Méthode des Champs Virtuels pour la caractérisation du comportement dynamique de matériaux métalliques sous chargement purement inertiel

HAL Id: tel-02132396

https://hal.archives-ouvertes.fr/tel-02132396

Submitted on 17 May 2019

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.

du comportement dynamique de matériaux métalliques sous chargement purement inertiel

Pascal Bouda

To cite this version:

Pascal Bouda. Méthode des Champs Virtuels pour la caractérisation du comportement dynamique de

matériaux métalliques sous chargement purement inertiel. Engineering Sciences [physics]. UNIVER-

SITE POLYTECHNIQUE HAUTS-DE-FRANCE, 2019. English. �tel-02132396�

Thèse de doctorat

Pour obtenir le grade de Docteur de

l’UNIVERSITÉ POLYTECHNIQUE HAUTS-DE-FRANCE

Discipline, spécialité selon la liste des spécialités pour lesquelles l’Ecole Doctorale est accréditée :

Mécanique

Présentée et soutenue publiquement par Pascal BOUDA Le 11/03/2019, à Lille

École doctorale :

Sciences Pour l’Ingénieur (ED SPI 072)

Équipe de recherche, Laboratoire :

Laboratoire d’Automatique, de Mécanique et d’Informatique Industrielles et Humaines (LAMIH - UMR 8201)

Méthode des Champs Virtuels

pour la caractérisation du comportement dynamique de matériaux métalliques sous chargement purement inertiel

JURY

M. Julien RÉTHORÉ, Directeur de Recherche CNRS au Laboratoire GEM de Nantes Président du Jury M. Michel GRÉDIAC, Professeur de l’Université Clermont Auvergne Rapporteur M. Jean-Philippe PONTHOT, Professeur de l’Université de Liège (Belgique) Rapporteur M. Eric MARKIEWICZ, Professeur de l’Université Polytechnique Hauts-de-France Directeur de thèse M. Fabrice PIERRON, Professeur de l’Université de Southampton (UK) Co-directeur de thèse Mme Patricia VERLEYSEN, Professeur de l’Université de Gand (Belgique) Examinateur

M. Bertrand LANGRAND, Directeur de Recherche à l’ONERA Examinateur

Mme Delphine NOTTA-CUVIER, Maître de Conférences de l’Université Polytechnique Hauts-de-France

Examinateur

t1R IUHH PHDO u 7HG %HO\WVFKNR

$ YRXV 0DPDQ HW 3DSD

$FNQRZOHGJHPHQWV

1RPHQFODWXUH

,QWURGXFWLRQ

0DWHULDO EHKDYLRXU PRGHOOLQJ DQG LGHQWLILFDWLRQ RI FRQVWLWXWLYH SDUDPHWHUV $Q RYHUYLHZ 0DWHULDO EHKDYLRXU PRGHOOLQJ ,GHQWLILFDWLRQ ZLWK VWDWLFDOO\ GHWHUPLQHG WHVWV )XOO ILHOG PHDVXUHPHQWV WHFKQLTXHV 'LVSODFHPHQW ILHOGV PHDVXUHPHQW WHFKQLTXHV 8OWUD+LJK6SHHG LPDJLQJ ,GHQWLILFDWLRQ ZLWK VWDWLFDOO\ XQGHWHUPLQHG DSSURDFKHV 2YHUYLHZ 7KH 9LUWXDO )LHOGV 0HWKRG ZLWK H[WHUQDO ORDG PHDVXUHPHQW 7KH 9LUWXDO )LHOGV 0HWKRG ZLWK DFFHOHUDWLRQ DV D ORDG FHOO 6XPPDU\

'HYHORSPHQW RI D QXPHULFDO WRROER[ IRU ,PDJH%DVHG ,QHUWLDO ,PSDFW WHVWV DQDO\VLV 5HWXUQ 0DSSLQJ $OJRULWKP IRU -RKQVRQ&RRN PRGHO DQG J

YLVFRSODVWLFLW\

7KHRUHWLFDO EDFNJURXQG $ 1HZWRQ5DSKVRQEDVHG VWUDWHJ\ IRU -RKQVRQ&RRN PRGHO LQWHJUDWLRQ 9DOLGDWLRQ RQ YLUWXDO WHVWV &RQFOXVLRQ 7KH JULG PHWKRG IURP JUH\VFDOH LPDJHV WR VWUDLQ DQG DFFHOHUDWLRQ ILHOGV $ )(EDVHG SURFHGXUH WR JHQHUDWH DQG SURFHVV UHOLDEOH VLPXODWHG WHVW FRQILJXUDWLRQV 2Q WKH XVH RI &DVW0‡ DQG (XURSOH[XV‡ IRU )($ RI ,%,, WHVWV )($ SRVWSURFHVVLQJ ,%,, YLUWXDO WHVWV UHOLDELOLW\ DVVHVVPHQW $ )(EDVHG VLPXODWLRQ RI IXOOILHOG PHDVXUHPHQWV 'HYHORSPHQW RI D V\QWKHWLF LPDJHV

JHQHUDWRU $ PDSSLQJEDVHG DOJRULWKP WR JHQHUDWH V\QWKHWLF LPDJHV 9DOLGDWLRQ RQ DQDO\WLF FDVHV 6XPPDU\

1XPHULFDO RSWLPL]DWLRQ RI ,PDJH%DVHG ,QHUWLDO ,PSDFW WHVWV IRU -RKQVRQ&RRN PRGHO

FKDUDFWHUL]DWLRQ

0RWLYDWLRQV 6HWWLQJ WKH FRQVWUDLQHG WHVW SDUDPHWHUV ,QIOXHQFH RI LPSDFW YHORFLW\ DQG VSHFLPHQ JHRPHWU\ RQ JHQHUDWHG YLVFRSODVWLF VSHFWUD 9)0EDVHG VHOHFWLRQ RI SURMHFWLOH YHORFLW\ DQG VSHFLPHQ JHRPHWU\

7KURXJKWKLFNQHVV VWUHVV DQDO\VLV

,QIOXHQFH RI WHPSRUDO UHVROXWLRQ DQG LPSDFW YHORFLW\ RQ YLVFRSODVWLF SDUDPHWHU H[WUDFWLRQ 2SWLPL]DWLRQ RI NLQHPDWLF ILHOGV H[WUDFWLRQ SURFHVV WKURXJK XQFHUWDLQWLHV TXDQWLILFDWLRQ 0HWKRGRORJ\

2SWLPL]DWLRQ RI UHJXODUL]DWLRQ SDUDPHWHUV &RQFOXVLRQ DQG IXUWKHU SURVSHFWV ,PDJH%DVHG ,QHUWLDO ,PSDFW WHVW FDPSDLJQ IRU 7L$O9 VWUDLQUDWH VHQVLWLYLW\ FKDUDFWHUL

]DWLRQ

'HVFULSWLRQ RI WKH WHVW FDPSDLJQ ,PSDFW WHVW SURWRFRO 0HWURORJLFDO SHUIRUPDQFH DVVHVVPHQW $SSOLFDWLRQ RI WKH G\QDPLF 9)0 WR YLVFRSODVWLFLW\ FKDUDFWHUL]DWLRQ 6WUDLQ DQG DFFHOHUDWLRQ H[WUDFWLRQ 9LVFRSODVWLF SDUDPHWHU LGHQWLILFDWLRQ 9LVFRSODVWLF VSHFWUD DQDO\VLV 6HQVLWLYLW\ RI WKH LGHQWLILFDWLRQ WR WKH FKRLFH RI YLUWXDO ILHOG $ FULWLFDO FRPSDULVRQ EHWZHHQ H[SHULPHQWDO UHVXOWV DQG QXPHULFDO SUHGLFWLRQV )XOOILHOG PDSV FRPSDULVRQ 9LVFRSODVWLFLW\ FKDUDFWHUL]DWLRQ FRPSDULVRQ 9LVFRSODVWLF VSHFWUD FRPSDULVRQ 6XPPDU\

*HQHUDO &RQFOXVLRQ

$ &RPSOHPHQWDU\ UHVXOWV IRU 5HWXUQ 0DSSLQJ $OJRULWKP YDOLGDWLRQ

% &DVW0 VFULSW IRU ,%,, WHVWV PHVK JHQHUDWLRQ

& &RPSOHPHQWDU\ UHVXOWV IRU ,%,, WHVWV WKURXJKWKLFNQHVV DQDO\VLV ' &RPSOHPHQWDU\ UHVXOWV IRU ,%,, WHVW FDPSDLJQ PHWURORJLFDO SHUIRUPDQFH DVVHVVPHQW ( &RPSOHPHQWDU\ UHVXOWV IRU FRPSDULVRQ EHWZHHQ H[SHULPHQWDO UHVXOWV DQG QXPHULFDO SUH

GLFWLRQV

$ YDQW GH G¶URXOHU FH PDQXVFULW MH VRXKDLWH UHPHUFLHU WRXV OHV DFWHXUV TXL RQW SHUPLV ­ FHWWH WKµVH GsDERXWLU FDU HOOH HVW OH IUXLW GH WURLV DQV GH WUDYDLO HW XQH FROODERUDWLRQ WULSDUWLWH HQWUH Os2IILFH 1DWLRQDO GsšWXGHV HW GH 5HFKHUFKHV $¶URVSDWLDOHV 21(5$ Os8QLYHUVLW¶ 3RO\WHFKQLTXH +DXWVGH )UDQFH HW Os8QLYHUVLW¶ GH 6RXWKDPSWRQ -H UHPHUFLH ¶JDOHPHQW Os21(5$ HW OD 5¶JLRQ +DXWVGH )UDQFH SRXU OH FRILQDQFHPHQW GH FHWWH WKµVH

7RXW GsDERUG MH UHPHUFLH OHV 3URIHVVHXUV 0LFKHO *U¶GLDF GH Os8QLYHUVLW¶ &OHUPRQW $XYHUJQH HW -HDQ3KLOLSSH 3RQWKRW GH Os8QLYHUVLW¶ GH /LµJH SRXU OsLQW¶U·W SRUW¶ ­ FH WUDYDLO HQ DFFHSWDQW GH OH UDS SRUWHU -H UHPHUFLH ¶JDOHPHQW OHV DXWUHV PHPEUHV H[W¶ULHXUV GX MXU\ TXH VRQW OH 3URIHVVHXU 3DWULFLD 9HUOH\VHQ GH Os8QLYHUVLW¶ GH *DQG HW 0RQVLHXU -XOLHQ 5¶WKRU¶ GX ODERUDWRLUH *(0 GH 1DQWHV SRXU OHXU LQW¶U·W SRUW¶ ­ FH WUDYDLO HW SRXU DYRLU DFFHSW¶ GH OsH[DPLQHU

