ∂ U ⎟⎠⎦⎥ ⎪ ⎣⎢ − ν ⎝ ∂ U ⎪ ⎪ ⎡ ( ⎛ ∂Z ⎪ (′ ∂ ⎛⎜ ∂ Z ∂ Z ⎞⎟⎤ ∂ Z ⎞⎟ ⎜ + + ν′ +ν ⎥ ⎨ σθ = −] ⎢ ⎜ U ∂U ⎟ ∂W ⎜ U ∂U ∂ [ ⎟⎠⎦⎥ ∂ U ⎠ − ν′ ⎪ ⎝ ⎣⎢ − ν ⎝ ⎪ ⎪ ⎪ τ = ⎪ Uθ ⎩

- 215 -




⎧ ⎡ ( ⎛ ∂ Z ν ∂ Z ⎞ ′ ∂ ⎛ ∂ Z ν′ ∂ Z ⎞⎤ ⎜ ⎟+ ( ⎜ ⎟⎥ ⎪ σU = −] ⎢ + + ⎜ U ∂ U ⎟⎠ − ν′ ∂ W ⎜⎝ ∂ U U ∂ U ⎟⎠⎦⎥ ⎪ ⎣⎢ − ν ⎝ ∂ U ⎪ ⎪ ⎡ ( ⎛ ∂Z ⎪ (′ ∂ ⎛⎜ ∂ Z ∂ Z ⎞⎟⎤ ∂ Z ⎞⎟ ⎜ + + ν′ +ν ⎥ ⎨ σθ = −] ⎢ ⎜ U ∂U ⎟ ∂W ⎜ U ∂U ∂ [ ⎟⎠⎦⎥ ∂ U ⎠ − ν′ ⎪ ⎝ ⎣⎢ − ν ⎝ ⎪ ⎪ ⎪ τ = ⎪ Uθ ⎩

- 215 -

⎧ ⎪ 8 U = = 8θ ⎪ ⎪ ⎛ ∂ Z ν ∂ Z ⎞ ∂ ⎛⎜ ∂ Z ν′ ∂ Z ⎞⎟ ⎪ ⎟ ⎜ ⎪ 0 U = −' ⎜ ∂ U + U ∂ U ⎟ − '′ ⋅ ∂ W ⎜ ∂ U + U ∂ U ⎟ ⎠ ⎝ ⎠ ⎝ ⎪ ⎪ ⎪ ⎛ ∂Z ∂ ⎛⎜ ∂ Z ∂ Z ⎞⎟ ∂ Z ⎞⎟ ⎪⎪ 0 θ = −' ⎜ ′ ′ − ⋅ + ν +ν ' ⎜ U ∂U ∂ W ⎜⎝ U ∂ U ∂ U ⎟⎠ ∂ U ⎟⎠ ⎨ ⎝ ⎪ ⎪ ⎪ 4 = −⎛⎜ ' + '′ ⋅ ∂ ⎞⎟ ⋅ ∂ ∆ Z ⎜ ⎪ U ∂ W ⎟⎠ ∂ U ⎝ ⎪ ⎪ HQQpJOLJHDQWOLQHUWLHGHURWDWLRQGHVILEUHV ⎪ ⎪ ⎪ 4θ = ⎩⎪

5DSSHORQVTXHO¶RSpUDWHXUODSODFLHQ ∆ V¶pFULW

∆=

⎧ ⎪ 8 U = = 8θ ⎪ ⎪ ⎛ ∂ Z ν ∂ Z ⎞ ∂ ⎛⎜ ∂ Z ν′ ∂ Z ⎞⎟ ⎪ ⎟ ⎜ ⎪ 0 U = −' ⎜ ∂ U + U ∂ U ⎟ − '′ ⋅ ∂ W ⎜ ∂ U + U ∂ U ⎟ ⎠ ⎝ ⎠ ⎝ ⎪ ⎪ ⎪ ⎛ ∂Z ∂ ⎛⎜ ∂ Z ∂ Z ⎞⎟ ∂ Z ⎞⎟ ⎪⎪ 0 θ = −' ⎜ ′ ′ − ⋅ + ν +ν ' ⎜ U ∂U ∂ W ⎜⎝ U ∂ U ∂ U ⎟⎠ ∂ U ⎟⎠ ⎨ ⎝ ⎪ ⎪ ⎪ 4 = −⎛⎜ ' + '′ ⋅ ∂ ⎞⎟ ⋅ ∂ ∆ Z ⎜ ⎪ U ∂ W ⎟⎠ ∂ U ⎝ ⎪ ⎪ HQQpJOLJHDQWOLQHUWLHGHURWDWLRQGHVILEUHV ⎪ ⎪ ⎪ 4θ = ⎩⎪


G G G ⎛ G ⎞ ⎜⎜ U ⎟⎟ + = G U U G U U G U ⎝ G U ⎠

∆=

7UDLWRQV O¶H[HPSOH GX GLVTXH HQFDVWUp VXLYDQW VRQ ERUG U = 5 HQ YLEUDWLRQV WUDQVYHUVDOHV D[LV\PpWULTXHV/LPLWRQVQRXVjO¶pWXGHGHVYLEUDWLRQVQDWXUHOOHVQRQDPRUWLHVHQO¶DEVHQFH GHFKDUJHPHQWOHUHSqUH {R [\]}pWDQWGHSOXVJDOLOpHQILJXUH



)LJXUHD 3HUVSHFWLYH

)LJXUHD 3HUVSHFWLYH

- 216 -

- 216 -

⎧ ⎪ 8 U = = 8θ ⎪ ⎪ ⎛ ∂ Z ν ∂ Z ⎞ ∂ ⎛⎜ ∂ Z ν′ ∂ Z ⎞⎟ ⎪ ⎟ ⎜ ′ 0 ' ' − ⋅ + = − + ⎪ U ⎜ ∂ U U ∂ U ⎟⎠ U ∂ U ⎟⎠ ∂ W ⎜⎝ ∂ U ⎝ ⎪ ⎪ ⎪ ⎛ ∂Z ∂ ⎛⎜ ∂ Z ∂ Z ⎞⎟ ∂ Z ⎞⎟ ⎪⎪ 0 θ = −' ⎜ ′ ′ − ⋅ + ν +ν ' ⎜ U ∂U ∂ W ⎜⎝ U ∂ U ∂ U ⎟⎠ ∂ U ⎟⎠ ⎨ ⎝ ⎪ ⎪ ⎪ 4 = −⎛⎜ ' + '′ ⋅ ∂ ⎞⎟ ⋅ ∂ ∆ Z ⎜ ⎪ U ∂ W ⎟⎠ ∂ U ⎝ ⎪ ⎪ HQQpJOLJHDQWOLQHUWLHGHURWDWLRQGHVILEUHV ⎪ ⎪ ⎪ 4θ = ⎪⎩

⎧ ⎪ 8 U = = 8θ ⎪ ⎪ ⎛ ∂ Z ν ∂ Z ⎞ ∂ ⎛⎜ ∂ Z ν′ ∂ Z ⎞⎟ ⎪ ⎟ ⎜ ′ 0 ' ' − ⋅ + = − + ⎪ U ⎜ ∂ U U ∂ U ⎟⎠ U ∂ U ⎟⎠ ∂ W ⎜⎝ ∂ U ⎝ ⎪ ⎪ ⎪ ⎛ ∂Z ∂ ⎛⎜ ∂ Z ∂ Z ⎞⎟ ∂ Z ⎞⎟ ⎪⎪ 0 θ = −' ⎜ ′ ′ − ⋅ + ν +ν ' ⎜ U ∂U ∂ W ⎜⎝ U ∂ U ∂ U ⎟⎠ ∂ U ⎟⎠ ⎨ ⎝ ⎪ ⎪ ⎪ 4 = −⎛⎜ ' + '′ ⋅ ∂ ⎞⎟ ⋅ ∂ ∆ Z ⎜ ⎪ U ∂ W ⎟⎠ ∂ U ⎝ ⎪ ⎪ HQQpJOLJHDQWOLQHUWLHGHURWDWLRQGHVILEUHV ⎪ ⎪ ⎪ 4θ = ⎪⎩


∆=



∆=




)LJXUHD 3HUVSHFWLYH

)LJXUHD 3HUVSHFWLYH

- 216 -

- 216 -

)LJXUHE &RXSHPpULGLHQQH


/¶pTXDWLRQGXPRXYHPHQW V¶pFULWLFL


' ⋅ ∆ ∆ Z + ρ K

∂ Z = ∂ W

' ⋅ ∆ ∆ Z + ρ K

8QPRGHSURSUHGHSXOVDWLRQ ω V¶pFULW

∂ Z = ∂ W


Z U W = I U ⋅ FRV ωW + ϕ


'¶RO¶pTXDWLRQGHIU RXpTXDWLRQDX[IRUPHVSURSUHV


⎧ ρ K ω ⎪ ∆ ∆ I = N I HQSRVDQW N = ' ⎪ ⎨ ⎪ DYHF ∆ I = G I + G I ⎪⎩ G U U G U


&HWWHpTXDWLRQHVWYpULILpHSDUOHVGHX[IRQFWLRQVGH%HVVHO - NU HW 1 NU IRQFWLRQVGH %HVVHOG¶RUGUH]pURGHqUHHWGHqPHHVSqFHUHVSHFWLYHPHQW


(Q HIIHW FRPPH QRXV O¶DYRQV YX DX GX FKDSLWUH 9,,, - NU HW 1 NU VRQW GHX[ IRQFWLRQVLQGpSHQGDQWHVVDWLVIDLVDQWjO¶pTXDWLRQGH%HVVHO


G - NU G - NU + = − - NU NU G NU G NU


G 1 NU G 1 NU + = − 1 NU NU G NU G NU


- 217 -

- 217 -





' ⋅ ∆ ∆ Z + ρ K

∂ Z = ∂ W


' ⋅ ∆ ∆ Z + ρ K

∂ Z = ∂ W
















- 217 -

- 217 -

6LO¶RQSUHQG I U = - NU RX I U = 1 NU RQD ∆I =

6LO¶RQSUHQG I U = - NU RX I U = 1 NU RQD

G I G I GI ⎤ ⎡ G I + = + N ⎢ ⎥ = −N ⋅ I U U G U NU G NU GU ⎥⎦ ⎣⎢ G NU

∆I =

G I G I GI ⎤ ⎡ G I + = + N ⎢ ⎥ = −N ⋅ I U U G U NU G NU GU ⎥⎦ ⎣⎢ G NU

HWSDUFRQVpTXHQW ∆ ∆ I = −N ∆ I = N ⋅ I


'¶RODVROXWLRQJpQpUDOH

'¶RODVROXWLRQJpQpUDOH I U = α - NU + β 1 NU

I U = α - NU + β 1 NU

/HVFRQGLWLRQVDX[OLPLWHVVXUOHERUGHQFDVWUp U = 5 VRQW I 5 = HW I ′5 = &RPSWH WHQX GHV UqJOHV GH GpULYDWLRQV GHV IRQFWLRQV GH %HVVHO IRUPXOHV GX FKDSLWUH ,,, FHV FRQGLWLRQVV¶pFULYHQW


