Modélisation multi-échelle d un assemblage riveté aéronautique ...
Transcript of Modélisation multi-échelle d un assemblage riveté aéronautique ...
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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, && # # 3#  ##8
: 2 0 3# 0# ##*
0 2 3 # / &1 ## # #,# M81 HJN
HJ %81# #
K
/ ! !) (( ! !) # '( 7-8
" #2 ##, %# 2#8 # 0##
## # & # 3# # /
0 3 0 1#2 % # 0,# 3 0 5#
# 3 %#!# # # MHKN #
# 3# ,# # # 3# F/
0#8#F 0!/!# ,# / V 0#G 3# M N 3#
#
Kσt =
σmax
σ∞et Kε
t =εmax
ε∞MHKN
) # #2 &1 # #,# ## #
, 5# 1, $ # 3#
1+ , 2 QR 0### 3 ##2 3
MHPN 4@ QJR M 2 1 2 0#8#N
% # 8# #* 3# <
# 2 " 3 QLR & ## #2
& # 3# # #2 2 3
## #8# "0,# & # MN #
MH N " # ### # ##* 1
3 &# M&# 2 ## /
## &1 #N #* ## 
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
MH N
V (r, θ) ##2 # #
P
,# ##8 * 31###
# #2
η =σθ(θ = π
2 )σ∞
=12(2 +
a2
r2+ 3
a4
r4) MHIN
0 5 10 15 20 250
0.5
1
1.5
2
2.5
3
3.5
Distance par rapport au centre de la perforation [mm]
Coe
ffici
ent d
e co
ncen
trat
ion
[.]
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
MHHN
V a 3# Es
ησ =σθ(θ = π
2 )σ∞
=12
(1 +
a2
r2+
Es
E∞s
(1 + 3
a4
r4
))MHHHN
ηε =εθ(θ = π
2 )ε∞
=14
((1 + 3
a2
r2
)E∞
s
Es+ 3 − 4
a2
r2+ 9
a4
r4
)MHHN
0 1 2 3 4 5
x 10−3
0
0.5
1
1.5
Déformation lointaine [.]
Es/
E [.
]
Es/E
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10−3
0
1
2
3
4
5
6
7
8
9
10
Déformation lointaine [.]
Coe
ffici
ent d
e co
ncen
trat
ion
[.]En déformationEn contrainte
H 6# # #2
% #2 0# ,# &
31### 0 2 3 # # &&* ### # 2
## #8# # 0# 3# #8# #2
$ 1 #1 0# # : / 3#
## # #2 3# 3#
< 9 3# # / & ## 2 !# 0 2 0
0.5% % # ## 1#1 0 #
3# M1 N
I
!) "#)&) '( ! 1 ! !) (( !
) 3 QHIR 0 2 2# 2#
3# #1 #1 0#G 0 3#
M # N 0 # 3# 0
% * ##2 0# ## #2 $ #
# / # &* 31### # 0#
, 3# 1 &# "0 &## #
&* 31### ηε 0
# #2 0 # #2 M HN % 8#
* # 31### ' & 2
### 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 #, #
!## / # U # 0###
# ## ### , 2 #
* 0 8# 2# T #
* ηie ηi
p # # #2 3= ##2
J
HK ## 2# &* 31### QHIR
!) (" ( * ! " 1 !)
) 3 QR 0 0# 1 / # 31###
## 1 # # 8 # 1
## ) , 1& QHIR 0 #
3= &* # / 31### , #
* ## "0& ## # , &* "0
# ## 2 # 8
M81 HPN % 0## 0
# 2 2 $ #3
# G,# #* ##1 3# 3
# < ## 8 0# ##
0 # 2 01 # 2 3 ## M 2
#N " ## # 3= #2 $ #
2 ,< 0 ## ## 2
A ## 3# MA 3# #2
3#N
)0 # # 0 ## # 3 ## 2
, % # # #!
# 2 3 # 2 # 2
" ## #
,#, ## 0 &1 #
&1 ## M2 0 # ,# 1
##N &1 FF 0* 3# "
2 0 5 "0###!
JH
# 01 ())! # *
M 2# N # !# # #*
## E 3 # < #1
## ) * 0 U, ) ##
1 # 8 0#1 0### * ##
1 1
HP * 2# 01 # QR
' #2 1 # E 0## 0 !
, #: #1 " &!
, ##, 1 # #
# E ) &# E # # /
## # 3# / # 3#
# / # " # &* <
## &
" ## #!/!# &* ##
8# 0&# # / " # # F#F ##
* & #2 : !
1 % ## # #20 #
0: 1 #2 )  ## #!
#8 # !## / 8,#
0&# & 3= #3# ## E 1 0
#1 # , * " *
5 3# ##1 01 !# / 0#1#
0=1 8
J
) , && #3 31##!
# ## 3# : 3#
E # 0# ## # # / ### #&1* &
# 3# F31### F #1 0
: / 3# #2 #* " !
# η 8# 3# 1 &#
3# # 3 η = εiεref
0# 31###
)0 #  31###
3# ## 0 5# # ) 3# 3
&## / 0#8# #2 2 2 # &1 )# 0 ## #8#
< #1 2#  ## 0
3#
"0#3 8 && # / & 0
8# #82 0 0 2 #2 3 %
/ # #2 M& N 31###
# / 3# E 2 #!# #
* ##2 , 0
2 3 $ 01# 0##8 # #
3# # 1 3# & # 0#
* #82 ) * 31### # / ##
& # # 0# 0#G # 3# 31###
JJ
QHR 4 B ; C %. ) $C9 B & : 3 & #A & #!
& 3#1 1& 3 L!B @ # #1 # #
$ # &" '"
QR '"" ,# # ##1# & 3#1 &# 3
31 # # / 0 0 123 $ 0 * 4
QJR ) )" B ) C"W 4 - X$9"'"" # 3 8#
#1 &1# 3 & #1 3 &@& # # #3 3!
1 " !% ! 9: ,- "
QLR . " - ) ! . W B W %4 '( ' )"B. 3 #!
&# % & 3 # #? #1 ! ## # # #3
* !; < & 4"= " "
QR ; .'4 # #1 3 G#1& &# ! ;##1 # !
. # #
QKR # # # 2 ##
0# % 0 0 >0
+ ="
QPR C %4 "$ B 0#1# < +444 =
Q R $"$B W B ) ) ## # 3 ## 3 #
@ ? ## #3 ! "
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
< =
QHR %.%' ; )"%$O " % " $ " "$B 8# !
# & 3# 3 @ # # * ) <
"= == " "
QHJR . " - ) %.%' # 3# &# 3 !@
1 8# ,# # #, $T$$ * 0 0
< ''''' "
QHLR !4 "$ C B BW ) 1 3# ## 3 !@
# #1 ## * 0 0 < 4444' "
QHR . " - ) )"B. 9$X$%Y ) YB$% %##
# 0 ## # 3$ 0 38 < 4 &" '
=
QHKR . " - ) )"B. 9$X$%Y ) YB$% # # !
#1 3 # # 3 #3 &@&# 7 ) %6 0
< = " "
QHPR . " - ) )"B. 9$X$%Y ) YB$% # !
& 3 3 # 1& 3 # # % 0 6
< & '
QH R . " - ) )"B. 9$X$%Y ) YB$% %&## 3
# 3# 3 # # # : 0 < < & "=
"
QHIR " B""$ %## 0 2 31##
# &* 31### # , 
#1 / 0 0 123 0 < "
QR B$$% " X - " X$ # 3 # #1 ##
# # #1 # $ 0 >123 0>.8*0 &!#%5<
"
QHR ! . W B . " - ) BE 3 31### # , &!
