Micro-Scale Modeling of Carbon-Fiber Reinforced Thermoplastic Materials

(1)

HAL Id: hal-00789382

https://hal.archives-ouvertes.fr/hal-00789382

Submitted on 18 Feb 2013

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published 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.

Micro-Scale Modeling of Carbon-Fiber Reinforced Thermoplastic Materials

Fethi Abbassi, Abderraouf Gherissi, Ali Zghal, Sébastien Mistou, Joël Alexis

To cite this version:

Fethi Abbassi, Abderraouf Gherissi, Ali Zghal, Sébastien Mistou, Joël Alexis. Micro-Scale Modeling of Carbon-Fiber Reinforced Thermoplastic Materials. Applied Mechanics and Materials, Trans Tech Publications, 2012, vol. 146, pp.1-11. �10.4028/www.scientific.net/AMM.146.1�. �hal-00789382�

(2)

This is an author-deposited version published in : http://oatao.univ-toulouse.fr/

Eprints ID : 6480

To link to this article : DOI:10.4028/www.scientific.net/AMM.146.1 URL : http://dx.doi.org/10.4028/www.scientific.net/AMM.146.1

Open Archive TOULOUSE Archive Ouverte (OATAO)

OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible.

To cite this version :

Abbassi, Fethi and Gherissi, Abderraouf and Zghal, Ali and Mistou, Sébastien and Alexis, Joël Micro-Scale Modeling of Carbon-Fiber Reinforced Thermoplastic Materials. (2012) Applied Mechanics and Materials, vol. 146 . pp. 1-11. ISSN 1662- 7482

Any correspondence concerning this service should be sent to the repository administrator: staff-oatao@listes.diff.inp-toulouse.fr!

(3)

0LFURVFDOHPRGHOLQJRIFDUERQILEHUUHLQIRUFHGWKHUPRSODVWLFPDWHULDOV )$%%$66,

^D

$*+(5,66,

^E

$=*+$/

^F

60,6728

^G

DQG-$/(;,6

^H

85066'7(6677XQLV$YHQXH7DKD+XVVHLQ%3%kE0DQDUD7XQLVLD

8QLYHUVLWpGH7RXORXVH,13(1,7063/*3$YHQXHG¶$]HUHL[7DUEHV)UDQFH ^DIHWKLDEEDVVL#LSHLEUQXWQ^EDEGHUUDRXIJKHULVVL#HQVDPHX^FDOL]JKDO#HVVWWUQXWQ

G0LVWRX#HQLWIU^H-RHO$OH[LV#HQLWIU

.H\ZRUGV &RPSRVLWH FDUERQ ILEHU 0LFURVFDOH PRGHOLQJ +RPRJHQL]DWLRQ DQG 0HFKDQLFDO SURSHUW\

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

$%$486 )URP WKH 6(0 FDUWRJUDSK\ RQH LGHQWLILHG WZR W\SHV RI UHSUHVHQWDWLYH YROXPH HOHPHQWDU\ 59( SHULRGLF DQG UDQGRP GLVWULEXWLRQ RI PLFURILEHUV LQ WKH \DUQ 5HIHUULQJ WR KRPRJHQL]DWLRQPHWKRGDQGE\DSSO\LQJWKHOLPLWVFRQGLWLRQVWRWKH59(ZHKDYHH[WUDFWHGWKH FRHIILFLHQWV RI WKH ULJLGLW\ PDWUL[ RI WKH VWXGLHG FRPSRVLWHV ,Q WKH ODVW SDUW RI WKLV ZRUN ZH FRPSDUHWKHUHVXOWVREWDLQHGE\UDQGRPDQGSHULRGLF59(PRGHORIFDUERQ336DQGZHFRPSXWH WKHUHODWLYHHUURUDVVXPLQJWKDWUDQGRPPRGHOJLYHVWKHULJKWYDOXH

,QWURGXFWLRQ

7KHGHWHUPLQDWLRQRIWKHPHFKDQLFDOSHUIRUPDQFHRIZRYHQIDEULFFRPSRVLWHVPDWHULDOVLVEDVHGRQ WKHVWXG\RIWKHEHKDYLRURIWKHWH[WXUHDQGWKHFRPSRVLWHXQGHUGLIIHUHQWVROLFLWDWLRQV&XUUHQWO\

WKH PXOWL VFDOH PRGHOLQJ RI FRPSRVLWHV ILJXUH LV RQH RI WKH PRVW XVHG PHWKRGV DQG LW ZDV DGDSWHG E\ VHYHUDO UHVHDUFKHUV ,Q IDFW XVLQJ WKLV DSSURDFK ) &RVWDQ]R DQG / *UD\ >@ KDYHV LPSODQWHGDVXUYH\RQSHULRGLFLW\DQGERXQGDU\FRQGLWLRQV3%RLVVH>@KDVUDLVHGWKHFRQVWUXFWLYH HTXDWLRQVRIWKHPHFKDQLFDOEHKDYLRURIWKHFRPSRVLWHVZRYHQGXULQJWKHIRUPLQJ*LOOHV+LYHW>@

