Modélisation multi-échelle d un assemblage riveté aéronautique ...

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HAL Id: tel-00512558https://tel.archives-ouvertes.fr/tel-00512558

Submitted on 30 Aug 2010

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Modélisation multi-échelle d un assemblage rivetéaéronautique - Vers un modèle de fragilisation

structuraleAnne-Sophie Bayart

To cite this version:Anne-Sophie Bayart. Modélisation multi-échelle d un assemblage riveté aéronautique - Vers un modèlede fragilisation structurale. Mécanique [physics.med-ph]. Université de Valenciennes et du Hainaut-Cambresis, 2005. Français. <tel-00512558>

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% # 8# #* 3# <

# 2 " 3 QLR & ## #2

& # 3# # #2 2 3

## #8# "0,# & # MN #

MH N " # ### # ##* 1

3 &# M&# 2 ## /

## &1 #N #* ## &#2

31### η 0## 0 5# # 1#

M2# HI θ = π2 N " 81 HL ### 5# 1 3

&# 3# 2 mm # * #

# ) # # 2 #: # #

5# # 1# 3# 2 * 1#2 / #

3# 1 2 # 0 # 0 10 mm

3# #* 0 0#G #1#8# &

# 3# η ∼= 1 σθ∼= σ∞

Kt = 2 + (1 − 2a

W)3 MHPN

V a 3# W 1 2

⎧⎪⎪⎨⎪⎪⎩

σr = σ∞2 (1 − a2

r2 ) + σ∞2 (1 − 4a2

r2 + 3a4

r4 ) cos 2θ

σθ = σ∞2 (1 + a2

r2 ) − σ∞2 (1 + 3a4

r4 ) cos 2θ

τrθ = −σ∞2 (1 + 2a2

r2 − 3a4

r4 ) sin 2θ

V (r, θ) ##2 # #

,# ##8 * 31###

η =σθ(θ = π

2 )σ∞

=12(2 +

r4) MHIN

0 5 10 15 20 250

Distance par rapport au centre de la perforation [mm]

HL 6# 5# #2 # 1 3 &#

" # & ### # 3 !##

#2 # 0* # # 5# # 1#

# #2 $ 01# 3# 3 1#2 M20

!## 1#2N # / !## # ) !

## # #1 / # 5# #

,# # 3# ) &# ##8 QR

# 5# # 3# # ## '

,# &# ### < 3 QK

P R " ### # 2 / # ###

# 0### QIR

# MHHN # 0 2 3 ## #8# # / &1

# $ # ,# 31### η

# MHHHN 3# MHHN # !## " 81 H

0# 3# &1 , 5# # ## 0!

/!# 3# # ## 2024 − T351 2

# ,# # 1 5# ,# 3# >

## 0#G 3#

2 1 2 ,# #2 3 ##

* 1#2 M 3#N # 1 *

#, M N

⎧⎪⎪⎨⎪⎪⎩

σr = σ∞2 (1 − a2

r2 + EsE∞

s(1 − 4a2

r2 + 3a4

r4 ) cos 2θ)

σθ = σ∞2 (1 + a2

r2 − EsE∞

s(1 + 3a4

r4 ) cos 2θ)

τrθ = −σ∞2

EsE∞

s(1 + 2a2

r2 − 3a4

r4 ) sin 2θ

V a 3# Es

ησ =σθ(θ = π

2 )σ∞

(1 + 3

))MHHHN

ηε =εθ(θ = π

2 )ε∞

((1 + 3

Es+ 3 − 4

0 1 2 3 4 5

x 10−3

Déformation lointaine [.]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10−3

Déformation lointaine [.]

[.]En déformationEn contrainte

H 6# # #2

% #2 0# ,# &

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## #8# # 0# 3# #8# #2

$ 1 #1 0# # : / 3#

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&* 31### ηε 0

# #2 0 # #2 M HN % 8#

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### 0 ## #& #

$# i H J L K P I

ri[mm] JP LI K PK I H HH HJJ

ηie J H HH H H H H H H

ηip K LI L P HI H HJ HH H H

& H ( #* 31### QHIR

# 0 #* ## 3

0 8# 2 2# # ## 10 8# 2

M81 HKN & #3 / # ### &# "0:

3# ## & FF ## # 8# !

2# 3# #* #

εi = ηi × εglobale " : # Fi #3 / &2 FF # #1T

# 1 0 2#

% , 0 ## 8 #, #

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p # # #2 3= ##2

HK ## 2# &* 31### QHIR

!) (" ( * ! " 1 !)

) 3 QR 0 0# 1 / # 31###

## 1 # # 8 # 1

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3= &* # / 31### , #

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# ## 2 # 8

M81 HPN % 0## 0

# 2 2 $ #3

# G,# #* ##1 3# 3

# < ## 8 0# ##

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#N " ## # 3= #2 $ #

2 ,< 0 ## ## 2

A ## 3# MA 3# #2

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# ## 3# : 3#

E # 0# ## # # / ### #&1* &

# 3# F31### F #1 0

: / 3# #2 #* " !

# η 8# 3# 1 &#

3# # 3 η = εiεref

0# 31###

)0 # &#2 31###

3# ## 0 5# # ) 3# 3

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1 " !% ! 9: ,- "

QLR . " - ) ! . W B W %4 '( ' )"B. 3 #!

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0# % 0 0 >0

QPR C %4 "$ B&#2 0#1# < +444 =

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QIR %4 (;; "0#2 < 4 "="== "

QHR . "4. . " - ) C!" %4 " C!; .W $ 3 &

3 1 # ""0 !% ! 5 ,%-

QHHR . " - ) %## / ## #2 ,# 0!

1 , # ### #2 / 0 0 123 0

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# & 3# 3 @ # # * ) <

"= == " "

QHJR . " - ) %.%' # 3# &# 3 !@

1 8# ,# # #, $T$$ * 0 0

< ''''' "

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# #1 ## * 0 0 < 4444' "

QHR . " - ) )"B. 9$X$%Y ) YB$% %##

# 0 ## # 3$ 0 38 < 4 &" '

QHKR . " - ) )"B. 9$X$%Y ) YB$% # # !

#1 3 # # 3 #3 &@&# 7 ) %6 0

< = " "

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# 3# 3 # # # : 0 < < & "=

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# &* 31### # , &#2

#1 / 0 0 123 0 < "

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QR 4 '. B& 3 # 3 & # ## # @#&

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QKR %4 ')B C! %'"$ # &# !

# # ### # %3 < ""4

QPR %4 ')B # # ### ! ## @!

## # 3 # "0 !$ 3

2 28 0 30 3$ 0 >3 3 ,!0-

Q R C!" %4 .%4 - % $""B ') # # # ###

# %3 < 4 =4

QIR Y BX"" # # # & # #8#

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# "0#3 8 # / 8# #82 #

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# , # 81 H 0## 1 / # &#

E 1 0 #1 #

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#3 &* #2 , 3# $ #

0##8 # * #82 ' 3# ##

3&# # & < # 8 0# *

0##8# * #

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%!# 0,# #* 3# η 8# 0 3!

# εi / 3# ## ε∞ " #* # &# 0#

0 #2 / 0 3# ## ## 20/ #1#!

8# &#2 #!/!# * & 2#

0 2 F31##F " # ,#

0 ### 3# 1 &# " 0

3 3 M#N #1 0## ML!"JN

# , MON " ## #* # G,#

## 0 * 31### # / 8# 2

F31##F

" 3# 3 8# 3# ,# / V 0#G

3# 3# # ) , &#2 !# ##8 / 0#!

8# #2 1 &1 ) 0

8# 0 # ## 2 3 8# "0##8# 0

3# 3 &#2 0* #5# # ## ) !

3# # 0 8# 2

31## #!# < # / 3# 1 !## ), 3!

# # # 3# 3 $ 01# 0

3# 1 0 8# εg = ∆LL0V ∆L 01 / 0# t

L0 1 ### 0 0 0 3#

2 % # #3 0# 2 , 3#

0 / 3# 3

" # 3# 3 # ##

8# 0 2 3 0### 0 ,## ), 2

3# #* 4 mm # 8 M# # 0

0.75 mm ##1 3# 2 mm ##N "

#* 2 80 × 80 mm2 !#1 ,# 2

160×160 mm2 3= / ## 0#G #1 M81 HN

" # , # # ## # 3

## * " ##2 #1

80 mm 160 mm

H #1 2 3

#$ " #$ $

)## 2 0

80 × 80mm2 1652 1736

160 × 160mm2 6448 6616

& H %##2 #1 2 3

" ## #2 # / 0# 8# # " !

## 2 0# 1 mm " # # 2#

: 0## L!BJH # !

#2 M* "@#1 MHNN B

# # & # / ## 3# #2

#2 # / ) 0## 16% % !

& * 1 0& &* , 01

# 5 0 # #

σ = A + B pn MHN

V σ # 2# # A ## 0## B n

* # 0#1

#2 0W1 %5# # * #2

ρ [g/mm3] E [MPa] ν A [MPa] B [MPa] n

0.0028 74000 0.33 350 600 0.5025

& %##2 #2 #

8 # : # #,# #3# 2

# 2 # 1# # # / &1

# / 1m/s " E 2 ## #

" 3# 1 8# 1 ###

0 # 3# 2 0 # , 8#

,# < #1 3# 3 )0 #

&#2 # 3# εr=a/εref 8# # MN $

# 0 # 0,# 3# 3 &#2 0!/!#

3# #8# ε∞ 3# 3# 3# εr=a

ηε =εr=a

ε∞=

Es+ 2 MN

V a 3# E∞s Es # 0W1

#2 3# 3# #

#3 0 & 3# " 81

# 3# 3 3= &#2 M3# #8#N

#2 M3# 1N " 3# 1 & ,

* 8# ### / 3# 3 &#2 20/

0 0.5% M81 N 0!/!# 20/ 2 ## #2 : # #

' 3# # # ## 0 3# 3 &#2 0

&#2 0#8# #8 " # 3# 1 &

* 2 / ### 0 ,# , 3#

5% &1 !# / #8 '## !

3# 1 3# 3 # 0#

,# 31### η

0 3# 3 #2 # 1 V !# #8 )

### < # 0 2

31## 3# 1 3# # 31###

0 0.05 0.1 0.15 0.2 0.250

Déformation au bord de la perforation [.]

ThéoriePlaque 80*80Plaque 160*160

)3# 3 &#2 3# 1

#$ " #$ $

" ## 0 3# 3 # #1 " ##!

# & 3# 2 1 80 mm , ,

0 0, # # 2 &# "0!

# 0< 0### #22 " 3# ,

# 0 # # #2 # #2 M81 JN

0 0.5 1 1.5 2 2.50

Déplacement [mm]

Domaine plastique Domaine élastique

J # 0# # 0## ##

& # M81 LN 3# # 3# ## #

0, , : 0 U / 3# 0

2 : 3# 0, # ##

3# M # < 1# ## # #

N # 1 20/ # # MA # 10 mm 25 mmN

# # 3# ## / 0: 2 ## )

# ### < 3# 1 &# " 3#

1 &# 0 ,# # 3 /

2 0 0 #1 3# ## 20/ # # MA #

/ 20 mmN 0: 0 " : 2 3#!# 2#

#,# $ ,# A A ## V 3# #

: ) A 3# 0, 3# 2#

2 3# 3 &#2 #2

2 3# 1 3# 3 #

A / #8 &1

0 5 10 15 20 25 30 35 400

Distance au centre de la plaque [mm]

Déformation globaleAxe verticalChemin de rupture

0 5 10 15 20 25 30 35 40−0.02

L )### 3# # 0## ##

"0# 81 L # 1 # ### #

3# 1 # % ## 3#

0* 0, &## : 1 &# !# # / #

30 mm 2 20 0 20/ 14 mm # 0, # #

M81 N ) #: # & # 3

## #:* #* ## # #2 #:

# 20/ 25% 3# 0, # " ##

0 3# 3 22 / #

0 0.4 0.8 1.2 1.60

Déplacement [mm]

0 0.4 0.8 1.2 1.60

4.5x 10

Déformation globale26mm30mm34mmForce

0 0.4 0.8 1.2 1.60

Déplacement [mm]

0 0.4 0.8 1.2 1.60

4.5x 10

Déformation globale10mm14mm18mmForce

6# 3# 1 3#

8 # 0 2 160 × 160 mm2 # "

## 0 3# 3 #!/!# ##

2 M : 0&N ## < #

#$ " #$ $

2 160 mm E , : #

2 2 0 # 0, # 1 &#

M81 KN , # F#F 3#!# 1

# % # 1 ## &1

< , , , " &* 20# 0 #1

# 3# 3 0, # 0 M81 P(a)N

&# M81 P(b)N $ # : # A

## 3# 3 3# ## 2

0 10 20 30 40 50 60 70 800

0 10 20 30 40 50 60 70 80−0.01

K )### 3# # #2 #2

0 10 20 30 40 50 60 70 80−0.01

−0.005

Plaque 80*80Plaque 160*160

0 10 20 30 40 50 60 70 800

Plaque 80*80Plaque 160*160

P )3# # #2 2 80 160 mm

% 0 ## 0 3#

3 22 # $ 0* 3 &## 3#

3 3# 1 8# εg = ∆LL0 ∆L 01 /

0# t L0 1 ### 0 # # εref

) # && / & 3# , ε

3# " &# # * #

&* 2# 20/ # #, 2 3# #

3# M4aN %#2 3# ## 1

,#2 B3# 20 3# #*

0 & 3# "0 1 1 3!