#2 01 #1 # $ .)#% &? '" )
2 " "
QR B # 3 ! %& # 3 # ?#1
1& # 3 &# % @6
QJR . 4WX) )#1#1 &## ! 0 5 0 4"
QLR B$49 B& 3 ## 7 )0 A;5 +: !
QR 4 '. B& 3 # 3 & # ## # @#&
# # # @ % < "= '
QKR %4 ')B C! %'"$ # &# !
# # ### # %3 < ""4
=4
JK
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#
# $ %!% & " 4
JP
# ! &!$
% &# &* 31### # / 0 3#
&* # 2# ' #* # ,#
# 2#!#2 # 3 3
' # ## 0 3# : ' *
31### # % * & #!&
#3 0 0 2 3# !
# 1#2 " #* # &# 0# / ,# ##
/ 8## 0 #* 31###
% < ,## &# ## 1
# 8# # ## *
##8 8,# 2 E 3 ) #
&& 0# 0&# / # 3# / #
# "0#3 8 # / 8# #82 #
0 2 #2 3 %#!# ## 0 &*
# , # 81 H 0## 1 / # &#
E 1 0 #1 #
"0 8# 2 # * !
#3 &* #2 , 3# $ #
0##8 # * #82 ' 3# ##
3&# # & < # 8 0# *
0##8# * #
# % 2 &#
# 1#2 # 1 # 20# 0 #1
/ ## ##2 < !,&# *
% &# 0## && &1#2 0 &!
* # ## # / 3# # 0##8
* & # # / / !
# # 31### 20 8# &#
%!# 0,# #* 3# η 8# 0 3!
# εi / 3# ## ε∞ " #* # &# 0#
0 #2 / 0 3# ## ## 20/ #1#!
8#  #!/!# * & 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# # ) ,  !# ##8 / 0#!
8# #2 1 &1 ) 0
LH
! "
8# 0 # ## 2 3 8# "0##8# 0
3# 3  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
H
80 mm 160 mm
H #1 2 3
L
#$ " #$ $
)## 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 #
LJ
! "
 # 3# εr=a/εref 8# # MN $
# 0 # 0,# 3# 3  0!/!#
3# #8# ε∞ 3# 3# 3# εr=a
ηε =εr=a
ε∞=
E∞s
Es+ 2 MN
V a 3# E∞s Es # 0W1
#
#2 3# 3# #
#3 0 & 3# " 81
# 3# 3 3=  M3# #8#N
#2 M3# 1N " 3# 1 & ,
* 8# ### / 3# 3  20/
0 0.5% M81 N 0!/!# 20/ 2 ## #2 : # #
' 3# # # ## 0 3# 3  0
 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###
&1 #
0 0.05 0.1 0.15 0.2 0.250
0.005
0.01
0.015
0.02
0.025
Déformation au bord de la perforation [.]
Déf
orm
atio
n de
réf
éren
ce [.
]
ThéoriePlaque 80*80Plaque 160*160
)3# 3  3# 1
LL
#$ " #$ $
" ## 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
1
2
3
4x 10
4
Déplacement [mm]
For
ce [N
]
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 3# 1 3# 3 #
A / #8 &1
L
! "
0 5 10 15 20 25 30 35 400
1
2
3
4
5
6x 10
−3
Distance au centre de la plaque [mm]
Déf
orm
atio
n [.]
Déformation globaleAxe verticalChemin de rupture
0 5 10 15 20 25 30 35 40−0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Distance au centre de la plaque [mm]
Déf
orm
atio
n [.]
Déformation globaleAxe verticalChemin de rupture
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
0.01
0.02
0.03
Déplacement [mm]
Déf
orm
atio
n [.]
0 0.4 0.8 1.2 1.60
1.5
3
4.5x 10
4
For
ce [N
]
Déformation globale26mm30mm34mmForce
!
0 0.4 0.8 1.2 1.60
0.01
0.02
0.03
Déplacement [mm]
Déf
orm
atio
n [.]
0 0.4 0.8 1.2 1.60
1.5
3
4.5x 10
4
For
ce [N
]
Déformation globale10mm14mm18mmForce
"
6# 3# 1 3#
8 # 0 2 160 × 160 mm2 # "
## 0 3# 3 #!/!# ##
2 M : 0&N ## < #
LK
#$ " #$ $
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
0.5
1
1.5
2
2.5
3x 10
−3
Distance au centre de la plaque [mm]
Déf
orm
atio
n [.]
Déformation globaleAxe verticalChemin de rupture
0 10 20 30 40 50 60 70 80−0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Distance au centre de la plaque [mm]
Déf
orm
atio
n [.]
Déformation globaleAxe verticalChemin de rupture
K )### 3# # #2 #2
0 10 20 30 40 50 60 70 80−0.01
−0.005
0
0.005
0.01
0.015
0.02
Distance au centre de la plaque [mm]
Déf
orm
atio
n [.]
Plaque 80*80Plaque 160*160
"
0 10 20 30 40 50 60 70 800
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Distance au centre de la plaque [mm]
Déf
orm
atio
n [.]
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
LP
! "
" #
) # && / & 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  0 # #
0 3 3# # #1 0##
" 3   # 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 ##
QHR
ux(x, y) = ax · x + bx · y + cx · xy + dx
uy(x, y) = ay · x + by · y + cy · xy + dy
MJN
" # # , #3 ###
3 1# # 0 8# 0#1
### " 1# 3 1+ / 0### 0 * #
, A Zi Zf M81 IN % * #
3 #
C =∫
∆S [f(x, y) · f∗(x∗, y∗)]2 · dxdy MLN
L
% & $
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
LI
! "
"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
#  M81 HHN # # &2 81# 8 0!
### " &>
7 0 3 #A#2 ### 9#
7 0 9
7 0 (?& # 0#1
7 0 1 3# 1 1 (#&
" & 3# 1## # ## # -
" &# # ## # & # # ' 1
# 3#
% & $
"
&
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
∑
i
Wi =N∑
i=0
12(δi+1 − δi)(F (δi+1) + F (δi)) MN
H %&#, 0#
H
! "
!"#$
" 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#
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
" :! 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
5000
10000
15000
Déplacement [mm]
For
ce [N
]
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##
J
! "
)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
5
10
15
20
25
30
Déplacement [mm]
Déf
orm
atio
n [%
]
Point 0Point 1Point 2Point 3Point 4Point 5
'
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80
5
10
15
20
25
30
Déplacement [mm]
Déf
orm
atio
n [%
]Point 1Point 2Point 3Point 4Point 5
'
HL # 3# #: # &#
! $
H 0 3
L
% & $
! $
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
5
10
15
20
25
30
35
Déplacement [mm]
Déf
orm
atio
n [%
]
Essai01 (2.26)Essai02 (2.23)Essai03 (2.22)
( )
0 1 2 3 4 5 6 70
5
10
15
20
25
30
35
Déplacement [mm]
Déf
orm
atio
n [%
]
Essai01 (2.74)Essai02 (2.79)Essai03 (2.68)
( *
0 1 2 3 4 5 6 70
5
10
15
20
25
30
35
Déplacement [mm]
Déf
orm
atio
n [%
]
Essai01 (10.34)Essai02 (10.34)Essai03 (10.24)
( +
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
5
10
15
20
25
30
Déplacement [mm]
Déf
orm
atio
n [%
]
Essai01 (2.23)Essai02 (2.55)Essai03 (2.46)
( )
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
5
10
15
20
25
30
Déplacement [mm]
Déf
orm
atio
n [%
]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
5
10
15
20
25
30
Déplacement [mm]
Déf
orm
atio
n [%
]
Essai01 (10.28)Essai02 (10.53)Essai03 (10.49)
( +
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
5
10
15
Distance au centre de l’éprouvette [mm]
Déf
orm
atio
n [%
]
Essai01Essai02Essai03
'
0 2 4 6 8 10 120
5
10
15
Distance au centre de l’éprouvette [mm]
Déf
orm
atio
n [%
]
Essai01Essai02Essai03
'
HI )3# 1 &# ##
K
% & $
0 2 4 6 8 10 120
5
10
15
20
25
30
Distance au centre de l’éprouvette [mm]
Déf
orm
atio
n [%
]Essai01Essai02Essai03
'
0 2 4 6 8 10 120
5
10
15
20
25
30
Distance au centre de l’éprouvette [mm]
Déf
orm
atio
n [%
]
Essai01Essai02Essai03
'
)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
0##
6 3 6 3
Fmax [N ] 9900 8780
δmax [mm] 15.95 7.97
Wmax [J ] 138 51
& K ,#, #
P
! "
0 2 4 6 8 10 12 14 160
2000
4000
6000
8000
10000
12000
Déplacement [mm]
For
ce [N
]
Eprouvette perforéeEprouvette non perforée
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/
" 81 0# 3# #: # &# !