&HWWH WKµVH DERXWLW QRWDPPHQW JU¯FH ­ XQ HQFDGUHPHQW HQYHUV TXL MH VXLV JUDQGHPHQW UHFRQQDLV VDQW $XVVL MH VRXKDLWDLV UHPHUFLHU PHV GHX[ GLUHFWHXUV GH WKµVH OHV 3URIHVVHXUV (ULF 0DUNLHZLF]

GH Os8QLYHUVLW¶ 3RO\WHFKQLTXH +DXWVGH)UDQFH HW )DEULFH 3LHUURQ GH Os8QLYHUVLW¶ GH 6RXWKDPSWRQ 0·PH VL OD GLVWDQFH J¶RJUDSKLTXH ¶WDLW LPSRUWDQWH OHV QRPEUHX[ ¶FKDQJHV TXH QRXV DYRQV HXV RQW

¶W¶ G¶WHUPLQDQWV GDQV OsDYDQF¶H HW OD FRQFOXVLRQ GH FHV WUDYDX[ GH WKµVH

$\DQW SDVV¶ OD PDMRULW¶ GH PRQ WHPSV ­ Os21(5$ MH VRXKDLWDLV DXVVL H[SULPHU PD JUDQGH JUDW LWXGH HQYHUV 0RQVLHXU %HUWUDQG /DQJUDQG TXL D ¶W¶ ELHQ SOXV TXsXQ HQFDGUDQW tORFDOu (Q HIIHW VD GLVSRQLELOLW¶ HW VRQ UHJDUG FULWLTXH PDLV WRXMRXUV REMHFWLI RQW ODUJHPHQW FRQWULEX¶ ­ PD IRUPDWLRQ SDU OD UHFKHUFKH &RQVWDPHQW ­ Os¶FRXWH FHV WUDYDX[ QsDXUDLHQW SDV SX DERXWLU VDQV OXL -H UHPHU FLH DXVVL PRQ HQFDGUDQWH 0DGDPH 'HOSKLQH 1RWWD&XYLHU GH Os8QLYHUVLW¶ 3RO\WHFKQLTXH +DXWVGH )UDQFH 1RV FRQYHUVDWLRQV QRXUULHV RQW HQ HIIHW SHUPLV OD VRUWLH GH VLWXDWLRQV SDUIRLV G¶OLFDWHV

0D S¶ULRGH ­ Os21(5$ D ¶JDOHPHQW ¶W¶ OsRFFDVLRQ GH V\PSDWKLVHU DYHF GH QRPEUHXVHV SHUVRQQHV

$X VHLQ GH OsXQLW¶ &RQFHSWLRQ HW 5¶VLVWDQFH '\QDPLTXH &5' TXL PsD FKDOHXUHXVHPHQW DFFXHLOOL MH VRXKDLWH GRQF UHPHUFLHU %HUQDUG 'DYLG 'LGLHU (ULF )DELHQ *¶UDOG -DFTXHV -HDQ/XF -XOLHQ /DXUHQW 0DWWKLHX 5RODQG 5RPDLQ 6WHYH HW 7KRPDV PDLV DXVVL OHV MHXQHV UHWUDLW¶V TXH VRQW $ODLQ HW -HDQ)UDQ´RLV -H UHPHUFLH ¶JDOHPHQW PHV FDPDUDGHV GH JDOµUH U¶FHPPHQW WK¶V¶V RX TXL OH VHURQW ELHQWÀW PRQ VXFFHVVHXU -HDQ'DYLG 0DJDOL 1LFRODV 1RHOLJ HW 7RYLJQRQ 4XH FH VRLW VXU OH SODQ VFLHQWLILTXH RX KXPDLQ OD UHODWLRQ HQWUHWHQXH DYHF FKDFXQ GsHQWUH YRXV D ¶W¶ JOREDOHPHQW DJU¶DEOH HW MH VXLV SHUVXDG¶ TXH OH FRQWH[WH FU¶¶H D ODUJHPHQW FRQWULEX¶ ­ OsDERXWLVVHPHQW GH FHV WUDYDX[

-H WHQDLV HQILQ ­ UHPHUFLHU WRXV OHV DFWHXUV OD FDPSDJQH GsHVVDLV P¶FDQLTXHV U¶DOLV¶H ­ Os8QLYHUVLW¶

GH 6RXWKDPSWRQ -H UHPHUFLH GsDERUG OHV PHPEUHV GX JURXSH GH WUDYDLO 3KRWRG\Q SRXU OHXU DFFXHLO HW OHXU ELHQYLHLOODQFH $OH[ )UDQFHV -DUHG /OR\G D WKRXVDQG WKDQNV IRU WKH SHUIHFW ,%,, WHVWVg HW ;DYLHU -H QsRXEOLH SDV DXVVL GH UHPHUFLHU OsXQLW¶ $WHOLHUV 0DTXHWWHV SRXU 6RXIIOHULHV $06 GH Os21(5$ SRXU OD TXDOLW¶ UHPDUTXDEOH GHV ¶SURXYHWWHV XVLQ¶HV

(QILQ MH QH SHX[ FRQFOXUH FHV UHPHUFLHPHQWV VDQV DWWULEXHU XQH PHQWLRQ VS¶FLDOH ­ PD IDPLOOH TXL PH VXSSRUWH GHSXLV ELHQWÀW YLQJWVHSW DQV 0DPDQ 3DSD 0DWKLHX )OHXU HW 5LFKDUG MH YRXV GLV WRXW VLPSOHPHQW PHUFL

0DWKHPDWLFDO TXDQWLWLHV RSHUDWRUV ( . ) ILUVWRUGHU WHQVRU YHFWRU ( . ) ( . ) VHFRQGRUGHU WHQVRU PDWUL[ ( . ) M IRXUWKRUGHU WHQVRU M

( . )

7UDQVSRVH RI PDWUL[ ( . ) I ,GHQWLW\ PDWUL[

7U >@ 7UDFH RI PDWUL[ ( . )

∇( . ) *UDGLHQW RI ( . ) ( ˙ . ) 7LPH GHULYDWLYH RI ( . )

a × b &URVV SURGXFW EHWZHHQ ' YHFWRUV a DQG b

0HFKDQLFDO TXDQWLWLHV

F 7UDQVIRUPDWLRQ JUDGLHQW WHQVRU u 'LVSODFHPHQW YHFWRU

v 9HORFLW\ YHFWRU γ $FFHOHUDWLRQ YHFWRU σ &DXFK\ VWUHVV WHQVRU ε 6WUDLQ WHQVRU

ε

(ODVWLF VWUDLQ WHQVRU ε

,QHODVWLF VWUDLQ WHQVRU ε

3ODVWLF VWUDLQ WHQVRU ε

9LVFRSODVWLF VWUDLQ WHQVRU C 6WLIIQHVV PDWUL[

E <RXQJsV PRGXOXV ν 3RLVVRQsV UDWLR

λ )LUVW /DP¶sV FRHIILFLHQW

μ 6HFRQG /DP¶sV FRHIILFLHQW VKHDU PRGXOXV

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

$Q H[DPSOH RI UHFHQW DGYDQFHV LQ WKLV ILHOG LV WKH VXFFHVVIXO GLWFKLQJ RQ WKH +XGVRQ ULYHU 2Q -DQ XDU\ DQ $LUEXV $ RI 86 $LUZD\V FRQQHFWLQJ 1HZ <RUN DQG &KDUORWWH FLWLHV 86$

XQGHUZHQW D FRPSOHWH ORVV RI WKUXVW LQ ERWK HQJLQHV HLJKW PLQXWHV DIWHU WDNHRII <HW WKH DLUFDIW ZDV VXFFHVVIXOO\ GLWFKHG RQ WKH ULYHU ZLWKRXW DQ\ FDVXDOWLHV )LJXUH 7KLV LQFLGHQW ZDV GXH WR HQJLQHV HQFRXQWHU ZLWK D IORFN RI ELUGV KHQFH WKH ODUJH LQJHVWLRQ RI IRUHLJQ ERGLHV $FFRUGLQJ WR WKH 1DWLRQDO 7UDQVSRUWDWLRQ 6DIHW\ %RDUG 176% UHSRUW >@ VHYHUDO IDFWRUV KDYH FRQWULEXWHG WR WKLV IDYRXUDEOH RXWFRPH ,Q DGGLWLRQ WR WKH FUHZ PHPEHUV GHFLVLRQV DQG DFWLRQV LW ZDV QRWDEO\ QRWLFHG WKDW WKH DLU SODQH tZDV HTXLSSHG IRU DQ H[WHQGHG RYHUZDWHU IOLJKW >g@ HYHQ WKRXJK LW ZDV QRW UHTXLUHG WR EH VR HTXLSSHGu %\ FRQWUDVW RWKHU DFFLGHQWV UHPLQG XV WKH FRQWLQXRXV QHFHVVLW\ WR HQKDQFH WKH RQJRLQJ UHVHDUFK 2Q $SULO D %RHLQJ RI 6RXWKZHVW $LUOLQHV ZDV FRQQHFWLQJ 1HZ <RUN DQG 'DOODV FLWLHV 86$ ZKHQ RQH RI WKH OHIW HQJLQH IDQ EODGH PDGH RI OLJKWZHLJKW DOOR\ VXGGHQO\ EURNH RII FDXVLQJ LQOHW IUDJPHQWV VWULNH D ZLQGRZ ZKLFK HYHQWXDOO\ OHG WR D IDWDOLW\ DPRQJ SDVVHQJHUV t7KH DHURVSDFH LQGXVWU\ >g@ KDV WR EH VDIH RI FRXUVH EXW LW DOVR KDV WR EH OLJKW DQG WKRVH WZR WKLQJV FRXQWHUDFW HDFK RWKHUu VDLG D SURIHVVRU DW DQ KHDULQJ IROORZLQJ WKH HQJLQH IDLOXUH

$IWHU WKRURXJK LQYHVWLJDWLRQV >@ WKH 176% UHSRUWHG WKH SUHVHQFH RI tIDWLJXH IUDFWXUH IHDWXUHVu DW WKH EURNHQ EODGH URRW FDXVLQJ KXJH GDPDJHV RQ WKH HQJLQH )LJXUH ZKHUHDV WKH EODGH VKRXOG QRW KDYH UHDFKHG WKH LQOHW

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

KWWSZZZSKLOO\FRPSKLOO\KHDOWKVFLHQFHVRXWKZHVWIOLJKWHQJLQHEODGHQWVEKWPO

Footnotes 1Numbers for 2017 are preliminary estimates. Aviation data are from the NTSB; marine data are reported by the US Department of Homeland Security; all other data are reported by the US Department of Transportation. 2Pedalcycles include bicycles and other cycles. 3Other refers to non-occupants (excluding pedestrians and pedalcyclists) and occupants in other or unknown vehicle types. 4Grade crossing fatalities are reported as a separate category but should not be added to the total because they are included in the highway and rail fatalities as appropriate. 5Freight, passenger, and commuter rail data are reported by the Federal Railroad Administration. 6Trespassing fatalities are reported as a separate category but should not be added to the total because they are included in the freight, passenger, and commuter rail fatalities. Trespassing fatalities are not available for rail transit. 7Rail transit data are reported by the Federal Transit Administration and include fatalities involving heavy rail, light rail, cable car, inclined plane, monorail/automated guideway, streetcar rail, and hybrid rail. 8'SQQIVGMEPƤWLMRKVIJIVWXSSTIVEXMSREPJEXEPMXMIW 9Total fatalities may not equal the sum of each category because accidents may involve multiple categories. 10 Foreign/unregistered includes non-US registered aircraft involved in acci- dents in the United States.