KDVHODERUDWHGDPDWKHPDWLFDODSSURDFKWRLGHQWLI\WKHWUDMHFWRU\DQGWKHGLIIHUHQWVHFWLRQVRIWKH

\DUQLQWH[WXUHWKHSURILOHVRIWKHFRQWDFWV¶FXUYHVDQGWKHFRQWDFW¶VVHFWLRQVDFFRUGLQJWRWKHFRQLF HTXDWLRQV/2UJpDV>@KDVVWXGLHGLQPHVRVFDOHWKHSHUPHDELOLW\RIWKHUHLQIRUFHPHQWVZRYHQRI VWUDWLILHG FRPSRVLWHV E\ VXUYH\LQJ WKH YHORFLW\ LQ VXFK FRPSRVLWHV - :DQJ >@ KDV VWXGLHG WKH SUHGLFWLYH PHFKDQLFDO EHKDYLRU PRGHOLQJ LQ ZRYHQ FRPSRVLWH VWUXFWXUH E\ DQDO\]LQJ ' ILQLWHV HOHPHQWV 3 %DGHO DQG 3 %RLVVH >@ KDYH GHWHUPLQHG ILEHUV RULHQWDWLRQV LQ UHLQIRUFHPHQWV ZRYHQGXULQJDQGDIWHUFRPSRVLWHV¶IRUPDWLRQ

(4)

)LJ0XOWLVFDOHPRGHOLQJWHFKQLTXHVLQZRYHQIDEULFFRPSRVLWHV

,Q RUGHU WR LGHQWLI\ WKH EHKDYLRU RI WKH VWXGLHG FRPSRVLWH XVLQJ PXOWLVFDOH DSSURDFK ZH KDYH GHYHORSHGLQWKLVSDSHUDVLPXODWLRQRIWKHUHLQIRUFHPHQW¶VZRYHQIDEULFFRPSRVLWHILJXUH :HKDYHVWDUWHGE\XVLQJDQH[SHULPHQWDOFKDUDFWHUL]DWLRQRIWKHWH[WXUHWRSUHSDUHDJHRPHWULFDO GHVFULSWLRQRIILEHUV¶GLDPHWHUVDQGGLVWULEXWLRQVLQWKH3RO\SKHQ\OHQHVXOILGH336PDWUL[7KHQ ZHLGHQWLILHGWZRW\SHVRI59(SHULRGLFDQGUDQGRPRQHLQRUGHUWRHVWLPDWHWKHHUURUV¶YDOXHVLQ WKHUHVXOWV7KHQEDVLQJRQWKHKRPRJHQL]DWLRQPHWKRGDQGDIWHUDSSO\LQJWKHERXQGDU\FRQGLWLRQV WRWKH59(RQHKDVH[WUDFWHGWKHFRHIILFLHQWVRIWKHULJLGLW\PDWUL[DQGWKHSDUDPHWHUVRIWKH\DUQ FRPSRVLWHV )LQHO\ ZH KDYH LGHQWLILHG WKH DEOH 59( WR FKDUDFWHUL]H DFFXUDWHO\ WKH \DUQ RI RXU ZRYHQIDEULFFRPSRVLWH

, &DUERQILEHUUHLQIRUFHGWKHUPRSODVWLFPDWHULDOV

7KHFRPSRVLWHWH[WXUHFRQVLVWVRIDFDUERQILEHUVDQGD336PDWUL[DQGWKHYROXPHWULFIUDFWLRQRI WKHILEHUVLQWKHFRPSRVLWHVLV 7KHFKDUDFWHULVWLFVRIWKHPDWHULDOVIRUPLQJWKHFRPSRVLWH DUHVXPPDUL]HGLQWKHIROORZLQJWDEOH

7DEOH7KHFKDUDFWHULVWLFRIWKHPDWHULDOVIRUPLQJWKHFRPSRVLWH

0DWHULDO

)LODPHQW GLDPHWHU ȝP

9ROXPHWULF 0DVVȡ NJP

<281*

PRGXOH (0SD

6KHDU 0RGXOH

*0SD

3RLVVRQ¶V UDWLR ȣ

&RQVWUDLQHG RIUXSWXUH WUDFWLRQ 03D

0D[LPXP (ORQJDWLRQ

7KHUPDO GLODWLRQ UDWLR&

&DUERQ )LEHU

336

BBBBB

(5)

7KHFKDUDFWHUL]DWLRQRIWKHWH[WXUHRIWKHFRPSRVLWHKDVEHHQFDUULHGRXWWKURZWZRPDLQVWHSV,Q WKH ILUVW RQH ZH GHWHUPLQH WKH WH[WXUH¶V FKDUDFWHU WKH WUDMHFWRU\ DQG WKH VHFWLRQV RI WKH \DUQV 7H[WXUH RI WKH FRPSRVLWH VDWLQ [ LQ WKUHH OD\HUV 7KHQ LQ D VHFRQG VWHS ZH ILQG RXW WKH PLFURJUDSKLFDUUDQJHPHQWRIWKHILEHUVLQD\DUQILJXUH

D

E

)LJDPLFURJUDSKLFRIWKUHHSO\IDEULFVSHFLPHQVPHVRVFDOHEVL]HDQG DUUDQJHPHQWRIWKHPLFURILEHUVLQWKH\DUQ

7KH\DUQLVFRPSRVHGE\WKRXVDQGVRIVPDOOILEHUVZKRVHGLDPHWHULQWKHRUGHURIPVILJXUH 7KLVYDOXHLVHVWDEOLVKHGE\FDOFXODWLQJWKHDYHUDJHRIILEHUV¶GLDPHWHU7KHGLVRUJDQL]HG DUUDQJHPHQW RI WKH ILEHUV LQ WKH \DUQ SUHVHQWHG LQ ILJXUH ZLOO SURGXFH D YDULDWLRQ LQ WKH ORFDO SURSHUWLHV LQIOXHQFHG E\ WKH GLVWDQFH EHWZHHQ WKHVH ILEHUV 7KHQ LW LV QHFHVVDU\ WR VWDUW E\