## :# # # P
3 3# ##2 0 # 0!
# 20/ 10 mm # # 3#
0 # # 3# &1*
0##8 2 ## 0=1 # 0&# M81 JN
3 ### / #
#1 0## " 3# > * &!
1 #, 3# #: 1 &# M81 (b)N %
#, 3# 0 2 # # & %
# 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
50
100
150
200
Déplacement [mm]
Déf
orm
atio
n [%
]
Point 0Point 1Point 2Point 3Point 4Point 5
'
0 1 2 3 4 50
50
100
150
200
Déplacement [mm]
Déf
orm
atio
n [%
]
Point 1Point 2Point 3Point 4Point 5
'
# 3# #: # 1 &#
! $
J 0 3 #
I
! "
! $
L 0 3 #
" 81 &* #, 3# , #:
# 0## ## 0: ,# & 0# M 4N
) , # # / # 3# 0!
# !# > &1 3# &1*
0 20/ 0: ,# # # 0
# / 0&# M 40% ,# 3#
0 81 (a)N 0 3 #
* &1 M81 (b)N
0 2 4 6 8 10 120
20
40
60
80
100
120
Distance au centre de l’éprouvette [mm]
Déf
orm
atio
n [%
]
domaine élastiquedomaine plastiqueeffort maximaladoucissement
'
0 2 4 6 8 10 120
20
40
60
80
100
120
Distance au centre de l’éprouvette [mm]
Déf
orm
atio
n [%
]
domaine élastiquedomaine plastiqueeffort maximaladoucissement
'
)### 3# 1 &#
K
% & $
( ')*
), ##8 ) 0 3 0!
# 3# 3# 3= &1* 0 # >
# 3# M# 0 3 1#2 ! 8N /
0&# ) 0 3 # 3#
* &1 M 3 1#2 ! # 3# !
* N %!# ## # 0 2 ,
0&# # # / # 3# # / 0#!
# 3# η &1 )0 # 1 3#
# 01# ## 01 / % #
3 ## #8# & 3# 2# #&1* *
### #2 0 3
" # 3# 3 3!
&2 # 1 &# 2 0#G 3#
# ##1 " , #1, ## * &1 M8!
1 KMaN PMaNN 3 / 2 0 0#1 3#
, #1, < 3 # # # &1!
M81 KMbN PMbNN # #1 3# M81 KMcN
PMcNN 3# ##2 0# 0!/!# 2
&1 # #2 #2 2 0 # 3 " 3#
1# 3#
0 1 2 3 4 5 6 70
5
10
15
20
25
Déplacement [mm]
Déf
orm
atio
n [%
]
Eprouvette perforéeEprouvette non perforée
( )
0 1 2 3 4 5 6 70
5
10
15
20
25
Déplacement [mm]
Déf
orm
atio
n [%
]
Eprouvette perforéeEprouvette non perforée
( ,
0 1 2 3 4 5 6 70
5
10
15
20
25
Déplacement [mm]
Déf
orm
atio
n [%
]
Eprouvette perforéeEprouvette non perforée
( +
K %# 3# 1 &#
3 3 #1 0##
KH
! "
0 2 4 6 8 10 12 14 160
20
40
60
80
100
120
140
160
180
200
Déplacement [mm]
Déf
orm
atio
n [%
]
Eprouvette perforéeEprouvette non perforée
( )
0 2 4 6 8 10 12 14 160
20
40
60
80
100
120
140
160
180
200
Déplacement [mm]D
éfor
mat
ion
[%]
Eprouvette perforéeEprouvette non perforée
( ,
0 2 4 6 8 10 12 14 160
20
40
60
80
100
120
140
160
180
200
Déplacement [mm]
Déf
orm
atio
n [%
]
Eprouvette perforéeEprouvette non perforée
( +
P %# 3# 1 &#
3 3 #
"0,## ### 3# #
0## ## / 0: ,# ) # #2
0 3 # #5# ,# ' #
3# !# / 3# # # M&!
#1 #5 # # #2N # 0
## 0 ### 3# 1 &#
# 2 3# 1# FF # 1
F##8F 3# 3# M81 N " 3!
# : #8 ### 3# 1 1 #
/ 0 3
0 2 4 6 8 10 120
0.5
1
1.5
Distance au centre de l’éprouvette [mm]
Déf
orm
atio
n [%
]
Eprouvette perforéeEprouvette non perforée
0 2 4 6 8 10 120
0.5
1
1.5
Distance au centre de l’éprouvette [mm]
Déf
orm
atio
n [%
]
Eprouvette perforéeEprouvette non perforée
)### 3# 3 3
/ 0 1 M##N
K
% & $
) # #2 / 0: ,# 0 * # #:
# # ) 0#1 0## #,
3# 0 3 / ,
3 M3# 0 12% 24% # #2 / 0:
,# # 81 I(a) J(a)N 2 3# #8
### 3# # 1 # 3#
1# FF 3# F##8F
) 0## ## * 0## 0: ,# "
3# # & 0&# 0* < 0 #
0 2 4 6 8 10 120
5
10
15
Distance au centre de l’éprouvette [mm]
Déf
orm
atio
n [%
]
Eprouvette perforéeEprouvette non perforée
0 2 4 6 8 10 120
5
10
15
20
25
30
Distance au centre de l’éprouvette [mm]
Déf
orm
atio
n [%
]
Eprouvette perforéeEprouvette non perforée
I )### 3# 3 3
/ 0 2 M##N
0 2 4 6 8 10 120
5
10
15
20
25
30
Distance au centre de l’éprouvette [mm]
Déf
orm
atio
n [%
]
Eprouvette perforéeEprouvette non perforée
0 2 4 6 8 10 120
25
50
75
100
Distance au centre de l’éprouvette [mm]
Déf
orm
atio
n [%
]
Eprouvette perforéeEprouvette non perforée
J )### 3# 3 3
/ 0 3 M: ,#N
KJ
! "
) 0# : 3# # # #2 /
0: ,# # 2 0##8# S 2 # #
3# # 3# M: FFN #
1 M: F##8FN M81 I(b) J(b)N
"0 0# 2 : 0##8# !
