HAL Id: hal-01495800
https://hal.archives-ouvertes.fr/hal-01495800
Submitted on 26 Mar 2017
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Public Domain
Identification de la loi de comportement cristallin du Zirconium à partir de la réponse locale et globale d’un
polycristal
Halim Haddadi, Abdellah Salahouelhadj
To cite this version:
Halim Haddadi, Abdellah Salahouelhadj. Identification de la loi de comportement cristallin du Zir- conium à partir de la réponse locale et globale d’un polycristal. 8e Colloque national en calcul des structures, CSMA, May 2007, Giens, France. �hal-01495800�
5HYXH9ROXPH;±Q[DQQpHSDJHVj;
FULVWDOOLQ GX =LUFRQLXP j SDUWLU GH OD UpSRQVHORFDOHHWJOREDOHG¶XQSRO\FULVWDO
++DGGDGLHW$6DODKRXHOKDGM
CNRS-LPMTM (UPR 9001) – Université Paris 13 – Institut Galilée 99, avenue J.-B. Clément, 93430 Villetaneuse, France
haddadi@lpmtm.univ-paris13.fr
RÉSUMÉ. Le but de ce travail est l’identification des paramètres de la loi de comportement cristallin du Zirconium grade 702 en minimisant une fonction coût globale qui représente la somme pondérée de deux fonctions coût. La première fonction représente l’écart entre la courbe σ - ε expérimentale macroscopique et celle provenant d’un calcul EF polycristallin réalisé sur un VER. La deuxième fonction coût représente l’écart entre les champs de déformations expérimentaux mesurés par corrélation d’images de microgrilles, et ceux simulés sur une microstructure réelle caractérisée par EBSD.
ABSTRACT. The aim of this study is the identification of the parameters of the crystalline behavior law of Zirconium by minimizing a global cost function which is the sum of two cost functions. The first function represents the difference between the experimental macroscopic stress-strain curve and the one coming from a crystalline finite element calculation realized on a RVE. The second cost function represents the difference between the experimental strain fields by digital image correlation technique of deformed micro-grids, and those simulated on a real microstructure characterized by EBSD.
MOTS-CLÉS: Identification, Polycristal, Méthode des Eléments Finis, Elastoviscoplasticité.
KEYWORDS: Identification; Polycrystals ; Finite Element Analysis; Elastoviscoplasticity.
,QWURGXFWLRQ
/¶XWLOLVDWLRQGHODSODVWLFLWpFULVWDOOLQHDXQJUDQGLQWpUrWSRXUODSUpGLFWLRQGHOD ORFDOLVDWLRQ GH OD GpIRUPDWLRQ FDU HOOH SUHQG HQ FRPSWH O¶DQLVRWURSLH GX FRPSRUWHPHQWORFDOGXPDWpULDX3OXVLHXUVWUDYDX[VRQWIDLWVGDQVFHVHQV+RFHWDO
>@RQWLGHQWLILpVOHVSDUDPqWUHVG¶XQHORLGHFRPSRUWHPHQWFULVWDOOLQHSRXUO¶DFLHU
$50&2 HQ XWLOLVDQW OD FRXUEH H[SpULPHQWDOH FRQWUDLQWHGpIRUPDWLRQ HW XQH IRQFWLRQ GH OD GLVWULEXWLRQ GH OD GpIRUPDWLRQ D[LDOH REWHQXH jSDUWLU GH PHVXUH GH FKDPSVSURYHQDQWG¶XQHVVDLGHWUDFWLRQG¶XQHpSURXYHWWHPXOWLFULVWDOOLQHFRQWHQDQW JUDLQV
5HYXH9ROXPH;±Q[DQQpH
/HEXWGHFHWUDYDLOHVWO¶LGHQWLILFDWLRQGHVSDUDPqWUHVGHODORLGHFRPSRUWHPHQW FULVWDOOLQGX=LUFRQLXPJUDGH&HVSDUDPqWUHVVRQWLGHQWLILpVHQPLQLPLVDQWXQH IRQFWLRQFRWUHSUpVHQWDQWO¶pFDUWHQWUHOHPRGqOHHWO¶H[SpULHQFHSRXUOHVUpSRQVHV JOREDOHHWORFDOH
.0RGqOHGHFRPSRUWHPHQWPRQRFULVWDOOLQ
/H=LUFRQLXPHVWXQPDWpULDXjVWUXFWXUHKH[DJRQDOHFRPSDFWHLOSUpVHQWHFLQT IDPLOOHVGHV\VWqPHVGHJOLVVHPHQWIDFLOH7URLVV\VWqPHVEDVDX[ ^` < >
WURLVV\VWqPHVSULVPDWLTXHV^` < >VL[V\VWqPHVS\UDPLGDX[GHW\SHD!