FKDUDFWHUL]LQJWKHILEHUV¶DUUDQJHPHQWLQRUGHUWRGHWHUPLQHWKHPLQLPDOVL]HRIWKHUHSUHVHQWDWLYH YROXPHHOHPHQWDU\59(RIWKH\DUQ7RGRVRZHFDQFKDUDFWHUL]HWKHGLVWULEXWLRQRIWKHILEHUV E\DQDO\]LQJWKH\DUQ¶VSLFWXUHDQGXVLQJWKHFRYDULDQFHFRQFHSWDGDSWHGDOUHDG\E\>@

7KHFRYDULDQFHLVGHILQHGDVWKHSUREDELOLW\RIDGKHUHQFHRIWZRSRLQWV´[³DQG³[K´LQWKHVDPH SKDVH G DQG LW FDQ EH YDOXHG E\ FDUU\LQJ RXW WKH )RXULHU¶V WUDQVIRUPDWLRQ RI WKH ILJXUH

$FFRUGLQJWRWKHZRUNVRI3%DGHODQGDO>@WKHSHULRGLFLW\RIWKHPLFURVWUXFWXUHLVSUHVHQWHGE\

WKHSHULRGLFLW\RIWKHFRYDULDQFH ,, 7KHPLFURVFDOHPRGHOLQJ

7KHJHRPHWULFPRGHORI59(

7KHFKRLFHRIWKH59(ZKLFKLVDFXELFVKDSHZDVEDVHGRQVHYHUDOUHVHDUFKHVZRUNV>

DQG@7KLV59(VKRXOGKDYHWKHVPDOOHVWVL]HZKLFKPDNHVLWUHSUHVHQWDWLYHRIWKH\DUQPDWHULDO :HRSWHGIRUWKLVVWHSRIWKHVLPXODWLRQIRUWZRFXELFFHOOVVKDSHVDQGZHFRQVLGHUHGWKHILEHUKDVD F\OLQGULFDOIRUP7KHILUVWFHOOILJXUHDLVSHULRGLFDQGWKHVHFRQGLVUDQGRPILJXUHE7KH YROXPHWULFIUDFWLRQRIWKHUHLQIRUFHPHQWLVFDOFXODWHGE\WKHUHSRUWEHWZHHQWKHYROXPHRIWKHILEHUV DQGWKHWRWDOYROXPHRIWKHEDVLFFHOO

!^"

#$^"

:KHUHGLVWKHGLDPHWHURIWKHILEHUDLVWKHVLGHRIWKHEDVLVFHOODQGQLVWKHQXPEHURIILEHUVE\

FHOO

(6)

&KRRVLQJWKHVL]HRIUHSUHVHQWDWLYHYROXPHHOHPHQWDU\PXVWVDWLVI\WKHIROORZLQJFULWHULD

,W PXVW EH VPDOO HQRXJK WR WDNH LQWR DFFRXQW WKH PLFURVFRSLF VWUXFWXUH RI PDWHULDO DQG VXIILFLHQWO\ODUJHWRGHVFULEHWKHRYHUDOOEHKDYLRURIPDWHULDO

7KHSURSHUWLHVPXVWEHLQGHSHQGHQWRIWKHORFDWLRQRIWKHPDWHULDOZKHUHLWZDVWDNHQ

:H KDYH FKRVHQ WR PDNH VWDWLVWLFDO FDVH WR LGHQWLI\ WKH \DUQ UDQGRP UHSUHVHQWDWLYH YROXPH HOHPHQWDU\2QHYDULHVWKHZLQGRZVL]H:HLGHQWLI\LQHDFKZLQGRZWKHPLQLPXPDQGPD[LPXP ILEHUVYROXPHIUDFWLRQE\VFDQQLQJWKHZLQGRZLQWKHSKRWRJUDSKRIWKHVWUXFWXUHVHH)LJXUHDQG

)LJ(YROXWLRQRIERXQGVIRUORFDOYROXPHIUDFWLRQZLWKZLQGRZVL]H

)LJ9DULDWLRQVLQORFDOYROXPHIUDFWLRQRIILEUHV

7ZRW\SHVRIUHSUHVHQWDWLYHYROXPHHOHPHQWDU\59(SHULRGLFDQGUDQGRPGLVWULEXWLRQRIPLFUR ILEHUVLQWKH\DUQKDVLGHQWLI\LQJILJXUH

(7)

D

E

)LJ7KHSHULRGLFUHSUHVHQWDWLYHHOHPHQWDU\YROXPHRIWKH\DUQQXPEHURIILEHU1 ILEHUVDDQGYI 7KHUDQGRPUHSUHVHQWDWLYHHOHPHQWDU\YROXPHRIWKH\DUQ

QXPEHURIILEHU1 ILEHUVDQGYI E

7KHHODVWLFFRQVWUXFWLYHHTXDWLRQVRIWKH\DUQ¶VKRPRJHQHVDWLRQ

7KH HODVWLF SURSHUWLHV DUH FDOFXODWHG E\ D SHULRGLF KRPRJHQL]DWLRQ YLD D ILQLWH HOHPHQW PHWKRG GHYHORSHGXVLQJ$%$486VRIWZDUH,WZLOOJLYHXVWKHRSSRUWXQLW\WRVWXG\WKHHODVWLFEHKDYLRURI WKH \DUQ DQG WR FDOFXODWH WKH HODVWLF FRHIILFLHQWV RI WKH FRPSRVLWH PDWHULDO )RU ' 59( FXELF VKDSHVXEPLWWHGWRDYROXPHWULFORDGLWVHODVWLFEHKDYLRUFDQEHSUHVHQWHGDVIROORZ