I 5 = ⇒ α - N5 + β 1 N5 =

I 5 = ⇒ α - N5 + β 1 N5 =

I ′5 = ⇒ − α - N5 + β 1 N5 =

I ′5 = ⇒ − α - N5 + β 1 N5 =

,O \ D PRXYHPHQW VL FH V\VWqPH DGPHW G¶DXWUHV VROXWLRQV TXH OD VROXWLRQ WULYLDOH α = β = F¶HVWjGLUHVLOHGpWHUPLQDQWHVWQXOFHTXLGRQQH


- N5 - N5 + = 1 N5 1 N5

- N5 - N5 + = 1 N5 1 N5

/HVWURLVSUHPLqUHVUDFLQHVVRQW

/HVWURLVSUHPLqUHVUDFLQHVVRQW

N5 = $YHF ω = N

N5 =

N5 =

ρK RQHQGpGXLWOHVWURLVSUHPLqUHVSXOVDWLRQVSURSUHV '

ω =

N5 = $YHF ω = N

' ' ' ω = ω = ρK 5 ρK 5 ρK 5


ω =

' ' ' ω = ω = ρK 5 ρK 5 ρK 5

/DVROXWLRQJpQpUDOHV¶pFULWILQDOHPHQW ∞

∑ (αQ - N Q U + βQ 1 N Q U )⋅ FRV ωQ W + ϕQ

Z U W =

Q =

DYHF α Q = . Q 1 N Q 5 HW βQ = −. Q - N Q 5

∑ (α

Q - N Q U + βQ


- 218 -

6LO¶RQSUHQG I U = - NU RX I U = 1 NU RQD G I G I + = N G U U G U

1 N Q U )⋅ FRV ωQ W + ϕQ

Q =

- 218 -

∆I =

N5 =


∞

Z U W =

N5 =

6LO¶RQSUHQG I U = - NU RX I U = 1 NU RQD

⎡ G I GI ⎤ + ⎢ ⎥ = −N ⋅ I U NU G NU G NU ⎥⎦ ⎣⎢

∆I =

G I G I + = N G U U G U

⎡ G I GI ⎤ + ⎢ ⎥ = −N ⋅ I U NU G NU G NU ⎥⎦ ⎣⎢



'¶RODVROXWLRQJpQpUDOH

'¶RODVROXWLRQJpQpUDOH I U = α - NU + β 1 NU

I U = α - NU + β 1 NU



I 5 = ⇒ α - N5 + β 1 N5 =

I 5 = ⇒ α - N5 + β 1 N5 =

I ′5 = ⇒ − α - N5 + β 1 N5 =

I ′5 = ⇒ − α - N5 + β 1 N5 =



- N5 - N5 + = 1 N5 1 N5

- N5 - N5 + = 1 N5 1 N5

/HVWURLVSUHPLqUHVUDFLQHVVRQW N5 = $YHF ω = N

/HVWURLVSUHPLqUHVUDFLQHVVRQW N5 =

N5 =


ω =

' ' ' ω = ω = ρK 5 ρK 5 ρK 5


N5 = $YHF ω = N

N5 =


ω =

' ' ' ω = ω = ρK 5 ρK 5 ρK 5


∞

Z U W =

N5 =

∑ (αQ - N Q U + βQ 1 N Q U )⋅ FRV ωQ W + ϕQ Q =


- 218 -

∞

Z U W =

∑ (α

Q - N Q U + βQ

1 N Q U )⋅ FRV ωQ W + ϕQ

Q =


- 218 -

±0e7+2'('(5$
±0e7+2'('(5$
1RXVO¶DYRQVLQWURGXLWHDXFKDSLWUHSUpFpGHQWSRXUOHVSRXWUHVRQSHXWDXVVLO¶XWLOLVHUSRXU OHV SODTXHV 3RXU FHOD pWDEOLVVRQV OD IRUPXOH GH 5D\OHLJK GDQV OH FDV GH YLEUDWLRQV WUDQVYHUVDOHVGHVSODTXHV


&RQVLGpURQV GRQF XQH SODTXH TXL YLEUH WUDQVYHUVDOHPHQW VDQV IRUFHV H[FLWDWULFHV HW VDQV DPRUWLVVHPHQWVXLYDQWOHPRGHSURSUHGHUDQJQ


/¶pQHUJLH PpFDQLTXH VRPPH GH O¶pQHUJLH FLQpWLTXH ( F HW GH O¶pQHUJLH SRWHQWLHOOH pODVWLTXH


( S HVW DORUV FRQVHUYpH F¶HVW j GLUH LQGpSHQGDQWH GX WHPSV ([SULPRQVOj HQ QpJOLJHDQW


O¶pQHUJLHFLQpWLTXHGHURWDWLRQGHVILEUHVHWO¶pQHUJLHpODVWLTXHGXFLVDLOOHPHQWWUDQVYHUVDO


(F =

(S =

− ν '

∫∫ 6

⎛∂Z ⎞ ⎟⎟ G6 ρK ⎜⎜ ⎝ ∂W ⎠

∫∫ [(0

[

− 0\

6=VXUIDFHPR\HQQH

) + + ν 0 [ 0 \ − 8 [ 8 \ ]G6

(F =

(S =

6

− ν '

∫∫ 6

⎛∂Z ⎞ ⎟⎟ G6 ρK ⎜⎜ ⎝ ∂W ⎠

∫∫ [(0

[

− 0\

6=VXUIDFHPR\HQQH

) + + ν 0 [ 0 \ − 8 [ 8 \ ]G6

6

&HWWHGHUQLqUHIRUPXOHDpWppWDEOLHDXWRPH,HQ6WDWLTXH


3RXUOHPRGHSURSUHGHUDQJQRQD Z [ \ W = I Q [ \ ⋅ VLQ ωQ W G¶RO¶RQGpGXLW


⎧ ∂Z = ωQ ⋅ FRV ωQ W ⋅ I Q [ \ ⎪ ⎪ ∂W ⎪ ⎪ ∂ Z ∂ IQ ⎪ 8 [ = − 8 \ = ' − ν = ' − ν ⋅ VLQ ωQ W ⋅ ∂[ ∂\ ∂[ ∂\ ⎪ ⎪ ⎨ ⎪ ⎞ ⎛ ∂ IQ ⎞ ⎛ ∂ Z Z I ∂ ∂ Q⎟ ⎟ = −' ⋅ VLQ ωQ W ⋅ ⎜ +ν +ν ⎪ 0 [ = −' ⎜⎜ ⎟ ⎟ ⎜ ∂ [ [ \ \ ∂ ∂ ∂ ⎪ ⎠ ⎝ ⎠ ⎝ ⎪ ⎪ ⎛ ∂ IQ ⎛ ∂ Z ∂ I Q ⎞⎟ ∂ Z ⎞⎟ ⎜ ⎪ 0\ = ' ⎜ ' VLQ W = ⋅ ω ⋅ + ν + ν Q ⎜ ∂ \ ⎜ ∂ \ ∂ [ ⎟⎠ ⎪⎩ ∂ [ ⎟⎠ ⎝ ⎝ '¶RHQSRUWDQWGDQVOHVH[SUHVVLRQVGH ( F HW ( S (F =

ωQ ⋅ FRV ωQ W ⋅ (S =

⋅ VLQ ωQ W ⋅

⎧ ∂Z = ωQ ⋅ FRV ωQ W ⋅ I Q [ \ ⎪ ⎪ ∂W ⎪ ⎪ ∂ Z ∂ IQ ⎪ 8 [ = − 8 \ = ' − ν = ' − ν ⋅ VLQ ωQ W ⋅ ∂[ ∂\ ∂[ ∂\ ⎪ ⎪ ⎨ ⎪ ⎞ ⎛ ∂ IQ ⎞ ⎛ ∂ Z Z I ∂ ∂ Q⎟ ⎟ = −' ⋅ VLQ ωQ W ⋅ ⎜ +ν +ν ⎪ 0 [ = −' ⎜⎜ ⎟ ⎟ ⎜ ∂ [ [ \ \ ∂ ∂ ∂ ⎪ ⎠ ⎝ ⎠ ⎝ ⎪ ⎪ ⎛ ∂ IQ ⎛ ∂ Z ∂ I Q ⎞⎟ ∂ Z ⎞⎟ ⎜ ⎪ 0\ = ' ⎜ ' VLQ W = ⋅ ω ⋅ + ν + ν Q ⎜ ∂ \ ⎜ ∂ \ ∂ [ ⎟⎠ ⎪⎩ ∂ [ ⎟⎠ ⎝ ⎝ '¶RHQSRUWDQWGDQVOHVH[SUHVVLRQVGH ( F HW ( S

∫∫ ρK ⋅ [I [ \ ] ⋅ G6 Q

(F =

6

∫∫ ' φ [ \ ⋅ G6


Q

6

⋅ VLQ ωQ W ⋅

- 219 -

∫∫ ρK ⋅ [I [ \ ] ⋅ G6 Q

6

∫∫ ' φ [ \ ⋅ G6 Q

6

- 219 -

±0e7+2'('(5$
±0e7+2'('(5$










(F =

(S =

− ν '

∫∫ 6

⎛∂Z ⎞ ⎟⎟ G6 ρK ⎜⎜ ⎝ ∂W ⎠

∫∫ [(0

[

− 0\

6=VXUIDFHPR\HQQH

) + + ν 0 [ 0 \ − 8 [ 8 \ ]G6

6

(F =

(S =

− ν '

∫∫ 6

⎛∂Z ⎞ ⎟⎟ G6 ρK ⎜⎜ ⎝ ∂W ⎠

∫∫ [(0

[

− 0\

6=VXUIDFHPR\HQQH

) + + ν 0 [ 0 \ − 8 [ 8 \ ]G6

6





⎧ ∂Z = ωQ ⋅ FRV ωQ W ⋅ I Q [ \ ⎪ ⎪ ∂W ⎪ ⎪ ∂ Z ∂ IQ ⎪ 8 [ = − 8 \ = ' − ν = ' − ν ⋅ VLQ ωQ W ⋅ ∂[ ∂\ ∂[ ∂\ ⎪ ⎪ ⎨ ⎪ ⎞ ⎛ ∂ IQ ⎞ ⎛ ∂ Z Z I ∂ ∂ Q⎟ ⎟ = −' ⋅ VLQ ωQ W ⋅ ⎜ +ν +ν ⎪ 0 [ = −' ⎜⎜ ⎜ ∂ [ ∂ \ ⎟⎠ ∂ \ ⎟⎠ ⎪ ⎝ ⎝ ∂[ ⎪ ⎪ ⎛ ∂ IQ ⎛ ∂ Z ∂ I Q ⎞⎟ ∂ Z ⎞⎟ ⎪ 0\ = ' ⎜ = ' ⋅ VLQ ωQ W ⋅ ⎜ +ν +ν ⎜ ∂\ ⎜ ∂\ ∂ [ ⎟⎠ ⎪⎩ ∂ [ ⎟⎠ ⎝ ⎝ '¶RHQSRUWDQWGDQVOHVH[SUHVVLRQVGH ( F HW ( S (F =