Highwa y 37,133

National T ranspor tation Saf ety Boar d US T ranspor tation F atalities in 2017 1 – b y Mode To tal: 38,958 Gr ade Cr ossings

273

Rail 761 Marine 694 A viation

350 Pipeline 20

Pedalcycles2. 783

Medium and hea

vy trucks. 841 Buses. . . 44 Other3. . . 765 Passenger cars 13,363

Light trucks and v ans 10,188

Pedestrians 5,977

Motorcycles 5,172

Rail transitȶ. . 77 Freight, passenger, and commuter railȴ 684 [512 were trespassingȴ] Liquid. . . 1

Gas 19

Cargo transport. . . 9 Commercial fishingȷ. . . 27 Commercial passenger. . . . 0

Recreational boating 658 Airlines. . . 0 Air Taxi. . . 15 Commuter. . . . 1 Foreign/ unregistered10. . 4

General aviation 330

)LJXUH 8QLWHG 6WDWHV WUDQVSRUWDWLRQ IDWDOLWLHV LQ 6RXUFH KWWSVZZZQWVEJRY

)LJXUH 6RXWKZHVW $LUOLQHV )OLJKW OHIW HQJLQH &)0% DIWHU IDLOXUH 6RXUFH 176% UHSRUW >@

)RU UDWHGHSHQGHQW PDWHULDOV WKH WHVW FDPSDLJQ LWVHOI LV PRUH FKDOOHQJLQJ EHFDXVH D KLJK QXPEHU RI H[SHULPHQWV LV QHFHVVDU\ ZLWK WKHVH DSSURDFKHV WR FKDUDFWHUL]H PDWHULDO EHKDYLRXUV RYHU ODUJH VSHFWUD RI VWUDLQ DQG VWUDLQ UDWH

2ZLQJ WR WKH GUDPDWLF LPSURYHPHQWV LQ LPDJLQJ WHFKQRORJ\ LW LV QRZ SRVVLEOH WR PHDVXUH NLQH PDWLF ILHOGV DW D ODUJH QXPEHU RI SRLQWV RYHU VSHFLPHQ VXUIDFH RIWHQ UHIHUUHG WR DV tIXOOILHOG PHDVXUH PHQWVu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

XVLQJ D OLPLWHG QXPEHU RI G\QDPLF WHVWV >@

,PDJH%DVHG ,QHUWLDO ,PSDFW ,%,, WHVWV KDYH EHHQ UHFHQWO\ SURSRVHG WR WDNH DGYDQWDJH RI WKH 9)0

IRU PDWHULDO EHKDYLRXU FKDUDFWHUL]DWLRQ DW KLJK VWUDLQ UDWH >@ XS WR DERXW V

,W FRQVLVWV LQ D

WKLQSODWH VSHFLPHQ KLW E\ D F\OLQGULFDO SURMHFWLOH RQ LWV HGJH DW VHYHUDO WHQV RI PHWHUV SHU VHFRQG ZKLOH

WKH RSSRVLWH HGJH LV IUHH WR PRYH WKLV OHDGLQJ WR D ORQJLWXGLQDO ZDYH SURSDJDWLRQ WKURXJK WKH WHVWHG VSHFLPHQ 7KLV PHWKRGRORJ\ PD\ EH VXLWDEOH WR GUDVWLFDOO\ UHGXFH WKH WHVW FDPSDLJQ ,QGHHG KHWHUR JHQHRXV VWUDLQ DQG VWUDLQ UDWH ILHOGV PD\ EH JHQHUDWHG RQ D VLQJOH VSHFLPHQ KHQFH D FKDUDFWHUL]DWLRQ RYHU D ZLGH UDQJH RI VWUDLQ DQG VWUDLQ UDWH ZRXOG EH SRVVLEOH 8S WR QRZ ,%,, WHVWV KDYH EHHQ XVHG PRVWO\ WR FKDUDFWHUL]H HODVWLF EHKDYLRXU >@ RU HODVWRSODVWLF EHKDYLRXU >@

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

7KH IROORZLQJ IRXU FKDSWHUV SUHVHQW WKH PHWKRGRORJ\ IROORZHG LQ WKLV ZRUN IRU WKLV H[WHQVLRQ w &KDSWHU SUHVHQWV D EULHI RYHUYLHZ DERXW WKH H[LVWLQJ OLWHUDWXUH IRU PDWHULDO PHFKDQLFDO EHKDYLRXU

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

w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

w &KDSWHU LV GHGLFDWHG WR WKH H[SORLWDWLRQ RI WKH QXPHULFDO WRROER[ WR RSWLPL]H ,%,, WHVWV IRU -RKQVRQ

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

w &KDSWHU SUHVHQWV WKH WHVW FDPSDLJQ ZKLFK KDYH EHHQ FDUULHG RXW ZLWK WKH ,%,, PHWKRGRORJ\ WR FKDUDFWHUL]H WKH VWUDLQUDWH VHQVLWLYLW\ RI D WLWDQLXP DOOR\ 7L$O9 XVHG LQ DHURVSDFH LQGXVWU\ $ VHULHV RI ,%,, WHVWV DUH SHUIRUPHG DW WKH 8QLYHUVLW\ RI 6RXWKDPSWRQ 8. FRQVLGHULQJ GLIIHUHQW JHRPHWULHV GHILQHG ZLWK WKH QXPHULFDO DQDO\VLV 7KH G\QDPLF 9)0 LV XVHG WR LGHQWLI\ WKH -RKQVRQ

&RRN PRGHO YLVFRSODVWLF SDUDPHWHU 3ULRU WR WKH LGHQWLILFDWLRQ PHWURORJLFDO SHUIRUPDQFH LV DOVR DVVHVVHG )LQDOO\ D FULWLFDO FRPSDULVRQ EHWZHHQ QXPHULFDO DQDO\VLV DQG ,%,, WHVWV UHVXOWV LV SHUIRUPHG 7KLV LQFOXGHV WKH NLQHPDWLF ILHOGV WKH LGHQWLILHG SDUDPHWHUV WKH UHFRQVWUXFWHG PHFKDQLFDO ILHOGV DQG WKH YLVFRSODVWLF VSHFWUD

0DWHULDO EHKDYLRXU PRGHOOLQJ DQG

LGHQWLILFDWLRQ RI FRQVWLWXWLYH SDUDPHWHUV $Q RYHUYLHZ

&RQWHQWV

0DWHULDO EHKDYLRXU PRGHOOLQJ ,GHQWLILFDWLRQ ZLWK VWDWLFDOO\ GHWHUPLQHG WHVWV )XOO ILHOG PHDVXUHPHQWV WHFKQLTXHV 'LVSODFHPHQW ILHOGV PHDVXUHPHQW WHFKQLTXHV 8OWUD+LJK6SHHG LPDJLQJ ,GHQWLILFDWLRQ ZLWK VWDWLFDOO\ XQGHWHUPLQHG DSSURDFKHV 2YHUYLHZ 7KH 9LUWXDO )LHOGV 0HWKRG ZLWK H[WHUQDO ORDG PHDVXUHPHQW 7KH 9LUWXDO )LHOGV 0HWKRG ZLWK DFFHOHUDWLRQ DV D ORDG FHOO 6XPPDU\

$EVWUDFW

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

0DWHULDO EHKDYLRXU PRGHOOLQJ

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

GXFWLOH RWKHUZLVH

1% 7KLV VHFWLRQ LV GHGLFDWHG WR PHWDOOLF PDWHULDOV ZLWK DQ LQFRPSUHVVLEOH LVRWURSLF PHFKDQLFDO EHKDYLRXU 7KH IUDPHZRUN RI VPDOO GHIRUPDWLRQV LV DOVR FRQVLGHUHG

&RQVWLWXWLYH PRGHOV KDYH EHHQ GHYHORSHG WR GHVFULEH WKH PHFKDQLFDO EHKDYLRXU RI PDWHULDOV 7KHQ D VWUDLQ WHQVRU GHQRWHG ε LV GHILQHG IURP GLVSODFHPHQW ILHOG u DFFRUGLQJ WR (TXDWLRQ

ε = 1 2

∇ u + (∇ u )

+\SRWKHVLV RI DGGLWLYH GHFRPSRVLWLRQ FRQVLGHUV ε DV WKH VXP RI DQ HODVWLF VWUDLQ ε

DQG DQ LQHODVWLF RQH ε

VR WKDW ε = ε

+ ε

ZKHUH ε

= ε

IRU SODVWLF EHKDYLRXU DQG ε

= ε

IRU YLVFRSODVWLF EHKDYLRXU 0RUHRYHU &DXFK\ VWUHVV WHQVRU σ LV DOVR FRQVLGHUHG 7KH ODWWHU LV OLQNHG WR ε E\ FRQVWLWXWLYH HTXDWLRQV ZKLFK GHVFULEH WKH PDWHULDO EHKDYLRXU

$VVXPLQJ OLQHDU HODVWLF PDWHULDO EHKDYLRXU XQGHU LVRWKHUP FRQGLWLRQV σ LV UHODWHG WR ε E\ WKH VR FDOOHG +RRNHsV ODZ VXFK DV σ = C : ε

ZKHUH C LV WKH VWLIIQHVV WHQVRU )RU LVRWURSLF EHKDYLRXU WKH FRQWUDFWHG SURGXFW FDQ EH GHYHORSHG WR JHW D PRUH VXLWDEOH IRUPXOD (TXDWLRQ