3# # / 3# 0# & 0#
"0# # 3# * 0: ,# 0: !
# 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
20
40
60
80
100
120
Distance au centre de l’éprouvette [mm]
Déf
orm
atio
n [%
]
Eprouvette perforéeEprouvette non perforée
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#
KL
% & $
,# # 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
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Distance au centre de l’éprouvette [mm]
Coe
ffici
ent d
e co
ncen
trat
ion
en d
éfor
mat
ion
[.] domaine élastiquedomaine plastiqueavant rupture
0 2 4 6 8 10 120
0.5
1
1.5
2
2.5
3
Distance au centre de l’éprouvette [mm]
Coe
ffici
ent d
e co
ncen
trat
ion
en d
éfor
mat
ion
[.] domaine élastiquedomaine plastiqueeffort maximaladoucissement
J 6# η 3
K
! "
0 2 4 6 8 10 120
1
2
3
4
5
6
7
8
9
10
Distance au centre de l’éprouvette [mm]
Coe
ffici
ent d
e co
ncen
trat
ion
en d
éfor
mat
ion
[.] domaine élastiquedomaine plastiqueavant rupture
0 2 4 6 8 10 120
1
2
3
4
5
6
7
8
9
10
Distance au centre de l’éprouvette [mm]
Coe
ffici
ent d
e co
ncen
trat
ion
en d
éfor
mat
ion
[.] domaine élastiquedomaine plastiqueeffort maximaladoucissement
JJ 6# η 3
0 2 4 6 8 10 120
1
2
3
4
5
6
7
8
9
10
Distance au centre de l’éprouvette [mm]
Coe
ffici
ent d
e co
ncen
trat
ion
en d
éfor
mat
ion
[.] AcierAlliage d’aluminium
0 2 4 6 8 10 120
1
2
3
4
5
6
7
8
9
10
Distance au centre de l’éprouvette [mm]
Coe
ffici
ent d
e co
ncen
trat
ion
en d
éfor
mat
ion
[.] 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#
KK
' ( $ η
## # FF & 0# M1N
/ 0&# " # &# ,# ##
&*
*
# 3# η 8# 3# ε / 3# #!
# #2 εref M ε∞ N [ # 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#
η
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###
KP
! "
+ η + # , -./
%# # Ω # ### S0 2 3 1#2
3 A(t) # "02##  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
EK2
I MHN
KI 3
KI = ασ∞√
πa MHHN
α 3 ##2 3 8 M#N ###
" & # ##1 # 8 0,#
# 3# KI 3 QHR
⎡⎢⎢⎣
σr
σθ
τrθ
⎤⎥⎥⎦ =
KI
4√
2πr
⎡⎢⎢⎣
5 cos θ2 − cos 3θ
2
3 cos θ2 + cos 3θ
2
sin θ2 + sin 3θ
2
⎤⎥⎥⎦ MHN
2 20 # 8 & # 3#
8## #8# Mr → 0N
K
' ( $ 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
√a
2ret ηε(r, a) =
εθ
ε∞= α
1 − ν
1 + ν
√a
2rMHLN
0V
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
E
1 − ν
1 + ν(ηε ε∞)2
MHKN
"01# # # 0 3 ## 3#
η(r, a) r σ∞ ε∞
0 η
" #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##
 ### MPN / 2 1# ## ##
# #8# Wplas 1 Wendo &1 & Wtemp MHPN
KI
! "
dWext = dWelas + dWcin + dWS + dWplas + dWendo + dWtemp MHPN
" #  0! 3# ε T ,2
# 3  σ s " #  !#
3# #2 εp # * 3 αj
M01 #: d ,N ,2 # 3 
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##  #8 / 0& # / 0&
#* ## 0 ## εref M] HHN
/ ε " 3# η 2 8# MKN #  &
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
dWi
dΩ∗ =dWiΩ∗
dΩ∗ +dWiS
dΩ∗ dWi
dS=
dWiΩ∗
dS+
dWiS
dSMHN
" ## 0 20#2 # ##8 #
H # F#F dWiΩ∗/dS = 0 dWiS/dΩ∗ = 0 M ##
P
' ( $ η
dS 01# ## Ω∗ #N
dWi
dΩ∗ =dWiΩ∗
dΩ∗ dWi
dS=
dWiS
dSMN
$ # 0,# 1#2 1# ## ## #8# !
1
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
+(
dWiS
dSdS
)t≤tA
+(
dWiS
dSdS
)t>tA
MN
dWi =(
dWiΩ∗
dΩ∗ dΩ∗)
η=1
+(
dWiΩ∗
dΩ∗ dΩ∗)
η>1
+(
dWiS
dSdS
)η=1
+(
dWiS
dSdS
)η>1
MKN
# MP!IN # 02# MKN &2 ### 1#2 #!
#
dWplas =(
dWplasΩ∗
dΩ∗ dΩ∗)
η=1
+(
dWplasΩ∗
dΩ∗ dΩ∗)
η>1
+(
dWplasS
dSdS
)η=1
+(
dWplasS
dSdS
)η>1
MPN
PH
! "
dWendo =(
dWendoΩ∗
dΩ∗ dΩ∗)
η=1
+(
dWendoΩ∗
dΩ∗ dΩ∗)
η>1
+(
dWendoS
dSdS
)η=1
+(
dWendoS
dSdS
)η>1
M N
dWS =(
dWSΩ∗
dΩ∗ dΩ∗)
η=1
+(
dWSΩ∗
dΩ∗ dΩ∗)
η>1
+(
dWSS
dSdS
)η=1
+(
dWSS
dSdS
)η>1
MIN
&&* 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∗)
η=1
+(
dWplasΩ∗
dΩ∗ dΩ∗)
η>1
MJN
dWendo =(
dWendoΩ∗
dΩ∗ dΩ∗)
η=1
+(
dWendoΩ∗
dΩ∗ dΩ∗)
η>1
MJHN
dWS =(
dWSS
dSdS
)η>1
MJN
− " (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###
P
' ( $ 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
#2
& &## 0 # Ω
J "# 0#  * 31###
PJ
! "
"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
01 #:
) 0 1 #: 2 # &1* Mη = 1N
20/ 0## 0 1 0 #8 " (
dWiΩ∗dΩ∗ dΩ∗
)η=1
# MJN MJHN / 01# #8# 01
#: #
3# 01 # Ω #8
1 > ## " & 3# #
#&1* 20/ 0## 0 8 ) 3!
# 01 & 31### 2 η > 1 "01#
PL
' ( $ 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∗)
η>1
MJJN
dWendo =(
dWendoΩ∗
dΩ∗ dΩ∗)
η>1
MJLN
dWS =(
dWSS
dSdS
)η>1
MJN
P
! "
) ## # 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
#  # ## # 0# " ##1&#
M%&# $N 0,# 0 # 0#  # !#2
# 0,# #2 3# ## # 3# 
ε∞ / 0#8# # #2 M%&# $$N " &#
0# # 0,# 0 # 0#
&# $$$ # #2 2# # # *
# # 0#
PK
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 )''"$ ,#  # 0#1
!$ * C 30 $ $ >0 +B ,7-
2 "
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 #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
εref
]
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!
H
) ( $
1### # 0 & #!& 2#
# & 3# # / & M## 01
#N / & #2 M## N )#:  #2
0 ) & #!& !