# ## / % # #5# # 3# M3

1# # N %0 2# #5# ##

## ### 3# 3

' & & 3# ## # 0#1

3 &#2 0 # #

0 3 3# # #1 0##

" 3 &#2 &#2 # 0#1

3 QHR 3# 3 ' ,# # ##!

# < 3 QR % & 3 &

# , #1 0,# #3 #1# #1

# %,!# 1+ / #, 1# # / 3

# $ 01# # # & 0

#, #3 0#1 ### / 0#1 3

M81 N $ && 3 MJN V dx dy #

#1# ax bx ay by 01# cx cy ##

ux(x, y) = ax · x + bx · y + cx · xy + dx

uy(x, y) = ay · x + by · y + cy · xy + dy

" # # , #3 ###

3 1# # 0 8# 0#1

### " 1# 3 1+ / 0### 0 * #

, A Zi Zf M81 IN % * #

C =∫

∆S [f(x, y) · f∗(x∗, y∗)]2 · dxdy MLN

V∆S 0# A # " 3# #* f(x, y) f∗(x∗, y∗)

2 / # 1# 0#1 ### 3 # % * C

# 0 1 2 3# "

3# # &

6# 0 #3 0#1 ### 0#1 8 QHR

" 3 QJR QLR # ## &

%!# # # & "0#!

# #3# / 0.05 #, % # 3#

#3# / 1% B3# #: # # #1 0# 2# # < !

02### 0#1 ## 0#1 / # # *

# # 2 &2 #3 0#1 # #: 0 /

0 " 2# " & 3# 0*

2# &# 11 ## 2 ##

##* 2D V 3# # 0, z

I ## # 0#1

"0 ,# # 3# "

## * # 2# )# 0 ## #8# < #1

# # ## ,# : &* 1

3# " ## &# 81 H #

# 1 mm 1.17 mm #1 0## M2024−PL3N

# , MON % ## 0

# #,# QR 0 #!1 2 # / 4a [ !

/ # : 3# &# 2# #,# QKR %

## # U ## # &# 0# ## /

H 6 # 3

" / # ### 2# #2 M5mm/mnN / 0# 0

# &#2 M81 HHN # # &2 81# 8 0!

### " &>

7 0 3 #A#2 ### 9#

7 0 (?& # 0#1

7 0 1 3# 1 1 (#&

" & 3# 1## # ## # -

" &# # ## # & # # ' 1

HH )###3 ,#

8 1 0: ,# Fmax 01 ,# δmax

01# W MN " 0# 3#

#: # &# "0 1 0!

# &* #2 # 0# # 0##

## M81 HN

W (δ) =∫ δmax

0F (δ) · dδ

Wi =N∑

12(δi+1 − δi)(F (δi+1) + F (δi)) MN

H %&#, 0#

" 3# ## ### # #

0 # 1 &# M, J LN " ## :#

# 3# 1# F#F M# / 0 3#

# 0#1N 2# #: 0 # / 0 %&2 81# 3# 0

# # "0 # 0# #: #

3# # &* # # # # 0##8 #

# 3# " # 0 M N 2

3# #1 0 0.5 mm

\ # # H # # J # L #

QR QR QR QR QR QR

H −0.12 2.26 2.74 3.68 5.59 10.34

0.06 2.23 2.79 3.74 5.63 10.34

J −0.15 2.22 2.68 3.62 5.54 10.24

& J ## # 3#

\ # H # # J # L #

QR QR QR QR QR

H 2.23 2.68 3.61 5.52 10.27

2.54 3.01 3.94 5.83 10.53

J 2.46 2.92 3.85 5.74 10.49

& L ## # 3#

" :! 3 # > F#F /

# 0 0 3 mm M81 HJN %!# / 0: #

" %&+# QPR &* 1#2 01 0## 0# A

## 1 0 Q R " 3#

# : M 3# # #&1*N

0 1 2 3 4 5 6 70

Déplacement [mm]

Eprouvette perforéeEprouvette non perforée

HJ :! 0 #1 0##

" # &# "0 0

1 ,# ##

1 " # ,# # 0 3 & !

##2 #2 # ## !

#, M#480MPaN # # :# 0 3 MS0 = 22 mm2

# 26 mm2N # ,# 0 440MPa # #3# /

10% / 0 # ' ### # # 0!/!

# / ## #2 Mσy = 385MPa # σy = 370MPa

3N " 3# # ## /

M31###N 0 M0.8 mm # 6 mm 81 HJN # 0!

# 01# 2 / # 90% M 2# < 1 2

## N ) , 81# 0# 3#

3& ## * 0: ,# M81 HJN

\ Fmax[N ] δmax[mm] Wmax[J ]

H 12416 6.15 83

12454 6.25 81

J 12608 6.55 92.5

\ Fmax[N ] δmax[mm] Wmax[J ]

H 9715 0.81 9.5

9690 0.82 10

J 9754 0.78 10.5

#' #4'

& ,#, #1 0##

)0 # 3# : ### 3# " 81 HL

0# 3# # 1 &#

3 3# &1* 1 &#

&1 M81 HL(a) H(a)N # ##

/ 0: # " %&+# 1 / 0 " #

# / 0## , + −45\M81 H(b)N " 3#

* / ### #&1* 3# 1 &# *

0# M81 HL(b)N " 81 HK # # #

3# 3# 0# # ### :

# " %&+#

0 1 2 3 4 5 6 70

Déplacement [mm]

Point 0Point 1Point 2Point 3Point 4Point 5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

Déplacement [mm]

]Point 1Point 2Point 3Point 4Point 5

HL # 3# #: # &#

HK 0 3

" 81 HP H 0# 3# 3#

, # 1 2 5 # # 3

3 3# &1* 1 &# ### &2

# #: # 20/ 3 mm ,## "

## 20/ 0## : # "

0 1 2 3 4 5 6 70

Déplacement [mm]

Essai01 (2.26)Essai02 (2.23)Essai03 (2.22)

0 1 2 3 4 5 6 70

Déplacement [mm]

0 1 2 3 4 5 6 70

Déplacement [mm]

HP )3# # 0, # ! 6 3

3 3# # 5 ### 0!

# % # # / # 5 #1 3#

20# 0 # 2 3# 0#G " ## # M 0 10 mm

3#N # / 4a # / # 2 3!

# 0#G ### 3# &# , # 1 2

, A < ##1 " 0 ### 20/

1 0 0.08 mm # #:* "0 :#

# #8 #: 3# " 3!

# : & ## / 2 : ##

# 1 0# 3# 0# 01 !

0# 03 !< ! 0# 02 # # # /

0.08 mm ε01 > ε03 > ε02 [ ### 3# 3# ## :#

3# & #1 3# M LN

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

Déplacement [mm]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

Déplacement [mm]

]Essai01 (2.69)Essai02 (3.01)Essai03 (2.92)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

Déplacement [mm]

H )3# # 0, # ! 6 3

" &* #, 3# , 2 3 0# #2

# # 3# ## M81 HI N 20/ 0!

## : # " %&+# M81 (a)N % #1 *

# : 1 3# $ #1 1 # # /

& 8 31### " ###

3# 1 &# 0 3 M81 (b)N ###

8 # &# &# H

) # &# 0 # &2 81!

0 2 4 6 8 10 120

Distance au centre de l’éprouvette [mm]

Essai01Essai02Essai03

0 2 4 6 8 10 120

HI )3# 1 &# ##

0 2 4 6 8 10 120

]Essai01Essai02Essai03

0 2 4 6 8 10 120

)3# 1 &# 0####

" &* 1, K # #3

0 3 ' # ### 0#1 0##

< 3# 3# # ## /

0 ## 2 # 0# 01# % ## # 0

90% 0#1 0## 3#!# 0 50%

60% 01# " #2 ,

### M 0# # 81 HN # 3# #:

< %!# 0,#2 F3#F # #

0# , M170MPaN : 0 8 # #

# 2# ##2 , 2 # # :#

# & 0 ) < # ,# # 0

3 & ##2 #2 # M330MPaN " #

,# 0 3 # 10% ,## M295MPaN

6 3 6 3

Fmax [N ] 9900 8780

δmax [mm] 15.95 7.97

Wmax [J ] 138 51

& K ,#, #

0 2 4 6 8 10 12 14 160

Déplacement [mm]

H :! 0 # ,

# / 0: ,# & 0# # *

% & 0 # ##* 0# 20

0# 2 22 1/10es ##* #1 0## '

2#* # &* & 0#

## # 8 0# &* #2 ,

)0 # 1 0: 3# < ##2 20/ 0##

0: ,# 2 2 # # # 0# #

0: 3# #:* 2# : # 01# 20/

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## :# # # P

3 3# ##2 0 # 0!

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3 ### / #

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1 #, 3# #: 1 &# M81 (b)N %

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# 81 L

# H J L

QR QR QR QR QR QR

6 3 JJ L HH

6 3 K JJ LH P HJ

& P ## # 3#

0 4 8 12 14 160

Déplacement [mm]

0 1 2 3 4 50

Déplacement [mm]

Point 1Point 2Point 3Point 4Point 5

# 3# #: # 1 &#

J 0 3 #

L 0 3 #

" 81 &* #, 3# , #:

# 0## ## 0: ,# & 0# M 4N

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# !# > &1 3# &1*

0 20/ 0: ,# # # 0

# / 0&# M 40% ,# 3#

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0 2 4 6 8 10 120

domaine élastiquedomaine plastiqueeffort maximaladoucissement

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)### 3# 1 &#

), ##8 ) 0 3 0!

# 3# 3# 3= &1* 0 # >

# 3# M# 0 3 1#2 ! 8N /

0&# ) 0 3 # 3#

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* N %!# ## # 0 2 ,

0&# # # / # 3# # / 0#!

# 3# η &1 )0 # 1 3#

# 01# ## 01 / % #

3 ## #8# & 3# 2# #&1* *

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&2 # 1 &# 2 0#G 3#

# ##1 " , #1, ## * &1 M8!

1 KMaN PMaNN 3 / 2 0 0#1 3#

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M81 KMbN PMbNN # #1 3# M81 KMcN

PMcNN 3# ##2 0# 0!/!# 2

&1 # #2 #2 2 0 # 3 " 3#

0 1 2 3 4 5 6 70

Déplacement [mm]

0 1 2 3 4 5 6 70

Déplacement [mm]

0 1 2 3 4 5 6 70

Déplacement [mm]

K %# 3# 1 &#

3 3 #1 0##

0 2 4 6 8 10 12 14 160

Déplacement [mm]

0 2 4 6 8 10 12 14 160

Déplacement [mm]D

0 2 4 6 8 10 12 14 160

Déplacement [mm]

P %# 3# 1 &#

"0,## ### 3# #

0## ## / 0: ,# ) # #2

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## 0 ### 3# 1 &#

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# : #8 ### 3# 1 1 #

0 2 4 6 8 10 120

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# # ) 0#1 0## #,

3# 0 3 / ,

3 M3# 0 12% 24% # #2 / 0:

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0 2 4 6 8 10 120

I )### 3# 3 3

/ 0 2 M##N

0 2 4 6 8 10 120

J )### 3# 3 3

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3# # 3# M: FFN #

1 M: F##8FN M81 I(b) J(b)N

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3# # / 3# 0# & 0#

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# 0 3 M# 01# 0 & ## 81 (a)N 2

#, 3# 1 # # 0 3 M!

#N , 3/4 & 0# 3# 01# 20

M81 JHN

0 2 4 6 8 10 120

JH )### 3# 3 3

# / 0 4 M& 0#N

## 0 #1 3# # 3#

0 # # # M## 1N #!

< # 0## 0 3 # / &* 31###

# / 0&# ) 0 # M3N 3!

# * * ### / &* 31### # /

F#F F##8#F 3# ##1 ) #

!## M#2 #N &* 0##8# 0#

" 3# 01# 2 3# #2 '

3# 3# # M#N # 0&# & "0#!

G # 0 / # & ## : 0:

31### # 0##8#

,# # 0# &* , 3!

# / 3# ## 3 εi/εref " &#, 3#

3 εref # !# 8# δ/L0 V δ

L0 # 1 ### 0

3# εi 3# 3 εref 2 8# 0#

0,# # ## η 0## 0#G # &1

M#2 #2 1 N

) 0 3 M81 JN 3# &1*

# 0## ## η 1 C 0####

1# η !# # 0## 0 3!

# 0!/!# 0 3 1#2 M 8N ) 0

3 M81 JJN ### 3# #&1* * &1!

η > 1 1 &# M3 ## * #1

3#N 2 0 # ## #2 #2 " # η

# 0#G # ) η #:

# #1 0## M81 JLN S0# # 0 3#

* ε/εref ## ## 0 0

B 2 η = 1 # 0 #2 # η > 1 # ## #2

0 2 4 6 8 10 120

[.] domaine élastiquedomaine plastiqueavant rupture

0 2 4 6 8 10 120

[.] domaine élastiquedomaine plastiqueeffort maximaladoucissement

J 6# η 3

0 2 4 6 8 10 120

[.] domaine élastiquedomaine plastiqueavant rupture

0 2 4 6 8 10 120

[.] domaine élastiquedomaine plastiqueeffort maximaladoucissement

JJ 6# η 3

0 2 4 6 8 10 120

[.] AcierAlliage d’aluminium

0 2 4 6 8 10 120

[.] Acier (effort maximal)Acier (adoucissement)Alliage d’aluminium (effort maximal)

JL 6# η # ##

$ % η

"0 ,# 2 01# ## 3 M#N

20/ W1 #3# / 01# ## <

## # 3 M#N W0 " ## M# ##N

2 W1/W0 3# # ## # F0#!

F " ## 1#2 / #1# 1#2 M#N ##!

### !< # / 0,# 0 ### &

3# #&1* " 3 1#2 M#N 1# F#!

#8F 3# ## ##8T 3#

' ( $ η

## # FF & 0# M1N

/ 0&# " # &# ,# ##

# 3# η 8# 3# ε / 3# #!

# #2 εref M ε∞ &#2N [ # 2 ###

η = η(r, θ) < 8# 1 3 &# Mη = η(r, 0)N 3= !