σ = 2 μ ( ε

) + λ 7U( ε

) I

$V VRRQ DV \LHOG VWUHVV LV UHDFKHG WKLV VROH HTXDWLRQ LV QRW DEOH WR PRGHO WKH PHFKDQLFDO EHKDYLRXU DQ\PRUH 3ODVWLF RU YLVFRSODVWLF PRGHOV SRVVLEO\ QRQOLQHDU DUH WKHQ VHW XS WR GHVFULEH PDWHULDO KDUGHQLQJ 7KH UHODWHG FRQVWLWXWLYH UHODWLRQV DUH PDLQO\ EDVHG RQ WKHUPRG\QDPLFV HTXDWLRQV >@

LH LQWHUQDO DQG REVHUYDEOH VWDWH YDULDEOHV DUH FRQVLGHUHG )RU H[DPSOH WRWDO VWUDLQ LV DQ REVHUYDEOH YDULDEOH :KHQ SODVWLFLW\ LV GHVFULEHG HODVWLF DQG SODVWLF VWUDLQ DUH FRQVLGHUHG DV LQWHUQDO YDULDEOHV DV ZHOO DV RWKHU YDULDEOHV GHQRWHG V

OLNH NLQHPDWLF KDUGHQLQJ YDULDEOHV IRU LQVWDQFH (DFK LQWHUQDO YDULDEOH V

LV UHODWHG WR DQ DVVRFLDWH RQH A

7KH H[LVWHQFH RI D GLVVLSDWLRQ SRWHQWLDO F ( A

) LV WKHQ DVVXPHG 7KH ODWWHU JRYHUQV WKH HYROXWLRQ RI LQWHUQDO YDULDEOHV DFFRUGLQJ WR WKH VWDWH HTXDWLRQV DQG QRUPDOLW\ DVVXPSWLRQ (TXDWLRQ

V ˙

= − γ ˙ ∂F

∂A

, γ ∈ R

)RU DVVRFLDWHG SODVWLFLW\ F FDQ EH FKRVHQ DV WKH \LHOG VXUIDFH GHQRWHG f :LWKLQ VWUHVV VSDFH

f GHOLPLWV WKH GRPDLQ LQ ZKLFK DQ\ VWUHVV YDULDWLRQ OHDGV WR HODVWLF VWUDLQV RQO\ 7KLV WKUHVKROG LV

UHDFKHG ZKHQ DQ HTXLYDOHQW VWUHVV GHQRWHG σ

HTXDOV WKH IORZ VWUHVV σ

ZKRVH HYROXWLRQ LV GHILQHG

DFFRUGLQJ WR WKH PRGHOHG EHKDYLRXU :LWKLQ J

SODVWLFLW\ IUDPHZRUN ZKLFK LV RIWHQ VXLWDEOH IRU

PHWDOOLF PDWHULDOV LW FRPHV f = σ

− σ

ZKHUH σ

LV WKH YRQ0LVHV HTXLYDOHQW VWUHVV 7KHQ \LHOG

VXUIDFH HTXDWLRQ FDQ EH GHWDLOHG (TXDWLRQ :KLOH D β YDULDWLRQ OHDGV WR \LHOG VXUIDFH WUDQVODWLRQ LH NLQHPDWLF KDUGHQLQJ VKDSH PRUSKLQJ LH LVRWURSLF KDUGHQLQJ LV GULYHQ E\ r )LJXUH

f = σ

− σ

( r ) , ZKHUH

σ

= 3

2

S − β :

S − β

WKH HTXLYDOHQW VWUHVV

S = σ − 1

3 7U( σ ) WKH VWUHVV GHYLDWRU

β, r WKH KDUGHQLQJ YDULDEOHV

β ¯ σ

σ

σ

df = 0

f < 0 f

σ

(¯ σ − β) ¯

r

)LJXUH <LHOG VXUIDFH UHSUHVHQWDWLRQ ZLWKLQ SULQFLSDO VWUHVV VSDFH GRWWHG OLQH LQLWLDO \LHOG VXUIDFH WKLFN OLQH XSGDWHG \LHOG VXUIDFH

)RU UDWHLQGHSHQGHQW SODVWLFLW\ DGPLVVLEOH VWUHVV VWDWHV PXVW EH ORFDWHG ZLWKLQ WKH \LHOG VXUIDFH LH f ≤ 0 NQRZLQJ WKDW f = 0 PHDQV D SODVWLF IORZ $QRWKHU FRQGLWLRQ LV WKDW f ˙ = 0 GXULQJ WKH ZKROH ORDGLQJ SURFHVV 7KXV DQ LQFUHDVH RI VWUHVV OHYHO LPSOLHV D VLPXOWDQHRXV LQFUHDVH RI WKH IORZ VWUHVV )LJXUH )XUWKHUPRUH f = 0 RU f < ˙ 0 FRUUHVSRQGV WR HODVWLF XQORDGLQJ VWDWHV ZKLOH SODVWLF YDULDEOHV UHPDLQ FRQVWDQW 7KH FRQGLWLRQ f ˙ = 0 FDQ EH GHYHORSHG WR EHFRPH WKH FRQVLVWHQF\ FRQGLWLRQ VXFK DV

df = ∂f

∂

σ − β : d

σ − β

+ ∂f

∂r : dr = 0

(TXDWLRQ LV XVHG WR FDOFXODWH SODVWLF PXOWLSOLHU γ XVLQJ WKH VWDWH HTXDWLRQV (TXDWLRQ DSSOLHG WR DVVRFLDWHG SODVWLFLW\ 7KHQ LW FRPHV

∂ε

∂t = ˙ γ ∂f

∂σ

∂r

∂t = − γ ˙ ∂f

∂σ

:LWKLQ J

YLVFRSODVWLFLW\ IUDPHZRUN DQG LQ DEVHQFH RI NLQHPDWLF KDUGHQLQJ β = 0 (TXDWLRQ FDQ

EH VLPSOLILHG ,QGHHG

∂ε

∂t =

γ ˙ ∂f

∂σ = 3

2 γ ˙ ∂

∂σ

S : S =

3 2 γ ˙

∂

∂σ

S : S 2

S : S

= 3

2 γ ˙ 2 S 2

S : S = 3

2 γ ˙ S 3

2 S : S

= 3 2 γ ˙ S

σ

( σ )

7KH VHW RI FRQGLWLRQV IRU DGPLVVLEOH VWUHVV VWDWHV ZLWKLQ UDWHLQGHSHQGHQW SODVWLFLW\ IUDPHZRUN DUH NQRZQ DV .XKQ7XFNHU FRQGLWLRQV (TXDWLRQ

˙ γ ≥ 0 f ≤ 0

˙

γf = 0

)LQDOO\ FXPXODWHG SODVWLF VWUDLQ p LV GHILQHG E\ (TXDWLRQ )RU D YLUJLQ LH QRW GDPDJHG PDWH ULDO p LV HTXDO WR LVRWURSLF KDUGHQLQJ YDULDEOH r DQG LV WKHUHIRUH OLQNHG WR γ DV GHILQHG E\ (TXDWLRQ

p =

0

2

3 ε ˙

: ˙ ε

dt

:LWKLQ J

YLVFRSODVWLFLW\ IUDPHZRUN DQG LQ DEVHQFH RI NLQHPDWLF KDUGHQLQJ p ˙ LV GLUHFWO\ OLQNHG WR γ ˙ ,QGHHG

˙ p =

2

3 ε ˙

: ˙ ε

=

3 2 γ ˙

S : S

σ

( σ ) = ˙ γ

6HOHFWLRQ RI VXLWDEOH FRQVWLWXWLYH ODZV WR GHVFULEH PDWHULDO EHKDYLRXU VWURQJO\ GHSHQGV RQ PDWHULDO LWVHOI DQG ORDGLQJ FRQGLWLRQV 7R GHVFULEH LVRWURSLF KDUGHQLQJ IRU UDWHLQGHSHQGHQW EHKDYLRXU VHYHUDO PRGHOV KDV EHHQ SURSRVHG LQ WKH OLWHUDWXUH *HQHUDOO\ VSHDNLQJ WKH\ DUH VXFK DV σ

( p ) = σ

+ R ( p ) ZLWK σ

WKH LQLWLDO \LHOG VWUHVV DQG R WKH KDUGHQLQJ VWUHVV )RU LQVWDQFH WKH 3UDQGWO5HXVV PRGHO >@

DVVXPHV D SRZHU KDUGHQLQJ ODZ R ( p ) = Kp

ZKHUH K DQG n DUH PDWHULDO SDUDPHWHUV NQRZQ DV LVRWURSLF KDUGHQLQJ PRGXOXV DQG H[SRQHQW UHVSHFWLYHO\ 9RFH PRGHO >@ FRPELQHV D OLQHDU KDUG HQLQJ DQG DQ H[SRQHQWLDO FRUUHFWLRQ VXFK DV R ( p ) = Kp + σ

[1 − e

] ZKHUH K σ

DQG b DUH PDWHULDO SDUDPHWHUV $UPVWURQJ)UHGHULFN PRGHO >@ LV DQ H[DPSOH RI D PRGHO IRU NLQHPDWLF KDUG HQLQJ ZKHUH

C

=

ε ˙

−

˙ ε

β % DQG & DUH PDWHULDO SDUDPHWHUV /LQHDU 3UDJHU PRGHO >@

IRU NLQHPDWLF KDUGHQLQJ LV RIWHQ HQFRXQWHUHG VXFK DV

=

H

∂t

ZKHUH H GHQRWHV WKH NLQHPDWLF KDUGHQLQJ PRGXOXV

:LWKLQ YLVFRSODVWLFLW\ IUDPHZRUN WKH VDPH YDULDEOHV FDQ EH QDWXUDOO\ GHILQHG VXEVWLWXWLQJ D YLV FRSODVWLF VWUDLQ WHQVRU GHQRWHG ε

WR WKH SODVWLF VWUDLQ WHQVRU ε

<HW FRQVLVWHQF\ FRQGLWLRQV (TXD WLRQ DUH QRW PDQGDWRU\ DQ\PRUH VLQFH f ˙ PLJKW EH QRQ]HUR *HQHUDOO\ VSHDNLQJ WKH ZKROH VHW RI .XKQ7XFNHU FRQGLWLRQV (TXDWLRQ PLJKW EH XQYHULILHG GHSHQGLQJ RQ KRZ YLVFRXV HIIHFWV DUH PRGHOOHG