% &'

:KHUHİLVWKHVWUDLQWHQVRUıLVWKHVWUHVVWHQVRUDQGĭWKHVXSSOHQHVV0DWUL[

7KHQWKHVWUHVVGLVWULEXWLRQLQWKHHOHPHQWDU\YROXPHFDQEHZULWWHQDVIROORZ

' %

:KHUHĭ &

7KHPHFKDQLFDOEHKDYLRURIWKH\DUQLVHTXLYDOHQWDQGLWGHSHQGVRQWKHPHFKDQLFDODQGJHRPHWULF SURSHUWLHVRIWKHGLIIHUHQWFRQVWLWXHQWWKHILEHUJHRPHWU\EHKDYLRUDQGGLVWULEXWLRQLQWKHPDWUL[

WKH PDWUL[ EHKDYLRU DQG WKH FKDUDFWHULVWLF RI WKH ILEHUPDWUL[ LQWHUIDFH 7KH SURFHVV RI KRPRJHQL]DWLRQFRQVLVWVLQDVVLPLODWLQJDPDWHULDOFKDUDFWHUL]HGE\DQLPSRUWDQWKHWHURJHQHLW\E\D KRPRJHQHRXVRQH7KLVSURFHVVZDVDSSOLHGWRWKH59(

7KHPDLQVWHSRIWKHKRPRJHQL]DWLRQFRQVLVWVLQWKHGHWHUPLQDWLRQRIWKHVWUHVVDQGGLVSODFHPHQW ILHOGVZLWKLQWKH59(

7KHDYHUDJHRIWKHPLFURVFRSLFVWUHVVRIWKLV59(FDQEHH[SUHVVHGDVIROORZ

(') ^*+ ' , -

,QWKHVDPHZD\WKHDYHUDJHRIWKHPLFURVFRSLFVWUDLQLVJLYHE\

(%) ^*+ % , .

:KHUH(LVWKHPDFURVFRSLFVWUDLQDQGLVWKHPDFURVFRSLFVWUHVV )URPHTXDWLRQDQGRQHFDQZULWHWKH+RRNHFULWHULD

/////////('0 %) (')0 (%) (

(8)

Î7KHPDFURVFRSLFVWUHVV- (')LVDOLQHDUIXQFWLRQRIWKHPDFURVFRSLFVWUDLQ( ¦!

- ¹²³.

:KHUH&^KRPUHSUHVHQWVWKHPDFURVFRSLFWHQVRUREWDLQHGE\WKHKRPRJHQL]DWLRQPHWKRG

7KH FDOFXODWLRQ RI WKH _4567/^89: FRHIILFLHQWV WDNHV SODFH ZKLOH FDOFXODWLQJ WKH VWUHVV ILHOG WKDW FRUUHVSRQGV WR DQ LPSRVHG PDFURVFRSLF GLVSODFHPHQW 6XSSRVLQJ WKDW WKH \DUQ UHSUHVHQWV D FRPSRVLWHZLWKRUWKRWURSLFFKDUDFWHULVWLFWKHPDFURVFRSLFHODVWLFLW\UHODWLRQLVH[SUHVVHGDVIROORZ

;<

<<

<=--^**_>>

-_??

-_>?

-_*?

-_*>@AAAAB /

;<

<<

=^****¹²³ _>>**¹²³ _??**¹²³ _**>>¹²³ >>>>123 _??>>¹²³ _**??¹²³ _>>??¹²³ _????¹²³

_>?>?¹²³

_*?*?¹²³

_*>*>¹²³@AAAAAB

/

;<

<<

<=._**

._>>

._??

._>?

._*?

._*>@AAAAB

)RUL M N OLMNOȯ^`WKH_4567/^89:FRHIILFLHQWVKDYHEHHQGHWHUPLQHGE\LPSRVLQJDVKHDU ORDGLQJZKRVHPDLQGLUHFWLRQVFRUUHVSRQGZLWKWKHV\PPHWU\¶VD[HVRIWKHFHOOWKDW¶VPHDQV

. ._**/C_*/D C_*/ ._>>/C_>/D C_>/ ._??C_?/D C_?/

)RUL NDQGM OLNȯ^`DQGMOȯ^`WKHFRHIILFLHQWV_4567/^89:KDYHEHHQGHWHUPLQHGE\

LPSRVLQJ WR WKH EDVLF FHOO D PDFURVFRSLF GLVSODFHPHQW RI W\SH VLPSOH VKHDU ZKLFK FDQ EH H[SUHVVHGDVIROORZ

. ._45/EF/C_4/D C_5/ C_5/D C_4/

,QWKHRUGHUWRKDYHDSHULRGLFDSSOLHGGLVSODFHPHQW¶VILOHGLWLVQHFHVVDU\WKDWHYHU\FHOOVDWLVILHV WKHIROORZLQJFRQGLWLRQV>@

7KHFRQWLQXLW\RIWKHYHFWRUıQ

7KH FRPSDWLELOLW\ RI WKH VWUDLQ ILHOGV İ WKHUHIRUH WKH QHLJKERULQJ VKRXOG QRW EH VHSDUDWHG RU VXSHUSRVHG

7KHSHULRGLFLW\RIWKHSDVVDJHIURPDFHOOWRLWVQHLJKERULVHTXLYDOHQWWRSDVVDIDFHIURPRQHIDFH RIWKHFHOOWKHFHOOWRWKHRSSRVLWHIDFH7KHFRQGLWLRQEHFRPHVıQPXVWEHRQWKHILUVWRSSRVLWH WRWKDWLQWKHRWKHUIDFH7KHVWUHVVILHOGıLVFDOOHGSHULRGLFRQWKHFHOOZKLOHWKHILHOGıQLVDQWL SHULRGLFRQLWVFRQWRXU