#2 ##2 3# 20/ 0

% 3# # ## 31### 31###

0,# 0 # 0 3 1#2 "2 η = 1 ##

# #2 # ## M3 !< 3# #N

# η > 1 ## # #2 ## 8 η < 1

#1#8 2 ### 3# ε #3# / 3# ##

εref 8## 0 #* 31### 3#

7 η > 1 ## 31### ε > εref

7 η = 1 31### ε = εref

7 η < 1 ## F#F ε < εref

" * # 3# η ## # / #

M !##N / 3 1#2 M1 a 0

3# 1 0 8N $ # η = η(r, σ, a) #2 σ(p, d)

"0 ##2 # M#1 3#N #

3# F##F εA !/ 2 #1# 1#2 >

0&# ### η 0# M 2 εref ≤ εA ⇒ η = 1N # 2

# εA # # 2# # ")! 0## 0

3 1#2 M1 # #N ) 0

3 1#2 ### M# 3#N * #

εA = 0 # 8

η = η(r, σ, a, εA) avec σ(p, d) et η =ε

εrefMKN

η 0# 0 31### U / 3#

0 * 31###

+ η + # , -./

%# # Ω # ### S0 2 3 1#2

3 A(t) # "02## &#2 2# 2

dWext = dWelas + dWcin + dWS MPN

V Wext # 3 ,# Welas 01# #2 ?

# Wcin 01# ##2 WS 01# ## #

#2 "## # # 2# 8

0,# S(t) = S0 + A(t) 01# dWS 0#

dWS = 2γdS M N

γ 01# # / # 0 3 ## )

0 ### 2#!#2 MWcin = 0N # 1#2 0,#

∂S(Wext − Welas) = G MIN

G 3 0,# 8 %!# 1 2 G #

##2 GC ##2 # G 0,# 1 3# 5#

0## # KI #2 # # M E νN

G =1 − ν2

KI = ασ∞√

πa MHHN

α 3 ##2 3 8 M#N ###

" & # ##1 # 8 0,#

# 3# KI 3 QHR

⎡⎢⎢⎣

⎤⎥⎥⎦ =

⎡⎢⎢⎣

5 cos θ2 − cos 3θ

3 cos θ2 + cos 3θ

sin θ2 + sin 3θ

⎤⎥⎥⎦ MHN

2 20 # 8 & # 3#

8## #8# Mr → 0N

' ( $ η

) ## 1# 8 Mθ = 0N &1 #,# #

3# ## &1 MeθN 0,#

σθ =KI√2πr

et εθ =KI

4µ√

2πr(K − 1) MHJN

K = 3−ν1+ν #

## MHHN MHJN # #

## / # 2 # # 0,# 0,# 3# ησ ηε

# # & ##

ησ(r, a) =σθ

σ∞= α

2ret ηε(r, a) =

ε∞= α

1 − ν

1 + ν

2rMHLN

KI = ησ

√2πr σ∞ ou KI = ηε

1 − ν

1 + ν

√2πr ε∞ MHN

;# 01# ## 0,#

∂S(Wext − Welas) = 2πr

1 − ν2

E(ησ σ∞)2 ou

∂S(Wext − Welas) = 2πr

(1 − ν)2

1 − ν

1 + ν(ηε ε∞)2

"01# # # 0 3 ## 3#

η(r, a) r σ∞ ε∞

" #2 # #2 1#

## 2# ## ## # 1#

8 # < # &# 2# M&&*N 0,# 0

8 ### MN " #2 ##, %# 0#

# # 0## ## 01 ' #5

, , &# ## 0 #2

#2 ##2 0# #2 0

# & 31###

# Ω # ### S0 # 20 3 1!

#2 3 A(t) # ### >

3# / 1 0,# 02##

&#2 ### MPN / 2 1# ## ##

# #8# Wplas 1 Wendo &1 & Wtemp MHPN

dWext = dWelas + dWcin + dWS + dWplas + dWendo + dWtemp MHPN

" # &#2 0! 3# ε T ,2

# 3 &#2 σ s " # &#2 !#

3# #2 εp # * 3 αj

M01 #: d ,N ,2 # 3 &#2

σp Aj #

%# # 2 01# ## &1 & 1#!

1 MWtemp = 0N 2 ### #2 ## / # #2 MWcin = 0N

" 1# ## 3= ## MWplas Wendo WSN 0,# 3

1#2 Wi

Wi = f(ε, αj) MH N

"02## &#2 #8 / 0& # / 0&

#* ## 0 ## εref M] HHN

/ ε " 3# η 2 8# MKN # &#2 &

2# # & 3# 8# & 31### M /

3N Ω #* &&* 2

01# ## 3= ## ## / #8# 01

3= 1#2

Wi = f(ε, αj) = f(ε, εp, d, η) MHIN

3# 0&&* 2 ## 0 0#2 #

, # # # 2 # Ω < ## #

S(t) 3 3 #2 2 S(t) = S0 + A(t)

# Ω∗ Ω # 2 0 # # &*

S % ## # & 1# ## 3=

dWi = dWiΩ∗ + dWiS dWi =dWi

dΩ∗ dΩ∗ +dWi

dSdS MN

dΩ∗ =dWiΩ∗

dΩ∗ +dWiS

dΩ∗ dWi

dWiΩ∗

" ## 0 20#2 # ##8 #

H # F#F dWiΩ∗/dS = 0 dWiS/dΩ∗ = 0 M ##

' ( $ η

dS 01# ## Ω∗ #N

dΩ∗ =dWiΩ∗

dΩ∗ dWi

$ # 0,# 1#2 1# ## ## #8# !

dWi =dWiΩ∗

dΩ∗ dΩ∗ +dWiS

dSdS MJN

dWi =dWi

dΩ∗ dΩ∗ +dWi

dSdS MLN

dWi/dΩ∗ ## 2# 01# 2# # 1# !

Ω # 0,# 3 M# #N dWi/dS ##

01# 1# # Ω 0#

3 3#2 S(t)

# Ω # / 3 / 01

M * # / 0# &# #2 ##, %#N # #

# 2# 1 / # * Ω M

1# # / 0# &# #2 N

/ # ,# # tA M #

# εAN 2 & 3# # #&1* M 3

#2 # >N 1# ## 3= ##

# 2 t ≤ tA t ∈ [0, tA] : η = 1 t > tA

31## 0 3 1#2 η # / 1 %&2

## ### 1#2 * M#3N Wi ## 3#

0 ##2 η

dWi =(

dWiΩ∗

dΩ∗ dΩ∗)

t≤tA

dWiΩ∗

dΩ∗ dΩ∗)

)t≤tA

dWi =(

dWiΩ∗

dΩ∗ dΩ∗)

dWiΩ∗

dΩ∗ dΩ∗)

# MP!IN # 02# MKN &2 ### 1#2 #!

dWplas =(

dWplasΩ∗

dΩ∗ dΩ∗)

dWplasΩ∗

dΩ∗ dΩ∗)

dWplasS

dWendo =(

dWendoΩ∗

dΩ∗ dΩ∗)

dWendoΩ∗

dΩ∗ dΩ∗)

dWendoS

dWS =(

dWSΩ∗

dΩ∗ dΩ∗)

dWSΩ∗

dΩ∗ dΩ∗)

&&* 01# dWS M #N 2 #!

# S Ω M&# ## 0 N $ # η

dWSΩ∗/dΩ∗ = 0 ) S # Ω∗ 3# 0&&* #*

,## # ##8# 2 01# #8# 0!

1 ## S 1#1 ## Ω∗ 2 2 #

η 0 0 01# # M# 31#N $ #

1# ## #8# 1 (dWiS/dS)∀η ≈ 0 8

# 0 # &&* 1 3 2 η = 1

(dWSS/dS)η=1 = 0

" 2# 1 MP!IN ##8 3# &&* 0

## # Ω M 2 S << Ω∗N 8

dWplas =(

dWplasΩ∗

dΩ∗ dΩ∗)

dWplasΩ∗

dΩ∗ dΩ∗)

dWendo =(

dWendoΩ∗

dΩ∗ dΩ∗)

dWendoΩ∗

dΩ∗ dΩ∗)

dWS =(

− " (dWplasΩ∗/dΩ∗)η=1

(dWendoΩ∗/dΩ∗)η=1 1# ##

## # # Ω #8# 1 &

3# &1* Mη = 1N % 1# ## # # &#

#2 ##, %# 0### # 01

− " (dWSS/dS)η>1 01# # ## 1#

# # 3 S # % 1# # #

&# #2 0### # 1# 2

− " (dWplasΩ∗/dΩ∗)η>1

(dWendoΩ∗/dΩ∗)η>1 1# ##

## # Ω∗ M## # ΩN 31###

' ( $ η

# / 1 M&1# & 3#N %

#3 / 01# ## ## Ω∗ ## / 0=1

3 ' 3# 1# 3

8# 1# 0=1 (dWiΩ∗/dΩ∗)η>1

#(dWplasΩ∗/dΩ∗)

η>1+(dWendoΩ∗/dΩ∗)η>1 % #* # # 0,#

0 * ##2 εS 2 η # 1 / # 2 3

/ 01# # dWS / < ## #!

# S " tA tS M# εA εSN * 0=1

0#### * η # $ # 8

# 3# * η

t ∈ [0, tA] η = 1 )### 1#2 #8# 1!

# Ω∗ %& 3# &1*

#2 #2 ##, %#

t ∈ [tA, tS ] η > 1 )### 1#2 #8# 1!

# Ω∗ %& 3# #&1*

=1 # 3

#2 ##, %#

t > tS η > 1 )### 1#2 # # S

%& 3# #&1* %# 3

& &## 0 # Ω

J "# 0# &#2 * 31###

"0#< # 0# * 31### M81 JN 20

8# F0=1F & / 0# * εA Mη #

# / 1N # 0#### / 0# * εS ' 3# # η(εref )

# 8# # 0##8

## 31### 0#### # " * ,##

0 0 # Ω 3 ###

! 5() Ω !) 1!( )!

JK &## 0 # 3 ###

" #2 ##, %# M #2 )# ## /

#2 ;1# N # 01#

## #8# # 1 / 0# 0 # 0 #2

) 0 1 #: 2 # &1* Mη = 1N

20/ 0## 0 1 0 #8 " (

dWiΩ∗dΩ∗ dΩ∗

# MJN MJHN / 01# #8# 01

3# 01 # Ω #8

1 > ## " & 3# #

#&1* 20/ 0## 0 8 ) 3!

# 01 & 31### 2 η > 1 "01#

' ( $ η

#8# 01 2 ## Ω∗ 2# ##

#2 M# 1# 3 N (dWiΩ∗

dΩ∗ dΩ∗)

η>1 # MJN MJHN

' 3# #2 η > 1 2 #

01# ## # 3 S(t) MJN

! 5() Ω !* 1!( )!

%# # 2 # Ω∗ # 3 ### M81 JPN

JP &## 0 # 3 ###

&2 ### 1#2 M#8# 1 N

,# # MJ!JN 3# η η > 1 Ω∗ #

3 1#2 ,# 8 #* 2 # 01#

## # 3 S(t)

) # MJ!JN ##8

dWplas =(

dWplasΩ∗

dΩ∗ dΩ∗)

dWendo =(

dWendoΩ∗

dΩ∗ dΩ∗)

dWS =(

) ## # Ω # F#F 1#2

dWiΩ∗dΩ∗ dΩ∗

)η=1 " # ## *

3# 01 Mη > 1 / t0N #E 1# 0=1(dWiΩ∗

dΩ∗ dΩ∗)

η>12# ## "01# ## # ##!

#3# / ##

& '()%

' ,# ### 2# #2 # #

&* # / # 31### %!# * / ###

# 0 3 3# # 0#

1# 3 # $ 1 2 3 1#2 M

3#N 1## F##8F FF 3#

# 0## ## # FF &

0# ##2 #

) #* # # 31### η 8# 3

0 3# ε / 3# ## 3 εref %

* 8# #* 31### η > 1 ' 3# #

#* # 2 # # Ω ### #

### 1#2 M## #2 N " # 31###

## # 1#2 &&* 0 ## #

# # Ω∗ # S 8 # 2

1# ## S #8# 1 1#1

Ω∗ & # &&*

8# 01# 0=1 3 # 0# # &

31### , * # 0=1 0####

εA εS #

" * 31### 3# ## 3# #2 3

2# # 0 31### 20/ # "0## # η

# &#2 # ## # 0# " ##1&#

M%&# $N 0,# 0 # 0# &#2 # !#2

# 0,# #2 3# ## # 3# &#2

ε∞ / 0#8# # #2 M%&# $$N " &#

0# # 0,# 0 # 0#

&# $$$ # #2 2# # # *

# # 0#

QHR ; B$ & & 3# & !

#2 # 0#1 #2 3 $ 3 !)

,7- . "

QR ) - %$ C!C B' J) )3# #1 !# #

& 3#1 3 & @: <0!0

0 !B 6 ,7- 2 "

QJR ( %4 )''"$ ,# &#2 # 0#1

!$ * C 30 $ $ >0 +B ,7-

QLR # (J ! ' A. . $ "

QR #, #2 ! # # % *D 0 %*

QKR B$49 B& 3 ## 7 )0 A;5 +: !

QPR B($ ; " %4^B"$ &* 0# #

0#1 3# ! # %0 < ( ' 4 4 "

Q R 4 "'%4 ( %4 $'O # ##2 #2

&* # " %&+# 0 # # 0#1

0## 0 (6$ < < &" " " "

QIR 4) .'$ #2 31# =

QHR C!. ".") #2 31# # 5 (

% &# # / & ## / 0&

* 31### # &1

)#: &#2 #2 0 )

& #!& 3# # #1 0### #

## )? #

,*- - #

& $ 02

& . 5"-

* ., / 0). 12

. 012 11

- 3 4 12 5+

& ()*%

" ## 0 #2 # 3# ## , & ##

0 0& 20# 3# ## # 01

0& #2 0 0& 20# # 0## 8 /

# U &###3 0& #2 ##

< &# H 1 # # / 0&

#2 M#N / 2

M# 01N " 3# / # # 0 /

# M 3= 1#2 # U N 2

!# #G 3 " 3# *

: / # 3# 2# 3# &* !

## # 0=1 2 0 :

#1# 3 # #2 ## 2

#2 / ## &

%# 8# 2 M3 3N ##

# / &1 M , $ 2 0 &#

# 81 JHN % ### # / 3# 1 0

2 εref 2# # &1 #2 / 0 8#

2 " #2 0 # #

01 31### < ## / 0#

# # & η 8# , 1& η =¯[ε

ηij = εij

εref ijM ##8 1 3 &# η = ε

εrefN V ε

0 3# εref &1 #2 / 0 ##

JH 6 8# 2 ## $

) && & #2 # #

0# 31### " 3# η # # &

3# , ε 3 1#2 M 3#N

### #2 ## # # εref " * 3!

1### # 0 & #!& 2#

# & 3# # / & M## 01

#N / & #2 M## N )#: &#2 #2

0 ) & #!& !