$ ILUVW DSSURDFK FRQVLVWV LQ GHVFULELQJ WKH GHYHORSPHQW RI YLVFRXV IORZ VWUHVV DQG WKH HYROXWLRQ

RI SODVWLF IORZ VHSDUDWHO\ DV LW LV GRQH IRU WKH 1RUWRQsV PRGHO > @ $QRWKHU DSSURDFK FRQVLVWV

LQ LQFRUSRUDWLQJ YLVFRXV HIIHFWV LQ WKH H[SUHVVLRQ RI σ

ZLWK PRGHOOLQJ RI YLVFRXV RYHUVWUHVV HIIHFWV

.XKQ7XFNHU FRQGLWLRQV (TXDWLRQ PD\ WKHQ EH YDOLG VR WKDW f = σ

− σ

≤ 0 6RPH DYDLODEOH

PRGHOV DUH EDVHG RQ WKLV PHWKRGRORJ\ >@ $Q H[DPSOH LV WKH -RKQVRQ&RRN -& PRGHO >@ ZKLFK LQWHJUDWHV ERWK YLVFRSODVWLFLW\ DQG WKHUPDO HIIHFWV (TXDWLRQ ,Q WKH IORZ UXOH WKH FRQWULEXWLRQ RI HDFK WHUP LV DFWLYDWHG IURP D WKUHVKROG YDOXH 7KXV WKHUPDO VRIWHQLQJ LV DFWLYDWHG ZKHQ WKH WHPSHU DWXUH T UHDFKHV T

ZKLFK LV D PDWHULDO SDUDPHWHU 3DUDPHWHUV m DQG T

PHOWLQJ WHPSHUDWXUH XQGHU 6&73 WKHQ GULYH WKH WKHUPDO VRIWHQLQJ /LNHZLVH YLVFRXV HIIHFWV DUH DFWLYDWHG ZKHQ FXPXODWHG SODVWLF VWUDLQUDWH GHQRWHG p ˙ LV KLJKHU WKDQ D WKUHVKROG ε ˙

ZKLOH -RKQVRQ&RRN YLVFRSODVWLF SDUDPHWHU M GULYHV WKH HYROXWLRQ RI WKH VWUHVV FRUUHFWLRQ FDXVHG E\ YLVFRXV KDUGHQLQJ )XUWKHUPRUH LW FDQ EH QRWHG WKDW EHORZ WKH YLVFRSODVWLF WKUHVKROG DQG LQ DEVHQFH RI WKHUPDO HIIHFWV -& PRGHO LV HTXLYDOHQW WR WKH 3UDQGWO5HXVV KDUGHQLQJ PRGHO ZLWK σ

WKH LQLWLDO \LHOG VWUHVV K DQG n WKH LVRWURSLF OLQHDU KDUGHQLQJ PRGXOXV DQG H[SRQHQW UHVSHFWLYHO\

σ

=

σ

+ Kp

1 + M OQ p ˙

˙ ε

1 −

T − T

T

− T

,Q SUDFWLFH -& PRGHO KDV EHHQ ZLGHO\ XVHG WR GHVFULEH WKH G\QDPLF EHKDYLRXU RI LVRWURSLF PHWDO OLF PDWHULDOV XVLQJ VWDWLFDOO\ GHWHUPLQDWH WHVWV >w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

SODVWLFLW\ IUDPHZRUN LH YRQ0LVHV GULYHQ KDUGHQLQJ

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

,GHQWLILFDWLRQ ZLWK VWDWLFDOO\ GHWHUPLQHG WHVWV

'XULQJ D PHFKDQLFDO WHVW VRPH RI SK\VLFDO YDULDEOHV FDQ EH TXDQWLILHG GLUHFWO\ HJ ORDGV XVLQJ G\

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

7KHRUHWLFDOO\ VSHDNLQJ DQ\ H[SHULPHQWDO WHVW PLJKW EH SURFHVVHG EDVHG RQ WKLV DSSURDFK 1HYHU WKHOHVV WKHVH DVVXPSWLRQV PD\ EH GLIILFXOW WR EH UHVSHFWHG LQ SUDFWLFH QRWDEO\ ZKHQ PDWHULDO EHKDYLRXU LV FKDUDFWHUL]HG XQGHU H[WUHPH FRQGLWLRQV HJ G\QDPLF ORDGLQJ YHU\ KLJKORZ DPELHQW WHPSHUDWXUH )RU LQVWDQFH JHQHUDWLRQ RI KRPRJHQHRXV ILHOGV RYHU JDXJH OHQJWK LV RIWHQ FKDOOHQJLQJ &RQVHTXHQWO\

RQH KDV WR WKLQN ZLVHO\ DERXW WHVWHG FRQILJXUDWLRQV IHDWXUHV HJ ORDGLQJ SDWK VSHFLPHQ JHRPHWU\

DQG FRQVWLWXWLYH HTXDWLRQV YDOLGLW\ GRPDLQ %DVHG RQ WKHVH LGHDV VWDQGDUG SURFHGXUHV KDYH EHHQ HV WDEOLVKHG WR tHQVXUHu WKH UHVSHFW RI VWDWLFLW\ K\SRWKHVHV ,Q SDUWLFXODU WKRVH VWDQGDUGV

VHW VSHFLPHQ JHRPHWU\ VHQVRUV VHWXS DQG ERXQGDU\ FRQGLWLRQV )XUWKHUPRUH RQH RI WKH PRVW FULWLFDO FRQVWUDLQWV IRU KLJK VWUDLQUDWH WHVWLQJ LV WKH VWUDLQUDWH ZKLFK KDV WR UHPDLQ FRQVWDQW DORQJ WKH JDXJH OHQJWK 7KHQ WKH PDWHULDO EHKDYLRXU LV FKDUDFWHUL]HG IRU D JLYHQ VWUDLQUDWH

7HQVLOH WHVWV DUH RIWHQ XVHG IRU PDWHULDO EHKDYLRXU FKDUDFWHUL]DWLRQ :KHQ VWDWLFLW\ K\SRWKHVHV DUH DVVXPHG PDWHULDO SDUDPHWHUV FDQ EH H[WUDFWHG XVLQJ JDXJH OHQJWK HORQJDWLRQ KHUH GHQRWHG Δ L DQG ORDG GHQRWHG F 1RPLQDO VWUDLQ ε

LV GHILQHG DV JDXJH OHQJWK UHODWLYH HORQJDWLRQ LH ε

=

ZKHUH t LV WKH WLPH DQG L

WKH LQLWLDO JDXJH OHQJWK 1RPLQDO VWUHVV σ

LV WKH UDWLR EHWZHHQ WKH ORDG F DQG WKH LQLWLDO FURVVVHFWLRQ S

+RZHYHU VWDWLFLW\ K\SRWKHVLV EHFRPH PHDQLQJOHVV ZKHQ PDWHULDO XQGHUJR SK\VLFDO ORFDOLVDWLRQ )RU WHQVLOH WHVWV DVVXPSWLRQV DUH WKHQ YDOLG XQWLO QHFNLQJ ,Q SUDFWLFH WUDQVLWLRQ LV VHW ZKHQ

= 0 )LJXUH

Strain

Stress —–

—–

Beginning of necking:

end of staticity hypothesis validity domain

)LJXUH 1RPLQDO7UXH WHQVLOH EHKDYLRXU

1RQHWKHOHVV WKH QRPLQDO EHKDYLRXU GRHV QRW PDWFK ZLWK PDWHULDO tWUXHu EHKDYLRXU ,Q IDFW WUXH VWUDLQ ε LV GHILQHG DV DFFXPXODWLRQ RI UHODWLYH HORQJDWLRQ (TXDWLRQ ,Q WKH FDVH RI PHWDOOLF PDWHULDOV DVVXPHG LQ 1% S WUXH VWUHVV σ PD\ EH FDOFXODWHG XVLQJ (TXDWLRQ

ε

=

L(t1)

dL L = OQ

L ( t

) L ( t

)

= OQ(1 + ε

)

σ ( t ) = F ( t )

S ( t ) = F ( t )

S

(1 + ε

)

)LJXUH VKRZV D VFKHPDWLF WUXH VWUHVVVWUDLQ FXUYH IRU DQ HODVWRSODVWLF LVRWURSLF PDWHULDO $VVXP LQJ XQLGLPHQVLRQDO VWUHVV VWDWH WKH <RXQJsV PRGXOXV E DQG LQLWLDO \LHOG VWUHVV σ

DUH FDOFXODWHG XVLQJ FXUYH OLQHDU SDUW DQG +RRNHsV ODZ σ = Eε 3ODVWLF VWUDLQ LV WKHQ GHGXFHG WKDQNV WR DGGLWLYH GHFRP SRVLWLRQ DVVXPSWLRQ VXFK DV ε

= ε −

+DUGHQLQJ PRGXOXV DQG H[SRQHQW K DQG n UHVSHFWLYHO\

FDQ EH GHWHUPLQHG XVLQJ D ORJDULWKPLF GLDJUDP DV LQ )LJXUH D :KHQ WKH PDWHULDO EHKDYLRXU LV UDWHGHSHQGHQW VHYHUDO WHVWV QHHGV WR EH SHUIRUPHG DW GLIIHUHQW VWUDLQUDWH ε ˙ >@ )RU HDFK WHVW VWUDLQ UDWH PXVW EH KRPRJHQHRXV RYHU WKH 52, WKLV EHLQJ PRUH DUH PRUH FKDOOHQJLQJ DV ε ˙ LV LQFUHDVLQJ >@

7UXH VWUHVVVWUDLQ FXUYHV DUH WKHQ DQDO\VHG IRU HDFK WHVW )RU LQVWDQFH YLVFRSODVWLF SDUDPHWHUV M HW

˙

ε

RI (TXDWLRQ FDQ WKHQ EH H[WUDFWHG DV VKRZQ LQ )LJXUH E 3UHYLRXVO\ VWDWLF SDUW PD\ KDYH EHHQ LGHQWLILHG XVLQJ WKH PHWKRG LOOXVWUDWHG LQ )LJXUH D

7KH PDLQ VWDQGDUGV DUH $)125 1) (1 WR )UDQFH $670 (0 86 RU ,62 ,QWHUQDWLRQDO 7KH\ DUH GHWDLOHG GHSHQGLQJ RQ WKH WHVWHG PDWHULDO DQG WKH WHVW FRQGLWLRQV )RU LQVWDQFH VWDQGDUG ,62 VHW D WHVW DW tDPELHQW WHPSHUDWXUHu VWDQGDUG ,62 ­ DW tKLJK WHPSHUDWXUHu IRU PHWDOOLF PDWHULDOV

D

E

)LJXUH 'HWHUPLQDWLRQ RI -& PRGHO SDUDPHWHUV IRU D WLWDQLXP DOOR\ >@ D VWDWLF SDUDPHWHUV E G\QDPLF SDUDPHWHUV

*HQHUDOO\ VSHDNLQJ PDWHULDO SDUDPHWHUV FDQ DOVR EH H[WUDFWHG E\ PLQLPL]LQJ D JOREDO FRVW IXQFWLRQ ϕ ZKLFK PD\ EH GHILQHG DV D VXP RI QRUPDOL]HG VTXDUH HUURUV EHWZHHQ WKH VWUHVV FXUYH FDOFXODWHG ZLWK WKH PRGHO σ