## 3# η = εεref
,## %!# 0 #
# V #2 # # !##
# ;# &# 2 ### :
& #!& # 2 !# # #5
# ## # ##
*)+ +
)#:  #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 #
# / ##  & 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# #
MJJN
φ(x) =∑
I
φINI(x) MJHN
φI = minxc∈Ωc‖x − xc‖ − rc MJN
V Ωc # / 3# xc rc #
3#
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#* ## #
 # ## 0# QLR "0,## 0#
3 1#2 MJLN ) ,# 1 # ## aJ #
3#* M81 JN " 3# F # <
&## #3 #,  * ) 0
3# 3# # φ(x) < ## MJN
Uh(x) =∑
I
UINI(x) +∑J
aJNJ(x)F (x) MJLN
V I 0 #1 J 0 , #3 / ##!
#
F (x) =
∣∣∣∣∣∑
I
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
) ( $
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### 0 1 # "
1# 3# / 0### 3# 3 &#
# / 0# E "1 ) 0 ###
3# 3 &# 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
L
*
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
2
∫ ξ
−1Pi−1(t)dt avec
P0(x) = 1
Pn(x) = 12nn!
dn
dxn
[(x2 − 1)n
], n = 1, 2, 3, . . .
" F&#F ## 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 Q R
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 # # #
&!  * 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 %
## #  &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# / # ( " # #: #!*
K
*
) 5# 9(& &)! " )!1 :9;
" #* # / 8# # #3 M(N %#!#
 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 <
L
Lw
MJ N
V d 1 ##2 &1# M1 N l 1
##2 ( L ## ##2 0!/!# 0&
#2 Lw 1 0 ###
JL &## #: &
P
) ( $
!) "!! 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 #
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
##2
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)
⟩V
MJHHN
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 # ,# % , & 
1 0# / # #2 , 1 #2 %
# 3# ,# ## ## ¯A(x) ¯B(x)
## # !##
I
) ( $
&)!) )( !!)
" ¯A(x) ¯B(x) 1# # !
#3 &1# ## 2 ##2 #2 #: #!
$ # #8 # MJHJN MJHLN %!# 0##
## #2 , # ## ## 0 ##8#
# 0&1##
⟨¯A(x)
⟩V
= ¯I MJHJN
⟨¯B(x)
⟩V
= ¯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##
&1#
" #: 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
I
*
## #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#
IH
) ( $
" # # 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# #
MJHIN
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 #
I
*
/ 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 # # &  *
QJPR # ## & ,  * 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
81 JK
IJ
) ( $
JK & 2 ! & #1
) ,* - (, ./
" # # # 3# ##
3# 2 # # # #2
2 # # # 0## ## ;
## 1#2 8 &
#:  #2 # #
##2 * #2 # /
## #!& % # #:
& ## ## 0 U ?1 &###3
% & ### # / 0 #, &1* # #
# 2 0## F##2F
# 1 && %0 2# #
&& / #2 & / ##,
## 3!# % & #
## ## &1* / ## 3 #
# 0#3# & , * ## # # ,#
 * #!& 31### εεref
" & B; , )? QJ JJR $ 3# !
1# 0# # ## # !## "
#8# & # 0# 0 ### #2 3!
IL
+*$ ) , -+ ./
# M# #N #2 #* 3#
 #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
! Vr
εr = ¯Ar : E +n∑
s=1
¯Drs :(εps + εth
s
)MJN
σr = ¯Br : Σ −n∑
s=1
¯Frs : ¯Ls :(εps + εth
s
)MJHN
) 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 <
# MJN
¯Frs = ¯Lr
[δrs : ¯I − cs
¯Ar : ¯Bts − ¯Drs
]: ¯L−1
s MJN
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
I
) ( $
∑r
cr¯Drs = 0 MJJN
∑r
cr¯Frs = 0 MJLN
∑r
¯Dsr = ¯I − ¯As MJN
∑r
¯Fsr = ¯I − ¯Bs MJKN
∑r
¯Dsr : ¯L−1r = 0 MJPN
∑r
¯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# ##
IK
$
0 # % & < #2 / ( 1
! B3# 2 & 3# 20
& : 3# # 1# 0 / 3#
# #20 3# ## # # #: # #
& #, # / # #2 QHL JLR 1#2
QJ R
εr = ¯Ar : E +n∑
s=1
¯Drs :(εins + εgl
s
)MJJHN
σr = ¯Br : Σ −n∑
s=1
¯Frs : ¯L0s :
(εins + εgl
s
)MJJN
+ +
+
' & #!& & #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 # FF
&* # 2 &#
IP
) ( $
d(perforation)
4mm<
l(V ER)
20mm<
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〉 =∑
r
crσr MJJLN
& #2 #2 2 3
# / # 2# / 0& #2 % &
&# 81 JP
JP ## & #!&
I
$
+ -./
" & #!& #! 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 &#
$N
x
y
z
J ( # #3
) 0## ## #
& #!& # & # #* #
0 3# #2 E && / # #2
Σ 1+ , # 1 ! 3#
εs 0 & ! 1+ / # ##
MJJN # 3# " # σs # / 0# #
MJJKN $ 01# # #2 # 4? " #
II
) ( $
#2 Σ # # ## # 0&1## MJJPN
,# #
εs = ¯As : E MJJN
σs = ¯Ls : εs MJJKN
V ¯Ls # #1## ! s
Σ =∑
s
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
1
2
3
4
5
6
x 10−3
Distance p/r au centre de la perforation [mm]
Déf
orm
atio
n [.]
Déformation macroscopiqueDéformation locale
0 1 2 3 4 5 6 7 8 9 100
50
100
150
200
250
300
350
400
450
500
550
Distance p/r au centre de la perforation [mm]
Con
trai
nte
[MP
a]
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 $$N
H
$
# / # 0, 81 JHMbN %#!# # # 20 ###
# * 0 3#!# 2 1.7 20# # 3
#
0 1 2 3 4 5 6 7 8 9 100
0.5
1
1.5
2
2.5
3
3.5
Distance par rapport au centre de la perforation [mm]
Coe
ffici
ent d
e co
ncen
trat
ion
des
defo
rmat
ions
[.]
ThéorieHomogénéisation
#! $
0 2 4 6 8 10 120
0.5
1
1.5
2
2.5
Distance par rapport au centre de la perforation [mm]C
oeffi
cien
t de
conc
entr
atio
n de
s de
form
atio
ns [.
]
#! $$
JH * 31### εεref
## # & #!&
# #2
" 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
0.5
1
1.5
2
2.5
3
3.5
Distance par rapport au centre de la perforation [mm]
Coe
ffici
ent d
e co
ncen
trat
ion
des
defo
rmat
ions
[.]
ThéorieE=1.E−03E=2.E−03E=3.E−03
JHH 6# * η 0 &1 #
HH
) ( $
" , # 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
0.5
1
1.5
2
2.5
3
3.5
Distance par rapport au centre de la perforation [mm]
Coe
ffici
ent d
e co
ncen
trat
ion
des
defo
rmat
ions
[.]
ThéorieAluminiumAcier
(
0 1 2 3 4 5 6 7 8 9 100
0.5
1
1.5
2
2.5
3
3.5
Distance par rapport au centre de la perforation [mm]
Coe
ffici
ent d
e co
ncen
trat
ion
des
defo
rmat
ions
[.]
Eprouvette perforéeEprouvette non perforée
(
JH ( # / &1 #
3&,
% < # #: !
