On numerical analyses of wovencomposite laminates

UNIVERSITATIS ACTA UPSALIENSIS

UPPSALA

Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1828

On numerical analyses of woven composite laminates

Homogenization, damage and fracture

JUAN JOSÉ ESPADAS ESCALANTE

ISSN 1651-6214

ISBN 978-91-513-0699-5

Dissertation presented at Uppsala University to be publicly examined in Siegbahnsalen, The Ȧngström Laboratory, Lägerhuddsvägen 1, Uppsala, Friday, 13 September 2019 at 09:15 for the degree of Doctor of Philosophy. The examination will be conducted in English. Faculty examiner: Martin Fagerström (Chalmers tekniska högskola).

Abstract

Espadas Escalante, J. J. 2019. On numerical analyses of woven composite laminates.

Homogenization, damage and fracture. Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1828. 53 pp. Uppsala: Acta Universitatis Upsaliensis. ISBN 978-91-513-0699-5.

This dissertation analyzes various mechanical properties of textile reinforced composite laminates.

The dissertation is based on a total of six published works, which are essentially numerical, although experimental elements are available. The numerical methods used are based on high- resolution finite element models in combination with sophisticated phase-field theories for brittle fracture. A key result is that important mechanical properties in engineering applications, such as fracture or damage resistance, can be substantially affected by the arrangement of the constituent materials at the meso level.

Juan José Espadas Escalante, Department of Engineering Sciences, Applied Mechanics, 516, Uppsala University, SE-751 20 Uppsala, Sweden.

© Juan José Espadas Escalante 2019 ISSN 1651-6214

ISBN 978-91-513-0699-5

urn:nbn:se:uu:diva-388808 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-388808)

7HPSRUDU\

2Q QXPHULFDO DQDO\VHV RI ZRYHQ FRPSRVLWH ODPLQDWHV KRPRJHQL]DWLRQ GDP

DJH DQG IUDFWXUH

&RYHU 3KRWR $ ILQLWH HOHPHQW GLVFUHWL]DWLRQ RI D SODLQ ZHDYH DUFKLWHFWXUH

‹ -XDQ -RVp (VSDGDV (VFDODQWH 

7KHVLV LQ SDUWLDO IXOILOOPHQW IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\

'HSDUWPHQW RI $SSOLHG 0HFKDQLFV 8SSVDOD 8QLYHUVLW\

6( 8SSVDOD

6ZHGHQ

3UHIDFH

7KH ZRUN SUHVHQWHG LQ WKLV WKHVLV KDV EHHQ FRQGXFWHG IURP 1RYHPEHU 

WR 0DUFK  DW WKH 'LYLVLRQ RI $SSOLHG 0HFKDQLFV DW 8SSVDOD 8QLYHUVLW\

7KH UHVHDUFK ZDV PDLQO\ VXSSRUWHG E\ WKH 1DWLRQDO 5HVHDUFK DQG 7HFKQRO

RJ\ &RXQFLO &21$&<7 0H[LFR XQGHU WKH VFKRODUVKLS QXPEHU  IRU ZKLFK , DP JUDWHIXO

, ZRXOG OLNH WR WKDQN P\ VXSHUYLVRU SURIHVVRU 3HU ,VDNVVRQ ZKR QRW RQO\

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 IRU VKRZLQJ KLV LQWHUHVW LQ P\ UHVHDUFK WRSLFV DQG IRU KLV IUXLWIXO FRPPHQWV

, DP JUDWHIXO WR P\ IRUPHU DQG FXUUHQW FROOHDJXHV DW 8SSVDOD 8QLYHUVLW\

IRU SURYLGLQJ D IULHQGO\ DWPRVSKHUH DQG IRU WKH LQWHUHVWLQJ GLVFXVVLRQV RQ GL

YHUVH WRSLFV 7KDQNV WR WKH PHPEHUV RI WKH /,*+7HU QHWZRUN IRU SURPRWLQJ DQ LQVSLULQJ DWPRVSKHUH RQ WRSLFV UHODWHG WR OLJKWZHLJKW VWUXFWXUHV

, ZRXOG OLNH WR WKDQN DOO P\ IULHQGV DQG IDPLO\ LQ 0H[LFR (VSHFLDOO\ WR P\

PRWKHU IRU KHU HIIRUWV WR VXSSRUW P\ DFDGHPLF HGXFDWLRQ LQ WKH HDUOLHU VWDJHV RI P\ OLIH

/DVW EXW QRW OHDVW , ZRXOG OLNH WR WKDQN P\ ZLIH $QDLV IRU KHU LQILQLWH SD

WLHQFH VXSSRUW DQG ORYH DQG ZKR ZLWKRXW WKLV ZRXOG QRW EH SRVVLEOH 6KH

WRJHWKHU ZLWK RXU VRQ -XDQ -RVp DUH P\ HYHU\GD\ GULYLQJ IRUFH

8SSVDOD 0DUFK 

-XDQ -RVp (VSDGDV (VFDODQWH

6DPPDQIDWWQLQJ Sn 6YHQVND

, GHQQD DYKDQGOLQJ DQDO\VHUDV ROLND PHNDQLVND HJHQVNDSHU KRV WH[WLOI|UVWlUNWD NRPSRVLWODPLQDW

$YKDQGOLQJHQ E\JJHU Sn WRWDOW VH[ SXEOLFHUDGH DUEHWHQ VRP L KXYXGVDN lU DY QXPHULVN NDUDNWlU lYHQ RP H[SHULPHQWHOOD LQVODJ ILQQV 'H QXPHULVND PHWRGHU VRP DQYlQGV lU EDVHUDGH Sn K|JXSSO|VWD ILQLWD HOHPHQWPRGHOOHU L NRPELQD

WLRQ PHG VRILVWLNHUDGH IDVIlOWWHRULHU I|U VSU|GD EURWW (WW Q\FNHOUHVXOWDW lU DWW YLNWLJD PHNDQLVND HJHQVNDSHU L LQJHQM|UWLOOlPSQLQJDU VnVRP EURWWVHJKHW HOOHU VNDGHWnOLJKHW YlVHQWOLJHQ NDQ SnYHUNDV JHQRP HQ NRPELQDWLRQ DY GH ROLND LQ