^ ` <> GRX]H V\VWqPHV S\UDPLGDX[ GH W\SH FD! GH SUHPLqUH HVSqFH
^ ` <> DLQVL TXH VL[ V\VWqPHV S\UDPLGDX[ FD! GH GHX[LqPH HVSqFH
^`<>'DQVFHWUDYDLOOHJOLVVHPHQWVXUOHVV\VWqPHVS\UDPLGDX[FD!
GHGHX[LqPHHVSqFHHVWQpJOLJpHWOHPDFODJHQ¶HVWSDVSULVHQFRPSWH2QVXSSRVH TXH OH FRPSRUWHPHQW GH FH PDWpULDX HVW pODVWRYLVFRSODVWLTXH /H PRGqOH GH FRPSRUWHPHQW FULVWDOOLQ UHWHQX HVW FHOXL XWLOLVp SDU %DODVXEUDPDQLDQ HW $QDQG>@
/DYLWHVVHGHJOLVVHPHQWVXUOHV\VWqPHα VXLWXQHORLSXLVVDQFH
VJQ
m
s
α
α α
α
γ γ τ τ
§ ·
¨ ¸
= ¨¨© ¸¸¹
R τα HW sα VRQW UHVSHFWLYHPHQW OD FLVVLRQ UpVROXH HW OD UpVLVWDQFH j OD GpIRUPDWLRQ GX V\VWqPH GH JOLVVHPHQW α γ HVW XQH YLWHVVH GH JOLVVHPHQW DUELWUDLUHHWmHVWOHFRHIILFLHQWGHVHQVLELOLWpjODYLWHVVHGHGpIRUPDWLRQ
/DORLGpYROXWLRQGHsαHVWGRQQpHSDUOHVUHODWLRQV
DYHF j
sα hαβ β sα sα t
β
γ
=
=
¦
= =
HW
hαβ = qαβ βh DYHF VL HW VRQWFRSODQDLUHV
VL HW QHVRQWSDVFRSODQDLUHV
q q
αβ α β
α β
= ®
¯
Rhβ HVW WDX[ G¶DXWRpFURXLVVDJH GX V\VWqPH GH JOLVVHPHQWβ HW qαβ HVW OD PDWULFHG¶pFURXLVVDJH
/HWDX[G¶DXWRpFURXLVVDJHhβHVWGRQQpSDUO¶pTXDWLRQ
VJQ
r
s s
s s
h h
s s
β β
β β β
§ ·
= − ¨¨© − ¸¸¹ DYHF
n
ss s
β γβ
γ
§ ·
= ¨ ¸
© ¹
RhHVWXQIDFWHXUGpFURXLVVDJHHWssβHVWODYDOHXUGHsβ jVDWXUDWLRQ
/HV FRQVWDQWHV pODVWLTXHV GX ]LUFRQLXP j XQH WHPSpUDWXUH GH & RQW pWp PHVXUpHV SDU )LVKHU HW 5HQNHQ >@ /HXUV YDOHXUV VRQW C *3D
C *3DC *3DC *3DC *3D
,GHQWLILFDWLRQGHVSDUDPqWUHVGXPRGqOHFULVWDOOLQ
$SDUWLUG¶XQHVVDLGHWUDFWLRQ>@QRXVGLVSRVRQVGHVGRQQpHVH[SpULPHQWDOHV VXLYDQWHV L OD WH[WXUH HW OD PLFURVWUXFWXUH LQLWLDOHV JUkFH j O¶(%6' (OHFWURQ
%DFN6FDWWHULQJ'LIIUDFWLRQLLODUpSRQVHPDFURVFRSLTXHσ εHWLLLOHVFKDPSV ORFDX[GHGpIRUPDWLRQREWHQXVSDUFRUUpODWLRQG¶LPDJHVQXPpULTXHVGHPLFURJULOOHV G¶RUGpSRVpHVjODVXUIDFHGHOD]RQHG¶pWXGHTXLPHVXUH×PHWFRQWLHQW JUDLQVFI)LJD/HVFRQGLWLRQVDX[OLPLWHVDSSOLTXpVVRQWGHW\SHGpSODFHPHQW GH EORF FRUUHVSRQGDQW j XQH WUDFWLRQ VLPSOH VXLYDQW OD GLUHFWLRQx MXVTXj XQH GpIRUPDWLRQORJDULWKPLTXHILQDOHGH/DFRQWUDLQWHPR\HQQHVXUOHVVXUIDFHV ODWpUDOHVpWDQWQXOOH/DYLWHVVHGHGpIRUPDWLRQPR\HQQHHVWFHOOHGHO¶HVVDLTXLYDXW × V /HV VLPXODWLRQV pOpPHQWV ILQLV RQW pWp UpDOLVpHV DYHF OH FRGH 0HVR' GpYHORSSp GDQV OH ODERUDWRLUH /3070 HQ FROODERUDWLRQ DYHF OH /06 GH O¶pFROH SRO\WHFKQLTXH>@