## 3# η = εεref

,## %!# 0 #

# V #2 # # !##

# ;# &# 2 ### :

& #!& # 2 !# # #5

# ## # ##

)#: &#2 #1 8 # * 31### !

η " && # & #

/ # #2 0 E 3 # 1#2

3# " & &## E 0#1T# 0#3!

# # # 0# # M#N 0

# #1 2 3 / 0&

" 8# # 0&# # 0!

# & 0* / 20# 01#

### 1#2 0 : 8 #

#1# M1 8 3#N #1 # 3 / ##

/ 1# !# .# 2 1# #1 # # >#

/ 0& # 0 ### 1# U

* ; &###3 ) #5 #

# / ## &#2 & F F QR 8#

QJR % & 0#< 3## #1 0 #20#

0 # 3#* # M1 8 3# N " !

## 3 ### 3# ## 0 # 3# #

8# #1 " & 3# # MF FN 2 /

1 3 & #

#2 3# # 3= ### : / &2

I #1 # φI / 3#* 3# M #1 ##3

1#3 20 # / 0,# / 0## 3#N #

# # 3# # ## 3# 3

8# #2 M# MJHNN "0#!A 3# # φ ## ##

3#* # ) 0 3# # 01# # MJN "0#

3#* ## 1+ / 02# 0# φ(x, t) 2# #

φ(x) =∑

φINI(x) MJHN

φI = minxc∈Ωc‖x − xc‖ − rc MJN

V Ωc # / 3# xc rc #

φt + V ‖∇φ‖ = 0

φ(x, 0) donnee MJJN

V V (x, t) # 0#3 # x # ## ,!

# / 0#3

) 0# O!; M, ;## &N #&# /

0,## & #2 8# ) 3# 3 # !

### 3# / 3#* ## #

&#2 # ## 0# QLR "0,## 0#

3 1#2 MJLN ) ,# 1 # ## aJ #

3#* M81 JN " 3# F # <

&## #3 #, &#2 * ) 0

3# 3# # φ(x) < ## MJN

Uh(x) =∑

UINI(x) +∑J

aJNJ(x)F (x) MJLN

V I 0 #1 J 0 , #3 / ##!

F (x) =

∣∣∣∣∣∑

φINI(x)

∣∣∣∣∣ MJN

" & O!; 8 0#&# 8#

### # 0 # # # # #1 A

8 ##1 3#* # 0!/!# # 3#

QR 8 0# ,## 1#2 "0###

& < #1 0 , M

< * U 20 & 8# #2N

J # # 3 # O!; QHR

# ! -1/

" 3# & 8# & 0

# #1# , ## # / ### 1# /

### 3# 2# # & ,

& ## ) 8# # 0,#

&& 3# # # 1

0## : #2 & 8#

&!& 8 #1 E 0 #2 1 0##

1 8, / 1 2 ) # 0 #1 h 1 ##2

1 2 0 0 # 2 h # "

&!& 3# 3 / &

## &! 8# 1# E 0

2# # / 3# # 1 0## #

## &! # E #1 1#

"0 E 1 # p

%0 2# ### 0 3# 01# 1 E

" &! 0### 0 1 # "

1# 3# / 0### 3# 3 &#&#2

# / 0# E "1 ) 0 ###

3# 3 &#&#2 8# # MJKN ) # 3#

N1 N2 3# 3 , ##2 / ##

& #2 8# " 3# Ni #1 2 /

3# 3 # " 3# 3 ,

0 ## # 2 3# 3

# 0#&# & / 0## 0

N1(ξ) =1 − ξ

2N2(ξ) =

1 + ξ

2Ni(ξ) = φi−1(ξ) i = 3, 4, . . . , p + 1 MJKN

V φi 8# / # E Pi "1

φi(ξ) =

√2i − 1

∫ ξ

−1Pi−1(t)dt avec

P0(x) = 1

Pn(x) = 12nn!

[(x2 − 1)n

], n = 1, 2, 3, . . .

" F&#&#2F ## 0 3# 3 1 p #

, 1 #3# p− 1, p− 2, . . . , 1 ! " &

## / 0## 0 # MJPN ) ,# #

, 3# N1 N2 q1 q2 ,

q3, . . . , qp+1 # E # 3#

3 # #3

Uh =p+1∑j=1

Nj(ξ)qj MJPN

"0# ! ### # / 0## 3# 3

# 3# 3 E 3D 2#* 3#

3# 3 3 ## " # 1# 0 2D

3D M 21# #1# &,#2 N <

3# &! 0* # 1 # 2

& #2 8# QPR ) +& ## #

" #1 # : 0 2# # 0* # /

## M0### 0 # #N #

3 QIR QHR # 0# 2# # < / #

#1 : 0### 0 1 # # # ,!

# # #* , 3#* # # 0##

#1 M#1 3# & QHHRN 2 #1

3# # )0 # U # 0 Tcomp #

# # Tsolv # / # ## * 02#

1 2 # / & #2 8# Tcomp Tsolv

< 1 # # #

# %0 2# 0 1 # # #

&! &#2 * QIR

"0## &! / ## 0 2 3 <

# QHR QPR $ # #1 0,# #

/ 0### 0 #1 1# #

& 0* # ## #A# ## 3# 2# #

# + 2*

"2 #2 0 # 3 &*

0& #3# M #N ## #2

#2 ##, %# #1 ## " ## / !

& #2 ##, 41* M4N 8 #

# #2 0 # 0#3 4

# ## #2 &1* 2# ## &1* M81 JJN %

## # &#2 &1#

0 # 01# # # , # !

# 0 / # #

JJ #3 #2 ##, 41*

" & 4 < # $ 0

# 0,# &* ##2 # & # 8

8# ( 6# #3 M(N # $ 01# # #

## &1* 2# %#!# < 1# 2 ( # #

1 ##2 / ( 20# # , < ### 2 #

# # 0,## # 1 !# #!

% # # #2 ## &1*

2# / # ( " # #: #!*

) 5# 9(& &)! " )!1 :9;

" #* # / 8# # #3 M(N %#!#

&#2 5 0&1# 8 #1

F8*F 2# $ / 1 , ,

&1* # # ##2 #2 1#2 # $

# #3 # 2# # 22

0 * 2 8## ( 0 # ##

#, # 2 0# ## %0 2# ##

1#2 #2 ( 3# 1 #* ##2

), &&* 0# / 8## ( " #* #

/ # # & " # #T(T M81 JLN #

# < 8# #1Z # MJ N # #: &

##2 * # < #3# " &&* 2 #2

##, # < #2 / 0& ( " #!

# < # # # &1#2 &# " # (

8# # 3 # # 1 # 1

#2 # , #: &

d < l <

V d 1 ##2 &1# M1 N l 1

##2 ( L ## ##2 0!/!# 0&

#2 Lw 1 0 ###

JL &## #: &

!) "!! 7,8

" ## # 1 #: & * !

0## ( / # #2 2# # # )

# 1 #2 #

/ #1# 1

" # # !!) # & , 3#

#2 # 0 #2 : (

0 # 01# # (σ, u) # / #

0 3# #2 * #2

( 2# 0# #* #

7 2## divσ = 0

7 ## ##

" ,# * # 3# 20# 0 1 *

8# $ : # 0 0 ## ##2 #!

* ##2 #2 ( ) 8## ## , ##

0* , : # ## ## ( 1

# ### #2 _ B# &&* 1 ## 8

/ 2# ## 3# M## )##&N

# M## N &1* ## !

### # 2 # " 81 J &# #: ##

%# &1*

td = σ0 · n)3# &1*

ud = ε0 · x%## ##!

# σ(M) = σ(N)

u = E · x + v v

J %## ##

" # * QHJR # / ## # 1 #!

%!# 0# 3 1#2 MJIN MJHN 20 ,#

# 3# ) ,# ¯A(x) ¯B(x) 0 L

## 3# # #

ε(x) = ¯A(x) : E MJIN

σ(x) = ¯B(x) : Σ MJHN

V E Σ # 3# # #2 ε σ

# &1 ##2

# #2 20# 01# # #

< # %!# # 0### &* -

M&* # #1 / #1 3N

4# QHJR ## , ## 2 & #

# < 1 / # #2 ' # ### #

3# 3# #2 % # ,##

,# MJHHN MJHN # 5#& )!)

0 # 01# & #2 ##2

E =⟨ ¯ε(x)

Σ = 〈σ〉V MJHN

V 〈x〉V = 1V

∫V xdx / x V

" # 1 # &1# <

# , & 3# #2 / 2 (

# # 1 MJIN # # ##!

8 # # #2 &1# MJHN % &

& " &! " & (! # 2 / # / # 3# !

#2 &1# / # # # #2 #

&# ##2 M### # MJHN MJHHNN #*

/ &2 3# 2 ### ,2 ( # #

# #2 M# 3# N # #2

0V # ,# % , & &#2

1 0# / # #2 , 1 #2 %

# 3# ,# ## ## ¯A(x) ¯B(x)

## # !##

&)!) )( !!)

" ¯A(x) ¯B(x) 1# # !

#3 &1# ## 2 ##2 #2 #: #!

$ # #8 # MJHJN MJHLN %!# 0##

## #2 , # ## ## 0 ##8#

# 0&1##

⟨¯A(x)

= ¯I MJHJN

⟨¯B(x)

= ¯I MJHLN

! #: 3# #2 ##2 #

# ## ## " # &

< 3 QHJR QHLR S2 , ## ' #*

1# & & ) " & (#1

, 3 0&&* ##8# 2 & 3# ##2 /

( M# ε = EN " ## # #

## 0 L % & # 3# &1

# , " & (#1

M& # 1N 8 3# 0

3# # ## # "0# QHR

3# &# F 4&# &#?F ##

## # 3# # #3# , 5# 0##

" #: 5# ## 1 <

& # !)!'( % & #1# , 0&

QHKR & # &1# ¯

Sesh 0 ##

&1* # 0 ## Ω 1 ## &1* #8# " #

Ω * < # #2 2 ## 0 # 3#

#2 " # 0,# ¯

Sesh < 3

QHPR # ## ; &# 0&

& # 0## 2# 0## # !& #!

B? QH R ) & 0## 2# 0&1# 3#!# #

#Z #1## #: ## # " & #* 2

/ # # n ## #1## Li ## #1## Lm

" # 0## # # #, # ' #

#: & 3 QHLR

' & ## 5# &1!

# 5#& )!) " '( 4##2 & 3

## #2 0&&* ###

# % &&* # < #1 #8 # 1!

# 0,## 2 ## A 1#* %

### 1#2 01 0 &&* ### & #

3# " # / ## #:

&1# < 3 QHI H JR

) # )))! ### #

# ## ##2 / ## # ## " & !

0### QL R 1 QK PR , '

,# * !& # !## 3 # Q R

QIR !## 0 # &1* #

0## 5 " 3 QJR QJHR

& 0 # # " !## #,

/ ##2 1 < # 3# )? QJ JJR %

& FB3# ;# #F # 3# #

M3# ,### / &1 ## 3# #2N

## : # , &* ##2 0### 0#G "

0 # &1# 0### &

< 3 QJLR QJR #, #

" # 1 !# 8# #: * # !

#2 ##, 41* < #8 , 1# " *

& '() / ## , & ) "

#: , & , 1& #

0 ( ! !)'( (4 < 0)

' #* * & '() # / ##

* #2 # / # # 1 #!

* # * &1#2 / 0&

#2 # ##2 "0#3 #

,# #2 # # / 3# #2 / #

# &* / & #3# ' ,#

3 1#2 # MJHN 0 && / #

# # &1* 2# / # # # #2

#: # # " 3# 0# / #

Σ = ¯Lhom : E MJHN

V ¯Lhom # # &1* 2#

" # # MJHKN V ¯L(x)

# / ### 1 # ##

Mε(x) = ¯A(x) : EN 0&1## MΣ = 〈σ(x)〉V N # #* # MJHKN # MJHPN "0### #

## # # / # # 3# #2

MJH N "0,# # # &1* 2# #

σ(x) = ¯L(x) : ε(x) MJHKN

Σ =⟨

¯L(x) : ε(x)⟩

VMJHPN

Σ =⟨

¯L(x) : ¯A(x)⟩

V: E MJH N

¯Lhom =⟨

¯L(x) : ¯A(x)⟩

VMJHIN

"0#< & 3# # #2 3#

# 8# % & # / ##

0 #2 &1# # #

#* # # # 3# 3 # M1

#3 8T# 0 #N

" * & ) 0###

# #! # # ## / 0&

&1# 3# 1 # ##

# 3# &2 ! "0#1# #

# 0# # 8 0###

# 0&1## / # #2 % &

M81 JKN 3# M& #N 1 < #

/ # 0 # #2 M& N ' &

& #!& #20 ## # 0,# #

#2 1+ / # #!& : 0 ##

/ & #3# # 1 / & #

" & ;2 QJKR 3# ## 0& #1

0### 0 ### 8# # #3

#1 # # 0& #2 " #2

0 8# # # #3 ##2 "

## & & 8# ##2 #

/ 0 8# (4 & ##

8# " #* 0& #2 #1 #

2 ## # &2 # 0#1# *

8# # / ## # #3 & <

&# # " # 0 3# / 0&

#2 ) 0 # 01# # # -

0V ,# " *1 ## # / 0&

##2 #1 ( " # * 8#

* & # ;# # #2

# - # # M# 0&1##N

" 8## < * ;2 0# # ! ##

#20 * 8# #2 3# / # *

8# ##2 2 0& #2 # K # 0#!