,,, +RPRJHQL]DWLRQRIWKH\DUQEDVHGRQPLFURVFDOHILQLWHHOHPHQWPRGHO 7KHPLFURVFDOHFRQVWUXFWLYHILQLWHHOHPHQWVPRGHOV

7KHDGDSWHGPHWKRGFRQVLVWVLQDSSO\LQJWKUHHVLPSOHWUDFWLRQORDGVIROORZLQJWKHWKUHHPDLQD[HV DQGDQGWKUHHVLPSOHVKHDUORDGVLQWKHGLUHFWLRQVDQGILJXUH,QRUGHUWR DSSO\WKLVPHWKRGZHVKRXOGLPSRVHDGLVSODFHPHQWORDGLQJDQGSXWDVSHFLILFERXQGDU\FRQGLWLRQV IRUHDFKORDGWKLVPHWKRGKDVEHHQDGRSWHGE\VHYHUDODXWKRUV>@

(9)

)LJ7KHVL[GLIIHUHQWFDVHVWREHVROYHGLQRUGHUWRFDOFXODWHWKHKRPRJHQL]HGHODVWLFSURSHUWLHV RIWKH59(

7KHFDOFXODWLRQRI_LMLVDSSUR[LPDWHGE\WKHVXPPDWLRQRIDOOWKHYROXPHWULFHOHPHQWVRIVWUXFWXUH DOUHDG\ FDOFXODWHG E\ HOHPHQWDULQHVV LQWHJUDWLRQV WKURZ HYHU\ ILQLWH HOHPHQW 7KHQ ZH KDYH WKH IROORZLQJHTXDWLRQ

- ₄₅ (') ^*+ ' , G^-^LÎMN_-^HÎ^J_H^KÎ

L I

IMN

:KHUH9NLVWKHYROXPHRIWKHN^WKHOHPHQWDQGıLMLVWKHFRPSRVLQJLMRIWKHPLFURVFRSLF FRQVWUDLQWRIWKHN^WKHOHPHQW

7KHVLPXODWLRQLVGRQHRQWKH$EDTXVVWDQGDUGVRIWZDUH7KHWLHERXQGDU\FRQGLWLRQVLWLVLPSRVHG EHWZHHQWKHILEHUVDQGWKHPDWUL[

7KH5(9PHVKIRUPLVWHWUDKHGUDOIRUWKHPDWUL[DQGKH[DKHGUDOIRUWKHILEHUV:HKDYHJHQHUDWHG KH[DKHGUDOPHVKHOHPHQWVIRUWKHILEHUVDQGWHWUDKHGUDOHOHPHQWVIRUWKHPDWUL[

3HULRGLFUHSUHVHQWDWLYHHOHPHQWDU\YROXPH

'XULQJWKHVLPXODWLRQLWLVQHFHVVDU\WRDSSO\WKHORDGVDVLPSRVHGGLVSODFHPHQWVDQGWRLPSRVH ERXQGDU\ FRQGLWLRQV WR WKH OLPLWV IRU HYHU\ ORDG $W ILUVW ZH KDYH VXSSRVHG WKDW WKH PDWHULDO LV RUWKRWURSLF7KHQWKHQXPHULFVLPXODWLRQDQGWKHFDOFXODWLRQVE\SHULRGLFKRPRJHQL]DWLRQJDYHWKH ULJLGLW\PDWUL[RIWKH\DUQ

&

7KHFDOFXODWLRQRIWKHLQYHUVHULJLGLW\PDWUL[ZLOOJLYHWKHYDOXHVRIWKHVXSSOHQHVVPDWUL[ĭVRZH FDQGHWHUPLQHWKHPDWHULDOSDUDPHWHUV7KHVHSDUDPHWHUVDUHVXPPDUL]HGLQWKHIROORZLQJWDEOH

7DEOH7KHSHULRGLF59(HODVWLFSDUDPHWHUV

7KH<RXQJ0RGXOHV03D 3RLVVRQ&RHIILFLHQWV 6KHDU0RGXOHV03D (

(

Ȟ

*

¸¸

¹

·

¨¨

©

§

(10)

Î7KH\DUQ¶VPDWHULDOLVXQLGLUHFWLRQDODQGWKHUHVXOWVRIWKHVLPXODWLRQRIWKHSHULRGLF59(XVLQJ 9RQ0LVHVFRQVWUDLQWDUHSURYLGHGLQILJXUH

7KH9RQ0LVHVFRQVWUDLQWLQWKH59(VWUXFWXUHLV

'O P^?_>- Q₄₅ ₄₅Q₄₅ P^?_>- '₄₅'_45R/^N

"