JnHQGH PDWHULDOHQ Sn PHVRQLYn

/LVW RI SDSHUV

7KLV WKHVLV LV EDVHG RQ WKH IROORZLQJ SDSHUV ZKLFK DUH UHIHUUHG WR LQ WKH WH[W E\ WKHLU 5RPDQ QXPHUDOV

, (VSDGDV(VFDODQWH -- YDQ 'LMN 13 ,VDNVVRQ 3 $ VWXG\ RQ WKH LQIOXHQFH RI ERXQGDU\ FRQGLWLRQV LQ FRPSXWDWLRQDO KRPRJHQL]DWLRQ RI SHULRGLF VWUXFWXUHV ZLWK DSSOLFDWLRQ WR ZRYHQ FRPSRVLWHV &RPSRVLWH 6WUXFWXUHV  

,, (VSDGDV(VFDODQWH -- %HGQDUF\N % $ ,VDNVVRQ 3 $QDO\VLV RI WKH LQIOXHQFH RI OD\HU VKLIWLQJ RQ WKH HODVWLF UHVSRQVH DQG GDPDJH QXFOHDWLRQ DQG JURZWK LQ ZRYHQ FRPSRVLWH ODPLQDWHV

1$6$70   -RXUQDO YHUVLRQ SXEOLVKHG LQ

&RPSRVLWH 6WUXFWXUHV   

,,, (VSDGDV(VFDODQWH -- YDQ 'LMN 13 ,VDNVVRQ 3 $ SKDVHILHOG PRGHO IRU VWUHQJWK DQG IUDFWXUH DQDO\VHV RI ILEHUUHLQIRUFHG FRPSRVLWHV

&RPSRVLWHV 6FLHQFH DQG 7HFKQRORJ\   

,9 (VSDGDV(VFDODQWH -- ,VDNVVRQ 3 0HVRVFDOH DQDO\VLV RI WKH WUDQVYHUVH FUDFNLQJ NLQHWLFV LQ ZRYHQ FRPSRVLWH ODPLQDWHV XVLQJ D SKDVHILHOG IUDFWXUH WKHRU\ (QJLQHHULQJ )UDFWXUH 0HFKDQLFV   

9 (VSDGDV(VFDODQWH -- YDQ 'LMN 13 ,VDNVVRQ 3 7KH HIIHFW RI IUHHHGJHV DQG OD\HU VKLIWLQJ RQ LQWUDODPLQDU DQG LQWHUODPLQDU VWUHVVHV LQ ZRYHQ FRPSRVLWHV &RPSRVLWH 6WUXFWXUHV   

9, (VSDGDV(VFDODQWH -- ,VDNVVRQ 3 $ VWXG\ RI LQGXFHG GHODPLQDWLRQ DQG IDLOXUH LQ ZRYHQ FRPSRVLWH ODPLQDWHV VXEMHFW WR VKRUWEHDP VKHDU WHVWLQJ (QJLQHHULQJ )UDFWXUH 0HFKDQLFV  

5HSULQWV ZHUH PDGH ZLWK SHUPLVVLRQ IURP WKH SXEOLVKHUV

7KH DXWKRU RI WKLV WKHVLV ZDV UHVSRQVLEOH IRU WKH PDMRU SURJUHVV RI WKH ZRUN

LH VXJJHVWLQJ WKH VWXGLHV SODQQLQJ SHUIRUPLQJ QXPHULFDO LPSOHPHQWDWLRQV

DQG VLPXODWLRQV DQG ZULWLQJ RI WKH SDSHUV ZLWK WKH DVVLVWDQFH RI WKH FRDXWKRUV

&RQWHQWV

 ,QWURGXFWLRQ                                                                                                   

 %DFNJURXQG DQG PRWLYDWLRQ                                                               

 )DEULF UHLQIRUFHPHQWV DQG ZRYHQ IDEULFV                                            

 7RZDUGV YLUWXDO WHVWLQJ RI ZRYHQ FRPSRVLWHV VFDOHV DQG PRGHOLQJ DVVXPSWLRQV                                                                                       

 $LPV RI WKH WKHVLV                                                                              

 7KHRUHWLFDO IRXQGDWLRQV                                                                                  

 &RPSXWDWLRQDO KRPRJHQL]DWLRQ RI KHWHURJHQHRXV PHGLD                     

 &RPSXWDWLRQDO GDPDJH DQG IUDFWXUH PHFKDQLFV                                  

 $ PXOWLVFDOH FRQWLQXXP GDPDJH PHFKDQLFV DSSURDFK        

 $ SKDVHILHOG DSSURDFK WR EULWWOH IUDFWXUH                            

 'HODPLQDWLRQ RQVHW LQ FRPSRVLWH ODPLQDWHV                                        

 1XPHULFDO DQG H[SHULPHQWDO SURFHGXUHV                                                         

 &RPSXWDWLRQDO KRPRJHQL]DWLRQ RI KHWHURJHQHRXV VWUXFWXUHV ZLWK

SUHIHUHQWLDO SHULRGLFLW\                                                                       

 /D\HU VKLIWLQJ HIIHFWV LQ WKH HODVWLF UHVSRQVH DQG GDPDJH QXFOHDWLRQ DQG JURZWK                                                                                         

 3KDVHILHOG PRGHOLQJ RI EULWWOH IUDFWXUH LQ FRPSRVLWH VWUXFWXUHV          

 &UDFN SURSDJDWLRQ DQG WUDQVYHUVH FUDFNLQJ NLQHWLFV RI ZRYHQ

FRPSRVLWH ODPLQDWHV                                                                           

 'HODPLQDWLRQ RQVHW LQGXFHG E\ IUHHHGJH HIIHFWV                                

 'HODPLQDWLRQ RQVHW LQGXFHG E\ VKRUWEHDP VKHDU WHVWLQJ                    

 0DQXIDFWXULQJ RI ZRYHQ FRPSRVLWH ODPLQDWHV                                    

 ;UD\ FRPSXWHG WRPRJUDSK\                                                              

 5HVXOWV VXPPDU\ RI DSSHQGHG SDSHUV                                                            

 &RQFOXVLRQV DQG RXWORRN                                                                                

5HIHUHQFHV                                                                                                           

 ,QWURGXFWLRQ

:RYHQ FRPSRVLWHV DUH PDWHULDOV RI VSHFLDO LQWHUHVW LQ VWUXFWXUDO DSSOLFDWLRQV GXH WR WKHLU H[FHOOHQW PHFKDQLFDO SURSHUWLHV DQG OLJKWZHLJKW ZKLFK KDV PDGH WKHP DQ DWWUDF