⋅ VLQ ωQ W ⋅

'¶RHQSRUWDQWGDQVOHVH[SUHVVLRQVGH ( F HW ( S

∫∫ ρK ⋅ [I [ \ ] ⋅ G6 Q

6

∫∫ ' φ [ \ ⋅ G6 Q

6

- 219 -

⎧ ∂Z = ωQ ⋅ FRV ωQ W ⋅ I Q [ \ ⎪ ⎪ ∂W ⎪ ⎪ ∂ Z ∂ IQ ⎪ 8 [ = − 8 \ = ' − ν = ' − ν ⋅ VLQ ωQ W ⋅ ∂[ ∂\ ∂[ ∂\ ⎪ ⎪ ⎨ ⎪ ⎞ ⎛ ∂ IQ ⎞ ⎛ ∂ Z Z I ∂ ∂ Q⎟ ⎟ = −' ⋅ VLQ ωQ W ⋅ ⎜ +ν +ν ⎪ 0 [ = −' ⎜⎜ ⎜ ∂ [ ∂ \ ⎟⎠ ∂ \ ⎟⎠ ⎪ ⎝ ⎝ ∂[ ⎪ ⎪ ⎛ ∂ IQ ⎛ ∂ Z ∂ I Q ⎞⎟ ∂ Z ⎞⎟ ⎪ 0\ = ' ⎜ = ' ⋅ VLQ ωQ W ⋅ ⎜ +ν +ν ⎜ ∂\ ⎜ ∂\ ∂ [ ⎟⎠ ⎪⎩ ∂ [ ⎟⎠ ⎝ ⎝

(F =


⋅ VLQ ωQ W ⋅

∫∫ ρK ⋅ [I [ \ ] ⋅ G6 Q

6

∫∫ ' φ [ \ ⋅ G6 Q

6

- 219 -

DYHF

φ Q [ \ =

+ ν ∆ I Q − − ν − ν

DYHF

⎛ ∂ IQ ⎛ ∂ IQ ⎞ ∂ I Q ⎟⎞ ⎛⎜ ∂ I Q ∂ I Q ⎟⎞ ⎜ ⎟ + − ν ⎜ +ν +ν ⎜ ⎜ ⎟ ∂ \ ⎟⎠ ⎝⎜ ∂ \ ∂ [ ⎠⎟ ⎝ ∂[ ⎝ ∂ [ ⋅∂ \ ⎠

(Q pJDODQW OHV GHX[ H[SUHVVLRQV GH ( P FRUUHVSRQGDQW DX[ LQVWDQWV W = HW W = REWLHQWODIRUPXOHGH5D\OHLJKFKHUFKpH

π RQ ωQ

∫∫ ' ⋅ φ ⋅ GV = ρ K ⋅ [I ] ⋅ G6 ∫∫

φ Q [ \ =

+ ν ∆ I Q − − ν − ν

⎛ ∂ IQ ⎛ ∂ IQ ⎞ ∂ I Q ⎟⎞ ⎛⎜ ∂ I Q ∂ I Q ⎟⎞ ⎜ ⎟ + − ν ⎜ +ν +ν ⎜ ⎜ ⎟ ∂ \ ⎟⎠ ⎝⎜ ∂ \ ∂ [ ⎠⎟ ⎝ ∂[ ⎝ ∂ [ ⋅∂ \ ⎠


Q

ωQ

Q

6

Q

∫∫ ' ⋅ φ ⋅ GV = ρ K ⋅ [I ] ⋅ G6 ∫∫

π RQ ωQ

ωQ

6

Q

6

6

7UqVVRXYHQWRQQHFRQQDvWSDVH[DFWHPHQWODIRUPHSURSUH I Q PDLVRQSHXWV¶HQGRQQHUXQH H[SUHVVLRQDSSURFKpHD\DQWPrPHDOOXUHHWVDWLVIDLVDQWDX[PrPHVFRQGLWLRQVDX[OLPLWHV2Q REWLHQWDLQVLXQHYDOHXUDSSURFKpHGH ωQ


$SSOLTXRQV FHWWH PpWKRGH DX[ GHX[ H[HPSOHV SUpFpGHQWV WUDLWpV DX[ HW SRXU FDOFXOHU ω


HUH[HPSOH9LEUDWLRQSDUIOH[LRQF\OLQGULTXH


)LJXUHSUHPLqUHIRUPHSURSUH

2QDLFL

2QDLFL

/

∫ [I ′′[ ] ⋅ G[

ω =


/

/

ω =

∫ [I [ ] ⋅ G[

∫ [I [ ] ⋅ G[

3UHQRQVFRPPHpTXDWLRQDSSURFKpHGHODGpIRUPpHPRGDOH


I [ = [ − / [ + / [

I [ = [ − / [ + / [

- 220 -

- 220 -

DYHF

+ ν ∆ I Q − − ν − ν

DYHF

⎛ ∂ IQ ⎛ ∂ IQ ⎞ ∂ I Q ⎟⎞ ⎛⎜ ∂ I Q ∂ I Q ⎟⎞ ⎜ ⎟ + − ν ⎜ +ν +ν ⎟⎜ ⎟ ⎜ ∂ [ ⎜ ⎟ ∂\ ⎠⎝ ∂\ ∂[ ⎠ ⎝ ⎝ ∂ [ ⋅∂ \ ⎠


∫∫ ' ⋅ φ ⋅ GV = ρ K ⋅ [I ] ⋅ G6 ∫∫

π RQ ωQ

φ Q [ \ =

+ ν ∆ I Q − − ν − ν

⎛ ∂ IQ ⎛ ∂ IQ ⎞ ∂ I Q ⎟⎞ ⎛⎜ ∂ I Q ∂ I Q ⎟⎞ ⎜ ⎜ ⎟ + − ν + ν + ν ⎜ ⎜ ⎟ ∂ \ ⎟⎠ ⎝⎜ ∂ \ ∂ [ ⎠⎟ ⎝ ∂[ ⎝ ∂ [ ⋅∂ \ ⎠


Q

/

φ Q [ \ =

∫ [I ′′[ ] ⋅ G[

ωQ

Q

6

Q

∫∫ ' ⋅ φ ⋅ GV = ρ K ⋅ [I ] ⋅ G6 ∫∫

π RQ ωQ

ωQ

6

Q

6

6








2QDLFL

/

ω =

∫


2QDLFL

/

[I′′ [ ] ⋅ G[

/

∫ [I [ ] ⋅ G[

∫ [I ′′[ ] ⋅ G[


ω =

/

∫ [I [ ] ⋅ G[


I [ = [ − / [ + / [

I [ = [ − / [ + / [

- 220 -

- 220 -

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Z R W = ∀W


⇒ I R =

⎛ ∂ Z ⎞ ⎜⎜ ⎟⎟ = ∀W ⇒ I′R = ⎝ ∂ [ ⎠[ =

Z R W = ∀W

⎛ ∂ Z ⎞ ⎜⎜ ⎟⎟ = ∀W ⇒ I′R = ⎝ ∂ [ ⎠[ =

(0 [ )[ = / = ∀W

⇒ I′′/ =

(0 [ )[ = / = ∀W

⇒ I′′/ =

(4 [ )[ = / = ∀W

⇒ I′′′/ =

(4 [ )[ = / = ∀W

⇒ I′′′/ =

2QWURXYHDORUV

' ω = ρK

∫ [ − /

/

⋅ G[

/

∫ ([

− / [ + / [

) ⋅ G[

' = ρ K /

' ω = ρK

2QWURXYHDORUV /

⇒ I R =

ω =

∫ [ − /

⋅ G[

/

∫ ([

− / [ + / [

) ⋅ G[

=

' ρ K /

'

⇒ ω =

ρK/

'

ρK/

ω =

'

⇒ ω =

ρK/

'

ρ K /

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- 221 -

- 221 -


Z R W = ∀W


⇒ I R =

⎛ ∂ Z ⎞ ⎜⎜ ⎟⎟ = ∀W ⇒ I′R = ⎝ ∂ [ ⎠[ =

Z R W = ∀W

⎛ ∂ Z ⎞ ⎜⎜ ⎟⎟ = ∀W ⇒ I′R = ⎝ ∂ [ ⎠[ =

(0 [ )[ = / = ∀W

⇒ I′′/ =

(0 [ )[ = / = ∀W

⇒ I′′/ =

(4 [ )[ = / = ∀W

⇒ I′′′/ =

(4 [ )[ = / = ∀W

⇒ I′′′/ =

2QWURXYHDORUV

' ω = ρK

∫

/

[ − / ⋅ G[

/

∫ ([

− / [ + / [

) ⋅ G[

' = ρ K /

2QWURXYHDORUV /

⇒ I R =

ω =

' ω = ρK

∫ [ − /

⋅ G[

/

∫ ([

− / [ + / [

) ⋅ G[

=

' ρ K /

'

ρ K /

⇒ ω =

'

ρ K /

ω =

'

ρ K /

⇒ ω =

'

ρ K /







- 221 -

- 221 -

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(

I U = 5 − U V\PpWULH


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(



I′R = I 5 = I′5 = 2QD I′U = U − 5 U HW I′′U = U − 5


5

'

∫

5

φ U ⋅ U ⋅ GU

5

ω =

'

∫ [I U ] ⋅ U ⋅ GU

ρK

DYHF φ U =

' ' G¶R ω = ρ K 5 ρ K 5

$XQRXVDYLRQVWURXYp ω =

' ρK5

+ ν ⎛ ⎞ ⎛ ν ⎞ ⎛ ⎞ ⎜ I′′ + I′ ⎟ − ⎜ I′′ + I′ ⎟ ⎜ I′ + ν I′′⎟ − ν ⎝ U ⎠ − ν ⎝ U ⎠⎝U ⎠

2QWURXYH ω =

' ' G¶R ω = ρ K 5 ρ K 5


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ρ K 5



(





5

'

∫ φ U ⋅ U ⋅ GU

ρK

5

'

5

ω =

∫ [I U ] ⋅ U ⋅ GU

⎞ ⎛ + ν ⎛ ν ⎞ ⎛ ⎞ ⎜ I′′ + I′ ⎟ − ⎜ I′′ + I′ ⎟ ⎜ I′ + ν I′′⎟ − ν ⎝ U ⎠ − ν ⎝ U ⎠⎝U ⎠