/D UpSRQVH PDFURVFRSLTXH HVW VLPXOpH VXU XQ 9(5 GRQW OD WDLOOH D pWp SUpDODEOHPHQWGpWHUPLQpHSDUXQHpWXGHVXUODGLVSHUVLRQGHODUpSRQVHJOREDOH>
@ /D UpSRQVH ORFDOH HVW VLPXOpH VXU XQH PLFURVWUXFWXUH GRQW OD PRUSKRORJLH LQLWLDOHSURYLHQWGHODFDUWRJUDSKLH(%6'
/¶LGHQWLILFDWLRQHVWUpDOLVpHHQPLQLPLVDQWXQHIRQFWLRQFRWJOREDOHGpILQLHSDU
G L
f 3 =ηf 3 + −η f 3
Rη HVW XQ SDUDPqWUH GH SRQGpUDWLRQ3 HVW OH MHX GH SDUDPqWUHV GX PRGqOH FULVWDOOLQ fG 3 UHSUpVHQWH O¶pFDUW HQWUH OD UpSRQVH JOREDOH H[SpULPHQWDOH HW FHOOH VLPXOpHGRQWO¶H[SUHVVLRQHVWGRQQpHSDU
H[S H[S
np sim i i
G
p i i
f n
σ ε σ ε
σ ε
=
§ − ·
=
¦
¨¨© 3 ¸¸¹3
R np HVW OH QRPEUH GH SRLQWV GH OD FRXUEH H[SpULPHQWDOH εi σH[S εi HW
sim
σ 3 εi VRQW UHVSHFWLYHPHQW OD GpIRUPDWLRQ OD FRQWUDLQWH PHVXUpH HW OD FRQWUDLQWH VLPXOpH FRUUHVSRQGDQW DX LqPH SRLQW GH PHVXUH GH OD FRXUEH H[SpULPHQWDOH
fL 3 HVWO¶pFDUW HQWUHOD UpSRQVHORFDOHH[SpULPHQWDOH HWFHOOHVLPXOpH&HWpFDUW HVWGpILQLSDU
H[S H[S
H[S H[S
mp xxsim j xx j mp simyy j yy j L
p j xx j p j yy j
f m m
ε ε ε ε
ε ε
= =
§ − · § − ·
¨ ¸ ¨ ¸
=
¦
¨© 3 [ [ ¸¹ +¦
¨© 3 [ [ ¸¹3 [ [
5HYXH9ROXPH;±Q[DQQpH
RmpHVWOHQRPEUHGHSRLQWVRODGpIRUPDWLRQDpWpPHVXUpH[jεxxsim 3 [j
H[S
xx j
ε [ εyysim 3 [j HW εH[Syy [j UHSUpVHQWHQW UHVSHFWLYHPHQW OD SRVLWLRQ OHV GpIRUPDWLRQVD[LDOHVVLPXOpHHWH[SpULPHQWDOHOHVGpIRUPDWLRQVWUDQVYHUVHVVLPXOpH HWH[SpULPHQWDOHFRUUHVSRQGDQWDXMqPHSRLQWGHPHVXUHGHVFKDPSVGHGpIRUPDWLRQ
/HV V\VWqPHV GH JOLVVHPHQWV DFWLIV REVHUYpV DSUqV O¶HVVDL VRQW OHV V\VWqPHV SULVPDWLTXHV HW OHV V\VWqPHV S\UDPLGDX[ D! 3RXU FHOD QRXV LGHQWLILRQV sprismHW
pyr a
s < >OHVDXWUHVYDOHXUVGHsVRQWPDLQWHQXHVj03D4XDQWDX[YDOHXUVGH
pyr c a
s < + > HW sbasal VRQW IL[pHV j 03D /¶DEVHQFH G¶LQIRUPDWLRQV GDQV OD
OLWWpUDWXUHFRQFHUQDQWO¶LQWHUDFWLRQHQWUHOHVGLIIpUHQWVV\VWqPHVGHJOLVVHPHQWQRXV DDPHQpjFRQVLGpUHUTXHq =/HSDUDPqWUHγHVWIL[pjV/HVYDOHXUV GHV KXLW DXWUHV SDUDPqWUHV LGHQWLILpV VRQW r = m= n =