WLYH RSWLRQ LQ HJ WKH DHURQDXWLFDO DQG DHURVSDFH LQGXVWULHV >@ 7KH\ DUH D VXLWDEOH DOWHUQDWLYH WR W\SLFDO XQLGLUHFWLRQDO WDSH ODPLQDWHV PDLQO\ GXH WR ORZHU SURGXFWLRQ FRVWV JRRG GUDSHDELOLW\ LPSURYHG RXWRISODQH SURSHUWLHV DQG D JRRG UHVLVWDQFH WR LPSDFW > @ +RZHYHU WKH DUFKLWHFWXUH RI WH[WLOH UHLQIRUFHPHQWV LV PRUH FRPSOH[

WKDQ WKH RQH RI WKH WDSH ODPLQDWHV VLQFH WKH DGGLWLRQDO VFDOH RI WKH ILEHU EXQGOHV RU

\DUQV LV DGGHG 7KLV PDNHV WKH DQDO\VLV RI ZRYHQ FRPSRVLWH ODPLQDWHV VLJQLILFDQWO\

PRUH FRPSOH[ DQG WHGLRXV WKDQ LQ WDSH ODPLQDWHV IURP WKH GHVLJQ SHUVSHFWLYH

'LIIHUHQW VFDOHV FDQ EH HQFRXQWHUHG LQ ZRYHQ FRPSRVLWHV IRU WKHLU DQDO\VLV 2QH FDQ LGHQWLI\ WKH PLFUR PHVR DQG PDFURVFDOHV 7KH PLFURVFDOH GHDOV ZLWK GHWDLOV DW WKH ILEHU DQG PDWUL[ FRQVWLWXHQW OHYHOV WKH PHVRVFDOH IRFXVHV RQ WKH UHSUHVHQWD

WLRQ RI KHWHURJHQHRXV VWUXFWXUHV E\ FRQVLGHULQJ H[SOLFLWO\ WKH \DUQV 7KH PDFURVFDOH FRQVLGHUV KRPRJHQHRXV VWUXFWXUHV RI RUWKRWURSLF QDWXUH ZKLFK SURSHUWLHV DUH LQKHU

LWHG IURP WKH PHVRVFRSLF VFDOH $QDO\VHV DUH RIWHQ PDGH ZLWK FRPSXWDWLRQDO PRGHOV ZLWK GLIIHUHQW OHYHOV RI GHWDLO +RPRJHQHRXV PRGHOV LGHDOL]HG KHWHURJHQHRXV PRGHOV DQG KLJKILGHOLW\ KHWHURJHQHRXV PRGHOV EDVHG RQ WKUHHGLPHQVLRQDO LPDJLQJ REWDLQHG IURP PDQXIDFWXUHG VDPSOHV FDQ RIWHQ EH HQFRXQWHUHG LQ WKH OLWHUDWXUH :KHUHDV RYHU

VLPSOLILHG DSSURDFKHV SURYLGH IDVW VFUHHQLQJ FDOFXODWLRQV WKH\ ODFN D GHVFULSWLRQ RI SK\VLFDO PHFKDQLVPV ,Q FRQWUDVW RYHUFRPSOH[ PRGHOV RIWHQ SURYLGH D PRUH DFFXUDWH UHSUHVHQWDWLRQ RI ZRYHQ FRPSRVLWH LQ WKH GRPDLQ RI DQDO\VLV HJ WKH VFDQQHG ]RQH DW H[SHQVHV RI KLJKHU FRVWV DQG FRPSXWDWLRQ WLPH 7KH\ FDQ DOVR ODFN GHWDLOV GHVFULE

LQJ WKH PHFKDQLVPV LQYROYHG VLQFH VHYHUDO PLFUR DQG PHVRVWUXFWXUDO LPSHUIHFWLRQV LQGXFHG DV D FRQVHTXHQFH RI WKH PDQXIDFWXULQJ SURFHVV DUH FRQVLGHUHG VLPXOWDQHRXVO\

LQ WKH DQDO\VLV 0RUHRYHU LW LV VWLOO TXHVWLRQDEOH LI WKH UHJLRQ RI DQDO\VLV FDQ EH D UHOLDEOH UHSUHVHQWDWLRQ RI WKH ZKROH VWUXFWXUH HVSHFLDOO\ LQ FXUYHG ]RQHV

7KH SUHVHQW WKHVLV LQYHVWLJDWHV ZRYHQ FRPSRVLWHV IURP WKH PHVRVFRSLF VFDOH SHU

VSHFWLYH 7KH PDLQ SXUSRVH LV WR JHQHUDWH LQVLJKWIXO NQRZOHGJH DERXW WKH SK\VLFDO PHFKDQLVPV LQKHUHQW WR WKH PHVRVFRSLF VFDOH UHODWHG WR WKH JOREDO PHFKDQLFDO UH

VSRQVH 7KH HODVWLF UHJLPH IDLOXUH RQVHW GHODPLQDWLRQ RQVHW DQG GDPDJH HYROXWLRQ DUH RI LQWHUHVW 7KH DSSURDFK DGRSWHG LQ WKH FXUUHQW LQYHVWLJDWLRQ UHOLHV PDLQO\ RQ VLPXODWLRQV EDVHG QXPHULFDO PRGHOV DQG LGHDOL]DWLRQV RI ´LPSHUIHFW´ ZRYHQ FRPSRV