2QWURXYH ω =

' ' G¶R ω = ρK5 ρ K 5


' ρK5

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∫ φ U ⋅ U ⋅ GU

5

ω =

ρK

DYHF φ U =

- 222 -


(

'

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- 222 -


+ ν ⎛ ⎞ ⎛ ν ⎞ ⎛ ⎞ ⎜ I′′ + I′ ⎟ − ⎜ I′′ + I′ ⎟ ⎜ I′ + ν I′′⎟ − ν ⎝ U ⎠ − ν ⎝ U ⎠⎝U ⎠

2QWURXYH ω =

∫ [I U ] ⋅ U ⋅ GU

ρK

DYHF φ U =

5

ω =

∫ φ U ⋅ U ⋅ GU

∫ [I U ] ⋅ U ⋅ GU

DYHF φ U =

⎞ ⎛ + ν ⎛ ν ⎞ ⎛ ⎞ ⎜ I′′ + I′ ⎟ − ⎜ I′′ + I′ ⎟ ⎜ I′ + ν I′′⎟ − ν ⎝ U ⎠ − ν ⎝ U ⎠⎝U ⎠

2QWURXYH ω =

' ' G¶R ω = ρK5 ρ K 5


'

ρ K 5

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- 223 -

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⎡ ⎤ ⎥ K ⎢ ⎢ ⎥ GDQVOHUHSqUH 3 H H . = ⎢ ⎥ ⎢⎣ ⎥⎦

{

[I ]

}

⋅ GV

−ρ

⋅ GV

{

[I ]

URWDWLRQGHODILEUHQRUPDOH

−ρ


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}


−ρ

⋅ GV

−ρ

⋅ GV


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3RXUOHVIRUFHV

3RXUOHVIRUFHV

∂ 1 ∂ 7 1 − 1 7 + 7 4 ⎧ − − = ⎪ S + β ∂ α + β ∂ α + 5J 5J 5 1 ⎪ ⎪ ⎛ ∂ X ⎛ ∂ Z ∂ X ⎞⎞ ⎟⎟ ρ K ⎜ + ΓH + ⎜⎜ θ − θ] ⎪ ⎜ ∂W ∂ W ⎟⎠ ⎟⎠ ⎪ ⎝ ⎝ ∂W ⎪ ⎪ ⎪ ⎪ ∂ 1 ∂ 7 1 − 1 7 + 7 4 + + − − = ⎪ S + 5J 5J 5 1 β β ∂ α ∂ α ⎪⎪ ⎨ ⎛ ∂ X ⎛ ∂ X ∂ Z ⎞ ⎞ ⎪ ⎟⎟ ρK⎜ + ΓH + ⎜⎜ θ ] − θ ⎜ ∂W ⎪ ∂W ∂ W ⎟⎠ ⎟⎠ ⎝ ⎝ ⎪ ⎪ ⎪ ⎪ ⎪ S + ∂ 4 + ∂ 4 + 1 + 1 + 4 − 4 = ⎪ ] β ∂ α β ∂ α 5 1 5 1 5J 5J ⎪ ⎛ ∂ Z ⎪ ⎛ ∂ X ∂ X ⎞ ⎞ ⎟⎟ ρ K ⎜ + ΓH ] + ⎜⎜ θ − θ ⎪ ⎜ ∂W ∂ W ⎟⎠ ⎟⎠ ⎪⎩ ⎝ ⎝ ∂W

∂ 1 ∂ 7 1 − 1 7 + 7 4 ⎧ − − = ⎪ S + β ∂ α + β ∂ α + 5J 5J 5 1 ⎪ ⎪ ⎛ ∂ X ⎛ ∂ Z ∂ X ⎞⎞ ⎟⎟ ρ K ⎜ + ΓH + ⎜⎜ θ − θ] ⎪ ⎜ ∂W ∂ W ⎟⎠ ⎟⎠ ⎪ ⎝ ⎝ ∂W ⎪ ⎪ ⎪ ⎪ ∂ 1 ∂ 7 1 − 1 7 + 7 4 + + − − = ⎪ S + 5J 5J 5 1 β β ∂ α ∂ α ⎪⎪ ⎨ ⎛ ∂ X ⎛ ∂ X ∂ Z ⎞ ⎞ ⎪ ⎟⎟ ρK⎜ + ΓH + ⎜⎜ θ ] − θ ⎜ ∂W ⎪ ∂W ∂ W ⎟⎠ ⎟⎠ ⎝ ⎝ ⎪ ⎪ ⎪ ⎪ ⎪ S + ∂ 4 + ∂ 4 + 1 + 1 + 4 − 4 = ⎪ ] β ∂ α β ∂ α 5 1 5 1 5J 5J ⎪ ⎛ ∂ Z ⎪ ⎛ ∂ X ∂ X ⎞ ⎞ ⎟⎟ ρ K ⎜ + ΓH ] + ⎜⎜ θ − θ ⎪ ⎜ ∂W ∂ W ⎟⎠ ⎟⎠ ⎪⎩ ⎝ ⎝ ∂W

3RXUOHVPRPHQW

3RXUOHVPRPHQW

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

⎞ ⎛ K ∂ ω ∂ 8 ∂ 0 8 − 8 0 + 0 + MH + MF ⎟ + + − + 4 = ρ ⎜ ⎟ ⎜ 5J 5J β ∂ α β ∂ α ⎠ ⎝ ∂ W

⎞ ⎛ K ∂ ω ∂ 8 ∂ 0 8 − 8 0 + 0 + MH + MF ⎟ + + − − 4 = ρ ⎜ ⎟ ⎜ 5J 5J β ∂ α β ∂ α ⎠ ⎝ ∂ W

{

}

⎞ ⎛ K ∂ ω ∂ 8 ∂ 0 8 − 8 0 + 0 + MH + MF ⎟ + + − − 4 = ρ ⎜ ⎟ ⎜ 5J 5J β ∂ α β ∂ α ⎠ ⎝ ∂ W

⋅ GV

−ρ

⋅ GV



{

[I ]


−ρ

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−ρ

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−ρ

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3RXUOHVIRUFHV

3RXUOHVIRUFHV

∂ 1 ∂ 7 1 − 1 7 + 7 4 ⎧ − − = ⎪ S + β ∂ α + β ∂ α + 5J 5J 5 1 ⎪ ⎪ ⎛ ∂ X ⎛ ∂ Z ∂ X ⎞⎞ ⎟⎟ ρ K ⎜ + ΓH + ⎜⎜ θ − θ] ⎪ ⎜ ∂W ∂ W ⎟⎠ ⎟⎠ ⎪ ⎝ ⎝ ∂W ⎪ ⎪ ⎪ ⎪ ∂ 1 ∂ 7 1 − 1 7 + 7 4 + + − − = ⎪ S + 5J 5J 5 1 β β ∂ α ∂ α ⎪⎪ ⎨ ⎛ ∂ X ⎛ ∂ X ∂ Z ⎞ ⎞ ⎪ ⎟⎟ ρK⎜ + ΓH + ⎜⎜ θ ] − θ ⎜ ⎪ ∂W ∂ W ⎟⎠ ⎟⎠ ∂W ⎝ ⎝ ⎪ ⎪ ⎪ ⎪ ⎪ S + ∂ 4 + ∂ 4 + 1 + 1 + 4 − 4 = ⎪ ] β ∂ α β ∂ α 5 1 5 1 5J 5J ⎪ ⎛ ∂ Z ⎪ ⎛ ∂ X ∂ X ⎞ ⎞ ⎟⎟ ⎟ ρ K ⎜ + ΓH ] + ⎜⎜ θ − θ ⎪ ⎟ ⎜ W W ∂ ∂ W ∂ ⎪⎩ ⎝ ⎠⎠ ⎝

∂ 1 ∂ 7 1 − 1 7 + 7 4 ⎧ − − = ⎪ S + β ∂ α + β ∂ α + 5J 5J 5 1 ⎪ ⎪ ⎛ ∂ X ⎛ ∂ Z ∂ X ⎞⎞ ⎟⎟ ρ K ⎜ + ΓH + ⎜⎜ θ − θ] ⎪ ⎜ ∂W ∂ W ⎟⎠ ⎟⎠ ⎪ ⎝ ⎝ ∂W ⎪ ⎪ ⎪ ⎪ ∂ 1 ∂ 7 1 − 1 7 + 7 4 + + − − = ⎪ S + 5J 5J 5 1 β β ∂ α ∂ α ⎪⎪ ⎨ ⎛ ∂ X ⎛ ∂ X ∂ Z ⎞ ⎞ ⎪ ⎟⎟ ρK⎜ + ΓH + ⎜⎜ θ ] − θ ⎜ ⎪ ∂W ∂ W ⎟⎠ ⎟⎠ ∂W ⎝ ⎝ ⎪ ⎪ ⎪ ⎪ ⎪ S + ∂ 4 + ∂ 4 + 1 + 1 + 4 − 4 = ⎪ ] β ∂ α β ∂ α 5 1 5 1 5J 5J ⎪ ⎛ ∂ Z ⎪ ⎛ ∂ X ∂ X ⎞ ⎞ ⎟⎟ ⎟ ρ K ⎜ + ΓH ] + ⎜⎜ θ − θ ⎪ ⎟ ⎜ W W ∂ ∂ W ∂ ⎪⎩ ⎝ ⎠⎠ ⎝

3RXUOHVPRPHQW

3RXUOHVPRPHQW

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

⎞ ⎛ K ∂ ω ∂ 8 ∂ 0 8 − 8 0 + 0 + MH + MF ⎟ + + − + 4 = ρ ⎜ ⎟ ⎜ 5J 5J β ∂ α β ∂ α ⎠ ⎝ ∂ W ⎞ ⎛ K ∂ ω ∂ 8 ∂ 0 8 − 8 0 + 0 + MH + MF ⎟ + + − − 4 = ρ ⎜ ⎟ ⎜ 5J 5J β ∂ α β ∂ α ⎠ ⎝ ∂ W

- 224 -

- 224 -


[I ]

⎞ ⎛ K ∂ ω ∂ 8 ∂ 0 8 − 8 0 + 0 + MH + MF ⎟ + + − + 4 = ρ ⎜ ⎟ ⎜ 5J 5J β ∂ α β ∂ α ⎠ ⎝ ∂ W

- 224 -

⎞ ⎛ K ∂ ω ∂ 8 ∂ 0 8 − 8 0 + 0 + MH + MF ⎟ + + − + 4 = ρ ⎜ ⎟ ⎜ 5J 5J β ∂ α β ∂ α ⎠ ⎝ ∂ W ⎞ ⎛ K ∂ ω ∂ 8 ∂ 0 8 − 8 0 + 0 + MH + MF ⎟ + + − − 4 = ρ ⎜ ⎟ ⎜ 5J 5J β ∂ α β ∂ α ⎠ ⎝ ∂ W