1# 2 ##2 ##2 k U 2# / K ×k

## " * ;2 3# #

* < * 1 # #

2 # 2# %0 2# 1 # #

# # & &# #2 8

# # # 1# M8 #!

##N 2 #2 M# / 1 #

#N # # & &#2 *

QJPR # ## & , &#2 * 0

5 #1 : 20# #

0# )T% U M1 #1 0 #

N ' # # U # / / 0& ##2 2

# 02# 02## 0 #8 (

## 0& #1 0* , / 2 0& 2

# 1 1#1 &2 & ,

# 0 #2 0### #2

% & # 3= #3# 0&1#

& 3# 0 3#

' &## & 0& 2 #1

JK & 2 ! & #1

) ,* - (, ./

" # # # 3# ##

3# 2 # # # #2

2 # # # 0## ## ;

## 1#2 8 &

#: &#2 #2 # #

##2 * #2 # /

## #!& % # #:

& ## ## 0 U ?1 &###3

% & ### # / 0 #, &1* # #

# 2 0## F##2F

# 1 && %0 2# #

&& / #2 & / ##,

## 3!# % & #

## ## &1* / ## 3 #

# 0#3# & , * ## # # ,#

&#2 * #!& 31### εεref

" & B; , )? QJ JJR $ 3# !

1# 0# # ## # !## "

#8# & # 0# 0 ### #2 3!

+*$ ) , -+ ./

# M# #N #2 #* 3#

&#2 #2 & , 3# #

# #3 V n ! Vr / 0## 2 &

#3 0 & ! / !

# &1* ( "0& B; # & , , 1 #2

1+ , # ## MJN MJHN ) ,# εr σr #1

#2 3# # ##2 ! Vr

) < ¯Ar ¯Br #2 #2 ## #!

3# # ! Vr " ¯Drs ¯Frs

0 L 0#G ¯Drs M# ¯FrsN 0#G

0 3# M# 0 #N 0#1# ##2 M#2 &!

#2N ! Vs & 3# M# #N

εr = ¯Ar : E +n∑

¯Drs :(εps + εth

σr = ¯Br : Σ −n∑

¯Frs : ¯Ls :(εps + εth

) 3= #2 ## ¯Ar ¯Br #

0&1## ##2 ,## 1& JJ "

0#G ¯Drs ¯Frs 3# 2 / # #2 % #

# 1#2 # # / 3!

# #2 ## &2 ! # 02## (

&1 ,# # # # 0#G ¯Drs #

# 6 × n #2 6

3# #2 n ! # (

# # 0#G ¯Drs ¯Frs <

¯Frs = ¯Lr

[δrs : ¯I − cs

¯Ar : ¯Bts − ¯Drs

]: ¯L−1

V ¯Lr # ! Vr δ #1 9?

cs = VsV 3# #2 ! 2#

" 0#G # #8 # MJJN / MJ N " #

# < , )? QJJR #8

0,# 5# $ < 1 #2

## 0 ## # #2 2 ! QJJR

cr¯Drs = 0 MJJN

cr¯Frs = 0 MJLN

¯Dsr = ¯I − ¯As MJN

¯Fsr = ¯I − ¯Bs MJKN

¯Dsr : ¯L−1r = 0 MJPN

¯Fsr : ¯Lr = 0 MJ N

" # ## 8# ,# MJN MJHN 3 ##

# ## 0#G % !

* ## # # #2 ! #

0#G 01 , # 0* # #:!

%# < * # ## / 3# 3= #2

( # , ! # * ,# !

& B; < 3 QHLR %!# & 3#

# 1# 8 0# ## ##

0#G # 1 ## # /

## # ### 3# # # :

0 ## # #2 " # # #* 1

# MJIN # 3 2# MJJN

σ = ¯L∗ : (ε − εin) ou εin = εp + εth MJIN

V ¯L∗ # FF : 01

εin 3# ##2

σ = ¯L0 : (ε − εin − εgl) MJJN

V ¯L0 ### # εgl 3# # 1#

" 3# # ## 8# ## # ## &

B; < # 20 3# ##2 M# MJJHN MJJNN ##

## 0#G 0 / : 20 3# ##

0 # % & < #2 / ( 1

! B3# 2 & 3# 20

& : 3# # 1# 0 / 3#

# #20 3# ## # # #: # #

& #, # / # #2 QHL JLR 1#2

εr = ¯Ar : E +n∑

¯Drs :(εins + εgl

)MJJHN

σr = ¯Br : Σ −n∑

¯Frs : ¯L0s :

(εins + εgl

' & #!& & #1 0### & B;

# 8 / # #2 ### &

3# # ## 0 3# 2 %

& ## # &1 * 31###

#!& ,# 3 η = εεref

# 0 # 8# & # * "0&

#2 3 3 / # 2 0& M#2N

# / 3# % & 0 # # V

2 #2 # 0 #2

" #* & # / #8 # & / !

# # #3 " ( 0 / 2 3

## 5 1 8 0# 0#G !

3# ## # MJJJN

0&1## ##2 ) ## 20 mm #1

2# *1 0 3# #* 4 mm % #

N ! 2 & # 3#

# &1* % 1 : #* / 1 # F&#2F

&* # 2 &#

d(perforation)

l(V ER)

L(structure)

≈ 5mMJJJN

" & # #1 #: & !

#2 #2 &2 t # 0# 3#

#2 E # !!) ## 8 &1 #

#3 3# / 0& # 0#1#

# 0# ### # ;#!

# #2 1+ / *1 5#& )!)

%#!# 8# / # &1 #2 #2

( ) 0&&* 0&1# & , & !

0,# # 0&1## #* #

Σ = 〈σ〉 =∑

crσr MJJLN

& #2 #2 2 3

# / # 2# / 0& #2 % &

&# 81 JP

JP ## & #!&

" & #!& #! 0 # !

# #2 ) # ## )? 0

/ #2 #!# ##2 3# ##2

## ¯A ¯B / " ## # #2

0&1## ##2 # 0

* #2 / &1 #

### & # 3# 0##

## , ## " 3# / 0### 1 !

3# # 02# )T"% ( #,

# 8!# ### QH HLR " 3# #

##2 #, 2# M 0 1MPa 0W1N

# # 8 F#F M #G ##2 0

##N " 81 J 1# ###

! # #3 ## M5 1 &#

J ( # #3

) 0## ## #

& #!& # & # #* #

0 3# #2 E && / # #2

Σ 1+ , # 1 ! 3#

εs 0 & ! 1+ / # ##

MJJN # 3# " # σs # / 0# #

MJJKN $ 01# # #2 # 4? " #

#2 Σ # # ## # 0&1## MJJPN

εs = ¯As : E MJJN

σs = ¯Ls : εs MJJKN

V ¯Ls # #1## ! s

Σ =∑

csσs MJJPN

V cs = VsV

" # #3 0 2 3 ## # /

3# #2 ## y M $N " #

3# ,# 81 JI #: # 3 &#

" # # 3# #

3# " & #!& ###

#&1* #: & 1 3 &#

0 1 2 3 4 5 6 7 8 9 100

x 10−3

Distance p/r au centre de la perforation [mm]

Déformation macroscopiqueDéformation locale

0 1 2 3 4 5 6 7 8 9 100

Contrainte macroscopiqueContrainte locale

JI )3# # # & #!&

% & # #* * 31###

η 2 ηi = εiεref

" 3# 3 εref 1 / 3#

#2 E (E22) " #: # &#

& #!& / &# M81 JHMaNN "0 ,#

0 5% 2 ( 5

2 0& #!& # < / &#

"0#< 0& #!& 20 0 ,

* η 2 2 # &1 # " 0 &1 ## M $$N

# / # 0, 81 JHMbN %#!# # # 20 ###

# * 0 3#!# 2 1.7 20# # 3

0 1 2 3 4 5 6 7 8 9 100

ThéorieHomogénéisation

0 2 4 6 8 10 120

Distance par rapport au centre de la perforation [mm]C

JH * 31### εεref

## # & #!&

" 81 JHH 0# * η = εεref

## &1 "

2 #2 * #!& 31###

0 &1 #2 #8 2 # #2

31### * 1#2 & #!&

# # #: # # 20/ :

* η 0,# 3# εs = As : E 5# As

### 1# ( #2 #

M 0W1 E 5# # νN

0 1 2 3 4 5 6 7 8 9 100

ThéorieE=1.E−03E=2.E−03E=3.E−03

JHH 6# * η 0 &1 #

" , # M81 JH(a)N 0* < * &

, ## &# "0& ##

#8 * 1#2 31### # #2 )

# # 2 & 1 0#

* η 0 2 3 ) ### &1*

3# # 0 2 " *

η 1 / 1 2 0 # # # M81 JH(b)N

0 1 2 3 4 5 6 7 8 9 100

ThéorieAluminiumAcier

0 1 2 3 4 5 6 7 8 9 100

JH ( # / &1 #

% < # #: !

### # #3 ) # #

0#G * # ( # 1 !

0& #!& * η

)=() ! ! ( 9 72>8

)0* 1& JLH ( &## 1 20 mm 3#

#* 4 mm " &#, ## 3 #1 ##

## #2 # # #

0#1# d << l Md #* 3# l 1 ##2

(N #1# # # &# # A #5#

0# / # 2 d < # # l

8 # 0#G d/l * #!& !

# #1 # #* 3# 4 mm #

1 1# 1 ( S ( 1 10 20 40 #

80 mm # 2# # / d/l 2/5 1/5 1/10 1/20 #!

" #1 # 2 &2 ( # #3# # <

F##2F # # # M81 JHJN %# ### 0#G

#1 0 ## 0## d/l %

##2 3# # ##2 #2 2#

# : ) 1 ! 5 # &#

#* / ## 5 0# &*

# 3# " 2 #1 81 J

##2 JH

"1 ( [mm] d/l 0

10 2/5 143 256

20 1/5 219 408

40 1/10 281 532

80 1/20 343 656

& JH %##2 #1 #

" ### ( # 20 mm 40 mm M81 JHLN 1

( 1 ## M80 mmN ) / # 20 mm ##

5 # # ( 2 0# 20!

/ 0 # 8 mm / # 3# # η 1 $

0 #G 3# ( # 10 mm

* &#2 &# H M81 JHLMaNN η

1 3= " # & # / 0### &

0&1## 0 #8 ( # 10 mm " 3#

## # / 2

"01# # ( * / # 2#

&# # & # & #!& #

$ # 2 3# # # #!/!# #

#3 " &#, 0 ( 20 mm # 2 /

0# ### 3# * η

./ 20 mm

./ 40 mm

./ 80 mm

JHJ #1 #: (

0 1 2 3 4 5 6 7 8 9 100

TheorieHomogeneisation

./ 10 mm

0 1 2 3 4 5 6 7 8 9 100

./ 20 mm

0 1 2 3 4 5 6 7 8 9 100

./ 40 mm

0 1 2 3 4 5 6 7 8 9 100

./ 80 mm

JHL & #!& &# #: #

)=() ( )& (*(&

"0 # 0# 1 !

# )0* 1& &#, 0 ( 1 20 mm

3# #* 4 mm 0# #3# )#: #1

( 1 20 mm # 8 # 0#G !

0# & , " #: #1 ## 81 JH

#1 ( 1 < # 8 # ( 3 1# " #1

5 1 &# " ##2 #1

" & 3# & ! ' 5#

# 3# # #: # F3F &#

3# 1 # 3# 3 " 81 JHK

5# #: #1 %,!# / &#

JH )#: #1 ( 1 20 mm

1 239 436

2 149 272

3 83 144

& J %##2 #1 # M( 20 mmN

" 1 01# ! !

( 1 2 / * ### "0### #1 ( 3 1*

# % ## 0#1 3# # 0,!

#2 3# 2 0#1 3# # 0&1# &

3# # " #1 JHK 0#G G1 #!/!

# 1 ! # # # 2 #

,< 2 ! 0 3# 0 #

# # # 3# 3#

0 $ ./

0 1 2 3 4 5 6 7 8 9 100

ThéorieVER 1VER 2VER 3

JHK %# & #!& &#

#: #1 ( 1 20 mm

" !## #2 ## & #!

& 1+ / # , # ##

## # / # $ 01# 3# #2

%!# # ## MJJ N MJJIN " #

3# #2 # # $

# 0 & # %#!#

81 JHP # ##8# 2# ¯ ¯ ##2 !

# 0 2 0 4 #

εs = As : E +n∑

Dsr : εpr MJJ N

σs = Bs : Σ −n∑

Fsr : Lr : εpr MJJIN

)## As Dsr

( ### εps(tini) = 0 ps(tini) = 0

%&1 ( 3# E(t)

εs(t) = As : E(t) +∑n

r=1 Dsr : εpr(t)

σs(t) = Ls : εes(t) εe

s(t) = εs(t) − εps(t)

# #* ##

fs(σs, R, t) = Js(σs, t) − Rs(ps, t)

Js(σs, t) =√

32 tr((ss(t))2) V ss(t) = σs(t) − 1

3 tr(σs(t))

R(ps(t)) = σy + B(ps(t))n

6# # 3# εs(t)

εs = As : E +∑N

r=1 Dsr : Lp−1r : Lr : εr

6# # 3# #2 εps(t)

hs(t) = hp, s(t) + ns(t) : Ls : ns(t)

hp, s(t) = 0 # fs(σs, R, t) < 0 hp, s(t) = R′(ps, t) # fs(σs, R, t) = 0

εps(t) = 1

hs(t)(ns(t) ⊗ ns(t)) : Ls : εs(t) V ns(t) = 3

2ss(t)Js(t)

6# # 3# #2 ps(t)

ps(t) =√

23 εp

s(t) : εps(t)

)## εps(t + 1) ps(t + 1) # / 0!"

$1# # & 0 εps(t) ps(t)

JHP ## # & #!& ##

0 $ ./

% ### # 8 #

## 0#G M3 0N &2 # &1 #

#3 # / &1 #2 # 3# M3 1N "

3# # # ## MJJ N " # #!