LWH PRGHOV SURYLGLQJ DQ LQWHUPHGLDWH VWHS EHWZHHQ WKH DSSURDFKHV DERYHPHQWLRQHG

1XPHULFDO VWXGLHV UHO\ RQ H[SHULPHQWDO HYLGHQFH HLWKHU FRQGXFWHG RU UHSRUWHG LQ WKH

OLWHUDWXUH 5HVXOWV UHSRUWHG DUH H[SHFWHG WR FRQWULEXWH WR D EHWWHU XQGHUVWDQGLQJ RI WKH

EHKDYLRU RI ZRYHQ FRPSRVLWHV OHDGLQJ WR HIILFLHQW \HW UHOLDEOH PRGHOLQJ SUDFWLFHV

 %DFNJURXQG DQG PRWLYDWLRQ

$ FRPSRVLWH PDWHULDO LV W\SLFDOO\ GHILQHG DV WKH FRPELQDWLRQ RI WZR RU PRUH PDWHULDOV WR SURGXFH QHZ PDWHULDOV ZLWK HQKDQFHG SURSHUWLHV EXW ZKHUH HDFK PDWHULDO FRPSR

QHQW UHWDLQV LWV VHSDUDWH FKHPLFDO SK\VLFDO DQG PHFKDQLFDO SURSHUWLHV >@ $ W\SH RI FRPSRVLWH PDWHULDOV DUH WKH ILEHUUHLQIRUFHG FRPSRVLWHV RU VRPHWLPHV VLPSO\ UHIHUUHG WR DV FRPSRVLWHV )LEHUUHLQIRUFHG FRPSRVLWHV FRQVLVW RI ILEHUV ERXQG ZLWKLQ D PDWUL[

)LEHUV FDQ EH PDGH RI JODVV FDUERQ DUDPLG RU PHWDO ZKHUHDV FRPPRQ FKRLFHV IRU WKH PDWUL[ DUH SRO\PHUV FHUDPLFV RU PHWDOV 'LVFRQWLQXRXV ILEHUV FRQWLQXRXV ILEHUV RU IDEULFV DUH FRPPRQO\ XVHG DV UHLQIRUFHPHQWV

a)

Greenhouse Gas Emission Regulations, CAFE, GHG, NOM-163-SEMARNAT- ENER-SCFI-2013

Euro 6 regulation, Regulation 2020: 95 g/km of CO₂

Phase IV standard

2020 regulation under discussion

b)

Figure 1.1. Worldwide awareness of the use of lightweight structures in the auto- motive industry. a) Correlation between CO ₂ emissions and car weight for different transmissions, b) established regulations worldwide aiming to reduce CO ₂ emis- sions. (After [5]).

(QJLQHHULQJ FRPSRVLWH PDWHULDOV DUH W\SLFDOO\ IRXQG LQ WKH DXWRPRWLYH ZLQG DHUR

QDXWLFDO DQG DHURVSDFH LQGXVWULHV WR PHQWLRQ D IHZ > @ 7KH FRPPRQ GHQRPLQDWRU RI DOO RI WKHP LV WKH VDPH WR SURGXFH OLJKWZHLJKW VWUXFWXUHV ZLWK JRRG PHFKDQLFDO DQG PRUH UHFHQWO\ PXOWLIXQFWLRQDO SURSHUWLHV >  @ ,Q WKH ZLQG LQGXVWU\ IRU LQVWDQFH

UHGXFHG ZHLJKW LQ FRPELQDWLRQ ZLWK JRRG IDWLJXH SURSHUWLHV RIIHUHG E\ FRPSRVLWHV LV RIWHQ VRXJKW LQ WKH ZLQG WXUELQH EODGHV > @ $ GLUHFW FRUUHODWLRQ EHWZHHQ WKH &2 ₂ HPLVVLRQV DQG WKH ZHLJKW RI WKH VWUXFWXUHV LQ WKH WUDQVSRUWDWLRQ LQGXVWU\ )LJ D LV D ZRUOGZLGH FRQFHUQ ZKLFK KDV OHG WR WKH FUHDWLRQ RI VHYHUDO UHJXODWLRQV DQG SURJUDPV DLPLQJ WR UHGXFH WKHVH HPLVVLRQV )LJ E $V D FRQVHTXHQFH UHVHDUFK SURJUDPV DLPLQJ IRU LPSURYHG GHVLJQV RI OLJKWZHLJKW VWUXFWXUHV KDYH DULVHQ LQ GLIIHUHQW ILHOGV

ZKHUH FRPSRVLWH VWUXFWXUHV RIWHQ SOD\ D NH\ UROH

 )DEULF UHLQIRUFHPHQWV DQG ZRYHQ IDEULFV

)DEULF UHLQIRUFHPHQWV FDQ EH FODVVLILHG DV LV VKRZQ LQ )LJ  7KH VLPSOHVW IDE

ULF UHLQIRUFHPHQW LV VRFDOOHG VWUDQG PDW ZKLFK FRQVLVWV RI FKRSSHG ILEHUV UDQGRPO\

GLVWULEXWHG DQG LV W\SLFDOO\ FRQVLGHUHG DV D QRQZRYHQ IDEULF $QRWKHU W\SH RI QRQ

ZRYHQ IDEULF FDQ EH EXLOW E\ VWLWFKLQJ D VWDFN RI XQLGLUHFWLRQDO SOLHV UHVXOWLQJ LQ D

VRFDOOHG QRQFULPS UHLQIRUFHPHQW :KHQ ILEHUV DUH EXQGOHG WRJHWKHU RQH JHWV D ILEHU

Fabric reinforcements

Braided Woven Knitted

Two- dimensional

Three-dimensional Non-woven

Stitched (non-crimp) Strand mat Three-dimensio

Knitted onal tched (non-cr

Braided Stit

d Strand mat

Plain weave Twill weave Satin weave (5-harness) Plain weave Twill wea ave Satin weave (5-harness) S atin weave (5-harn

Non woven raided

( ) ) S

Figure 1.2. A classification of textile reinforcements.