- 224 -

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⎧ ⎛ ∂Z X ⎞ + ⎟⎟ ] ⎪ X = X − ⎜⎜ ⎝ β ∂ α 5 1 ⎠ ⎪ ⎪ ⎛ ∂Z G X ⎞ ⎪ + ⎟⎟ ] X = 3R3 ⎨ X = X − ⎜⎜ β ∂ α 5 1 ⎠ ⎝ ⎪ ⎪ ⎪Z=Z ⎪ ⎩

⎧ ε = ε + ] ε′ ⎪⎪ ⎨ ε = ε + ] ε′ ⎪ ′ ⎪⎩ γ = γ + ] γ

HpWDSHGpIRUPDWLRQV

DYHF

⎧ ∂ X X Z − − ⎪ ε = β ∂ α 5J 5 1 ⎪ ⎪ ∂ X X Z ⎪ + − ⎨ ε = β ∂ α 5J 5 1 ⎪ ⎪ ⎪ γ = ⎛⎜ ∂ X + ∂ X + X − X ⎞⎟ ⎪ ⎜ β ∂ α β ∂ α 5J 5J ⎟⎠ ⎝ ⎩

⎧ ⎛ ∂Z X ⎞ + ⎟⎟ ] ⎪ X = X − ⎜⎜ ⎝ β ∂ α 5 1 ⎠ ⎪ ⎪ ⎛ ∂Z G X ⎞ ⎪ + ⎟⎟ ] X = 3R3 ⎨ X = X − ⎜⎜ β ∂ α 5 1 ⎠ ⎝ ⎪ ⎪ ⎪Z=Z ⎪ ⎩

⎧ ε = ε + ] ε′ ⎪⎪ ⎨ ε = ε + ] ε′ ⎪ ′ ⎪⎩ γ = γ + ] γ

DYHF


- 225 -

- 225 -







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DYHF


- 225 -

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DYHF


- 225 -

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

ε′ =

X 5J

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ε′ =

X 5J

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∂ X ∂ − β 5 1 ∂ α β ∂ α

′ = γ −

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⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

ε′ =

X 5J

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ε′ =

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∂ X ∂ − β 5 1 ∂ α β ∂ α

′ = γ −

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⎧ ( (′ ε + ν ε + ⎪ σ = − ν − ν′ ⎪ ⎪ ( (′ ⎪ ε + ν ε + ⎨ σ = − ν − ν′ ⎪ ⎪ ⎪ τ = ( γ + (′ ∂ γ ⎪ − ν − ν′ ∂ W ⎩



∂ ε + ν′ ε ∂W ∂ ε + ν′ ε ∂W

′ HWOHVH[SUHVVLRQVDQDORJXHVSRXU σ′ σ′ τ


HpWDSHYLVVHXUV

HpWDSHYLVVHXUV

2QHQGpGXLWHQSRVDQW , =

K


- 226 -

ε′ =

X 5J

⎛ ∂Z ⎞ X ∂ ⎛ ⎞ Z ∂ ⎛ ∂Z ⎞ ⎜⎜ ⎟⎟ − ⎜⎜ ⎟⎟ − − ⎜⎜ ⎟⎟ + − β ∂ α β ∂ α β ∂ α β ∂ 5 5 5 5J 5 1 ⎠ ⎝ 1 ⎠ ⎝ ⎠ α ⎝ 1 1

ε′ =

X 5J

⎛ ⎞ X ∂ ⎜⎜ ⎟⎟ − − 5 5 1 ⎠ β ∂ α ⎝ 1

′ = γ −

∂ X ∂ − β 5 1 ∂ α β ∂ α

∂ ε + ν′ ε ∂W

K

- 226 -

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

∂ ε + ν′ ε ∂W

⎛ ⎞ Z ∂ ⎜⎜ ⎟⎟ − − β 5 5 ∂ α ⎝ 1 ⎠ 1

⎛ ∂Z ⎞ ∂Z ⎜⎜ ⎟⎟ − β ∂ α β ∂ 5J ⎝ ⎠ α

⎛ X ⎞ ∂ X ∂ ⎜⎜ ⎟⎟ + − β ∂ α β ∂ α 5 5 1 ⎝ 1 ⎠

⎛ X ⎞ ⎜⎜ ⎟⎟ ⎝ 5 1 ⎠

∂Z ∂ Z ∂Z + −− β 5J ∂ α β 5J ∂ α β β ∂ α ⋅ ∂ α

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

ε′ =

X 5J

⎛ ∂Z ⎞ X ∂ ⎛ ⎞ Z ∂ ⎛ ∂Z ⎞ ⎜⎜ ⎟⎟ − ⎜⎜ ⎟⎟ − − ⎜⎜ ⎟⎟ + − β ∂ α β ∂ α β ∂ α β ∂ 5 5 5 5J 5 1 ⎠ ⎝ 1 ⎠ ⎝ ⎠ α ⎝ 1 1

ε′ =

X 5J

⎛ ⎞ X ∂ ⎜⎜ ⎟⎟ − − 5 5 1 ⎠ β ∂ α ⎝ 1

′ = γ −

∂ X ∂ − β 5 1 ∂ α β ∂ α

⎛ ⎞ Z ∂ ⎜⎜ ⎟⎟ − − β 5 5 ∂ α ⎝ 1 ⎠ 1

⎛ ∂Z ⎞ ∂Z ⎜⎜ ⎟⎟ − β ∂ α β ∂ 5J ⎝ ⎠ α

⎛ X ⎞ ∂ X ∂ ⎜⎜ ⎟⎟ + − β ∂ α β ∂ α 5 5 1 ⎝ 1 ⎠

⎛ X ⎞ ⎜⎜ ⎟⎟ ⎝ 5 1 ⎠

∂Z ∂ Z ∂Z + −− β 5J ∂ α β 5J ∂ α β β ∂ α ⋅ ∂ α

HpWDSHFRQWUDLQWHV

HpWDSHFRQWUDLQWHV







∂ ε + ν′ ε ∂W ∂ ε + ν′ ε ∂W



HpWDSHYLVVHXUV

HpWDSHYLVVHXUV


K


- 226 -

K

- 226 -

∂ ε + ν′ ε ∂W ∂ ε + ν′ ε ∂W

⎧ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩

⎧ ⎪ ⎪ ⎪ τ′ ⎪ 7 = K τ − , 5 1 ⎪⎪ ⎨ ⎛ τ ⎞ ⎪ ′ ⎟⎟ 8 = , ⎜⎜ − τ ⎪ 5 ⎝ 1 ⎠ ⎪ ⎪ ⎛ σ ⎞ + σ′ ⎟⎟ 0 = , ⎜⎜ − ⎪ ⎪⎩ ⎝ 5 1 ⎠

1 = K σ − ,

σ′ 5 1

1 = K σ − ,

σ′ 5 1

7 = K τ − ,

′ τ 5 1

⎛ τ ⎞ ′ ⎟⎟ 8 = , ⎜⎜ − τ 5 ⎝ 1 ⎠ ⎛ σ ⎞ + σ′ ⎟⎟ 0 = , ⎜⎜ − ⎝ 5 1 ⎠

⎧ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩

⎧ ⎪ ⎪ ⎪ τ′ ⎪ 7 = K τ − , 5 1 ⎪⎪ ⎨ ⎛ τ ⎞ ⎪ ′ ⎟⎟ 8 = , ⎜⎜ − τ ⎪ 5 ⎝ 1 ⎠ ⎪ ⎪ ⎛ σ ⎞ + σ′ ⎟⎟ 0 = , ⎜⎜ − ⎪ ⎪⎩ ⎝ 5 1 ⎠

1 = K σ − ,

σ′ 5 1

1 = K σ − ,

σ′ 5 1

7 = K τ − ,

′ τ 5 1

⎛ τ ⎞ ′ ⎟⎟ 8 = , ⎜⎜ − τ 5 ⎝ 1 ⎠ ⎛ σ ⎞ + σ′ ⎟⎟ 0 = , ⎜⎜ − ⎝ 5 1 ⎠

2QSHXWDORUVH[SULPHUOHVHIIRUWVWUDQFKDQWWUDQVYHUVDX[ 4 HW 4 jSDUWLUGHVpTXDWLRQV G¶pTXLOLEUHGHVPRPHQWV


HpWDSHpTXDWLRQVGXPRXYHPHQW


2Q OHV REWLHQW HQ UHPSODoDQW GDQV OHV WURLV pTXDWLRQV G¶pTXLOLEUH GHV IRUFHV OHV FRPSRVDQWHV GHV YLVVHXUV SDU OHXUV H[SUHVVLRQV REWHQXV j OD H pWDSH HQ IRQFWLRQ GH X X Z


/¶LQWpJUDWLRQ GH FHV pTXDWLRQV GRQQH DORUV OHV H[SUHVVLRQV GH X α α W X α α W Z α α W DYHFGHVFRQVWDQWHVTXHO¶RQGpWHUPLQHjO¶DLGHGHVFRQGLWLRQVDX[OLPLWHVGHV FRQGLWLRQVGHOLDLVRQHWGHVFRQGLWLRQVLQLWLDOHV


/HVIRUPXOHV j GRQQHQWHQILQOHVH[SUHVVLRQVGHVDXWUHVSDUDPqWUHVGXSUREOqPH


±7+e25,('(60(0%5$1(6

±7+e25,('(60(0%5$1(6

(OOH VH GpGXLW GH OD WKpRULH OLQpDLUH HQ DQQXODQW OHV FRHIILFLHQWV GH WRXV OHV WHUPHV HQ ] SXLVTXH OH ORQJ GH WRXWH ILEUH QRUPDOH GpSODFHPHQWV GpIRUPDWLRQV HW FRQWUDLQWHV VRQW FRQVWDQWV


UHpWDSHGpSODFHPHQWV

UHpWDSHGpSODFHPHQWV

⎧ X = X = X α α W ⎪⎪ G X = 3R3 ⎨ X = X = X α α W ⎪ ⎪⎩ Z = Z = Z α α W

HpWDSHGpIRUPDWLRQV

⎧ X = X = X α α W ⎪⎪ G X = 3R3 ⎨ X = X = X α α W ⎪ ⎪⎩ Z = Z = Z α α W

⎧ ∂ X X Z − − ⎪ ε = ε = ε α α W = β ∂ α 5J 5 1 ⎪ ⎪ ∂ X X Z ⎪ + − ⎨ ε = ε = ε α α W = β ∂ α 5J 5 1 ⎪ ⎪ ⎛ ⎞ ⎪ γ = γ = γ α α W = ⎜ ∂ X + ∂ X + X − X ⎟ ⎜ ⎝ β ∂ α β ∂ α 5J 5J ⎟⎠ ⎪⎩