h = 03D sprism =03D spyr a< > =03D sprism =03D
pyr a
s < > = 03D
/D ILJXUH PRQWUH TXH OD UpSRQVH JOREDOH HVW ELHQ UHSURGXLWH 4XDQW j OD ILJXUHHOOHLQGLTXHXQHPHLOOHXUHSUpGLFWLRQGXFKDPSεSDUUDSSRUWjFHOOHGX FKDPS ε /¶XWLOLVDWLRQG¶XQHEDVHGHPHVXUHGHOD GpIRUPDWLRQ GH PQHQRXV SHUPHW SDV GH GLUH VL OHV IRUWHV IOXFWXDWLRQV REVHUYpHV VXU OHV FDUWRJUDSKLHV SURYLHQQHQWG¶XQEUXLWGHPHVXUHRXVLOHPDLOODJHpOpPHQWVILQLVQ¶HVWSDVDVVH]ILQ SRXUUHSURGXLUHFHVPrPHVIOXFWXDWLRQV
)LJ&RXUEHGHODUpSRQVHJOREDOHH[SpULPHQWDOHσεHWFHOOHVLPXOpHVXU XQ9(5DYHFOHMHXGHSDUDPqWUHVLGHQWLILp
FRQWUDLQWHPR\HQQH>03D@
GpIRUPDWLRQORJDULWKPLTXH>@
H[SHULHQFH VLPXODWLRQVXUXQ9(5
&RQFOXVLRQ
2Q D LGHQWLILp OH MHX GH SDUDPqWUHV SRXU OH PRGqOH FULVWDOOLQ G¶XQ =LUFRQLXP JUDGH HQ XWLOLVDQW OD UpSRQVH PDFURVFRSLTXH HW OHV FKDPSV GHV GpIRUPDWLRQV PHVXUpVSDUFRUUpODWLRQG¶LPDJHVGHPLFURJULOOHV/DFRQIURQWDWLRQGHVUpVXOWDWVGH OD VLPXODWLRQ QXPpULTXH SDU pOpPHQWV ILQLV j FHX[ GH O¶H[SpULHQFH GRQQH XQ ERQ DFFRUG VXU OD FRXUEH GH WUDFWLRQ &RQFHUQDQW OHV FKDPSV ORFDX[ OHV UpVXOWDWV VRQW HQFRXUDJHDQWV 8QH DPpOLRUDWLRQ GHYUDLW rWUH REWHQXH SDU XQH PHLOOHXUH SULVH HQ FRPSWH GHV FRQGLWLRQV DX[ OLPLWHV UpHOOHV HQ XWLOLVDQW OHV GpSODFHPHQWV H[SpULPHQWDX[VXUOHVERUGVGXPRWLIpWXGLpHWGHVPDLOODJHVSOXVILQV
5HPHUFLHPHQWV
/HV DXWHXUV WLHQQHQW j UHPHUFLHU - &UpSLQ /06(FROH 3RO\WHFKQLTXH HW 0'H[HW/06HW650$&($SRXUOHVUpVXOWDWVH[SpULPHQWDX[TX¶LOVQRXVRQW IRXUQLV
)LJ=RQHG¶pWXGHîPFRPSRVpHGHJUDLQVDPRUSKRORJLHGHV JUDLQV(%6'&KDPSVGHODGpIRUPDWLRQεEH[SpULPHQWDOHWFVLPXOp
&KDPSVGHODGpIRUPDWLRQεGH[SpULPHQWDOHWHVLPXOp
@
>
ε
@
>
ε
D
G I
E F
5HYXH9ROXPH;±Q[DQQpH
%LEOLRJUDSKLH
>@7+RF-&UpSLQ/*pOpEDUWDQG$=DRXL$SURFHGXUHIRULGHQWLI\LQJWKH SODVWLF EHKDYLRU RI VLQJOH FU\VWDOV IURP WKH ORFDO UHVSRQVH RI SRO\FU\VWDOV $FWD 0DWHULDOLD
>@6%DODVXEUDPDQLDQDQG/$QDQG3ODVWLFLW\RILQLWLDOO\WH[WXUHGKH[DJRQDO SRO\FU\VWDOV DW KLJK KRPRORJRXV WHPSHUDWXUHV DSSOLFDWLRQ WR WLWDQLXP $FWD 0DWHULDOLD
>@(6)LVKHUDQG&-5HQNHQ6LQJOH&U\VWDOHODVWLFPRGXOLDQGWKHKFS!