### # #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 #
H
$
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
*  &# 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
HJ
$
0 1 2 3 4 5 6 7 8 9 100
0.5
1
1.5
2
2.5
3
3.5
Distance p/r au centre de la perforation [mm]
Coe
ffici
ent d
e co
ncen
trat
ion
des
defo
rmat
ions
[.]
TheorieHomogeneisation
./ 10 mm
0 1 2 3 4 5 6 7 8 9 100
0.5
1
1.5
2
2.5
3
3.5
Distance p/r au centre de la perforation [mm]
Coe
ffici
ent d
e co
ncen
trat
ion
des
defo
rmat
ions
[.]
TheorieHomogeneisation
./ 20 mm
0 1 2 3 4 5 6 7 8 9 100
0.5
1
1.5
2
2.5
3
3.5
Distance p/r au centre de la perforation [mm]
Coe
ffici
ent d
e co
ncen
trat
ion
des
defo
rmat
ions
[.]
TheorieHomogeneisation
./ 40 mm
0 1 2 3 4 5 6 7 8 9 100
0.5
1
1.5
2
2.5
3
3.5
Distance p/r au centre de la perforation [mm]
Coe
ffici
ent d
e co
ncen
trat
ion
des
defo
rmat
ions
[.]
TheorieHomogeneisation
./ 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
J
" & 3# & ! ' 5#
# 3# # #: # F3F &#
3# 1 # 3# 3 " 81 JHK
5# #: #1 %,!# / &#
H
) ( $
./ 1
x
y
z
./ 2
./ 3
JH )#: #1 ( 1 20 mm
\ ( 0
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#
HK
0 $ ./
0 1 2 3 4 5 6 7 8 9 100
0.5
1
1.5
2
2.5
3
3.5
Distance p/r au centre de la perforation [mm]
Coe
ffici
ent d
e co
ncen
trat
ion
des
défo
rmat
ions
[.]
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∑
r=1
Dsr : εpr MJJ N
σs = Bs : Σ −n∑
r=1
Fsr : Lr : εpr MJJIN
HP
) ( $
)## 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 ## # & #!& ##
H
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∑
r=1
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∑
r=1
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## # # ! / #!
HI
) ( $
1 # ## #* ## # ' ##
 # < 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
p =
√23
˙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
HH
0 $ ./
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10−3
0
1
2
3
4
5
6
7
8
9
10
Déformation macroscopique [.]
Coe
ffici
ent d
e co
ncen
trat
ion
des
defo
rmat
ions
[.]
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
100
200
300
400
500
600
700
800
900
1000
Déformation plastique cumulée [.]
Con
trai
nte
de V
on M
ises
[MP
a]
Théorier=2.2mm
0 1 2 3 4 5 6
x 10−3
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
Déformation macroscopique [.]
Def
orm
atio
n lo
cale
[.]
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
HHH
) ( $
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010
100
200
300
400
500
600
700
800
900
1000
Déformation plastique cumulée [.]
Con
trai
nte
de V
on M
ises
[MP
a]
Théorier=2.2mm
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010
100
200
300
400
500
600
700
Déformation plastique cumulée [.]
Con
trai
nte
équi
vale
nte
[MP
a]
dt=1E−4dt=1E−5dt=1E−6
# !
J #8# 01#& #
0 1 2 3 4 5 6
x 10−3
0
100
200
300
400
500
600
700
800
900
1000
Déformation plastique cumulée [.]
Con
trai
nte
de V
on M
ises
[MP
a]
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###
HH
1 2-(
< 1+ / & #!&
) , # 1 # #!
# # #2 ' ### #!/!# #
2 / # # 0& ( 1 #
#1 " # 0## ##
( 20 mm 0 2# 2 / 0## /
## #2 # ( &## & #!
& > : 1 ### 1#2 &
, # 3# ## # "
# #2 / # *
!1 & #2 #  "0#!
# 0 # # 0# # #
# 3# #2
" & #!& # # & #2
& < # # , *
31### η ) 0& # 0 ,
& 31### M0!/!# # N
8# η =¯[ε
εref
] 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###
HHJ
) ( $
% * 31### # / ## & 0#!
# # 3# 3# 1#2 * / #1
# # ## # # # 0#
0#G #2 &* 31### ) #2
,# &* # ### #2
&# L
HHL
QHR ` B ."WB%49 O!; 3#* 8# #2
3 0 33 E &A? " "
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"= '
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50 %0 =
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="
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QR 'S'B # 3 # # #8 # &
# #? @#& %0 # ## !# %0 (
3 &" 4'4 4
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0 < = '4
QPR CX 4'B%4$ #!# &# 3 ## #
(0 * #6 6 0 < % & """
HHK
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 < &'
="='" '
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# 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 # !
< ' 4
QJ R %4$;; ## #!& #2 #
/ # 1#2 : ###  / 0 0 123 0
0 6 "
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
30 =4
HHP
' # $ %
!(#
% &# # 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
) &# * 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# #
&1 E
εs = As : E +n∑
r=1
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∑
r=1
Dsr : εvpr MLN
HH
) $
"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
" &# #  # # ###
1 / V (m/s) = 8.3 10−5, 8.3 10−3, 0.1, 1, 2 ) 0 # # 3#  < 8# # ###
/ 1 ### 0 MLN # 1 ### 0
# M 3N # # εref (s−1) = 3.3 10−3, 0.33, 4, 40, 80
εref =δ
L0=
V
L0MLN
H
&
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 ##
##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 #
HJ
) $
L %&#, # 0#
12!34
, ) +
" ### 3# 1 &# " ## :!
# #: # 0# # , LH L #
# 3
(# ### # # 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
0##
S 2 # # ### / 0
##2 # / 1m/s " :!
81 LJ ' M 0 20%N
# " / 8 # !
0 "0 1, &# LJ #,
HL
34"5
(# ### # 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
0##
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
5000
10000
15000
Déplacement [mm]
For
ce [N
]
Eprouvette perforéeEprouvette non perforée
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
H
) $
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
5
10
15
20
25
30
Déplacement [mm]
Déf
orm
atio
n [%
]
Point 0Point 1Point 2Point 3Point 4Point 5
'
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80
5
10
15
20
25
30
Déplacement [mm]
Déf
orm
atio
n [%
]
Point 2Point 3Point 4Point 5
'
LL 6# 3# #: # &#
#1 0## ! V = 1m/s
' '
L #1 0## ! V = 1m/s
HK
34"5
" &* # 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 # #5 #
&* < 8 3# # ### ) #
#2 / 0: ,# M81 LP(b − c)N F0,#F
3# # #, 3# ### , 8!