EXQGOH RU \DUQ <DUQV FDQ EH ZLQGHG DURXQG DQ D[LV WR JHW D EUDLGHG IDEULF WKH\ FDQ EH ³LQWHUORRSHG´ WR JHW D NQLWWHG IDEULF RU WKH\ FDQ EH LQWHUODFHG WR JHW D ZRYHQ IDEULF

)LJ 

:RYHQ IDEULFV FDQ EH LQWHUODFHG E\ DOWHUQDWLQJ WKH \DUQV LQ GLIIHUHQW RUGHUV DFKLHY

LQJ GLIIHUHQW ZRYHQ IDEULF DUFKLWHFWXUHV ,I WKH LQWHUODFLQJ LV GRQH DURXQG WZR D[HV

RQH JHWV D WZRGLPHQVLRQDO ZRYHQ IDEULF ZKHUHDV LI DQ DGGLWLRQDO \DUQ LV LQWHUODFHG RXWRISODQH RQH JHWV D WKUHHGLPHQVLRQDO ZRYHQ IDEULF $PRQJ WKH PRVW FRPPRQ DU

FKLWHFWXUHV IRU WZRGLPHQVLRQDO ZRYHQ IDEULFV DUH VRFDOOHG SODLQ ZHDYH WZLOO ZHDYH RU VDWLQ ZHDYH )LJ 

3HUKDSV WKH PRVW W\SLFDO XVH RI FRPSRVLWHV KDV EHHQ KLVWRULFDOO\ WKH XVH RI XQLGL

UHFWLRQDO WDSH ODPLQDWHV D FRQWLQXRXV ILEHU FRPSRVLWH 7KHVH WDSH ODPLQDWHV FRQVLVW RI SOLHV VWDFNHG RQ WRS RI HDFK RWKHU ZLWK GLIIHUHQW RULHQWDWLRQV WR DFKLHYH SURSHUWLHV VSHFLILHG E\ WKH GHVLJQHU :KHQ WKH GUDSHDELOLW\ LV D FRQFHUQ LH WKH DELOLW\ WR FRQ

IRUP WR FRPSOH[ PXOWLFXUYHG VXUIDFHV QHHGHG IRU WKH PDQXIDFWXULQJ RI FRPSOH[ SDUWV

>@ D PRUH VXLWDEOH RSWLRQ LV WKH XVH RI ZRYHQ IDEULF UHLQIRUFHG FRPSRVLWHV > @

$ GUDZEDFN LV WKDW WKH XQGXODWLRQ RI WKH \DUQV FDXVHV D UHGXFHG VWLIIQHVV LQ WKH LQSODQH GLUHFWLRQV LQ FRPSDULVRQ WR WDSH ODPLQDWHV +RZHYHU LPSURYHG RXWRISODQH SURSHU

WLHV DQG JRRG UHVLVWDQFH WR LPSDFW ORDGV DUH FRPPRQO\ UHJDUGHG DV EHQHILWV > @

 7RZDUGV YLUWXDO WHVWLQJ RI ZRYHQ FRPSRVLWHV VFDOHV DQG PRGHOLQJ DVVXPSWLRQV

:RYHQ FRPSRVLWH VWUXFWXUHV SRVVHVV LQKHUHQWO\ D KLHUDUFKLFDO DUFKLWHFWXUH UDQJLQJ IURP D PLFURVFRSLF VFDOH WR D VWUXFWXUDO VFDOH OHYHO )LJ  7KHUH DUH QRW VWULFW OLPLWV ERXQGLQJ WKHVH VFDOHV +RZHYHU RQH FDQ W\SLFDOO\ VLWXDWH WKH PLFURVFRSLF VFDOH RU PLFURVFDOH LQ WKH RUGHU RI 1 · 10 ⁻⁶ P ZKHUH WKH VLQJOH ILEHUV WKH VXUURXQGLQJ PDWUL[

DQG DQ LQWHUSKDVH ]RQH EHWZHHQ WKHP DUH RI FRQFHUQ 7KH PHVRVFRSLF VFDOH RIWHQ UHIHUUHG WR DV WKH PHVRVFDOH FRQVLGHUV KRPRJHQHRXV \DUQV LQWHUODFHG WR EXLOG D KHW

HURJHQHRXV ODPLQD /DPLQDH DUH WKHQ VWDFNHG WRJHWKHU WR EXLOG D ODPLQDWH 7KLV VFDOH

LV DSSUR[LPDWHO\ LQ WKH RUGHU RI 1 · 10 ⁻³ ± 1 · 10 ⁻² P ,Q WKH PDFURVFRSLF VFDOH RU PDFURVFDOH WKH ZRYHQ FRPSRVLWH LV D KRPRJHQHRXV HQWLW\ ZKLFK FDQ EH D SDUW RI D VXEFRPSRQHQW D FRPSRQHQW RU D VWUXFWXUH LQ WKH RUGHU RI 1 · 10 ⁰ ± 1 · 10 ¹ P

Mesoscopic scale Macroscopic scale

(subcomponent and component) Macroscopic

scale (structure)

Microscopic scale

Figure 1.3. Different scales encountered in the analysis of woven composite struc- tures.

3LRQHHULQJ PHFKDQLFDO DQDO\VHV RI ZRYHQ FRPSRVLWHV ZHUH EDVHG RQ VLPSOLILHG DQ

DO\WLFDO PRGHOV WR HVWLPDWH WKHLU KRPRJHQL]HG HODVWLF SURSHUWLHV > @ 0RUH PRGHUQ DQDO\WLFDO PRGHOV KDYH EHHQ UHFHQWO\ GHYHORSHG IRU IDVW VFUHHQLQJ >@ :KHQ PRUH FRPSOH[ JHRPHWULHV DQG PRUH GHWDLOHG LQIRUPDWLRQ DERXW WKH VWUXFWXUDO LQWHJULW\ RI ZRYHQ FRPSRVLWH ODPLQDWHV LV QHHGHG WKH XVH RI FRPSXWDWLRQDO PRGHOV LV PRUH VXLW

DEOH 0RUH VSHFLILFDOO\ ILQLWH HOHPHQW DQDO\VLV LV SHUKDSV WKH PRVW SRSXODU WHFKQLTXH IRU SHUIRUPLQJ FRPSXWHU DQDO\VHV :KHUHDV WKLV WHFKQLTXH KDV EHHQ XVHG VLQFH D IHZ