HpWDSHGpIRUPDWLRQV

⎧ ∂ X X Z − − ⎪ ε = ε = ε α α W = β ∂ α 5J 5 1 ⎪ ⎪ ∂ X X Z ⎪ + − ⎨ ε = ε = ε α α W = β ∂ α 5J 5 1 ⎪ ⎪ ⎛ ⎞ ⎪ γ = γ = γ α α W = ⎜ ∂ X + ∂ X + X − X ⎟ ⎜ ⎝ β ∂ α β ∂ α 5J 5J ⎟⎠ ⎪⎩

- 227 -

⎧ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪

⎧ ⎪ ⎪ ⎪ τ′ ⎪ 7 = K τ − , 5 1 ⎪⎪ ⎨ ⎛ τ ⎞ ⎪ ′ ⎟⎟ 8 = , ⎜⎜ − τ ⎪ 5 1 ⎝ ⎠ ⎪ ⎪ ⎛ σ ⎞ + σ′ ⎟⎟ 0 = , ⎜⎜ − ⎪ ⎝ 5 1 ⎠ ⎩⎪

1 = K σ − ,

σ′ 5 1

- 227 -

1 = K σ − ,

σ′ 5 1

7 = K τ − ,

′ τ 5 1

⎛ τ ⎞ ′ ⎟⎟ 8 = , ⎜⎜ − τ ⎝ 5 1 ⎠ ⎛ σ ⎞ + σ′ ⎟⎟ 0 = , ⎜⎜ − ⎝ 5 1 ⎠

⎧ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩⎪

⎧ ⎪ ⎪ ⎪ τ′ ⎪ 7 = K τ − , 5 1 ⎪⎪ ⎨ ⎛ τ ⎞ ⎪ ′ ⎟⎟ 8 = , ⎜⎜ − τ ⎪ 5 1 ⎝ ⎠ ⎪ ⎪ ⎛ σ ⎞ + σ′ ⎟⎟ 0 = , ⎜⎜ − ⎪ ⎝ 5 1 ⎠ ⎩⎪

1 = K σ − ,

σ′ 5 1

1 = K σ − ,

σ′ 5 1

7 = K τ − ,

′ τ 5 1

⎛ τ ⎞ ′ ⎟⎟ 8 = , ⎜⎜ − τ ⎝ 5 1 ⎠ ⎛ σ ⎞ + σ′ ⎟⎟ 0 = , ⎜⎜ − ⎝ 5 1 ⎠











±7+e25,('(60(0%5$1(6

±7+e25,('(60(0%5$1(6



UHpWDSHGpSODFHPHQWV

UHpWDSHGpSODFHPHQWV

⎧ X = X = X α α W ⎪⎪ G X = 3R3 ⎨ X = X = X α α W ⎪ ⎩⎪ Z = Z = Z α α W

HpWDSHGpIRUPDWLRQV

⎧ X = X = X α α W ⎪⎪ G X = 3R3 ⎨ X = X = X α α W ⎪ ⎩⎪ Z = Z = Z α α W

⎧ ∂ X X Z − − ⎪ ε = ε = ε α α W = β ∂ α 5J 5 1 ⎪ ⎪ ∂ X X Z ⎪ + − ⎨ ε = ε = ε α α W = β ∂ α 5J 5 1 ⎪ ⎪ ⎛ ⎞ ⎪ γ = γ = γ α α W = ⎜ ∂ X + ∂ X + X − X ⎟ ⎜⎝ β ∂ α β ∂ α 5J 5J ⎟⎠ ⎪⎩

HpWDSHGpIRUPDWLRQV

⎧ ∂ X X Z − − ⎪ ε = ε = ε α α W = β ∂ α 5J 5 1 ⎪ ⎪ ∂ X X Z ⎪ + − ⎨ ε = ε = ε α α W = β ∂ α 5J 5 1 ⎪ ⎪ ⎛ ⎞ ⎪ γ = γ = γ α α W = ⎜ ∂ X + ∂ X + X − X ⎟ ⎜⎝ β ∂ α β ∂ α 5J 5J ⎟⎠ ⎪⎩

- 227 -

- 227 -

HpWDSHFRQWUDLQWHV

HpWDSHFRQWUDLQWHV

⎧ ( (′ ∂ ε + ν ε + ε + ν′ ε ⎪ σ = σ = σ α α W = ∂W ′ − ν − ν ⎪ ⎪⎪ ( (′ ∂ ε + ν ε + ε + ν′ ε ⎨ σ = σ = σ α α W = − ν − ν′ ∂ W ⎪ ⎪ ′ ⎪ τ = τ = τ α α W = ( γ + ( ∂ γ ⎪⎩ − ν − ν′ ∂ W

HpWDSHYLVVHXUV

HpWDSHYLVVHXUV ⎧ 1 = K σ ⎧ 4 = 4 = ⎪⎪ ⎪⎪ ⎨ 8 = = 8 ⎨ 1 = K σ ⎪ ⎪ ⎪⎩ 7 = 7 = 7 = K τ ⎪⎩ 0 = = 0

⎧ ( (′ ∂ ε + ν ε + ε + ν′ ε ⎪ σ = σ = σ α α W = ∂W ′ − ν − ν ⎪ ⎪⎪ ( (′ ∂ ε + ν ε + ε + ν′ ε ⎨ σ = σ = σ α α W = − ν − ν′ ∂ W ⎪ ⎪ ′ ⎪ τ = τ = τ α α W = ( γ + ( ∂ γ ⎪⎩ − ν − ν′ ∂ W

⎧ 1 = K σ ⎧ 4 = 4 = ⎪⎪ ⎪⎪ ⎨ 8 = = 8 ⎨ 1 = K σ ⎪ ⎪ ⎪⎩ 7 = 7 = 7 = K τ ⎪⎩ 0 = = 0



2QOHVREWLHQWFRPPHSRXUODWKpRULHOLQpDLUHHQUHPSODoDQWGDQVOHVpTXDWLRQVGHO¶pTXLOLEUH ORFDO GHV IRUFHV OHV FRPSRVDQWHV 1 1 HW 7 = 7 = 7 GHV YLVVHXUV SDU OHXUV H[SUHVVLRQV HQ IRQFWLRQ GH X X HW Z REWHQXHV j OD TXDWULqPH pWDSH DYHF GH SOXV 4 = 4 =


±352%/Ê0(6$;,6<0e75,48(60e5,',(16

±352%/Ê0(6$;,6<0e75,48(60e5,',(16

&RQVLGpURQV OHV YLEUDWLRQV D[LV\PpWULTXHV G¶XQH FRTXH D[LV\PpWULTXH FKDTXH SDUWLFXOH YLEUDQW GDQV VRQSODQPpULGLHQ R] GpVLJQH O¶D[H GH UpYROXWLRQ /HV OLJQHV GH FRXUEXUH L


VRQWOHVFHUFOHVSDUDOOqOHVDYHF α = θ OHVOLJQHVGHFRXUEXUH L VRQWOHVPpULGLHQQHVDYHF


α = V

α = V

7RXWHVOHVIRUPXOHVQpFHVVDLUHVDXWUDLWHPHQWGHFHSUREOqPHRQWpWppWDEOLHVHQ6WDWLTXHDX WRPH,


2QDLFLOHVSDUDPqWUHVpWDQWLQGpSHQGDQWVGH θ


⎧ X = GpSODFHPHQWFLUFRQIpUHQWLHO ⎪⎪ 3R3 ⎨ X = X V W ⎪ ⎩⎪ Z = Z V W

⎧ X = GpSODFHPHQWFLUFRQIpUHQWLHO ⎪⎪ 3R3 ⎨ X = X V W ⎪ ⎩⎪ Z = Z V W

- 228 -

- 228 -

HpWDSHFRQWUDLQWHV

HpWDSHFRQWUDLQWHV

⎧ ( (′ ∂ ε + ν ε + ε + ν′ ε ⎪ σ = σ = σ α α W = − ν − ν′ ∂ W ⎪ ⎪⎪ ( (′ ∂ ε + ν ε + ε + ν′ ε ⎨ σ = σ = σ α α W = − ν − ν′ ∂ W ⎪ ⎪ ′ ⎪ τ = τ = τ α α W = ( γ + ( ∂ γ − ν − ν′ ∂ W ⎩⎪

HpWDSHYLVVHXUV

⎧ ( (′ ∂ ε + ν ε + ε + ν′ ε ⎪ σ = σ = σ α α W = − ν − ν′ ∂ W ⎪ ⎪⎪ ( (′ ∂ ε + ν ε + ε + ν′ ε ⎨ σ = σ = σ α α W = − ν − ν′ ∂ W ⎪ ⎪ ′ ⎪ τ = τ = τ α α W = ( γ + ( ∂ γ − ν − ν′ ∂ W ⎩⎪

HpWDSHYLVVHXUV ⎧ 1 = K σ ⎧ 4 = 4 = ⎪⎪ ⎪⎪ ⎨ 8 = = 8 ⎨ 1 = K σ ⎪ ⎪ ⎪⎩ 7 = 7 = 7 = K τ ⎪⎩ 0 = = 0

⎧ 1 = K σ ⎧ 4 = 4 = ⎪⎪ ⎪⎪ ⎨ 8 = = 8 ⎨ 1 = K σ ⎪ ⎪ ⎪⎩ 7 = 7 = 7 = K τ ⎪⎩ 0 = = 0





±352%/Ê0(6$;,6<0e75,48(60e5,',(16

±352%/Ê0(6$;,6<0e75,48(60e5,',(16





α = V

α = V





⎧ X = GpSODFHPHQWFLUFRQIpUHQWLHO ⎪⎪ 3R3 ⎨ X = X V W ⎪ ⎪⎩ Z = Z V W

⎧ X = GpSODFHPHQWFLUFRQIpUHQWLHO ⎪⎪ 3R3 ⎨ X = X V W ⎪ ⎪⎩ Z = Z V W

- 228 -

- 228 -

±UH$33/,&$7,219,%5$7,21675$169(56$/(6 '¶81(0(0%5$1(5(&7$1*8/$,5(3/$1(7(1'8(

±UH$33/,&$7,219,%5$7,21675$169(56$/(6 '¶81(0(0%5$1(5(&7$1*8/$,5(3/$1(7(1'8(

8QHPHPEUDQHUHFWDQJXODLUHHVWIL[pHVXLYDQWVHVERUGVjXQFDGUHLQGpIRUPDEOHFRQVWLWXDQW XQ VROLGH JDOLOpHQ /HV TXDWUH F{WpV GH FH FDGUH H[HUFHQW VXU OD PHPEUDQH GHV HIIRUWV GH WHQVLRQXQLIRUPpPHQWUpSDUWLVHWG¶LQWHQVLWp1SDUXQLWpGHORQJXHXU