1# 2 # 3# ### , # #2
#2
0 1 2 3 4 5 6 70
10
20
30
40
50
Déplacement [mm]
Déf
orm
atio
n au
poi
nt 2
[%]
Eprouvette perforéeEprouvette non perforée
( *
0 1 2 3 4 5 6 70
10
20
30
40
50
Déplacement [mm]
Déf
orm
atio
n au
poi
nt 3
[%]
Eprouvette perforéeEprouvette non perforée
( ,
0 1 2 3 4 5 6 70
10
20
30
40
50
Déplacement [mm]
Déf
orm
atio
n au
poi
nt 5
[%]
Eprouvette perforéeEprouvette non perforée
( +
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
0.1
0.2
0.3
0.4
0.5
Distance au centre de l’éprouvette [mm]
Déf
orm
atio
n [%
]
Eprouvette perforéeEprouvette non perforée
0 2 4 6 8 10 120
5
10
15
Distance au centre de l’éprouvette [mm]
Déf
orm
atio
n [%
]
Eprouvette perforéeEprouvette non perforée
0 2 4 6 8 10 120
5
10
15
20
25
30
Distance au centre de l’éprouvette [mm]
Déf
orm
atio
n [%
]
Eprouvette perforéeEprouvette non perforée
- "
LP )### 3# 1 &#
#1 0## ! V = 1m/s
HP
) $
45
" :! # # #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 # #5
# #2 # 3# ,# #
#2 #
# ### # 3# &1* 1
&# M81 L (a)N " # 0.003s−1 #
2# #2 70s−1 # ### 2m/s % &
#  $ # εi/εref ≈ 1 M81 LI(a)N
10−4
10−3
10−2
10−1
100
101
102
10−4
10−3
10−2
10−1
100
101
102
103
Vitesse [m/s]
Vite
sse
de d
éfor
mat
ion
[1/s
]
Point 0 Point 1Point 2Point 3Point 4Point 5
'
10−4
10−3
10−2
10−1
100
101
102
10−4
10−3
10−2
10−1
100
101
102
103
Vitesse [m/s]
Vite
sse
de d
éfor
mat
ion
[1/s
]
Point 1Point 2Point 3Point 4Point 5
'
L (# 3# #2 #1 0##
H
34"5
0 2 4 6 8 10 120
0.5
1
1.5
2
2.5
3
Distance au centre de l’éprouvette [mm]Vite
sse
de d
éfor
mat
ion
loca
le /
Vite
sse
de d
éfor
mat
ion
réfé
renc
e [.]
8.3E−05m/s8.3E−03m/s0.1m/s1m/s2m/s
'
0 2 4 6 8 10 120
1
2
3
4
5
6
7
8
9
10
Distance au centre de l’éprouvette [mm]Vite
sse
de d
éfor
mat
ion
loca
le /
Vite
sse
de d
éfor
mat
ion
réfé
renc
e [.]
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 N 0 3# #
&1 M81 LI(b)N
#1 0## # #
εi
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
HI
) $
8 3# 3 εref 0 1 3# # &1!
0 2 4 6 8 10 120
5
10
15
Distance au centre de l’éprouvette [mm]
Déf
orm
atio
n [%
]
8.3E−05m/s8.3E−03m/s0.1m/s1m/s2m/s
'
0 2 4 6 8 10 120
5
10
15
Distance au centre de l’éprouvette [mm]
Déf
orm
atio
n [%
]
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
5
10
15
20
25
30
Distance au centre de l’éprouvette [mm]
Déf
orm
atio
n [%
]
8.3E−05m/s8.3E−03m/s0.1m/s1m/s2m/s
'
0 2 4 6 8 10 120
5
10
15
20
25
Distance au centre de l’éprouvette [mm]
Déf
orm
atio
n [%
]
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
HJ
& 36%25
0 2 4 6 8 10 120
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Distance au centre de l’éprouvette [mm]
Coe
ffici
ent d
e co
ncen
trat
ion
en d
éfor
mat
ion
[.] 8.3E−05m/s8.3E−03m/s0.1m/s1m/s2m/s
' *
0 2 4 6 8 10 120
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Distance au centre de l’éprouvette [mm]
Coe
ffici
ent d
e co
ncen
trat
ion
en d
éfor
mat
ion
[.] 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
1
2
3
4
5
6
7
8
9
10
Distance au centre de l’éprouvette [mm]
Coe
ffici
ent d
e co
ncen
trat
ion
en d
éfor
mat
ion
[.] 8.3E−05m/s8.3E−03m/s0.1m/s1m/s2m/s
' *
0 2 4 6 8 10 120
1
2
3
4
5
6
7
8
9
10
Distance au centre de l’éprouvette [mm]
Coe
ffici
ent d
e co
ncen
trat
ion
en d
éfor
mat
ion
[.] 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, &#
L
HJH
) $
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
5000
10000
15000
Déplacement [mm]
For
ce [N
]
Eprouvette perforéeEprouvette non perforée
LHL :! 0 # ! V = 1m/s
)0 # 0# & 3# #:
# &# M LKN " #1 3# #
0#1 * # #2 M1 ### 0 #
#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
HJ
& 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
10
20
30
40
50
60
70
80
90
100
Déplacement [mm]
Déf
orm
atio
n [%
]
Point 0Point 1Point 3Point 4Point 5
'
0 0.5 1 1.5 2 2.5 3 3.5 40
10
20
30
40
50
60
Déplacement [mm]
Déf
orm
atio
n [%
]
Point 1Point 2Point 3Point 4Point 5
'
LH 6# 3# #: # &#
# ! V = 1m/s
0 2 4 6 8 10 120
10
20
30
40
50
60
70
80
Distance au centre de l’éprouvette [mm]
Déf
orm
atio
n [%
]
domaine élastiquedomaine plastiqueeffort maximaladoucissement
'
0 2 4 6 8 10 120
5
10
15
20
25
30
35
40
45
50
Distance au centre de l’éprouvette [mm]
Déf
orm
atio
n [%
]
domaine élastiquedomaine plastiqueeffort maximaladoucissement
'
LHK )### 3# 1 &#
# ! V = 1m/s
HJJ
) $
" 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
2
4
6
8
10
12
14
16
18
20
Distance au centre de l’éprouvette [mm]
Déf
orm
atio
n [%
]
Eprouvette perforéeEprouvette non perforée
0 2 4 6 8 10 120
5
10
15
20
25
30
35
40
Distance au centre de l’éprouvette [mm]
Déf
orm
atio
n [%
]
Eprouvette perforéeEprouvette non perforée
- "
0 2 4 6 8 10 120
10
20
30
40
50
60
70
80
Distance au centre de l’éprouvette [mm]
Déf
orm
atio
n [%
]
Eprouvette perforéeEprouvette non perforée
(!
LHP )### 3# 3 3
# ! V = 1m/s
45
" :! #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
HJL
& 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
5000
10000
15000
Déplacement [mm]
For
ce [N
]
8.3E−05m/s1m/s
'
0 1 2 3 4 5 6 7 8 90
5000
10000
15000
Déplacement [mm]
For
ce [N
]
8.3E−05m/s1m/s
'
LH :! #
V = 5mm/min
V = 1m/s
LHI ;#* 3 #
HJ
) $
V = 5mm/min
V = 1m/s
L ;#* 3 #
"0# # 3# #2 81 LH
%!# &1* 0 # ### / #
# ## 3=  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
100
101
10−3
10−2
10−1
100
101
102
103
Vitesse [m/s]
Vite
sse
de d
éfor
mat
ion
[1/s
]
Point 0Point 1Point 3Point 4Point 5
'
10−4
10−3
10−2
10−1
100
101
10−3
10−2
10−1
100
101
102
103
Vitesse [m/s]
Vite
sse
de d
éfor
mat
ion
[1/s
]
Point 1Point 2Point 3Point 4Point 5
'
LH (# 3# #2 #
HJK
& 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,
εi
εref= f(r, a, σ, εref , εref ) avec σ = σ(εref , εref ) MLIN
0 2 4 6 8 10 120
1
2
3
4
5
6
Distance au centre de l’éprouvette [mm]Vite
sse
de d
éfor
mat
ion
loca
le /
Vite
sse
de d
éfor
mat
ion
réfé
renc
e [.]
8.3E−05m/s1m/s
'
0 2 4 6 8 10 120
2
4
6
8
10
12
14
16
18
20
Distance au centre de l’éprouvette [mm]Vite
sse
de d
éfor
mat
ion
loca
le /
Vite
sse
de d
éfor
mat
ion
réfé
renc
e [.]