\HDUV EDFN > @ UHVHDUFK LV VWLOO EHLQJ FRQGXFWHG RQ RQH KDQG WR FUHDWH PRUH UH

OLDEOH PRGHOV ZLWK D EHWWHU XQGHUVWDQGLQJ RI WKH XQGHUO\LQJ SK\VLFDO PHFKDQLVPV DQG RQ WKH RWKHU KDQG WR UHGXFH WKH GHSHQGHQFH RQ H[SHULPHQWDO WHVWLQJ >    @

7KH VWXG\ RI WKH ZRYHQ FRPSRVLWH¶V FRQVWLWXHQWV DQG VWUXFWXUHV LV W\SLFDOO\ GRQH DW GLIIHUHQW VFDOHV 7KLV FDQ EH GRQH HLWKHU H[SHULPHQWDO WHVWLQJ RU E\ YLUWXDO WHVWLQJ )LJ  XVLQJ FRPSXWHU WRROV XVLQJ DSSURSULDWH JHRPHWULFDO DQG FRQVWLWXWLYH PRGHOV

,Q WKH DEVHQFH RI DFFXUDWH PRGHOLQJ DSSURDFKHV ZLWK D SURYHQ DELOLW\ WR SUHGLFW IDLO

XUH DQG VWUHQJWK RI FRPSRVLWHV D VLJQLILFDQW DPRXQW RI H[SHULPHQWDO WHVWLQJ LV QHHGHG

>  @ $V DQ H[DPSOH DQ DLUIUDPH VWUXFWXUH UHTXLUHV DERXW 1 · 10 ⁴ WHVWV DW GLI

IHUHQW VFDOHV VSHFLPHQV FRPSRQHQWV DQG VWUXFWXUHV IRU FHUWLILFDWLRQ > @ ZKLFK

LV DQRWKHU PRWLYDWLRQ IRU FRQGXFWLQJ UHVHDUFK LQ WKH DXWRPRWLYH > @ DHURQDXWL

FDO > @ DHURVSDFH > @ LQGXVWULHV GHYRWHG WR WKH VLPXODWLRQ RI WKH PHFKDQLFDO UHVSRQVH RI ZRYHQ FRPSRVLWHV

Virtual testing Experimental testing

Experimental testing Virtual testing

Macro

Micro Meso

Figure 1.4. Experimental and virtual testing of woven composite structures at dif- ferent scales.

,Q WKH KLHUDUFKLFDO DUFKLWHFWXUH RI ZRYHQ FRPSRVLWHV DVVXPSWLRQV DUH PDGH LQ HDFK VFDOH WR VLPSOLI\ WKH DQDO\VHV LQYROYHG LQ WKH GHVLJQ SURFHVV $W WKH PLFURVFRSLF VFDOH

WKH ILEHU DQG PDWUL[ LQWHUDFWLRQV DUH VWXGLHG 7KH GLVSRVLWLRQ RI WKH ILEHUV RU SDFNLQJ RI WKH ILEHUV ZLWKLQ WKH ILEHU EXQGOHV LV D VWRFKDVWLF SURFHVV ZKLFK FDQQRW EH FRQWUROOHG

7KH SDFNLQJ RI WKH ILEHUV LQ WKH ILEHU EXQGOHV LV RIWHQ LGHDOL]HG LQ KH[DJRQDO RU VTXDUH SDFNHG DUUD\ PRGHOV WR DQDO\]H WKH PLFURVFRSLF VFDOH > @ 7KHVH LGHDOL]DWLRQV SURYLGH UHDVRQDEOH DSSUR[LPDWLRQV RI WKH OLQHDU HODVWLF EHKDYLRU RI WKH ILEHU EXQGOHV

>  @ DQG VXEWOH GLIIHUHQFHV FDQ EH IRXQG LQ WKH QRQOLQHDU UHJLPH ZKHQ GDPDJH HYROXWLRQ LV FRQVLGHUHG > @ 7R IXOO\ FDSWXUH WKH IDLOXUH PHFKDQLVPV LQYROYHG

PRUH GHWDLOHG PRGHOLQJ DSSURDFKHV FRQVLGHU WKH VWRFKDVWLF QDWXUH RI ILEHU SDFNLQJ DQG LQWHUIDFHV EHWZHHQ WKH PDWUL[ DQG ILEHUV KDYH WR EH FRQVLGHUHG VHH HJ > @

DW H[SHQVHV RI VLJQLILFDQWO\ KLJKHU FRPSXWDWLRQDO FRVWV 7KH LGHDOL]HG SDFNHG DUUD\

PRGHOV KRZHYHU UHSUHVHQW IDLUO\ ZHOO WKH VWUXFWXUDO EHKDYLRU RI WKH ILEHU EXQGOHV ZKHQ WKH\ DUH FRQVLGHUHG LQ WKH ZRYHQ FRPSRVLWH DW WKH PHVRVFRSLF VFDOH HJ > @

$W WKH PHVRVFRSLF VFDOH ILEHU EXQGOHV DUH DOUHDG\ FRQVLGHUHG DV KRPRJHQHRXV HQ

WLWLHV 0HFKDQLFDO SURSHUWLHV DUH LQKHULWHG IURP DQDO\VHV DW WKH PLFURVFDOH $ FRP

PRQ DVVXPSWLRQ PDGH DW WKLV VFDOH LV WKDW D VLQJOHOD\HU UHSHDWLQJ XQLW FHOO 58&

ZLWK WKH ZRYHQ DUFKLWHFWXUH FDQ UHSUHVHQW WKH JOREDO EHKDYLRU RI WKH ODPLQDWH HJ

>    @ $QRWKHU DSSURDFK ZLWK WKH DLG RI VFDQV REWDLQHG ZLWK ;UD\ FRP

SXWHG WRPRJUDSK\ > @ LV WR SHUIRUP LPDJHEDVHG VLPXODWLRQV RI ZRYHQ FRPSRV

LWHV DW WKH PHVRVFDOH >    @ 7KHVH DQDO\VHV FRQVLGHU PRGHOV H[WUDFWHG IURP WKH VFDQV ZKHUH DOO WKH OD\HUV LQ WKH ODPLQDWH DQG PRVW RI WKH LPSHUIHFWLRQV DULV