DQG H[SHULPHQWDO WUXH VWUHVV FXUYH σ

(TXDWLRQ ZKHUH X GHQRWHV WKH VHW RI PDWHULDO SDUDPHWHUV HJ X = [ σ

, K, n, M, ε ˙

] IRU -& PRGHO DQG N WKH QXPEHU RI VDPSOLQJ SRLQWV )RU QRQOLQHDU EHKDYLRXU RQH ZD\ WR FDOFXODWH WKH VWUHVV PLJKW EH EDVHG RQ DOJRULWKPV XVHG WR SHUIRUP WKH QXPHULFDO WLPHLQWHJUDWLRQ RI FRQVWLWXWLYH HTXDWLRQV > @

ϕ ( X ) =

i

σ

( i ) − σ

( i )( X ) σ

( i )

1HOGHU0HDG KHXULVWLF DOJRULWKP >@ LV RIWHQ VXLWDEOH WR PLQLPL]H IXQFWLRQ ϕ LQVRIDU DV GLUHFWLRQ RI UHVHDUFK FRPSXWDWLRQ GRHV QRW UHTXLUH FRVWIXQFWLRQ GHULYDWLYHV ZUW PDWHULDO SDUDPHWHUV 'HVSLWH WKH GUDZEDFNV XQFRQVWUDLQHG UHVHDUFK OHVV UREXVW IRU LOOFRQGLWLRQHG IXQFWLRQV WKH PDLQ DGYDQWDJH LV WKH OLPLWHG QXPEHU RI LWHUDWLRQV FRPSDUHG WR JHQHWLF DOJRULWKPV >w@ ZKLFK DUH PRUH UREXVW HQVXUHG FRQYHUJHQFH WRZDUGV JOREDO PLQLPD EXW OHVV FRVW HIILFLHQW DQG OHVV DFFXUDWH IRU PXOWLGLPHQ VLRQDO SUREOHPV

,Q FRQFOXVLRQ H[WUDFWLQJ PDWHULDO SDUDPHWHUV ZLWK VWDQGDUG WHFKQLTXHV tRQO\u UHTXLUHV WR PHDVXUH ORDG DQG GLVSODFHPHQW RU VWUDLQ 1RQHWKHOHVV WKH XVH RI UHODWHG SURFHGXUHV LV FKDOOHQJLQJ EHFDXVH RI WKH VWURQJ DVVXPSWLRQV $V PHFKDQLFDO WHVW VHWWLQJV DUH WKHQ FRQVWUDLQHG VHYHUDO WHVWV DUH JHQHUDOO\

QHHGHG IRU LGHQWLILFDWLRQ RI FRPSOH[ EHKDYLRXU DV VKRZQ KHUH IRU WKH FDVH RI YLVFRSODVWLFLW\ EXW DOVR WR LGHQWLI\ GDPDJH PRGHOV DV GRQH E\ %HQQDQL HW DO >@ IRU LQVWDQFH 7KH QXPEHU RI UHTXLUHG WHVWV

LQFUHDVHV ERWK ZLWK FRQVWLWXWLYH HTXDWLRQV FRPSOH[LW\ DQG SUREOHP GLPHQVLRQ 0RUHRYHU WKH LGHQWL ILHG PRGHO LV RQO\ YDOLG ZLWKLQ WKH GRPDLQ IUDPHG E\ WKH VWDWLFLW\ DVVXPSWLRQV LH D ODUJH SRUWLRQ RI H[SHULPHQWDO FXUYHV FDQQRW EH H[SORLWHG

,Q D QXWVKHOO WKH FROOHFWHG GDWD SRWHQWLDO LV FOHDUO\ XQGHUH[SORLWHG ZKHQ XVLQJ WKRVH DSSURDFKHV ,PSURYHPHQW FRXOG FRPH IURP WKH XVH RI KHWHURJHQHRXV ILHOGV LH VWDWLFDOO\ XQGHWHUPLQDWH DS SURDFKHV ,QGHHG WKDQNV WR WKH FRQWLQXRXV LPSURYHPHQW RI VHQVRUV IXOOILHOG PHDVXUHPHQW WHFK QLTXHV DUH PRUH DQG PRUH VXLWDEOH IRU NLQHPDWLF ILHOGV H[WUDFWLRQ LQ PDWHULDO WHVWLQJ &RQVHTXHQWO\

RQH PD\ WDNH DGYDQWDJH RI WKH FRQVLGHUDEO\ HQULFKHG H[SHULPHQWDO GDWD DW EHVW )RU LQVWDQFH ZLGHU VSHFWUD RI VWUDLQ DQG VSHFLDOO\ VWUDLQUDWH PLJKW EH HQFRXQWHUHG ZKLOH SURFHVVLQJ D VLQJOH VSHFLPHQ 7KXV WKLV PD\ VWURQJO\ UHGXFH WKH QXPEHU RI WHVWV QHHGHG IRU PDWHULDO FKDUDFWHUL]DWLRQ

7KH QH[W VHFWLRQ SUHVHQWV VRPH RI VXFK PHDVXUHPHQW WHFKQLTXHV DQG UHFHQW DGYDQFHV LQ WKHLU SUDF WLFH LQ PDWHULDO FKDUDFWHULVDWLRQ 0HWKRGV VXLWDEOH WR WKHLU SURFHVV LH VWDWLFDOO\ XQGHWHUPLQDWH ZLOO DOVR EH SUHVHQWHG

)XOO ILHOG PHDVXUHPHQWV WHFKQLTXHV

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

LQIUDUHG WKHUPRJUDSK\ ZDV DOVR XVHG IRU FUDFN GHQVLW\ DQDO\VLV ZLWKLQ WKH PDWUL[ RI D FRPSRVLWH >@

RU GLVVLSDWLYH SKHQRPHQD PRQLWRULQJ RQ D FRPSRVLWH MRLQW >@ DW LQWHUPHGLDWH VWUDLQUDWH 6WUDLQ ILHOGV FDQ EH H[WUDFWHG XVLQJ VKHDURJUDSK\ >@ VKHDU VSHFNOH SKRWRJUDSK\ >@ 0RLU¶ IULQJH GLV SODFHPHQW >@ RU WKH JULG PHWKRG > @ 1HYHUWKHOHVV VWUDLQ ILHOGV DUH RIWHQ FDOFXODWHG IURP WKH QXPHULFDO GLIIHUHQWLDWLRQ RI GLVSODFHPHQWV ZKLFK DUH WKH PRVW IUHTXHQWO\ PHDVXUHG PHFKDQLFDO ILHOGV 7KH IROORZLQJ SDUW GHVFULEHV VXLWDEOH WHFKQLTXHV IRU WKLV

'LVSODFHPHQW ILHOGV PHDVXUHPHQW WHFKQLTXHV

,QWHUIHURPHWULF PHWKRGV

,QWHUIHURPHWULF PHWKRGV FRQVLVWV LQ FRPSDULQJ WZR PRQRFKURPDWLF EHDPV FRPLQJ IURP WKH VDPH VRXUFH EXW ZLWK GLIIHUHQW RSWLFDO SDWKV :KLOH WKH ILUVW RQH HQFRXQWHUV QR REVWDFOHV UHIHUHQFH WKH VHFRQG RQH OLJKWV WKH WUDFNHG VSHFLPHQ VXUIDFH VR WKDW WKH GLVSODFHPHQW ILHOGV FDQ EH OLQNHG WR WKH GLIIHUHQFHV EHWZHHQ RSWLFDO SDWKV $V WKH YDULDWLRQ FRXOG EH YHU\ VPDOO HJ D IHZ QDQRPHWUHV LQ WHUIHURPHWULF PHWKRGV DUH VXLWDEOH WR H[WUDFW YHU\ VPDOO GLVSODFHPHQWV 1RQHWKHOHVV WKH\ DUH YHU\

VHQVLWLYH WR H[WHUQDO SHUWXUEDWLRQV DV YLEUDWLRQV IRU LQVWDQFH $PRQJ WKHP 0RLU¶ DQG VSHFNOH LQWHU IHURPHWU\ DUH WKH PRVW IUHTXHQWO\ XVHG E\ WKH FRPPXQLW\

)RU WKH ILUVW WHFKQLTXH D UHIHUHQFH JULG SDWWHUQ LV FRPSDUHG WR D FXUUHQW FRQILJXUDWLRQ LH GH IRUPHG GXULQJ WKH PHFKDQLFDO WHVW >@ 7KH JULG SLWFK FRXOG EH YHU\ VPDOO OLQHVPP >@

0RLU¶ IULQJHV DUH WKHQ REWDLQHG RYHUODSSLQJ WKH WZR JULG SDWWHUQV KHQFH WKH GLVSODFHPHQWV RQ WKH

VXUIDFH FDQ EH NQRZQ >@ 7KH VDPH PHWKRG FDQ DOVR EH XVHG WR FDOFXODWH RXW RI SODQH GLVSODFH

PHQWV >@ ,Q WKDW FDVH WKH JULG LV ERXQGHG SHUSHQGLFXODUO\ WR WKH VSHFLPHQ VXUIDFH +RZHYHU

0RLU¶ LQWHUIHURPHWU\ FDQ EH GLIILFXOW WR XVH LQ SUDFWLFH ,QGHHG WKH UHVXOWV DUH YHU\ VHQVLEOH WR RSWLFDO

QRLVHV ,Q DGGLWLRQ WR EH VWURQJO\ VHQVLWLYH WR H[WHUQDO SHUWXUEDWLRQV 0RLU¶ IULQJHV KLJKO\ GHSHQGV RQ

WKH PDWHULDO VXUIDFH VR WKDW VSHFLPHQ VXUIDFH QHHGV WR EH SROLVKHG 6SHFNOH LQWHUIHURPHWU\ >@ PLJKW

EH OHVV GHOLFDWH WR XVH LQVRIDU QR JULG KDV WR EH WUDQVIHUUHG RQWR WKH VSHFLPHQ VR QR SROLVKLQJ LV UH

TXLUHG +RZHYHU WKH PDWHULDO KDV WR GLVSHUVH OLJKW ,QGHHG WKH FRPELQDWLRQ RI OLJKW EHDPV UHVXOWLQJ

IURP WKH UHIOHFWLRQ RI FRKHUHQW VRXUFH RQ VSHFLPHQ VXUIDFH ZLWK D ODVHU EHDP JLYHV D GHVFULSWLRQ RI WKH VXUIDFH VSHFNOH SDWWHUQ RZLQJ WR SKDVH VKLIWLQJ 7KHQ GLVSODFHPHQWV FDQ EH H[WUDFWHG FRUUHODWLQJ WKH VSHFNOH SDWWHUQ EHIRUH DQG GXULQJ ORDGLQJ