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
HJP
) $
0 2 4 6 8 10 120
2
4
6
8
10
12
14
16
18
20
Distance au centre de l’éprouvette [mm]
Déf
orm
atio
n [%
]
8.3E−05m/s1m/s
0 2 4 6 8 10 120
5
10
15
20
25
30
35
40
Distance au centre de l’éprouvette [mm]D
éfor
mat
ion
[%]
8.3E−05m/s1m/s
- "
0 2 4 6 8 10 120
10
20
30
40
50
60
70
80
90
100
110
Distance au centre de l’éprouvette [mm]
Déf
orm
atio
n [%
]
8.3E−05m/s1m/s
(!
LJ )### 3# 0 3 #
0 2 4 6 8 10 120
5
10
15
20
25
30
Distance au centre de l’éprouvette [mm]
Déf
orm
atio
n [%
]
8.3E−05m/s1m/s
0 2 4 6 8 10 120
10
20
30
40
50
60
70
80
90
Distance au centre de l’éprouvette [mm]
Déf
orm
atio
n [%
]
8.3E−05m/s1m/s
- "
0 2 4 6 8 10 120
10
20
30
40
50
60
70
80
90
100
110
Distance au centre de l’éprouvette [mm]
Déf
orm
atio
n [%
]
8.3E−05m/s1m/s
(!
LL )### 3# 0 3 #
" * 31### ηi = εiεref
8!
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
HJ
0 2-(
0 2 4 6 8 10 120
0.5
1
1.5
2
2.5
3
Distance au centre de l’éprouvette [mm]
Coe
ffici
ent d
e co
ncen
trat
ion
en d
éfor
mat
ion
[.] 8.3E−05m/s1m/s
0 2 4 6 8 10 120
0.5
1
1.5
2
2.5
3
Distance au centre de l’éprouvette [mm]
Coe
ffici
ent d
e co
ncen
trat
ion
en d
éfor
mat
ion
[.] 8.3E−05m/s1m/s
- "
0 2 4 6 8 10 120
0.5
1
1.5
2
2.5
3
Distance au centre de l’éprouvette [mm]
Coe
ffici
ent d
e co
ncen
trat
ion
en d
éfor
mat
ion
[.] 8.3E−05m/s1m/s
(!
L 6# 31### 3 #
0 2 4 6 8 10 120
1
2
3
4
5
6
7
Distance au centre de l’éprouvette [mm]
Coe
ffici
ent d
e co
ncen
trat
ion
en d
éfor
mat
ion
[.] 8.3E−05m/s1m/s
0 2 4 6 8 10 120
1
2
3
4
5
6
7
Distance au centre de l’éprouvette [mm]
Coe
ffici
ent d
e co
ncen
trat
ion
en d
éfor
mat
ion
[.] 8.3E−05m/s1m/s
- "
0 2 4 6 8 10 120
1
2
3
4
5
6
7
Distance au centre de l’éprouvette [mm]
Coe
ffici
ent d
e co
ncen
trat
ion
en d
éfor
mat
ion
[.] 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 $ #
HJI
) $
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#
7 0# W |dyn > W |sta η|dyn < η|sta7 #2 W |acier > W |alu η|acier < η|alu M&# N
"0#G #2 * 31### / #!
0 # # / # 3# % #G
# # < # 3# 1#2 *
# " 1 #
0## , *
HL
QHR . " - ) %## / ## #2 ,# 0!
1 , # ### #2 / 0 0 123 0
< =
QR B BB$ ## #!& 01 !
# / # #2 / 0 0 F 3 0 (
!3 =
QJR % d 0 #!& #, # / # #2
## / 0 0 F 6$ "
QLR %4$;; ## #!& #2 #
/ # 1#2 : ###  / 0 0 123 0
0 6 "
QR " .)"B #2 ,# #!& ! ## , !
1 # #2 3 0 )% 123 0 "
QKR C ; .$ %## / ## #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# # /
0& *
) # &# ##1 0 #:
 ## # 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
& #2
" * 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
2
HLL
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#  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#
HL
0## M , 8 3#2 # #N ) & #!
& ,# #, # &1* )
< 3= # # # 0##2 & #!& F#F
F1F ## 8# * 31### / # &
01 #
HLK
%8 %
##  # 3#
(# ### # # 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
& H ## :# # #
#1 0##
(# ### # 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
& ## :# # 3
#1 0##
HL
%8 %
( 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
( εpoint 0 εpoint 1 εpoint 2 εpoint 3 εpoint 4 εpoint 5 εref
[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
( εpoint 0 εpoint 1 εpoint 2 εpoint 3 εpoint 4 εpoint 5 εref
[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
HLI
%8 %
( εpoint 0 εpoint 1 εpoint 2 εpoint 3 εpoint 4 εpoint 5 εref
[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
3
( εpoint 0 εpoint 1 εpoint 2 εpoint 3 εpoint 4 εpoint 5 εref
[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
3
( 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
H
%8 %
( εpoint 1 εpoint 2 εpoint 3 εpoint 4 εpoint 5 εref
[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
( εpoint 1 εpoint 2 εpoint 3 εpoint 4 εpoint 5 εref
[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
( εpoint 1 εpoint 2 εpoint 3 εpoint 4 εpoint 5 εref
[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
3
HH
%8 %
( εpoint 1 εpoint 2 εpoint 3 εpoint 4 εpoint 5 εref
[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
3
H
%8 +
. ##  # 3#
(# ### # # H # # J # L #
[m/s] [mm] [mm] [mm] [mm] [mm] [mm]
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 #
[m/s] [mm] [mm] [mm] [mm] [mm]
8.310−5 2.60 3.35 4.15 5.70 10.30
1 2.85 3.75 5 7.65 10.75
& . ## :# # 3
#
( εpoint 0 εpoint 1 εpoint 2 εpoint 3 εpoint 4 εpoint 5 εref
[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
HL
%8 +
( εpoint 0 εpoint 1 εpoint 2 εpoint 3 εpoint 4 εpoint 5 εref
[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
( εpoint 0 εpoint 1 εpoint 2 εpoint 3 εpoint 4 εpoint 5 εref
[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
( εpoint 0 εpoint 1 εpoint 2 εpoint 3 εpoint 4 εpoint 5 εref
[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
( εpoint 0 εpoint 1 εpoint 2 εpoint 3 εpoint 4 εpoint 5 εref
[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
3
H
%8 +
( εpoint 0 εpoint 1 εpoint 2 εpoint 3 εpoint 4 εpoint 5 εref
[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
3
( εpoint 0 εpoint 1 εpoint 2 εpoint 3 εpoint 4 εpoint 5 εref
[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#
3
( εpoint 1 εpoint 2 εpoint 3 εpoint 4 εpoint 5 εref
[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
( εpoint 1 εpoint 2 εpoint 3 εpoint 4 εpoint 5 εref
[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
( εpoint 1 εpoint 2 εpoint 3 εpoint 4 εpoint 5 εref
[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
HK
%8 +
( εpoint 1 εpoint 2 εpoint 3 εpoint 4 εpoint 5 εref
[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
( εpoint 1 εpoint 2 εpoint 3 εpoint 4 εpoint 5 εref
[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
3
( εpoint 1 εpoint 2 εpoint 3 εpoint 4 εpoint 5 εref
[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
3
( εpoint 1 εpoint 2 εpoint 3 εpoint 4 εpoint 5 εref
[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# !
3
HP