&1 &## 5 # 8 #* ##

#2 ### 3# #2 < #

1 / 0 " # 2 / # # σs = Ls : (εs − εps)

M3 2N " # # 0 #* ## f &

! M3 3N

"03 4 0* , $ 01# # # 3# εs

# 02# MJJ N 2 A D

# M 2# &&* #8 * ##

# N # # 0# 0,# εs 3#

# 3# #2 εps M# MJLNN

εs = As : E +n∑

Dsr : εpr MJLN

# &# #2 ## 0# # ##

3 1 MJLHN 0# εps # 3# εs 1+ ,

1 M2# MJLNN " ## εs # / *

6× n 2# Mn !N 3∑N

r=1[δsrI −Dsr :

Lp−1r : Lr] : εr = As : E

εs = As : E +N∑

Dsr : Lp−1r : Lr : εr MJLHN

εps = Lp−1

s : Ls : εs MJLN

V Ls Lps # 0 1 0 1 #2 #

! s σs = Ls : εs εps = Lp−1

s : σs

' 3# εs εps ps M3 5 6N ' & 0#1#

# # εps ps / t + 1 # # 0# #

2 # #2 0,# * 31##!

# η 3# &1 * 1#2 #, " #:

* # &# " 3# 3 #

&1 0!/!# / 3# #2 " * 1#2 !

# ##2 3 2 # ##

( 8 * #, ## #

" & #!& #! # 0 #1

0## # # ! / #!

1 # ## #* ## # ' ##

&#2 # < 3 QLR ##8# 2 /

< 3 QLJR " 2# # / #: # !

JJ ' & 0#1# 0 ## 8 0#1 3#

#2 ## 2 3# #2

## 3# ε = εe + εp

%#* ## f = J2(σ) − R(p)

J2(σ) =√

32 tr(s2)

s = σ − 13 δij trσ

6#1 # R(p) = σy + B pn

σy = 350MPa B = 600MPa

n = 0.5025

)3# #2

˙εp : ˙εp

& JJ "# #2

" 81 JH & " # 3!

# 3# !# < 20 #

&# * ## : * 2 ## #2 #

# # 0 #8 M81 JHI(a)N # 1

# " * < #1# # #

3= # # 3# 2 " 3#

3# M81 JHI(b)N

0 $ ./

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10−3

Déformation macroscopique [.]

ThéorieHomogénéisation

JH %# 3# 3# # !

& #!& # #2

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010

Déformation plastique cumulée [.]

Théorier=2.2mm

0 1 2 3 4 5 6

x 10−3

Déformation macroscopique [.]

r=2.2mmr=3.6mmr=5.5mm

JHI ( # / &1 # # #2

8 0# # U #* & 03 4

##8 " # 3# # 0,#

& MJLJN % * ## #

# 0 2 1 & 0#1# 0 #

#8 M81 JN 1 ##2 0#1 MA = 350MPa B =

3000MPa n = 0.4N 2 # # #,

& " ## # M81 JHN

# 1 #* # # # / # #

3 &1 / ## 8 #8

1 0# 3# #2

εs(t) =εs(t) − εs(t − 1)

∆tMJLJN

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010

Théorier=2.2mm

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010

dt=1E−4dt=1E−5dt=1E−6

J #8# 01#& #

0 1 2 3 4 5 6

x 10−3

Théorier=2.2mm

JH #8# 0#1

0 '()%

) &# #: & 0 / !

## * 31### # &1

8 # 0&1## % & 3# #

#: & * / # 0& #2 # /

0& #2 # / 3#

" &# 20# # #

* #2 31### / 0# ## # 0 &

#3# * #2 2 3 &

0&1## #1 0### # ## )? % &

/ 0& &* ##

#1# 1#2 0# & , 3# #

' ## #* * 31###

< 1+ / & #!&

) , # 1 # #!

# # #2 ' ### #!/!# #

2 / # # 0& ( 1 #

#1 " # 0## ##

( 20 mm 0 2# 2 / 0## /

## #2 # ( &## & #!

& > : 1 ### 1#2 &

, # 3# ## # "

# #2 / # *

!1 & #2 # &#2 "0#!

# 0 # # 0# # #

# 3# #2

" & #!& # # & #2

& < # # , *

31### η ) 0& # 0 ,

& 31### M0!/!# # N

8# η =¯[ε

] ηij = εij

εref ij % # 1

8# F#1F 3#2 ##2 * 31###

1 3 &# , % & # ## 0 /

#* #2 31### Πλ MJLLN M#2 3#2 ##2

3# λN # < / #* M81 JN

Πλ =∫

ληij dλ MJLLN

J %## #!& * 31###

% * 31### # / ## & 0#!

# # 3# 3# 1#2 * / #1

# # ## # # # 0#

0#G #2 &* 31### ) #2

,# &* # ### #2

QHR ` B ."WB%49 O!; 3#* 8# #2

3 0 33 E &A? " "

QR C B4$ " & 3 &#1 & #1 #3 #

# 1 G# &# ## # # !0

QJR ` C )".X B ."WB%49 8# & 3 ? 1@&

@#& &#1 * 0 ) < ' 4

QLR C "9 $ . .'9 B& ## 3 # 8# & # &

## ! 0 %0 0 ) <

QR '9' )" %4 ` B ."WB%49 #1 & ##

# & , 8# & ! 0 %0

0 ) < '='" "

QKR $ . .'9 '$ B& &! # 3 & 8# & # ##

# % * %6 < ' & 4='"

QPR ; %'- .%9 %# @ & ! &!# 3 & 8#

& &# ## &# 5# &!# !

0 %0 2 ,+$-

Q R ; %'- ## 8# &!

/ 0 0 123 0 / ,+$- "

QIR 9 a%9 )aB 4 .b9 B& 5# 3 & !# 8#

& # ## #1 # * 0

) < 4" 4=' "

QHR O!C "' 44 ) C!; %" 0. X . " . Y .

%B$ !# & 1# #

QHHR - 9$ "W; "($ . Y . S#!1# #1 3 & !# 3 & 8# !

& %6 0 < " =4

QHR )aB 9 B& !# 3 & 8# & !

# &!# 3 & 3# & 3 ## ! 0 %0

0 ) < "44 "

QHJR .B B .B4 ' -$"$$ 41## #2 !

#, H 5 "

QHLR B BB$ ## #!& 01 !

# / # #2 / 0 0 F 3 0 (

QHR Y 4 4$ 4B$9 ## & & & 3 & # &#

3 #& # * 0 (6 * 0 < " '

QHKR C) 4".W B& ## 3 & # 8 3 ## ##

(0 * #6 6 0 < % &" '' 4

QHPR ) ; c$ $ ' Y '$ % #2 #, #!

# ## 5/ ( "

QH R 9 B 9 1 # #, 1 # 1 3 # @#& #8#1

## % < " 44 4

QHIR .' C" "$ - $%" ' # # 3 ##

50 %0 =

QR - )'( 'B 3# #2 ##, # ## / 0

#, # #2 / ##2 &1## ! 5

0 %0 '

QHR 9'%4 % 1 #, # ##2 / 0 0

123 0 ( ' ==

QR ) "-'$"" %4Y! "%$ & &# 3 ? #

@#& M @#&N 3## 0 3$ 3$ %$3 < &" 4"

QJR 'S'B ## &1## / 0 0 123 0 ( ' ="

QLR .($"" Y '$ ,# 3 & 3!# & #

G@#1 # * 0 (6 * 0 < "' "4

QR 'S'B # 3 # # #8 # &

# #? @#& %0 # ## !# %0 (

3 &" 4'4 4

QKR 4$"" %# #!&# 3 # # * (6

0 < = '4

QPR CX 4'B%4$ #!# &# 3 ## #

(0 * #6 6 0 < % & """

Q R "$ $ - % ( 4Y$ 3!# & 3 & 1 3!

# # ### % < 4 "=" =

QIR - B ) -C X- & 3 #!#3 ## * %0

< 44 "'4 ==

QJR B % B ) , # 3 & :# &# !

# 3 !# # * 0 (6 * 0 < &'

="='" '

QJHR .B 'S'B Y '$ 5 3# 3 & #!

# 3 & :# # 3 !# # # *

0 (6 * 0 < = " "" "

QJR - )( 9 #3 8 # &1 # (0 * #6 6

0 < % & =

QJJR - )( 9 W .($B 3# # #3 8 # #!

& # # (0 * #6 6 0 < % & "

QJLR % d 0 #!& #, # / # #2

## / 0 0 F 6$ "

QJR C!" %4 .%4 9'%4 C!; $ B BB$ B@ #&#

## 1 #1 3 # * (6 <

QJKR ; ;W" ## * 1 # # / 0

0 F 0 0 ( =

QJPR ; ;W" # %2 ## # 3 # !

QJ R %4$;; ## #!& #2 #

/ # 1#2 : ### &#2 / 0 0 123 0

QJIR ! . W B ## #2 0 2 3 & #!

& # $ .)#% &? ' ) 2 "

QLR " .)"B #2 ,# #!& ! ## , !

1 # #2 3 0 )% 123 0 "

QLHR ! . W B . " - ) ) YB$% # #1 3 !# !

&# 3 3 # # #3 &@&# ) !

! 0 %0 0 ) 62G:6G ,70-

QLR C " $B C!" %4 .%4 #2 #, # 0 ( "0

QLJR . " - ) %## * 01 - 0#1

0## L!&JH ! #2 0 3 #

# $ .)#% &=? " ) =

' # $ %

% &# # 0#G # 3# 31###

' G,# 0 2 , ### :

#2, 3# 1#2 * 31### ' 1

,# # ### #2 3

3 3# 0 # # / # 3# / # # ,

#1 0## ' #!& &# #

% #3 0# / # 3#

### 3# % ## 0 # 0#1

#2, * 31### # #

# 2# #2

35 6 4

" 37!(4

' ()*%

) &# * 31### % *

0 #2 # / &1 ,<

M& # ,#N # # 0 2 3 M#N /

& #2 $ 8# εεref

V ε εref #!

3# 3# # 3 $ ## 3# #

&* , #: & * / # 0& #2

# / 01 / 3# 0& #2 # /

% < &# 3# 1#

FF T F##8F 3# 3# # !

3# : # 3# 0 7000 s−1 ,

< # A 3 ## 0 # 3 /

# 5 m/s QHR 0## & # 0* # 0#

0#G 01# # 3# M# 01# #

###N ## 2 # * 31### <

2 & # 0 /

) 3# 1#2 * 31### # 0

#2 < # # 3# ' #* &

# # / # 3# εs ,## 3# #

εs = As : E +n∑

Dsr : εpr + Gs : E MLHN

$ # * ## Gs M# 3#

&1 ( EN ## 2 !## # #!

2, M 1# & ## As 2#

#2 ; ##N % #* # 0 2 #1

' & # 0## : #2, 3#

##2 # ## )? Q J LR ) #

3# ##2 εvpr MLN ' #* ### # 0!

##2 ## 8 & &#

M# JN "0 0 #2, > 2 #8#

/ 3# # 3#

εs = As : E +n∑

Dsr : εvpr MLN

"0## 0 #2 3# 1#2 * 31##!

# # # # 0 #

# # / # &1 * η 0<

3# 1#2 $ # # 2 η # 2

η = f(r, a, σ, εref ) si σ = σ(εref ) MLJN

) < #* # # # / # &1 ###

/ # ### # < * η εref ##2

&1 #2 ## # 3# εref < &## 8

# &1 8# εref = dεref

dt # #

/ # 3# # #

η = f(r, a, σ, εref , εref ) si σ = σ(εref , εref ) MLLN

8 # 0#G # &1 31###

# #2 0 3# 0 #1 0##

" # # / # 3# ### *

31### / # &1 εref < #

: #2 # " 1 ,# # #

#2 # M# # / # 3#N

QR 8 0# 0#G #2 0 # * 31###

" 1 # ## & 1#2

## 3# η &1 #2

" ## M3 N ##2 / &#

&1 2# #2 M&# N 81 LH #!

# 1 mm 1.17 mm #1 0##

ML!"JN # , MON

" &# # &#2 # # ###

1 / V (m/s) = 8.3 10−5, 8.3 10−3, 0.1, 1, 2 ) 0 # # 3# &#2 < 8# # ###

/ 1 ### 0 MLN # 1 ### 0

# M 3N # # εref (s−1) = 3.3 10−3, 0.33, 4, 40, 80

εref =δ

LH 6 # 3

" 3 # 3 #A!#2 ##!

# 9# 9 " & 3# #

# 0#1 ' # (## ## 8 # # 0#1

# / 0# & # ,# 02### 10000 #1

"01# # #1 # / #

?1 * 02### ## # ### # #1

## # 0 2# # 0 0.5 mm 2# #2

0 1.5 mm # / 1m/s ) 0&#1 #1 ##

##2 # ## 0 0 # 0 3

#1 0## # / # 02### 55 #1 2# #2 M#

∆δmoyen = 0.015 mm , #1 #N 2 12 #1 #!

# # ### 1m/s M# ∆δmoyen = 0.07 mm , #1

#N 2 #2 1## 0 & 3#

22 #5 0,## , #1

&2 # 3# #: # &#

% 1 0# &* #2

# 0# # 0## ##

# 2#* 0# & 0# M81 LN

#2 ## > :! % &*

# , ## 1 ,# 2# 0:

1 0# QK PR 01# # ###

## # 0 #

L %&#, # 0#

" ### 3# 1 &# " ## :!

# #: # 0# # , LH L #

(# ### # # H # # J # L #

[m/s] [mm] [mm] [mm] [mm] [mm] [mm]

8.310−5 0.06 2.23 2.79 3.74 5.63 10.34

8.310−3 0.05 2.70 3.25 4.30 5.40 10.30

0.1 0.01 2.15 3.05 3.90 5.65 10.45

1 −0.15 2.10 3.25 4.45 5.55 10.15

2 −0.35 3 4.15 5.20 6.20 10.65

& LH ## # # #1

S 2 # # ### / 0

##2 # / 1m/s " :!