45 - '₆ ₆₆^>

)LJ5HVXOWVRIVLPXODWLRQRIWKH59(9RQ0LVHVFRQVWUDLQWLQWKHGLIIHUHQWORDGVSODQ\]

5DQGRPUHSUHVHQWDWLYHHOHPHQWDU\YROXPH

%\ RQH DSSO\LQJ WKH VDPH ERXQGDU\ FRQGLWLRQV DQG WKH VDPH ORDGV RQ WKH UDQGRP FHOO ZH FDQ GHWHUPLQH WKH FRQVWDQWV RI WKH ULJLGLW\ PDWUL[ & RI WKH \DUQ DQG WKH VXSSOHQHVV PDWUL[ ĭ DQG FRQVHTXHQWO\ZHFDQGHWHUPLQDWHWKHPDWHULDOUDQGRPSDUDPHWHUVZKLFKDUHVKRZQLQWKHIROORZLQJ WDEOH

7DEOH7KHUDQGRP59(HODVWLFSDUDPHWHUV

7KH<RXQJ0RGXOHV03D 3RLVVRQ¶VUDWLR 6KHDU0RGXOHV03D (

(

Ȟ

*

7KHUHVXOWVRIWKHVLPXODWLRQRIWKHUDQGRP59(XVLQJ9RQ0LVHVFRQVWUDLQWDUHSURYLGHGLQILJXUH

)LJ7KHUHVXOWVRIVLPXODWLRQRIWKH59(9RQ0LVHVFRQVWUDLQWLQWKHGLIIHUHQWORDGVSODQ\]

z

H Hz H z

z

H H z _H_z

z

H Hz H z

z

H H z _H_z

(11)

&RPSDULVRQVEHWZHHQSHULRGLFDQGUDQGRPPRGHO

7KH UHVXOWV JRWWHQ IRU WKH SHULRGLF DQG UDQGRP PRGHO DUH UHSRUWHG LQ WDEOH :H FDQ LGHQWLI\ D IOXFWXDWLRQLQWKH<RXQJPRGXOHVDQGWKH3RLVVRQFRHIILFLHQWVDPRQJWKHWZRPRGHOVWKHUHODWLYH HUURUIRU(UHDFKHVDQGIRUWKH3RLVVRQFRHIILFLHQWVȞDQGȞLWLVUHVSHFWLYHO\DQG

7KHVHUHVXOWVFRQYHUJHZLWKWKH'VWXGLHVLQVLPSOHWUDFWLRQIROORZLQJWKH2<D[LVDFKLHYHGE\

'7ULDV >@ ZKHUH WKH <RXQJ PRGXOH SUHVHQW D GLIIHUHQWLDWLRQ RI DQG IRU WKH 3RLVVRQ FRHIILFLHQW

7DEOH&RPSXWDWLRQRIHIIHFWLYHSURSHUWLHVIRUWKHSHULRGLFDQGUDQGRPPRGHORIWKH\DUQ

2XU VXUYH\ LQ ' VLPXODWLRQ ZLOO JLYH VRPH UHVXOWV PRUH DGYDQFHG WKDQ >@ 7KH GLIIHUHQFH EHWZHHQ WKH UDQGRP DQG WKH SHULRGLF 59( LQ 6KHDU 0RGXOHV * DQG *LV URXJKO\ DQG IRU*&RQFHUQLQJWKH3RLVVRQFRHIILFLHQWȞWKHUHODWLYHHUURUEHWZHHQWKHWZRPRGHOVLV DURXQG

,QWKHQXPHULFDOUHVXOWVIRUWKHSHULRGLF5(9ZHREVHUYHDOLNHYDOXHRI<281*PRGXOHV(DQG ( ( 03D DQG ( 03D DQG D UHJXODU EHKDYLRU LQ WKH WRZ WUDQVYHUVH GLUHFWLRQV%XW IRUWKH UDQGRPPRGHODVPDOOGLIIHUHQFHEHWZHHQWKHYDOXHRIWKHWZR <281*

PRGXOHV ( 03D DQG ( 03D WKLV GLIIHUHQFH LV JHQHUDOO\ GXH WR WKH SURSRVHGDUUDQJHPHQWRIILEHUVDQGWKHLUUHJXODUGLVWDQFHVLQWHUILEHUVLQWKH5(9VHHILJXUHDE

$OVRDYDULDWLRQRIWKHYDOXHRI(RIDQGWKHYDOXHRI*RIKDVEHHQREVHUYHGLQ WKHWZRFDVHVUDQGRPDQGSHULRGLF5(97KLVGHIHUHQFHLVGXHWRWKHFORVHQHVVEHWZHHQILEHUVLQ WKHUDQGRP5(9ZKRZLOOJLYHDPRUHUHVLVWDQFH

7KHUHVXOWVRIWKHGLVWULEXWLRQXVLQJ9RQ0LVHVFRQVWUDLQWLQWKHPDWUL[DQGWKHILEHUVILJXUHDQG SUHVHQWDKXJHGLIIHUHQFHEHWZHHQWKHWZRW\SHVRI5(9,QGHHGWKHUDQGRPPRGHOJLYHVDPRUH UHDOUHVSRQVHWKDQWKHSHULRGLFPRGHO