LQJ DV D FRQVHTXHQFH RI WKH PDQXIDFWXULQJ SURFHVVHV DUH FRQVLGHUHG ([DPSOHV RI WKHVH LPSHUIHFWLRQV DUH WKH UHODWLYH OD\HU VKLIWLQJ EHWZHHQ OD\HUV YDULDELOLWLHV LQ WKH FURVVVHFWLRQDO DUHDV RI WKH \DUQV FKDQJHV YROXPH IUDFWLRQV LQ WKH \DUQV DQG YDULDELOL

WLHV LQ WKH FHQWHUV RI WKH \DUQV WR PHQWLRQ D IHZ $OWKRXJK LPDJHEDVHG VLPXODWLRQV LV

D WHFKQLTXH ZKLFK LV LQ SULQFLSOH DSSHDOLQJ LW LV VWLOO TXHVWLRQDEOH LI WKH VDPH LPSHUIHF

WLRQV FDQ EH IRXQG WKURXJK WKH ZKROH VWUXFWXUH HVSHFLDOO\ LQ VWUXFWXUHV ZLWK FRPSOH[

FXUYDWXUHV ZKHUH WKH UHODWLYH OD\HU VKLIWLQJ FDQ FKDQJH 7KHUHIRUH LW LV VWLOO LPSRUWDQW WR XQGHUVWDQG WKH XQGHUO\LQJ PHFKDQLVPV KDSSHQLQJ DW WKH PHVRVFDOH DQG WR LGHQWLI\

ZKLFK DUH WKH PDLQ UHVSRQVLEOH IRU WKH RYHUDOO VWUXFWXUDO LQWHJULW\ RI WKH ZRYHQ FRP

SRVLWH VWUXFWXUHV

7KH JHRPHWULFDO LGHDOL]DWLRQV DW WKH PHVRVFDOH FDQ KDYH D PRUH SURQRXQFHG HIIHFW RQ WKH RYHUDOO UHVSRQVH RI WKH FRPSRVLWH WKDQ VRPH LGHDOL]DWLRQV DW WKH PLFURVFDOH )RU LQVWDQFH ZKHUHDV LQ WKH PLFURVFRSLF VFDOH UDQGRPO\ GLVWULEXWHG ILEHUV DQG LGHDOL]HG SDFNHG DUUD\ PRGHOV SURYLGH D IDLUO\ VLPLODU UHVSRQVH LQ WKH HODVWLF UHJLPH > @

LQ WKH PHVRVFRSLF VFDOH E\ RQO\ FRQVLGHULQJ UHODWLYH OD\HU VKLIWLQJ DV DQ LGHDOL]HG PHVRVWUXFWXUDO LPSHUIHFWLRQ DQ LQSODQH VWLIIHQLQJ DV KLJK DV  % KDV EHHQ REVHUYHG H[SHULPHQWDOO\ DQG DQDO\WLFDOO\ >  @ 7KLV VXJJHVWV WKDW OD\HU VKLIWLQJ PLJKW EH D GRPLQDQW HIIHFW ZKLFK FRXOG EH DOVR VWURQJO\ LQYROYHG LQ WKH IDLOXUH SURFHVV DQG WKH RYHUDOO UHVSRQVH RI WKH ODPLQDWH 7KLV KDV PRWLYDWHG PRUH UHFHQW UHVHDUFK WR FRQVLGHU OD\HU VKLIWLQJ DV DQ LGHDOL]HG LPSHUIHFWLRQ >   @

Gap!

Innovation Integrated approach

Traditional (experimental) testing approach Virtual testing approach

Innovati oach egrated appro Int

al (experimental) testing Traditiona

tual testing approach tual testing ap Virt

Virt Switching point oint int int int int nt t t t t

Effort (elapsed time, cost)

T ec hnical p erformance

Figure 1.5. Integration of virtual and experimental testing approaches. Adapted from [54, 55].

9LUWXDO WHVWLQJ FDQQRW EH FRPSOHWHO\ LVRODWHG IURP H[SHULPHQWDO WHVWLQJ &RQVWLWX

WLYH PRGHOV DW WKH PHVRVFRSLF VFDOH DUH W\SLFDOO\ SKHQRPHQRORJLFDO DSSURDFKHV WKDW UHO\ RQ VHYHUDO SDUDPHWHUV REWDLQHG E\ H[SHULPHQWDO WHVWLQJ 7KHVH SDUDPHWHUV RIWHQ UHTXLUH UHFDOLEUDWLRQ HDFK WLPH WKH FRQVWLWXHQWV FKDQJH DQG DV D UHVXOW H[SHQVLYH LQ

GXVWULDO H[SHULPHQWDO FDPSDLJQV DUH FDUULHG RXW >  @ 1RQHWKHOHVV WKH QXPEHU RI H[SHULPHQWDO WHVWV PD\ EH UHGXFHG LI VRPH RI WKH SDUDPHWHUV FDQ EH GHWHUPLQHG E\

YLUWXDO WHVWLQJ ZLWK DSSURSULDWH PRGHOLQJ RI WKH IDLOXUH PHFKDQLVPV DW WKH PHVRVFRSLF VFDOH >@

)URP DQ LQGXVWULDO SHUVSHFWLYH RI WKH SURGXFW GHYHORSPHQW SURFHVV LW LV QRW H[

SHFWHG WKDW YLUWXDO WHVWLQJ ZLOO RQO\ VXSSOHPHQW VRPH H[SHULPHQWDO WHVWLQJ EXW UDWKHU WR VHUYH DV D FRPSOHPHQW WR SURPRWH DQ HQKDQFHG SHUIRUPDQFH RI VWUXFWXUDO FRP

SRQHQWV >  @ 7KLV LV KDV EHHQ VXJJHVWHG DIWHU H[DPLQLQJ KRZ WKH WHFKQLFDO

SHUIRUPDQFH EHKDYHV ZLWK UHVSHFW WR WKH HIIRUW HODSVHG WLPH FRVW )LJ  7KH

WUDGLWLRQDO H[SHULPHQWDO WHVWLQJ FXUYH VKRZV WKH SHUIRUPDQFH DFKLHYHG ZLWK H[SHU

LPHQWDO WHVWLQJ ZKHUHDV WKH YLUWXDO WHVWLQJ FXUYH LQGLFDWHV WKH SHUIRUPDQFH DFKLHYHG ZLWK VLPXODWLRQ WRROV 7\SLFDOO\ RQO\ DERXW  % RI WKH WHFKQLFDO SHUIRUPDQFH FDQ EH DFKLHYHG LQ D FRQVLGHUDEO\ OHVV DPRXQW RI WLPH DQG FRVW VKRZLQJ VLJQLILFDQW VDYLQJV