81 LJ ' M 0 20%N

# " / 8 # !

0 "0 1, &# LJ #,

(# ### # H # # J # L #

[m/s] [mm] [mm] [mm] [mm] [mm]

8.310−5 2.46 2.92 3.85 5.74 10.49

8.310−3 2.75 3.25 3.80 6 10.80

0.1 2.50 2.90 3.80 5.95 10.45

1 × 2.90 4 5.25 11.05

2 2.40 3.10 3.90 5.65 10.45

& L ## # 3 #1

01# / # ###3 % < 2# #2

3# 0 2 0#G # ,# # #

#2 2# #2 3# # ##

/ 0 ) 3# # 0# 01# 0

3 3 ## % ## /

/ M 0 85%N

0 1 2 3 4 5 6 7 80

Déplacement [mm]

LJ :! 0 #1 0##

Fmax [N ] δmax [mm] Wmax [J ]

6 3 12050 6.43 77

6 3 9200 0.81 10

& LJ ,#, #1 0## ! V = 1m/s

3 3# &1* 1 &#

2# 0# M81 LL(a)N # 0&# #

# / # 3# M81 L(a)N 3

3# * * # / #8# ###

3# M81 LL(b)N %!# #&1* # #

3# 3# M81 L(b)N

0 1 2 3 4 5 6 70

Déplacement [mm]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

Déplacement [mm]

Point 2Point 3Point 4Point 5

LL 6# 3# #: # &#

#1 0## ! V = 1m/s

L #1 0## ! V = 1m/s

" &* # 0##8# 2# #2 1!

# #2 : / ,## 3# #1, 3#

3 3 #1 * # M81 LK(a)N "2 0

0#1 3# #1, 3 # 1

3# ## # e # ,## 10 mm ,

#1, 3 &1 M81 LK(c)N 3#

#:## #1, * 0# M&*

# ###N 2 3# 1#

3# ##1 1 0# ) #

#2 1 E 0##8

3# M81 LP(a)N 0&#1 # #5 #

&* < 8 3# # ### ) #

#2 / 0: ,# M81 LP(b − c)N F0,#F

3# # #, 3# ### , 8!

1# 2 # 3# ### , # #2

0 1 2 3 4 5 6 70

Déplacement [mm]

0 1 2 3 4 5 6 70

Déplacement [mm]

0 1 2 3 4 5 6 70

Déplacement [mm]

LK %# 3# 1 &#

3 3 #1 0## ! V = 1m/s

0 2 4 6 8 10 12−0.5

−0.4

−0.3

−0.2

−0.1

0 2 4 6 8 10 120

LP )### 3# 1 &#

#1 0## ! V = 1m/s

" :! # # #G #1#8# #

&1 # 3 S 0 # 2#

#2 #2 3# # #* ####

0 ' ### 0 85% / ##

2 # 0# 01# 0 M LLN

(# ### [m/s] 8.3 10−5 8.3 10−3 0.1 1 2

6 # [J ] 81 73.5 71.5 77 79.5

6 3 [J ] 10.5 9 9 10 9.5

& LL S# 01# #1 0##

" # 3# #2 0

# # 3# F#F # 3#

< # , A ## / # A

#2 A #2 # 0&#1 # #5

# #2 # 3# ,# #

# ### # 3# &1* 1

&# M81 L (a)N " # 0.003s−1 #

2# #2 70s−1 # ### 2m/s % &

# &#2 $ # εi/εref ≈ 1 M81 LI(a)N

10−4

10−3

10−2

10−1

10−4

10−3

10−2

10−1

Vitesse [m/s]

Point 0 Point 1Point 2Point 3Point 4Point 5

10−4

10−3

10−2

10−1

10−4

10−3

10−2

10−1

Vitesse [m/s]

L (# 3# #2 #1 0##

0 2 4 6 8 10 120

Distance au centre de l’éprouvette [mm]Vite

réfé

8.3E−05m/s8.3E−03m/s0.1m/s1m/s2m/s

0 2 4 6 8 10 120

réfé

8.3E−05m/s8.3E−03m/s0.1m/s1m/s2m/s

LI (# 3# #2 # #1 0#!

3 # 3# 3# # 0!

# !# 3# M20/ 420s−1 # /

2m/sN 20 2 M81 L (b)N # # # 1 &#

εi/εref M##2 0# 3#

/ # 3# 3 &#2N 0 3# #

&1 M81 LI(b)N

#1 0## # #

εref= f(r, a, σ, εref ) avec σ = σ(εref ) MLKN

V r ## 1 &# a εref # #

3 1#2 &1

" &* #, 3# # #2 / 0: ,#

# 3# ## M81 LH LHHN , &#

3# #, 3# 3 &1* 1

&# 2 2 # # &1 3

### 3# 8 ### 2 0 # 2# #2

#2 2 # 2 # 3# # M ##

3#N 01# # &1 #

0#G ### 3# 2 0 # 3 $

# 0#1 0##

εi = f(r, a, σ, εref ) avec σ = σ(εref ) MLPN

8 3# 3 εref 0 1 3# # &1!

0 2 4 6 8 10 120

8.3E−05m/s8.3E−03m/s0.1m/s1m/s2m/s

0 2 4 6 8 10 120

8.3E−05m/s8.3E−03m/s0.1m/s1m/s2m/s

LH )### 3# / 0 M##N

#1 0##

0 2 4 6 8 10 120

8.3E−05m/s8.3E−03m/s0.1m/s1m/s2m/s

0 2 4 6 8 10 120

8.3E−05m/s8.3E−03m/s0.1m/s1m/s2m/s

LHH )### 3# / 0 J M: ,#N

#1 0##

2 * 31### η 0 #

&1 εref # 81 LH LHJ $ # 8

# ## / # 3#

η = f(r, a, σ, εref ) avec σ = σ(εref ) ML N

& 36%25

0 2 4 6 8 10 120

[.] 8.3E−05m/s8.3E−03m/s0.1m/s1m/s2m/s

0 2 4 6 8 10 120

[.] 8.3E−05m/s8.3E−03m/s0.1m/s1m/s2m/s

' , - "

LH 6# 31### # &1 M

3 #1 0##N

0 2 4 6 8 10 120

[.] 8.3E−05m/s8.3E−03m/s0.1m/s1m/s2m/s

0 2 4 6 8 10 120

[.] 8.3E−05m/s8.3E−03m/s0.1m/s1m/s2m/s

' , - "

LHJ 6# 31### # &1 M

3 #1 0##N

# 15"'4

" 81 LHL :! # 3

3 / # # 1m/s % # <

2# #2 0#1 0## 3# # ##

/ 0 ## 2 # 0# 01# %

## < 1 2 2# #2 / # 45%

/ 60% 01# " 1, &#

Fmax [N ] δmax [mm] Wmax [J ]

6 3 12160 14.75 157

6 3 10790 8.1 64

& L ,#, # ! V = 1m/s

0 2 4 6 8 10 12 14 160

Déplacement [mm]

LHL :! 0 # ! V = 1m/s

)0 # 0# & 3# #:

# &# M LKN " #1 3# #

0#1 * # #2 M1 ### 0&#1 #

#5#N 3 3# 0 &1* 1

&# M81 LH(a)N # 3# 01# "0#

3# / #: # &1 # # &*

# M81 LHK(a)N 2# & 0# '

15% 3# 2 M# 0N

2 M# 5N 3 ### 3# #&1*

* &1 M81 LH(b)N " 3# , # 1 2

* ### M ## 3# 81 LHK(b)N " #

# 0.9 mm 2 # 0 0 # 1 / 1.5 mm $

2 3# # 1 # !# M

0 1 2N %0 2# ## 8 ## , #

1 %,!# / # ###3 2 2##3

& 36%25

# H J L

QR QR QR QR QR QR

6 3 L JK J × H

6 3 JP PK HP

& LK ## # 3#

# ! V = 1ms

0 2 4 6 8 10 12 140

Déplacement [mm]

0 0.5 1 1.5 2 2.5 3 3.5 40

Déplacement [mm]

LH 6# 3# #: # &#

# ! V = 1m/s

0 2 4 6 8 10 120

LHK )### 3# 1 &#

# ! V = 1m/s

" 81 LHP 1 &* 0##8# 1 !

3# 2 &* # 2# #2 # !

#2 M81 (b)N #2 M81 I(b)N / 0: ,# M81 J(b)N ,!#

< 2 # 0## ## M81 LHP(a)N

### #2 1m/s 0: ,# #, 3# !

3# ### / , 0 # ) &

0# #, 3# 0 3 1 # !

/ , 0 # #3# ) 0 # # /

# 3# 01# # ### # E

F##8F 3#

0 2 4 6 8 10 120

LHP )### 3# 3 3

# ! V = 1m/s

" :! #1 0 ### / # 3# M8!

1 LH N "01# # ### * / # #, 0:

/ ### / "0# #, 0: <

### 2 0 # 3 & 81#

20% 0: ,# "01# 2 /

#2 20 #2 M LPN ' 1# 0 15%

25% # 3 3 )

& 81# 3#* ## / 1m/s #:

# ### 2# #2 M81 LHI LN

& 36%25

(# ### 6 # 6 3

[m/s] [J ] [J ]

8.3 10−5 138 51

1 157 64

& LP 61# &2 #

0 2 4 6 8 10 12 14 160

Déplacement [mm]

8.3E−05m/s1m/s

0 1 2 3 4 5 6 7 8 90

Déplacement [mm]

8.3E−05m/s1m/s

LH :! #

V = 5mm/min

V = 1m/s

LHI ;#* 3 #

V = 5mm/min

V = 1m/s

L ;#* 3 #

"0# # 3# #2 81 LH

%!# &1* 0 # ### / #

# ## 3= &#2 0.003s−1 2# #2 38s−1

# / 1m/s % 0#1 0## εi/εref ≈ 1 20/ 0##

0 3 1#2 & 0# # εi/εref

3# #2 20 #2 M81 L(a)N "0# 3#

/ # &1 2 3# #2 20 #2

10−4

10−3

10−2

10−1

10−3

10−2

10−1

Vitesse [m/s]

10−4

10−3

10−2

10−1

10−3

10−2

10−1

Vitesse [m/s]

LH (# 3# #2 #

& 36%25

) 0 3 # 3# 3# ##

# 20 # #: , # 1 2 2# #2 !#

### # / 1m/s #2 #1, 3# , #

* & " #, # 3# # 0# #3#

/ , 3 3# #1 0## M #

2N " 81 L(b) # 3# εi / # 3#

3 εref 3 # 0* < 3# ##

# M#&1*N # &1 εref %

0 3 & 0# 2 εi/εref

3# #2 20 #2 ) # ### /

# 3# # #

Pourεi

εref= 1,

εref= f(r, a, σ, εref , εref ) avec σ = σ(εref , εref ) MLIN

0 2 4 6 8 10 120

réfé

8.3E−05m/s1m/s

0 2 4 6 8 10 120

réfé

8.3E−05m/s1m/s

L (# 3# # # M#N

" 81 LJ LL &* #, 3# !

3 3 # "01# # ###

## #, 3# #: 0#

εi|sta > εi|dyn ) < 3# 3 # #:

3# # &1 εref |sta > εref |dyn ;#

0 # # / # 3# # #

εi = f(r, a, σ, εref , εref ) avec σ = σ(εref , εref ) MLHN

εref = f(εref ) MLHHN

0 2 4 6 8 10 120

8.3E−05m/s1m/s

0 2 4 6 8 10 120

Distance au centre de l’éprouvette [mm]D

8.3E−05m/s1m/s

0 2 4 6 8 10 120

8.3E−05m/s1m/s

LJ )### 3# 0 3 #

0 2 4 6 8 10 120

8.3E−05m/s1m/s

0 2 4 6 8 10 120

8.3E−05m/s1m/s

0 2 4 6 8 10 120

8.3E−05m/s1m/s

LL )### 3# 0 3 #

" * 31### ηi = εiεref

1 L LK ) 0 3 ηi 1 / 1 2 2 #

# &1 εref 20/ 0## 0 3 1#2 & 0!

# "0#G # &1 ### 3# εi

εref / # &1 ;# # #

∀εref , η = 1 MLHN

0 3 & 0# 0 !

3 3 1#2 η > 1 ) 2 ηi|sta > ηi|dyn

2# #1#8 2 01# # &1 ## *

31### 0#G 3 1#2 2 8 2

ηi|dyn

ηi|sta < 1 2 W |sta

W |dyn # 15% 0

3 25% 0 3 # # / #

3# # # 8

Pour η = 1, η = f(r, a, σ, εref , εref ) avec σ = σ(εref , εref ) MLHJN

0 2 4 6 8 10 120

[.] 8.3E−05m/s1m/s

0 2 4 6 8 10 120

[.] 8.3E−05m/s1m/s

0 2 4 6 8 10 120

[.] 8.3E−05m/s1m/s

L 6# 31### 3 #

0 2 4 6 8 10 120

[.] 8.3E−05m/s1m/s

0 2 4 6 8 10 120

[.] 8.3E−05m/s1m/s

0 2 4 6 8 10 120

[.] 8.3E−05m/s1m/s

LK 6# 31### 3 #

& '()%

" # ### #2 # 0# 0#G #

&1 &* 31### ' #!&

/ 3 3 2

# / #2 #2

# # / # 3# #G #1#8!

# 0 < 01# ### 3#

* 31### η 2 # ## /

# &1 # #

∀εref , εi = f(r, a, σ, εref ) avec σ = σ(εref ) MLHLN

∀εref , η = f(r, a, σ, εref ) avec σ = σ(εref ) MLHN

# # / # 3# 1* #G #

&1 01# 3 3

01# # &1 #8 3=

### 3# εi 3 εref $ #

εi = f(r, a, σ, εref , εref ) avec σ = σ(εref , εref ) MLHKN

εref = f(εref , εref ) MLHPN

) 0 ## # #G 20/ & 0!

# ∀εref , η = 1 0 3 #2 31###

# # &1

Pour η = 1, η = f(r, a, σ, εref , εref ) avec σ = σ(εref , εref ) MLH N

2 8 2 η < # / 01#

"0#G #2 * 31### / #!

0 # # / # 3# % #G

# # < # 3# 1#2 *

# " 1 #

0## , *

QHR . " - ) %## / ## #2 ,# 0!

1 , # ### #2 / 0 0 123 0

QR B BB$ ## #!& 01 !