(WXGLRQVOHVYLEUDWLRQVWUDQVYHUVDOHVQDWXUHOOHV {Z [ \ W }QRQDPRUWLHVHQVXSSRVDQWFHWWH WHQVLRQ1FRQVWDQWHGDQVOHWHPSVF¶HVWjGLUHQRQDIIHFWpHSDUOHPRXYHPHQWFHTXLHVWYUDL SRXUOHVIDLEOHVDPSOLWXGHV


2$ = D

)LJXUH

2% = E

2$ = D

)LJXUH

2% = E

/H WHQVHXU GHV HIIRUWV GH PHPEUDQH HVW XQLIRUPH FRQVWDQW HW LVRWURSH LO HQ HVW GRQF GH PrPH GHV FRQWUDLQWHV HW GpIRUPDWLRQV HW O¶RQ D G¶DSUqV HW − ν − ν σ= 1 = 1 = 1 = K σ = K σ = K σ SXLV ε = ε = ε = 1 ( (K


/DWURLVLqPHpTXDWLRQ G¶pTXLOLEUHGHVIRUFHVWUDQVYHUVDOHVV¶pFULWDORUV


⎞ ∂ Z 1 ⎛ ⎜⎜ ⎟ = + ρ K ⎝ 5 1 5 1 ⎟⎠ ∂W

⎞ ∂ Z 1 ⎛ ⎜⎜ ⎟ = + ρ K ⎝ 5 1 5 1 ⎟⎠ ∂W

&RPPH GDQV OH FDV GHV FRUGHV YLEUDQWHV QRXV DYRQV G FRQVLGpUHU FHW pTXLOLEUH HQ FRQILJXUDWLRQGpIRUPpHHWHQWHQDQWFRPSWHGHFHWWHGpIRUPDWLRQ2QDDLQVL


∂ Z ∂ Z + = + = ∆Z 5 1 5 1 ∂ [ ∂ \

∂ Z ∂ Z + = + = ∆Z 5 1 5 1 ∂ [ ∂ \

DLQVLTXHQRXVO¶DYRQVPRQWUpDXFKDSLWUH9,,GXWRPH,GDQVOHFDVGHVVXUIDFHV©SUHVTXH SODQHVªSDJH


- 229 -

- 229 -

±UH$33/,&$7,219,%5$7,21675$169(56$/(6 '¶81(0(0%5$1(5(&7$1*8/$,5(3/$1(7(1'8(

±UH$33/,&$7,219,%5$7,21675$169(56$/(6 '¶81(0(0%5$1(5(&7$1*8/$,5(3/$1(7(1'8(





2$ = D

)LJXUH

2% = E

2$ = D

)LJXUH

2% = E





∂ Z 1 = ∂ W ρ K

⎛ ⎞ ⎜⎜ ⎟⎟ + ⎝ 5 1 5 1 ⎠

∂ Z 1 = ∂ W ρ K

⎛ ⎞ ⎜⎜ ⎟⎟ + ⎝ 5 1 5 1 ⎠



∂ Z ∂ Z + = + = ∆Z 5 1 5 1 ∂ [ ∂ \

∂ Z ∂ Z + = + = ∆Z 5 1 5 1 ∂ [ ∂ \



- 229 -

- 229 -

)LQDOHPHQWO¶pTXDWLRQGXPRXYHPHQWV¶pFULW

∆Z =


ρ K ∂ Z ∂ Z 1 RX DYHF & = ∆ Z = 1 ∂ W ρK & ∂ W

∆Z =

ρ K ∂ Z ∂ Z 1 RX DYHF & = ∆ Z = 1 ∂ W ρK & ∂ W

,OV¶DJLWHQFRUHG¶XQHpTXDWLRQGXW\SHGHG¶$OHPEHUWTXHQRXVDOORQVUpVRXGUHHQFKHUFKDQW GHVVROXWLRQVGHODIRUPH Z [ \ W = S [ ⋅ T \ ⋅ J W


2QDQpFHVVDLUHPHQW

2QDQpFHVVDLUHPHQW J ⎛ S′′ T′′ ⎞ = & ⎜⎜ + ⎟⎟ = − ω J ⎝S T⎠

J ⎛ S′′ T′′ ⎞ = & ⎜⎜ + ⎟⎟ = − ω J ⎝S T⎠

G¶RO¶RQGpGXLWLPPpGLDWHPHQW


J W = $ FRV ωW + % VLQ ωW


/HVIRQFWLRQVS[ HWT\ YpULILHQWDORUVO¶pTXDWLRQ


S′′ T′′ = − − & ω S T

S′′ T′′ = − − & ω S T

/HSUHPLHUPHPEUHQHSRXYDQWGpSHQGUHTXHGH[HWOHVHFRQGTXHGH\LOVQHGpSHQGHQWQL GH [ QL GH \ HW pJDOHQW GRQF XQH FRQVWDQWH 2Q YpULILH DLVpPHQW TXH VHXOH XQH FRQVWDQWH VWULFWHPHQWQpJDWLYHSHUPHWG¶REWHQLUGHVVROXWLRQV QRQ QXOOHVS[ HW T\ VDWLVIDLVDQW DX[ FRQGLWLRQVVXUOHVERUGV2QDSSHOOHUD −N FHWWHFRQVWDQWH


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(

)

S′′ + N S = HW T′′ + & ω − N T =

(

)

S′′ + N S = HW T′′ + & ω − N T =

/DSUHPLqUHGRQQH S = α FRV N[ + β VLQ N[


/DFRQGLWLRQ Z = VXUOHVERUGV [ = HW [ = D LPSRVH α = HW β VLQ ND = RQSHXW


+

SUHQGUH β = HW ND = Q π Q ∈ 1 +

SUHQGUH β = HW ND = Q π Q ∈ 1 2QDGRQFSRXUS[ XQHVXLWHGHVROXWLRQV

2QDGRQFSRXUS[ XQHVXLWHGHVROXWLRQV

[ S Q [ = VLQ Q π D

S Q [ = VLQ Q π

6HPEODEOHPHQWRQWURXYH

[ D


T \ = γ FRV ⎜⎛ & ω − N ⋅ \ ⎞⎟ + δ VLQ ⎜⎛ & ω − N ⋅ \ ⎞⎟ ⎝ ⎠ ⎝ ⎠

T \ = γ FRV ⎜⎛ & ω − N ⋅ \ ⎞⎟ + δ VLQ ⎜⎛ & ω − N ⋅ \ ⎞⎟ ⎝ ⎠ ⎝ ⎠

- 230 -

- 230 -


∆Z =


ρ K ∂ Z ∂ Z 1 RX DYHF & = ∆ Z = 1 ∂W ρK & ∂W

∆Z =

ρ K ∂ Z ∂ Z 1 RX DYHF & = ∆ Z = 1 ∂W ρK & ∂W



2QDQpFHVVDLUHPHQW

2QDQpFHVVDLUHPHQW J ⎛ S′′ T′′ ⎞ = & ⎜⎜ + ⎟⎟ = − ω J ⎝S T⎠

J ⎛ S′′ T′′ ⎞ = & ⎜⎜ + ⎟⎟ = − ω J ⎝S T⎠







S′′ T′′ = − − & ω S T

S′′ T′′ = − − & ω S T





(

)

S′′ + N S = HW T′′ + & ω − N T =

(

)

S′′ + N S = HW T′′ + & ω − N T =





+

SUHQGUH β = HW ND = Q π Q ∈ 1 +

SUHQGUH β = HW ND = Q π Q ∈ 1 2QDGRQFSRXUS[ XQHVXLWHGHVROXWLRQV

2QDGRQFSRXUS[ XQHVXLWHGHVROXWLRQV

[ S Q [ = VLQ Q π D


S Q [ = VLQ Q π

[ D


T \ = γ FRV ⎜⎛ & ω − N ⋅ \ ⎞⎟ + δ VLQ ⎜⎛ & ω − N ⋅ \ ⎞⎟ ⎝ ⎠ ⎝ ⎠

T \ = γ FRV ⎜⎛ & ω − N ⋅ \ ⎞⎟ + δ VLQ ⎜⎛ & ω − N ⋅ \ ⎞⎟ ⎝ ⎠ ⎝ ⎠

- 230 -

- 230 -

/D FRQGLWLRQ

Z = VXU OHV ERUGV

\ = HW

\ = E LPSRVH

γ = HW

/D FRQGLWLRQ

Z = VXU OHV ERUGV

\ = HW

δ VLQ & ω − N ⋅ E =

δ VLQ & ω − N ⋅ E =

2QSUHQGDORUV δ = HW & ω − N ⋅ E = P π P ∈ 1 +


3RXU N = Q

π RQWURXYH D

3RXU N = Q

⎛ Q P ⎞ ω = π & ⎜ − ⎟ = ωQ P ⎜D E ⎟⎠ ⎝

2QDDLQVLSRXUT\ ODVXLWHGHVROXWLRQV


T P \ = VLQ P π

\ E

T P \ = VLQ P π

/HVVROXWLRQVSDUWLFXOLqUHVSRXUODIOqFKHZV¶pFULYHQWGHFHIDLW

(

)

:Q P = VLQ Q

HWO¶RQHVWFRQGXLWjpFULUHODVROXWLRQJpQpUDOHVRXVODIRUPHG¶XQHGRXEOHVpULHGH)RXULHU ∞

(

)

π[ π[ VLQ P $ Q P FRV ωQ P W + %Q P VLQ ωQ P W D E

HWO¶RQHVWFRQGXLWjpFULUHODVROXWLRQJpQpUDOHVRXVODIRUPHG¶XQHGRXEOHVpULHGH)RXULHU

∞

∑∑

\ E



Z=

γ = HW

π RQWURXYH D

⎛ Q P ⎞ ω = π & ⎜ − ⎟ = ωQ P ⎜D E ⎟⎠ ⎝

:Q P = VLQ Q

\ = E LPSRVH

∞

Z=

:Q P [ \ W

Q = P =

∞

∑ ∑:

Q P [

\ W

Q = P =

/HVFRQVWDQWHV $ Q P HW %Q P VHGpWHUPLQHQWjO¶DLGHGHVFRQGLWLRQVLQLWLDOHVRXFRQGLWLRQVGH


&DXFK\

&DXFK\ ⎧ ⎪ ϕ [ \ = Z [ \ R IOqFKHLQLWLDOH ⎪ ⎨ ⎪ ψ [ \ = ⎛⎜ ∂ Z ⎞⎟ YLWHVVH LQLWLDOH ⎜ ∂W ⎟ ⎪ ⎝ ⎠ W = ⎩

⎧ ⎪ ϕ [ \ = Z [ \ R IOqFKHLQLWLDOH ⎪ ⎨ ⎪ ψ [ \ = ⎛⎜ ∂ Z ⎞⎟ YLWHVVH LQLWLDOH ⎜ ∂W ⎟ ⎪ ⎝ ⎠ W = ⎩

2QWURXYHOHVLQWpJUDOHVpWDQWSULVHVVXUOHUHFWDQJOHRFFXSpSDUODPHPEUDQH


π[ π\ ⎧ ϕ [ \ ⋅ VLQ Q ⋅ VLQ P ⋅ G[ ⋅ G\ ⎪ $ Q P = DE D E ⎪ ⎨ [ \ ⎪% = ψ [ \ ⋅ VLQ Q π ⋅ VLQ P π ⋅ G[ ⋅ G\ ⎪ Q P ω DE D E QP ⎩