7R FRQFOXGH LQWHUIHURPHWULF WHFKQLTXHV PD\ EH YHU\ HIILFLHQW +RZHYHU WKHLU XVH LQ SUDFWLFH PD\

EH FKDOOHQJLQJ IRU WHVWLQJ XQGHU H[WUHPH FRQGLWLRQV HJ G\QDPLF ORDGLQJ 7KXV ZKLWHOLJKWEDVHG WHFKQLTXHV ZKRVH VHWWLQJ LV HDVLHU KDYH EHHQ GHYHORSHG 7KH QH[W VHFWLRQ GHVFULEHV WZR RI WKHP :KLWH OLJKW PHWKRGV

¯ 7KH 'LJLWDO ,PDJH &RUUHODWLRQ PHWKRG

7KH 'LJLWDO ,PDJH &RUUHODWLRQ PHWKRG FRQVLVWV LQ WUDFNLQJ PRWLRQV DQG GHIRUPDWLRQV RI D UDQGRP VSHFNOH SDWWHUQ >w@ ,Q SUDFWLFH WKH ODWWHU PD\ EH REWDLQHG VSUD\LQJ EODFN GURSV RQWR D ZKLWH SDLQWHG VSHFLPHQ )LJXUH 5HFRUGHG LPDJHV DUH WKHQ DQDO\VHG QXPHULFDOO\ DV JUH\ OHYHO PDSV

$QRWKHU VROXWLRQ LV DOVR WR XVH PDWHULDO PLFURVWUXFWXUH LWVHOI DV D VSHFNOH LI JUH\ OHYHO JUDGLHQWV DUH VWURQJ HQRXJK IRU WKH SRVWSURFHVVLQJ >@ )LQDOO\ WKH VHW RI LPDJHV LV DQDO\VHG ZLWK D FRUUHODWLRQ DOJRULWKP

)LJXUH ([DPSOH RI D VSHFNOH SDWWHUQ

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¯ DQG G ¯ DYHUDJH YDOXHV RYHU S DQG i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w@

¯ 7KH JULG PHWKRG

7KH JULG PHWKRG >w@ LV D IXOOILHOG PHDVXUHPHQW PHWKRG VXLWDEOH IRU H[WUDFWLQJ ' RU ' NLQH PDWLF ILHOGV $V D ZKLWH OLJKW WHFKQLTXH LW LV EDVHG RQ LQWHQVLW\ DQDO\VLV RI D JULG ERQGHG RQWR WKH WHVWHG VSHFLPHQ *ULG SLWFK ZKLFK FRXOG EH 1 WR 1000+ OLQHV SHU PLOOLPHWUH LV FKRVHQ LQ DFFRUGDQFH ZLWK RSWLFDO VHWXS IHDWXUHV 7KH JULG FDQ EH HQJUDYHG SULQWHG RU JOXHG )LJXUH RQ WKH VXUIDFH >w@

7KH SURFHGXUH WR H[WUDFW GLVSODFHPHQWV IURP JULG LPDJHV LV GHVFULEHG DQG DQDO\VHG E\ *U¶GLDF HW DO >@

7DEOH 66'EDVHG ',& FULWHULD 1DPH &RVW IXQFWLRQ

66'

i∈S

F

− G

166'

i∈S

⎧ ⎨

⎩ F

j∈S

F

− G

j∈S

G

⎫ ⎬

⎭

2

=166'

i∈S

⎧ ⎨

⎩ F ¯

j∈S

F ¯

− G ¯

j∈S

G ¯

⎫ ⎬

⎭

2

)LJXUH ([DPSOH RI D JOXHG JULG SDWWHUQ 6RXUFH KWWSZZZWKHJULGPHWKRGQHW

$ TXDVLSHULRGLF IXQFWLRQ (TXDWLRQ LV DVVXPHG WR GHVFULEH WKH IULQJHV SDWWHUQ ZKHUH A DQG γ JRYHUQV JULG FRQWUDVW tIUQJu LV D SHULRGLF IXQFWLRQ ω LV WKH JULG SXOVDWLRQ DQG φ WKH YHFWRU FDUU\LQJ JULG PRGXODWLRQV :LWKLQ DQ (XOHULDQ IUDPHZRUN DV JULG PRYHPHQWV DUH UHVSRQVLEOH IRU YDULDWLRQV RI SKDVH PRGXODWLRQV RQH FDQ GHPRQVWUDWH WKDW GLVSODFHPHQW DQG SKDVH ILHOGV DUH UHODWHG DFFRUGLQJ WR DQ LPSOLFLW UHODWLRQ (TXDWLRQ ZKHUH H[SRQHQWV

DQG

GHQRWH UHIHUHQFH DQG GHIRUPHG VWDWH UHVSHFWLYHO\ )RU VPDOO SHUWXUEDWLRQV (TXDWLRQ PD\ EH OLQHDULVHG 7KHUHIRUH LW FRPHV u ( x ) = −

[ φ

( x ) − φ

( x )] ,Q FDVH RI ODUJH GLVSODFHPHQWV HJ ZKHQ ODUJH ULJLG ERG\ PRWLRQV RFFXUV GLVSODFHPHQWV FDQ EH FDOFXODWHG ZLWK DQ LWHUDWLYH DSSURDFK >@

I ( x, y ) = A

1 + γ IUQJ

ωx + φ

( x, y )

+ IUQJ

ωy + φ

( x, y )

u ( x ) = − 1 ω

φ

x + u ( x )

− φ

x

)XUWKHUPRUH SKDVH PRGXODWLRQV FDQ EH UHWULHYHG WKURXJK D /RFDO 6SHFWUXP $QDO\VLV /6$ >

@ $VVXPLQJ WKDW SKDVH WKH JULG GHIRUPDWLRQV FDUULHU LQIRUPDWLRQ LV FRQFHQWUDWHG ZLWKLQ VSLNHV LQ

WKH )RXULHU VSHFWUXP RI WKH PRQLWRUHG JULG /6$ FRQVLVWV LQ FRQYROYLQJ WKH JULG VLJQDO s ZLWK D ORFDO

DQDO\VLV ZLQGRZ g (TXDWLRQ WR H[WUDFW WKLV LQIRUPDWLRQ 7KXV LW FRPHV φ = $UJ[ˆ s ] + 2 nπ, n ∈

Z ,Q SUDFWLFH WKLV LV RIWHQ GRQH XVLQJ :LQGRZHG 'LVFUHWH )RXULHU 7UDQVIRUP :')7 >@ +RZ

HYHU SKDVH PDSV DUH DW WKLV PRPHQW GHILQHG PRGXOR 2 π $V SKDVH H[WUDFWLRQ LV SHUIRUPHG ORFDOO\ LQ

VSDFH DQDO\VLV ZLQGRZ DQG LQ WLPH GHIRUPHG LPDJHV DUH SURFHVVHG VHSDUDWHO\ SKDVH MXPSV RIWHQ

RFFXU 7KHQ SKDVH PDSV QHHGV WR EH XQZUDSSHG ERWK VSDWLDOO\ DQG WHPSRUDOO\ 7R GR VR ELGLPHQ

VLRQQDO >@ RU WULGLPHQVLRQDO >@ XQZUDSSLQJ DOJRULWKPV DUH XVHG WR FDQFHO SKDVH MXPSV DORQJ x s ˆ

( x, y, ω ) =

R²

s ( u, v ) g ( x − u, y − v ) H[S[− ωx ] dudv DORQJ y s ˆ

( x, y, ω ) =

R²

s ( u, v ) g ( x − u, y − v ) H[S[− ωy ] dudv

,Q SUDFWLFH WKH JULG PHWKRG LV OHVV VLPSOH WR XVH WKDQ WKH ',& PHWKRG EHFDXVH RI VHYHUDO FRQVWUDLQWV HJ UHJXODU SDWWHUQ JULG WUDQVIHU >@ +RZHYHU LWV PHWURORJLFDO SHUIRUPDQFH LH PHDVXUHPHQW UHVROXWLRQV PDNHV LW YHU\ LQWHUHVWLQJ IRU DSSOLFDWLRQV LQ FULWLFDO FRQGLWLRQV HJ KLJK VSHHG WHVWV ,QGHHG LW KDV EHHQ UHFHQWO\ DGGUHVVHG DV EHWWHU WKDQ WKH ',& LI D FRPSURPLVH EHWZHHQ VSDWLDO UHVROX WLRQ QXPEHU RI PHDVXUHPHQW SRLQWV DQG GLVSODFHPHQW UHVROXWLRQ VPDOOHVW WUDFNDEOH GLVSODFHPHQW LV ORRNHG IRU >@

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

8OWUD+LJK6SHHG LPDJLQJ

+LJK 6SHHG LPDJLQJ LV JHWWLQJ PRUH DQG PRUH DWWHQWLRQ LQ H[SHULPHQWDO PHFKDQLFV QRWDEO\ VLQFH WKH VSHFWDFXODU GHYHORSPHQW RI &KDUJHG &RXSOH 'HYLFHV &&' DQG &RPSOHPHQWDU\ 0HWDO 2[LGH 6HPL FRQGXFWRU &026 VHQVRUV DQG WKHLU LQWHJUDWLRQ LQ FRPPHUFLDO FDPHUDV $FWXDOO\ +LJKVSHHG +6 LPDJLQJ IUDPHUDWH !