EFFWUDQVIRUPDWLRQLQ7L=UDQG+I3K\VLFDO5HYLHZ$
>@ 0 'H[HW 0pWKRGH GH FRXSODJH H[SpULPHQWDWLRQVVLPXODWLRQV QXPpULTXHV HQYXHGHO¶LGHQWLILFDWLRQGHORLGHFRPSRUWHPHQWFULVWDOOLQHDSSOLFDWLRQDX[DOOLDJHV GH]LUFRQLXP7KqVHGHGRFWRUDW(FROH3RO\WHFKQLTXH3DODLVHDX)UDQFH
>@6+pUDXG/$OODLV++DGGDGL&7HRGRVLXDQG$=DRXL'XSRO\FULVWDO DXPXOWLFULVWDOYHUVXQPHVRVFRSHQXPpULTXH-RXUQDOGH3K\VLTXH,935
>@++DGGDGL6+pUDXG/$OODLV&7HRGRVLXDQG$=DRXL$QXPHULFDO PHVRVFRSH IRU WKH LQYHVWLJDWLRQ RI ORFDO ILHOGV LQ UDWHGHSHQGHQW HODVWRSODVWLF PDWHULDOV DW ILQLWH VWUDLQ ,Q HGLWRU & 0LHKH HGLWRU ,87$0 6\PSRVLXP RQ
&RPSXWDWLRQDO0HFKDQLFVRI6ROLG0DWHULDOVDW/DUJH6WUDLQV6WXWWJDUW
*HUPDQ\
>@ $ 6DODKRXHOKDGM + +DGGDGL 0 *DVSpULQL YHUV XQH DPpOLRUDWLRQ GH OD SUpYLVLRQGHVFKDPSVPpFDQLTXHVORFDX[GDQVOHVSRO\FULVWDX[&RQJUqV)UDQoDLV GHOD0pFDQLTXHDRWDXVHSWHPEUH7UR\HV
>@ $ 6DODKRXHOKDGM + +DGGDGL 0 *DVSpULQL 8WLOLVDWLRQ GX SULQFLSH GX PpVRVFRSH QXPpULTXH SRXU O¶DPpOLRUDWLRQ GH OD SUpYLVLRQGHVFKDPSV PpFDQLTXHV ORFDX[GDQVOHVSRO\FULVWDX[&ROORTXH3ODVWLFLWpPDUV$QQHF\
>@ $ 6DODKRXHOKDGM + +DGGDGL 0 *DVSpULQL +RPRJpQpLVDWLRQ SDU (OpPHQWV )LQLV GX &RPSRUWHPHQW 0pFDQLTXH GH 3RO\FULVWDX[ 0pWDOOLTXHV 0DWpULDX[QRYHPEUH'LMRQ
>@ $ 6DODKRXHOKDGM + +DGGDGL 0 *DVSpULQL +RPRJHQL]DWLRQ RI WKH 0HFKDQLFDO %HKDYLRU RI 0HWDOOLF 3RO\FU\VWDOV E\ )LQLWH (OHPHQW 0HWKRG ,QWHUQDWLRQDO &RQIHUHQFH RQ $GYDQFHV LQ 0HFKDQLFDO (QJLQHHULQJ DQG 0HFKDQLFV ,&$0(0GpFHPEUH7XQLVLH