$QDO\WLFUHVXOWV

,QRUGHUWRFDOFXODWHWKHDQDO\WLFYDOXHRIKRPRJHQL]HGFRHIILFLHQWVE\WKHPLFURPHFKDQLFVWKHRU\

PDQ\ WKHRUHWLFDO DQG H[SHULPHQWDO UHVXOWV DUH DYDLODEOH WR HVWLPDWH DQG GHVFULEH SURSHUWLHV RI XQLGLUHFWLRQDO ORQJ ILEHU FRPSRVLWHV UHVXOWLQJ IURP WKH ZRUN RI +DVKLQ DQG +LOO KRPRJHQL]HGFRHIILFLHQWVREWDLQHGE\DVLPSOLILHGDSSURDFKUHVXOWLQJIURPWKHLPSOHPHQWDWLRQLQ SDUDOOHORULQVHULHVILEHUVDQGPDWUL[

7KH9RLJWPRGHODQGWKH5HXVVPRGHODUHH[SUHVVHGE\>DQG@

5HXVVPRGHOWUDQVYHUVHPRGHO

._S TE./ ₃E.₃/U^R*

9DULDEOHV <RXQJ0RGXOH 7KHVKDUH0RGXOH 7KHSRLVRQVUDWLR

( ( ( * * * Ȟ Ȟ Ȟ

3HULRGLF59(

5DQGRP59(

5HODWLYHHUURU

(12)

7KH9RLJWPRGHOORQJLWXGLQDOPRGHO

.₇ . .₃ ₃

6KHDU0RGXOHDQGSRLVVRQ¶VUDWLRDUHFDOFXODWHGE\WKHPL[WXUHVODZDVIROORZ

V_7S V V₃ ₃

W_7S TEW/ ₃EW₃/U^R*

+DOSLQ DQG 7VDL KDYH SURSRVHG HTXDWLRQV GHVLJQHG WR DFKLHYH D VLPSOH V\QWKHWLF IRUPXODWLRQWKLVHPSLULFDOODZKDVWRDGPLWWKDWDJRRGDSSUR[LPDWLRQRIWKHDFWXDOPRGXOHVRID XQLGLUHFWLRQDOFRPSRVLWHPDWHULDOLVJLYHQE\

9 <RXQJPRGXOXVX_Y/Z[\/SRLVVRQ¶VUDWLR]_Y^WKH\DUHJLYHQE\PL[WXUHVODZ 9 <RXQJPRGXOXVX_{_}`_{_}DQG]_{__}WKH\DUHJLYHQE\WKHHPSLULFDOIRUPXOD

a

ab

^*cd_*R_K^K_/H^/H_e^e

K

âê^fâ^b^/R/*

aefab/c/d

:KHUH 0 0P DQG 0I GHQRWH WKH PRGXOXV RI WKH FRPSRVLWH VWUHQJWK PRGXOXV ILEHUV DQG WKH PRGXOXVRIWKHPDWUL[

7KHg/IDFWRU LV D PHDVXUH RI UHLQIRUFHPHQW SURYLGHG E\ WKH ILEHUV ,WV YDOXH GHSHQGV RQ WKH JHRPHWU\DQGDUUDQJHPHQWRIILEHUV

7KHDQDO\WLFDOUHVXOWVEDVHVLQHPSLULFDOODZDUH

( 03D

( ( 03D

ȣ

3DUWLFXODUO\ WKH DQDO\WLF UHVXOWV SURYH WKH QXPHULFDO SUHGLFWLRQ LQ SHULRGLF PRGHO RI (SRLVRQV FRHIILFLHQWȞEXWZHREVHUYHDVPDOOYDULDWLRQRIWKHYDOXHVRI((ȞDQGȞWKLVGLIIHUHQFHLV GXH WR WKH WKHRUHWLFDO PRUSKRORJLFDO DSSUR[LPDWLRQ RI WKH PDWHULDO PRGHO WKDW QRW DSSURSULDWHO\

UHSUHVHQWWKHUHDOLW\RIWKHFRPSRVLWHPRUSKRORJLFDO

&RQFOXVLRQV

7KHPLFURVFDOHPRGHOLQJDGRSWHGLQWKLVZRUNKDVSHUPLWWHGWRH[WUDFWWKHHODVWLFIHDWXUHVRIWKH FRPSRVLWH\DUQDQGWKHVLPXODWLRQRIWKHSHULRGLFDQGUDQGRP59(JDYHWKDWWKH\DUQPDWHULDOLV XQLGLUHFWLRQDO$FFRUGLQJWRWKHZRUNVRI'7ULDV>@ZKHUHWZRW\SHVRI'UHSUHVHQWDWLYHPRGHOV UDQGRPDQGSHULRGLFZHUHFRPSDUHGZHFDQFRQFOXGHWKDWWKHSHULRGLFPRGHOVFRXOGEHXVHGLQ VRPH FDVHV ZKHQ WKH REVHUYHG HUURU LV FRQVLGHUHG OLNH QHJOLJLEOH DQG QR DVVHVVPHQW IRU WKH PDWHULDO¶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

(13)

VKRUWO\LQWKHPHVRVFDOHPRGHOLQJRIRXUZRYHQIDEULFFRPSRVLWH7KLVVWXG\LVSURPRWLYHDQGLW UHTXLUHVDQDGYDQFHPRGHOLQGDPDJHDQGWKHUXSWXUHSUREOHPVDQGWRGHILQHWKHFRQVWLWXWLYHODZRI WKH\DUQ