# / # #2 / 0 0 F 3 0 (

QJR % d 0 #!& #, # / # #2

## / 0 0 F 6$ "

QLR %4$;; ## #!& #2 #

/ # 1#2 : ### &#2 / 0 0 123 0

QR " .)"B #2 ,# #!& ! ## , !

1 # #2 3 0 )% 123 0 "

QKR C ; .$ %## / ## #2 # #

#, # # $ .)#% &? )?

QPR - 4 '-' 0# ## # #

#2 # / 0 0 123 0 < "

" && # 0# * ## !

1 # ## & # # / #

# ## # 1 # ##1 8,#

&# 0 E 1 0 #1 # # 0#3 ,

# 3## 0 * 31### #

/ 3# 8## 0 &1# # /

) # &# ##1&#2 0 #:

&#2 ## # 1 # ##

& , %!# # # #

, # / # &* # / 3# %

&* 8# ## F31### F # # # η

8# 3# 1 &#

3# # 3 η = εεref

8 # #2

0 2 3 # # #8

# # 31### 3 * #

" &# 0# / &# &* #

31### 0 ,# #2 " # 3!

1### 8# ### #&1* 3# ε

3# 3 εref "0 #2 0 # / 3#

3 " 20 < 8# < #

### # 20 # < # 1 0

/ ## ##2 ## " ### 3# #

# ,# &# # 3# #: #, $

2 &* 31### ## 0

# # 0# 1# 3 # %

## 3# # # ## # 0# 2#

# S 0 # ### 3 # / &2

3# # / ### #&1* 3# η = εεref

0* # :

0 3 1#2 ## # ## 31###

$ 1 20# 8# #* 0=1 Si η = 1, #2

Si η > 1, ## #2

" # η # * 31### 8#

η = f(r, a, σ, εref ) avec σ = σ(p, d)

" ##* &# 0 3# 1#2 !

## * 31### # 0#

&1 " < * 31### #

0 & #!& !# 0# #

& #2 & ' 3# & 0&!

1## #1 0### # ## )? 8

' # # #2 $ #

3### 0 & ) ### 1 !

# & " # ## # 0

# " # / / # 8 0# 1

" * 31### # # E / ##

& %0 2# 2#* &# &*

&* 31### #!/!# # &1 #G

< ## 3# 1#2 * η M # ##

* B; N # # # M/ 0& N /

# 3# 3# 1#2 * η

#5# # / # &1 "0#G # 3#

# 1 ,# #2 3# 0

# # / # 3# " # /

&# < ##2 2 0 # #2 #2

0 3 #2 ## / 01# # &1

0 # # / # 3# $ 8

Si σ = σ(εref ), alors η = f(r, a, σ, εref ) ∀εref

Si σ = σ(εref , εref ), alors

η = f(r, a, σ, εref ) si η = 1

η = f(r, a, σ, εref , εref ) si η > 1

# , # # 0# /

3## 0 * 31### 0 3#

1#2 ## # 0# &1 #

, < ## #

# # # # # 0& 3# 1#2

* η # #2 ) 0

2 31## # 0# * η # # < # 20/

* ## # " # 0 1 #

/ #1 %#!# # < ## # # 1+ / 0###

0 3# # 1#

' 3# # 01 : # # !

* # / 1# 8 # #3 "0##

8 #8# 1#2 ( 1# ###

3# 20# # # #

8 # # 0# * 31### ,

&& # 0# 8## 0 3# &#2 T !

#2 * η < # !# < #, 3 0

2 31## 0 # / 8# ,# &*

# 0 #1# ## ### #2

8 * 31### # / ## 1 #

# # 0# 1 * # % 1

* : #2 3# 3# #

# # F&#F ) , # # 1 # 0## M #N

0#G 3# #2 # # 2

#1 & #!&

% && #2 1 # $ # # 0!

# 1## * η & #!&

0 1 , 2 # #2

3## ,1

) && # 2 # &1* #

# ' 1## #1 1 # #* 3#

0 # / 0&&* ## #* 0&1#

0## M , 8 3#2 # #N ) & #!

& ,# #, # &1* )

< 3= # # # 0##2 & #!& F#F

F1F ## 8# * 31### / # &

!% # *#

H 6 # 3

## &#2 # 3#

(# ### # # H # # J # L #

8.310−5 0.06 2.23 2.79 3.74 5.63 10.34

8.310−3 0.05 2.70 3.25 4.30 5.40 10.30

0.1 0.01 2.15 3.05 3.90 5.65 10.45

1 −0.15 2.10 3.25 4.45 5.55 10.15

2 −0.35 3 4.15 5.20 6.20 10.65

& H ## :# # #

#1 0##

(# ### # H # # J # L #

8.310−5 2.46 2.92 3.85 5.74 10.49

8.310−3 2.75 3.25 3.80 6 10.80

0.1 2.50 2.90 3.80 5.95 10.45

1 × 2.90 4 5.25 11.05

2 2.40 3.10 3.90 5.65 10.45

& ## :# # 3

#1 0##

( εpoint 0 εpoint 1 εpoint 2 εpoint 3 εpoint 4 εpoint 5 εref

[m/s] [%] [%] [%] [%] [%] [%] [%]

8.3 10−5 0.13 0.32 0.35 0.35 0.3 0.23 0.3

8.3 10−3 −0.04 0.89 0.88 0.45 0.06 0.38 0.27

0.1 0.84 −0.32 −0.16 −0.15 0.13 0.14 0.29

1 −0.004 −0.09 −0.11 −0.05 −0.02 −0.02 −0.004

2 0.04 0.04 0.006 −0.09 −0.06 0.08 0.009

& J )3# / 0 H 3

[m/s] [%] [%] [%] [%] [%] [%] [%]

8.3 10−5 9.15 9.05 9.48 9.63 9.36 8.05 10.27

8.3 10−3 9.26 10.01 10.01 9.44 9.4 8.07 10.81

0.1 11.43 10.29 9.41 9.74 10.15 9.3 11.96

1 9.28 10.25 11.21 9.72 9.36 7.5 11.35

2 10.9 10.81 10.75 9.89 9.66 10.18 11.12

& L )3# / 0 3

[m/s] [%] [%] [%] [%] [%] [%] [%]

8.3 10−5 26.39 21.64 21.85 21.95 21.41 21.37 23.86

8.3 10−3 27.39 21.67 21.3 20.58 21.49 23.05 23.58

0.1 25.73 22.6 22.53 23.31 24.03 20.16 23.42

1 22.58 23.17 24.88 24.35 24.87 23.09 23.81

2 26.2 26.29 25.87 25.25 24.15 21.56 22.23

& )3# / 0 J 3

[m/s] [1/s] [1/s] [1/s] [1/s] [1/s] [1/s] [1/s]

8.3 10−5 0.00051 0.00046 0.00051 0.00056 0.00048 0.0005 0.0005

8.3 10−3 0.015 0.041 0.04 0.027 0.025 0.021 0.034

0.1 1.74 0.8 0.69 0.69 0.7 0.77 0.99

1 − 7.78 9.26 0.0005 − 31.27 8

2 33.3 34.38 36.4 13.3 5.19 23.82 18

& K (# 3# # #2

[m/s] [1/s] [1/s] [1/s] [1/s] [1/s] [1/s] [1/s]

8.3 10−5 0.0033 0.0027 0.0027 0.0027 0.0026 0.0027 0.003

8.3 10−3 0.36 0.27 0.27 0.26 0.28 0.3 0.3

0.1 3.73 3.46 3.48 3.61 3.72 3.07 3.55

1 31.21 32.34 33.57 33.36 34.64 29.92 32.18

2 71.06 71.18 69.75 70.64 68.41 58.88 61.45

& P (# 3# # #2

( εpoint 1 εpoint 2 εpoint 3 εpoint 4 εpoint 5 εref

[m/s] [%] [%] [%] [%] [%] [%]

8.3 10−5 0.41 0.4 0.21 0.33 0.16 0.19

8.3 10−3 0.45 0.16 0.12 0.54 0.005 0.21

0.1 0.38 0.49 0.31 0.41 0.45 0.19

1 × 0.23 0.2 −0.07 0.24 −0.05

2 −0.04 0.08 −0.02 −0.14 −0.08 0.008

& )3# / 0 H 3

[m/s] [%] [%] [%] [%] [%] [%]

8.3 10−5 12.25 8.9 5.31 3.3 1.45 1.81

8.3 10−3 9.73 7.46 5.92 2.76 1.27 1.87

0.1 10.23 7.89 4.76 2.1 0.78 1.58

1 × 8.01 6.05 3.32 0.85 1.99

2 13.01 6.25 3.89 5.46 2.5 1.86

& I )3# / 0 3

[m/s] [%] [%] [%] [%] [%] [%]

8.3 10−5 23.39 17.35 10.28 5.62 2.45 2.96

8.3 10−3 18.57 14.38 11.36 5.92 2.46 3.17

0.1 20.5 15.42 9.21 4.7 1.44 2.72

1 × 10.71 7.85 5.09 0.95 2.46

2 16.58 12.93 8.12 4.97 1.08 3.1

& H )3# / 0 J 3

[m/s] [1/s] [1/s] [1/s] [1/s] [1/s] [1/s]

8.3 10−5 0.00095 0.00077 0.00042 0.00045 0.00019 0.0003

8.3 10−3 0.165 0.121 0.089 0.059 0.021 0.039

0.1 2.81 2.25 1.61 0.73 0.48 0.73

1 × 2.51 4.75 2.38 4.43 0.3

2 − 2.54 − − − 0.3

& HH (# 3# # #2

[m/s] [1/s] [1/s] [1/s] [1/s] [1/s] [1/s]

8.3 10−5 0.0125 0.0091 0.0055 0.0028 0.0012 0.0014

8.3 10−3 1.34 1.05 0.84 0.41 0.18 0.204

0.1 23.2 17.26 9.92 5.2 1.21 2.56

1 × 141.6 102.3 66.73 10.44 32.65

2 419.31 321.57 203.97 124.84 27.89 77.27

& H (# 3# # #2

!% # $

.H 6 # 3

. ## &#2 # 3#

(# ### # # H # # J # L #

8.310−5 0.20 2.50 3.30 4 5.55 10.15

1 0.45 2.05 3.65 5.30 × 10

& .H ## :# # #

(# ### # H # # J # L #

8.310−5 2.60 3.35 4.15 5.70 10.30

1 2.85 3.75 5 7.65 10.75

& . ## :# # 3

[m/s] [%] [%] [%] [%] [%] [%] [%]

8.3 10−5 0.49 0.71 0.78 0.79 0.58 0.28 0.47

1 0.14 −0.03 × −0.05 0.31 0.63 0.16

& .J )3# / 0 H 3

[m/s] [%] [%] [%] [%] [%] [%] [%]

8.3 10−5 15.45 15.21 15.2 15.2 14.96 12.44 15.19

1 14.53 15.41 × 13.61 12.11 11.46 12.73

& .L )3# / 0 3

[m/s] [%] [%] [%] [%] [%] [%] [%]

8.3 10−5 34.58 34.58 34.8 34.85 33.85 29.92 30.62

1 31.33 31.09 × 30.81 28.91 29.13 25.46

& . )3# / 0 J 3

[m/s] [%] [%] [%] [%] [%] [%] [%]

8.3 10−5 103.06 99.06 97.22 95.04 87.31 65.57 49.23

1 67.17 68.42 × 67.06 63.9 57.73 38.73

& .K )3# / 0 L 3

[m/s] [1/s] [1/s] [1/s] [1/s] [1/s] [1/s] [1/s]

8.3 10−5 0.0018 0.0017 0.0017 0.0017 0.0018 0.0011 0.0017

1 24.4 25.82 × 10.17 − 31.6 14.55

& .P (# 3# # #2

[m/s] [1/s] [1/s] [1/s] [1/s] [1/s] [1/s] [1/s]

8.3 10−5 0.003 0.003 0.003 0.003 0.0029 0.0026 0.0027

1 37.83 37.36 × 38.82 38.36 34.14 31.36

& . (# 3# # #2

[m/s] [1/s] [1/s] [1/s] [1/s] [1/s] [1/s] [1/s]

8.3 10−5 0.011 0.01 0.0097 0.0086 0.0057 0.011 0.003

1 103.67 100.71 × 97.18 79.45 99.56 43.58

& .I (# 3# & 0#

[m/s] [%] [%] [%] [%] [%] [%]

8.3 10−5 1.12 0.41 0.27 0.31 0.39 0.23

1 0.095 0.38 0.9 1.15 −0.052 0.31

& .H )3# / 0 H 3

[m/s] [%] [%] [%] [%] [%] [%]

8.3 10−5 29.12 18.31 12.94 8.82 5.87 8.15

1 18.44 16.21 13.08 9.41 4.21 6.31

& .HH )3# / 0 3

[m/s] [%] [%] [%] [%] [%] [%]

8.3 10−5 88.15 64.99 48.19 28.67 13.16 15.88

1 30.57 30.92 23.32 18.16 11.9 11.69

& .H )3# / 0 J 3

[m/s] [%] [%] [%] [%] [%] [%]

8.3 10−5 104.57 79.73 60.62 35.68 14.82 17.07

1 43.76 44.32 31.41 21.24 10.53 14.46

& .HJ )3# / 0 L 3

[m/s] [1/s] [1/s] [1/s] [1/s] [1/s] [1/s]

8.3 10−5 0.0063 0.0034 0.0022 0.0018 0.13 0.0016

1 1.05 4.23 10.05 12.81 − 3.45

& .HL (# 3# # #2

[m/s] [1/s] [1/s] [1/s] [1/s] [1/s] [1/s]

8.3 10−5 0.015 0.014 0.0085 0.005 0.002 0.0026

1 70.87 71.02 52.12 39.55 27.78 26.46

& .H (# 3# # #2

[m/s] [1/s] [1/s] [1/s] [1/s] [1/s] [1/s]

8.3 10−5 0.041 0.036 0.031 0.0174 0.0041 0.003

1 164.88 167.48 101.1 38.46 − 34.63

& .HK (# 3# & 0# !

Modélisation multi-échelle d un assemblage riveté aéronautique ...

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