ISV VHH )LJXUH DUH XVHG VLQFH VHYHUDO GHFDGHV +RZHYHU LPDJH VWRUDJH LV RIWHQ SHUIRUPHG RXWVLGH WKH FKLS GXULQJ LPDJLQJ VR WKDW GDWD RIIORDG UDWH LV WKH OLPLWLQJ IDFWRU WR LQFUHDVH IUDPH UDWH %\ FRQWUDVW 8OWUD+LJK6SHHG 8+6 LPDJLQJ KDV UHFHQWO\ HQDEOHG WR LQFUHDVH WKH IUDPH UDWH WKDQNV WR QHZ WHFKQLFDO LPSURYHPHQWV WKHUHIRUH DOORZLQJ WKHLU XVH IRU KLJK VSHHG WHVWV PRQLWRULQJ 7KRVH VROXWLRQV DUH SUHVHQWHG KHUHDIWHU

¯ 5RWDWLQJ PLUURUV FDPHUDV

,Q WKLV FDVH VHYHUDO VHQVRUV DUH HPEHGGHG LQ WKH FDPHUD )LJXUH 'XULQJ WKH UHFRUGLQJ SURFHVV OLJKW LV VSOLW WRZDUGV PXOWLSOH VHQVRUV RZLQJ WR D URWDWLQJ PLUURU )LJXUH F 7KLV LQFUHDVHV GUDVWL FDOO\ WKH IUDPH UDWH ZKLOH NHHSLQJ D VDWLVI\LQJ VSDWLDO UHVROXWLRQ )LQDOO\ IUDPH UDWH LV RQO\ OLPLWHG E\

WKH URWDWLQJ PLUURU V\VWHP +RZHYHU VHYHUDO GUDZEDFNV DULVH VXFK DV VHQVRUV V\QFKURQL]DWLRQ OHQVHV GLVWRUWLRQ RSWLFDO SDWK YDULDWLRQV DQG ORZ OLJKW H[SRVXUH

¯ %HDP VSOLWWLQJ FDPHUDV

%HDP VSOLWWLQJ FDPHUDV DUH EDVHG RQ D YHU\ FORVHG SULQFLSOH EXW OLJKW LQWHQVLILHUV )LJXUH DUH LQ FRUSRUDWHG WR LQFUHDVH OLJKW H[SRVXUH WKLV OHDGLQJ WR KLJKHU WHPSRUDO UHVROXWLRQ EXW LQFUHDVLQJ RSWLFDO SDWK FRPSOH[LW\

¯ 2Q FKLS VWRUDJH FDPHUDV

$QRWKHU ZD\ WR LQFUHDVH WHPSRUDO UHVROXWLRQ LV WR XVH RQFKLSVWRUDJH FDPHUDV )LJXUH D $V HQFRGHG LPDJHV DUH VWRUHG ZLWKLQ WKH LPDJLQJ V\VWHP UHDGRXW UDWH GRHV QRW OLPLW WHPSRUDO UHVROXWLRQ +RZHYHU DFWLYH SDUW SKRWRGLRGH RI WKH VHQVRU LV UHGXFHG EHFDXVH RI PHPRU\ FKLS )LJXUH E

$V D FRQVHTXHQFH VSDWLDO UHVROXWLRQ DQG PD[LPXP QXPEHU RI UHFRUGHG LPDJHV DUH OLPLWHG E\ WKH FKLSV VL]H DQG VWRUDJH FDSDFLW\ UHVSHFWLYHO\ 7KH 6KLPDG]X +39; FDPHUD LV EDVHG RQ WKLV WHFKQRORJ\

)LJXUH

1 10

10 0

1, 00 0

10 ,0 00

10 0, 00 0

1, 00 0,0 00

Record Length (Frames)

Frame Rate (fps)

Cordin 222-4G 12-bit ICCD Exp = 5 ns 40 lp/mm

Cordin 580 14-bit RM CCD Exp = 220 ns 91 lp/mm

1000 10k 100k 1M 10M 100M 1B 10B 100B

Hi gh Speed Ul tr a-Hi gh Speed

Key: •kF = 1000 Frames •kFPS = 1000 frames/sec •RM = Rotating Mirror •ICCD = Intensified CCD •Area = resolution •Pie chart = minimum exposure •Ultra-high speed resolution constant at all rates. •High-speed resolution changes at higher rates. •Resolution from specs or pixel size (1:1 Imaging)

Photron SA-Z 12-bit/64 GB CMOS Exp = 159 ns 25 lp/mm

0

NAC Memrecam HX-1/3 12-bit/128 GB CMOS Exp = 200 ns 36 lp/mm

Olympus I-Speed FS 10-bit/16 GB CMOS Exp = 200 ns 24 lp/mm PCO-Tech dimax HS 12-bit/36 GB CMOS Exp = 1500 ns 45 lp/mm Cordin 521 12-bit RM CCD Exp = 400 ns 91 lp/mm

IMACON 12-bit ICCD Exp = 3 ns Stanford Comp. Optics 12-bit ICCD Exp = .2 ns 60 lp/mm

Specialised Imaging SIMD 12-bit ICCD Exp = 3 ns 36 lp/mm

Invisible Vision UHSi 12-bit ICCD Exp = 5 ns

Cordin 510 14-bit RM CCD Exp = 40 ns 71 lp/mm

Phantom2010 12-bit/96 GB CMOS Exp = 500 ns 18 lp/mm IDT Y7-S3 10-bit/32 GB CMOS Rate scales only with rows Exp = 100 ns 69 lp/mm ShimadzuHPV-2 10-bit IS-CCD Exp = 250 ns 8 lp/mm

Specialised Imaging Kirana 10-bitμCMOS Exp = 100 ns 17 lp/mm Shimadzu HPV-X 10-bit FTCMOS Exp = 110 ns 16 lp/mm

6RXUFH 5HX HW DO >@

)LJXUH &KDUW RI DYDLODEOH 8+6 FDPHUDV RQ WKH PDUNHW

D E

6RXUFH 5RVVL HW DO >@

F

)LJXUH &RUGLQ DYDLODEOH DW 21(5$ D 2YHUYLHZ E ,QVLGH YLHZ F 6FKHPDWLF UHSUHVHQWDWLRQ

6RXUFH 5RVVL HW DO >@

)LJXUH %HDP VSOLWWLQJ WHFKQRORJ\

6RXUFH 5RVVL HW DO >@

D E

)LJXUH 2Q FKLS VWRUDJH &026 WHFKQRORJ\ D *HQHUDO SULQFLSOH E &026 VHQVRU

,GHQWLILFDWLRQ ZLWK VWDWLFDOO\ XQGHWHUPLQHG DSSURDFKHV

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

7KH IROORZLQJ VHFWLRQ EULHIO\ SUHVHQWV ILYH WHFKQLTXHV WR SURFHVV QRQ VWDQGDUG WHVWV EDVHG RQ OLWHU DWXUH UHYLHZV >@ QDPHO\ WKH &RQVWLWXWLYH (TXDWLRQ *DS 0HWKRG &(*0 WKH 5HFLSURFLW\ *DS 0HWKRG 5*0 WKH (TXLOLEULXP *DS 0HWKRG (*0 )LQLWH (OHPHQW 0RGHO 8SGDWLQJ )(08 DS SURDFKHV DQG WKH 9LUWXDO )LHOGV 0HWKRG 9)0 $IWHUZDUGV D GHHSHU UHYLHZ RI WKH 9)0 LV SUHVHQWHG

2YHUYLHZ

&RQVWLWXWLYH (TXDWLRQ *DS 0HWKRG &(*0

&(*0 KDYH EHHQ LQLWLDOO\ LQWURGXFHG WR YDOLGDWH )($ VLPXODWLRQV >@ :LWKLQ VPDOO SHUWXUEDWLRQ IUDPHZRUN DQG IRU HODVWLF EHKDYLRXU WKH VHW RI XQNQRZQ SDUDPHWHUV LV H[WUDFWHG PLQLPL]LQJ (TXD WLRQ ZKHUH v WKH GLVSODFHPHQW ILHOG ε WKH VWUDLQ WHQVRU τ D SK\VLFDOO\ DGPLVVLEOH VWUHVV WHQVRU DQG C WKH VWLIIQHVV PDWUL[ &RVW IXQFWLRQ C QRUP LV GHULYDWHG IURP )($ PRGDO DQDO\VLV 5D\OHLJKsV TXRWLHQWV ZKHUH PRGHV QRUPDOL]DWLRQ LV SHUIRUPHG WKURXJK ZHLJKWHG FURVV SURGXFW XVLQJ VWLIIQHVV RU PDVV PDWUL[ >@ 7KXV DV v τ PLQLPL]HV 39: LQ )($ WKH VROXWLRQ PLQLPL]HV (TXDWLRQ LQ &(*0

E ( v, τ , C ) = τ − C : ε ( v )

,Q DGGLWLRQ WR HODVWLF EHKDYLRXU HODVWRSODVWLF FRQVWLWXWLYH PRGHOV KDV DOVR EHHQ FKDUDFWHUL]HG >@

)RU WKLV QRQOLQHDU LGHQWLILFDWLRQ J

SODVWLFLW\ DQG NLQHPDWLF KDUGHQLQJ D UHWXUQPDSSLQJ DOJRULWKP KDV EHHQ GHYHORSHG WR HQDEOH VWUHVV UHFRQVWUXFWLRQ

5HFLSURFLW\ *DS 0HWKRG 5*0

,WV WKHRUHWLFDO EDFNJURXQG LV EDVHG RQ WZR HTXDWLRQV RI FRQWLQXXP PHFKDQLFV 3ULQFLSOH RI 9LUWXDO :RUN 39: DQG 0D[ZHOO%HWWL WKHRUHP &RQVLGHULQJ 39: IRU D PDWHULDO FXUUHQW VWDWH DQG DQRWKHU RQH IRU D YLUWXDO DGMRLQW VWDWH LW FRPHV

∀ u

,

V

ε ( u ) : [C − C

] : ε ( u

) dV =

∂Ω

(ˆ t.u

− t

. u ˆ ) dS ZKHUH V WKH PRQLWRUHG YROXPH RI PDWHULDO u WKH GLVSODFHPHQW ILHOG ˆ t WKH WUDFWLRQ IRUFHV u ˆ WKH GLV SODFHPHQW ILHOG DW ERXQGDULHV t

WKH DGMRLQW VWDWH GLVSODFHPHQW ILHOG RI u

WKH DGMRLQW GLVSODFHPHQW ILHOG $V WKH UHODWHG VWUHVV VWDWH VDWLVILHV WKH UHFLSURFLW\ WKHRUHP WKH VHW RI PDWHULDO SDUDPHWHUV FDQ EH H[WUDFWHG GLUHFWO\ FRPELQLQJ (TXDWLRQ IRU VHYHUDO DGMRLQW VWDWHV 7KRXJK LW KDV ILUVW EHHQ DSSOLHG IRU HODVWLF SDUDPHWHUV LW KDV EHHQ QRWDEO\ H[WHQGHG WR IUDFWXUH PHFKDQLFV WR PRQLWRU FUDFN SURSDJD WLRQV >@

(TXLOLEULXP *DS 0HWKRG (*0

(*0 LV D LGHQWLILFDWLRQ PHWKRG ZKLFK LV QRW VXLWDEOH IRU DOO LQYHUVH SUREOHPV ,QGHHG LW FRQVLVWV LQ FRPSDULQJ D UHIHUHQFH NQRZQ VWDWH WR DQ XQNQRZQ RQH $VVXPLQJ WKH NQRZOHGJH RI DQ HODVWLF EHKDYLRXU IRU LQVWDQFH GDPDJH DIIHFWV VWLIIQHVV FRPSRQHQWV YDOXHV 7KXV PRGLILHG VWLIIQHVV PDWUL[

C ( x ) = D ( x )C

PD\ EH LGHQWLILHG ZKHUH C

FRUUHVSRQGV WR WKH SXUHO\ HODVWLF EHKDYLRXU 7R FRPSXWH