5HIHUHQFHV

>@ ) &RVWDQ]R DQG / *UD\ 0XOWLVFDOH 0RGHOLQJ DQG 6LPXODWLRQ RI &RPSRVLWH 0DWHULDOV DQG6WUXFWXUHV&KDSWHU$0LFURPHFKDQLFV%DVHG1RWLRQRI6WUHVVIRU8VHLQWKH'HWHUPLQDWLRQRI

&RQWLQXXP/HYHO0HFKDQLFDO3URSHUWLHVYLD0ROHFXODU'\QDPLFV6SULQJHU6FLHQFH

>@ 3 %RLVVH DQG *LOOHV +LYHW &RQVLVWHQW PHVRVFRSLF PHFKDQLFDO EHKDYLRXU PRGHO IRU ZRYHQ FRPSRVLWHUHLQIRUFHPHQWVLQELD[LDOWHQVLRQ&RPSRVLWHV3DUW%±

>@ *LOOHV +LYHW &RQVLVWHQW G JHRPHWULFDO PRGHO RI IDEULF HOHPHQWDU\ FHOO DSSOLFDWLRQ WR D PHVKLQJ SUHSURFHVVRU IRU G ILQLWH HOHPHQW DQDO\VLV )LQLWH (OHPHQWV LQ $QDO\VLV DQG 'HVLJQ

±

>@ / 2UJpDV :RYHQ IDEULF SHUPHDELOLW\ )URP WH[WLOH GHIRUPDWLRQ WR IOXLG IORZ PHVRVFDOH VLPXODWLRQV&RPSRVLWHV6FLHQFHDQG7HFKQRORJ\±

>@ -LQKXR :DQJ 3UHGLFWLYH 0RGHOLQJ DQG ([SHULPHQWDO 0HDVXUHPHQW RI &RPSRVLWH )RUPLQJ

%HKDYLRU 7KHVLV VXEPLWWHG WR 7KH 8QLYHUVLW\ RI 1RWWLQJKDP IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\

>@ 3 %RLVVH &RPSXWDWLRQDO GHWHUPLQDWLRQ RI LQSODQH VKHDU PHFKDQLFDO EHKDYLRU RI WH[WLOH FRPSRVLWHUHLQIRUFHPHQWV&RPSXWDWLRQDO0DWHULDOV6FLHQFH±

>@3%DGHO6*DXWKLHU(9LGDO6DOOp3%RLVVH5DWHFRQVWLWXWLYHHTXDWLRQVIRUFRPSXWDWLRQDO DQDO\VHV RI WH[WLOH FRPSRVLWH UHLQIRUFHPHQW PHFKDQLFDO EHKDYLRXU GXULQJ IRUPLQJ &RPSRVLWHV 3DUW$±

>@ *XLOODXPH &28e*1$7 $SSURFKH PXOWLpFKHOOH GX FRPSRUWHPHQW PpFDQLTXH GH PDWpULDX[

FRPSRVLWHVjUHQIRUWWLVVpWKqVH8QLYHUVLWpGH%RUGHDX[

>@-0ROLPDUG0pFDQLTXHGHV0DWpULDX[FRPSRVLWHV(06(9HUVLRQ

>@ )DL]D %HQ $EGDOODK0DDPRXQ (ODERUDWLRQ FDUDFWHULVDWLRQ HW PRGHOLVDWLRQ G¶XQ PDWHULDX FRPSRVLWHjEDVHG¶XQHPDWULFHWKHUPRSODVWLTXHUHQIRUFHHSDUGXOLHJHWKqVH(1,7

>@$$O]LQD0XOWLVFDOHPRGHOOLQJRIWKHUPDOFRQGXFWLYLW\LQFRPSRVLWHPDWHULDOVIRUFU\RJHQLF VWUXFWXUHV&RPSRVLWH6WUXFWXUHV±

>@ -D\HVK 5 -DLQ DQG 6RPQDWK *KRVK 'DPDJH (YROXWLRQ LQ &RPSRVLWHV ZLWK D +RPRJHQL]DWLRQEDVHG &RQWLQXXP 'DPDJH 0HFKDQLFV 0RGHO ,QWHUQDWLRQDO -RXUQDO RI 'DPDJH 0HFKDQLFV

>@ ' 7ULDV 5DQGRP PRGHOV YHUVXV SHULRGLF PRGHOV IRU ILEUH UHLQIRUFHG FRPSRVLWHV

&RPSXWDWLRQDO0DWHULDOV6FLHQFH±

>@ %9DQ 'HQ %URXFNH 'HWHUPLQDWLRQ RI WKH PHFKDQLFDO SURSUHWLHV RI WH[WLOHUHLQIRUFHG FRPSRVLWHVWDNLQJLQWRDFFRXQWWH[WLOHIRUPLQJSDUDPHWHUV6SULQJHU(6$)250

>@ 0RUDQ :DQJ 1LQJ 3DQ (ODVWLF SURSHUW\ RI PXOWLSKDVH FRPSRVLWHV ZLWK UDQGRP PLFURVWUXFWXUHV-RXUQDORI&RPSXWDWLRQDO3K\VLFV±

>@:9RLJWhEHUGLH%H]LHKXQJ]ZLVFKHQGHQEHLGHQ(ODVWL]LWlWVNRQVWDQWHQLVRWURSHU.|˶ଉUSHU :LHG$QQ±

>@$5HXVV%HUHFKQXQJGHU)OLHVVJUHQ]YRQ0LVFKNULVWDOOHQDXI*UXQGGHU3ODVWL]LWlWVEHGLQJXQJ IU(LQNULVWDOOH=$QJHZ0DWK0HFK