- 231 -

- 231 -

∫∫

∫∫

∫∫

/D FRQGLWLRQ

Z = VXU OHV ERUGV

\ = HW

\ = E LPSRVH

∫∫

γ = HW

/D FRQGLWLRQ

Z = VXU OHV ERUGV

\ = HW

δ VLQ & ω − N ⋅ E =

δ VLQ & ω − N ⋅ E =



3RXU N = Q

π RQWURXYH D

3RXU N = Q

⎛ Q P ⎞ ω = π & ⎜ − ⎟ = ωQ P ⎜D E ⎟⎠ ⎝



T P \ = VLQ P π

\ E

T P \ = VLQ P π


(

)

HWO¶RQHVWFRQGXLWjpFULUHODVROXWLRQJpQpUDOHVRXVODIRUPHG¶XQHGRXEOHVpULHGH)RXULHU ∞

:Q P = VLQ Q

(

)


HWO¶RQHVWFRQGXLWjpFULUHODVROXWLRQJpQpUDOHVRXVODIRUPHG¶XQHGRXEOHVpULHGH)RXULHU

∞

∑∑

\ E



Z=

γ = HW

π RQWURXYH D

⎛ Q P ⎞ ω = π & ⎜ − ⎟ = ωQ P ⎜D E ⎟⎠ ⎝

:Q P = VLQ Q

\ = E LPSRVH

∞

Z=

:Q P [ \ W

Q = P =

∞

∑ ∑:

Q P [

\ W

Q = P =



&DXFK\

&DXFK\ ⎧ ⎪ ϕ [ \ = Z [ \ R IOqFKHLQLWLDOH ⎪ ⎨ ⎪ ψ [ \ = ⎛⎜ ∂ Z ⎞⎟ YLWHVVH LQLWLDOH ⎜ ∂W ⎟ ⎪ ⎝ ⎠ W = ⎩


⎧ ⎪ ϕ [ \ = Z [ \ R IOqFKHLQLWLDOH ⎪ ⎨ ⎪ ψ [ \ = ⎛⎜ ∂ Z ⎞⎟ YLWHVVH LQLWLDOH ⎜ ∂W ⎟ ⎪ ⎝ ⎠ W = ⎩




- 231 -

- 231 -

∫∫

∫∫

∫∫

∫∫

±H$33/,&$7,215e3216('¶81(0(0%5$1(63+e5,48(¬81&+2&

±H$33/,&$7,215e3216('¶81(0(0%5$1(63+e5,48(¬81&+2&

/DPHPEUDQHVSKpULTXHGHUD\RQDHWG¶pSDLVVHXUKHVWDXUHSRVMXVTX¶jO¶LQVWDQW W = $ FHW LQVWDQW D OLHX XQH H[SORVLRQ HQ VRQ FHQWUH FUpDQW VXU OD SDURL LQWHUQH XQH SUHVVLRQ S = S H −αW R S HW α GpVLJQHQWGHVFRQVWDQWHVSRVLWLYHVGRQQpHV2QVHSURSRVHG¶pWXGLHU OHPRXYHPHQWGHODVSKqUHVRXVO¶HIIHWGHFHFKRFHQO¶DEVHQFHGHWRXWDXWUHFKDUJHPHQWHW HQ WHQDQW FRPSWH GH O¶DPRUWLVVHPHQW YLVTXHX[ GX PDWpULDX /HV D[HV R[ R\ R] OLpV j OD VSKqUHVRQWVXSSRVpVJDOLOpHQV


)LJXUHD

)LJXUHE

)LJXUHD

)LJXUHE

/HSUREOqPHSRVVqGHODV\PpWULHVSKpULTXHHWO¶RQDOHVH[SUHVVLRQV


± SRXUOHVGpSODFHPHQWV X = X = Z = Z W


± SRXUOHVGpIRUPDWLRQV ε = ε = ± SRXUOHVFRQWUDLQWHV σ = σ =

Z D

± SRXUOHVGpIRUPDWLRQV ε = ε =

( Z (′ Z + − ν D − ν′ D

± SRXUOHVFRQWUDLQWHV σ = σ =

Z D

( Z (′ Z + − ν D − ν′ D

± SRXUOHVYLVVHXUV 1 = 1 = K σ


$SDUWLUGHODLqPHpTXDWLRQ G¶pTXLOLEUHGHVIRUFHVVXLYDQWODQRUPDOH . jODFRTXH RQ REWLHQWO¶pTXDWLRQGXPRXYHPHQW


⎛ ⎞ G Z ⎟⎟ = ρ K + S H − αW + 1 ⎜⎜ G W ⎝ 5 1 5 1 ⎠

⎛ ⎞ G Z ⎟⎟ = ρ K + S H − αW + 1 ⎜⎜ G W ⎝ 5 1 5 1 ⎠

6RLWSXLVTX¶LFL 5 1 = 5 1 = −D

G Z GW

+


G Z

(′

GZ ( S Z = H − αW + G W K ρ ′ ρ D − ν ρ D − ν

GW

+

(′

GZ ( S Z = H − αW + G W K ρ ′ ρ D − ν ρ D − ν

- 232 -

- 232 -

±H$33/,&$7,215e3216('¶81(0(0%5$1(63+e5,48(¬81&+2&

±H$33/,&$7,215e3216('¶81(0(0%5$1(63+e5,48(¬81&+2&



)LJXUHD

)LJXUHE

)LJXUHD

)LJXUHE





± SRXUOHVGpIRUPDWLRQV ε = ε = ± SRXUOHVFRQWUDLQWHV σ = σ =

Z D

± SRXUOHVGpIRUPDWLRQV ε = ε =

( Z (′ Z + − ν D − ν′ D

± SRXUOHVFRQWUDLQWHV σ = σ =

Z D

( Z (′ Z + − ν D − ν′ D





⎛ ⎞ G Z ⎟⎟ = ρ K + S H − αW + 1 ⎜⎜ G W ⎝ 5 1 5 1 ⎠

⎛ ⎞ G Z ⎟⎟ = ρ K + S H − αW + 1 ⎜⎜ G W ⎝ 5 1 5 1 ⎠


G Z GW

+


(′

GZ ( S Z = H − αW + ρK ρ D − ν′ G W ρ D − ν

- 232 -

G Z GW

+

(′

GZ ( S Z = H − αW + ρK ρ D − ν′ G W ρ D − ν

- 232 -

(QSRVDQW ω=

N=

(

(QSRVDQW

− ν D ρ

(′ D − ν′

ω=

SXOVDWLRQOLEUHQRQDPRUWLH

− ν DPRUWLVVHPHQWUpGXLW (ρ

N=

HOOHV¶pFULWVRXVIRUPHFDQRQLTXH

G Z GW

(

− ν D ρ

(′ D − ν′




+ N ω

G Z

GZ S + ω ⋅ Z = H − α W GW ρK

GW

+ N ω


3RXU OD VROXWLRQ JpQpUDOH GH O¶pTXDWLRQ KRPRJqQH VDQV VHFRQG PHPEUH RQ GRLW HQYLVDJHU WURLVFDV


HUFDV N Q > DPRUWLVVHPHQWIRUW


(QSRVDQW U = ω N − RQD


Z W = H − NωW ($ FK UW + % VK UW )


HFDV N Q = DPRUWLVVHPHQWFULWLTXH


HFDV N Q < DPRUWLVVHPHQWIDLEOH


(QSRVDQW Ω = ω − N RQD


Z W = H −ωW ($ + %W )

Z W = H −ωW ($ + %W )

Z W = H − NωW ($ FRV ΩW + % VLQ ΩW ) 8QHVROXWLRQSDUWLFXOLqUHGHO¶pTXDWLRQFRPSOqWHUpSRQVHIRUFpH HVW

Z ∗ W =

S

ρ K α − Nω α + ω

8QHVROXWLRQSDUWLFXOLqUHGHO¶pTXDWLRQFRPSOqWHUpSRQVHIRUFpH HVW

H − αW

VL α Q¶DQQXOHSDVOHGpQRPLQDWHXU HW ω∗ W =

Z W = H − NωW ($ FRV ΩW + % VLQ ΩW )

Z ∗ W =


S W H − αW ρ K Nω − α

HW ω∗ W =


GDQVOHFDVFRQWUDLUH - 233 -

- 233 -

(QSRVDQW

N=

(

(QSRVDQW

− ν D ρ

(′ D − ν′

H − αW

VL α Q¶DQQXOHSDVOHGpQRPLQDWHXU

GDQVOHFDVFRQWUDLUH

ω=

S

ω=



N=


G Z GW

(

− ν D ρ

(′ D − ν′




+ N ω

G Z


GW

+ N ω
















Z W = H −ωW ($ + %W )

Z W = H −ωW ($ + %W )


Z ∗ W =

S


VL α Q¶DQQXOHSDVOHGpQRPLQDWHXU HW ω∗ W =


H − αW

Z ∗ W =

S


VL α Q¶DQQXOHSDVOHGpQRPLQDWHXU


HW ω∗ W =

GDQVOHFDVFRQWUDLUH


GDQVOHFDVFRQWUDLUH - 233 -

- 233 -

H − αW

$FKHYRQVOHSUREOqPHGDQVOHFDVG¶XQDPRUWLVVHPHQWIDLEOH2QDDORUV


S H − αW Z W = H − NωW ($ FRV ΩW + % VLQ ΩW )+ ρ K α − Nω α + ω YLEUDWLRQQDWXUHOOH


UpSRQVHIRUFpH

UpSRQVHIRUFpH

= GRQQHQW /HVFRQGLWLRQVLQLWLDOHV Z = HW Z

$=

− S


HW % =


α − Nω S

ρ K ω − N α − Nω α + ω

$=

− S


- 234 -

UpSRQVHIRUFpH



HW % =


α − Nω S


$=

- 234 -


UpSRQVHIRUFpH

− S




α − Nω S

- 234 -


$=

HW % =

− S


HW % =

- 234 -

α − Nω S ρ K ω − N α − Nω α + ω

- 235 -

- 235 -

- 235 -

- 235 -

Vous pouvez faire part de vos remarques, critiques, suggestions aux auteurs à cette adresse : [email protected]

Achevé d’imprimer en France en juillet 2011 chez Messages SAS 111, rue Nicolas Vauquelin • 31100 Toulouse Tél. : 05 61 41 24 14 • Fax : 05 61 19 00 43 [email protected]

Mécanique Des Structures Tom3 Thermique Des Structures ,Dynamique Des Structures

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