+RZHYHU WKH UHPDLQLQJ  % FDQQRW EH DFKLHYHG GXH WR DVVXPSWLRQV LQDFFXUDFLHV DQG RWKHU OLPLWDWLRQV RI PRGHOV GXULQJ YLUWXDO WHVWLQJ OHDYLQJ D JDS LQ SHUIRUPDQFH

7RZDUGV WKH SURSHU LQWHJUDWLRQ RI ERWK YLUWXDO DQG H[SHULPHQWDO WHVWLQJ DSSURDFKHV LQ D ³VZLWFKLQJ SRLQW´ WKH SHUIRUPDQFH JDS LV QRW RQO\ HOLPLQDWHG EXW HYHQ DQ LQFUHDVHG LQQRYDWLRQ LV H[SHFWHG

 $LPV RI WKH WKHVLV

7KH RYHUDOO SXUSRVH RI WKH SUHVHQW WKHVLV LV WR LQYHVWLJDWH UHODWLRQVKLSV EHWZHHQ WKH JOREDO PHFKDQLFDO UHVSRQVH RI ZRYHQ FRPSRVLWHV DQG XQGHUO\LQJ SK\VLFDO PHFKDQLVPV DW WKH PHVRVFRSLF VFDOH 7KH VSHFLILF REMHFWLYHV DUH

‡ ,PSOHPHQW D FRPSXWDWLRQDO KRPRJHQL]DWLRQ WHFKQLTXH WR LQYHVWLJDWH WKH HIIHFWV RI XVLQJ GLIIHUHQW W\SHV RI ERXQGDU\ FRQGLWLRQV DQG WKH QXPEHU RI OD\HUV LQ WKH HVWLPDWLRQ RI HODVWLF SURSHUWLHV DQG VWUDLQ ILHOGV

‡ ,QYHVWLJDWH VWUXFWXUHSURSHUW\ UHODWLRQVKLSV RI HIIHFWV REVHUYHG H[SHULPHQWDOO\

LQ WKH OLQHDU HODVWLF DQG QRQOLQHDU UHVSRQVHV GXH WR GDPDJH QXFOHDWLRQ DQG JURZWK

‡ &RPSDUH GHWDLOHG IUDFWXUH SDWKV LQ PRGHOV ZLWK LGHDOL]HG LPSHUIHFWLRQV ZLWK PRUH JHRPHWULFDOO\ FRPSOH[ PRGHOV DQG H[SHULPHQWV UHSRUWHG LQ WKH OLWHUDWXUH

‡ ,GHQWLI\ IDYRUDEOH UHJLRQV IRU GHODPLQDWLRQ RQVHW DV D FRQVHTXHQFH RI IUHHHGJH HIIHFWV

‡ ,QYHVWLJDWH UHODWLRQVKLSV EHWZHHQ WKH PHVRVWUXFWXUH DQG WKH DSSDUHQW LQWHUODP

LQDU VKHDU VWUHQJWK

 7KHRUHWLFDO IRXQGDWLRQV

 &RPSXWDWLRQDO KRPRJHQL]DWLRQ RI KHWHURJHQHRXV PHGLD

Ω

C _ijkl ⁰

C _ijkl ¹

C _ijkl ²

C _ijkl ^...n C ijkl

Figure 2.1. Schematic representation of a homogenized stiffness tensor C _{i jkl} of a heterogeneous material containing n inclusions with different stiffness.

&RPSXWDWLRQDO KRPRJHQL]DWLRQ LV D WHFKQLTXH FRPPRQO\ XVHG WR HVWLPDWH HIIHFWLYH SURSHUWLHV RI KHWHURJHQHRXV PDWHULDOV > @ )LJ  $OWKRXJK WKLV WHFKQLTXH DS

SOLHV WR DQ\ KHWHURJHQHRXV PDWHULDO LQ WKH FRQWH[W RI SHULRGLF VWUXFWXUHV WKLV LV GRQH E\ VHOHFWLQJ D UHSHDWLQJ XQLW FHOO 58& ZLWK D GRPDLQ V  )RU D OLQHDU HODVWLF FRQVWL

WXWLYH EHKDYLRU DQG DVVXPLQJ VPDOO GHIRUPDWLRQV WKH PDFURVFRSLF VWUHVVHV σ i j  DQG VWUDLQV  i j  FDQ EH H[SUHVVHG DV DYHUDJHV RI WKH FRUUHVSRQGLQJ PLFURVFRSLF SDUWV σ i j

DQG  i j  UHVSHFWLYHO\ DV >@

σ i j = 1 V



V

σ i j d V  (2.1a)

 i j = 1 V



V

 i j d V  (2.1b)

&RPSXWDWLRQDO KRPRJHQL]DWLRQ LV EDVHG RQ WKH +LOO0DQGHO SULQFLSOH >@ ZKLFK FDQ EH XQGHUVWRRG DV D SULQFLSOH RI HQHUJ\ HTXLYDOHQFH LQ WKH PDFUR DQG WKH PLFUR

VFDOHV

σ i j  i j = 1 V



V

σ i j  i j d V  (2.2)

7KH +LOO0DQGHO SULQFLSOH FDQ EH UHZULWWHQ FRQVLGHULQJ WKH EDODQFH EHWZHHQ VWUDLQ DQG SRWHQWLDO HQHUJLHV DQG LQ WKH DEVHQFH RI ERG\ IRUFHV DV >@



Γ

(t i − σ i j n _j )(u i −  ik x _k )dS = 0 (2.3)

ZKHUH t _i  u _i DQG x _k UHSUHVHQW WUDFWLRQV GLVSODFHPHQWV DQG SRVLWLRQV LQ WKH PLFURVFDOH 58& RQ LWV ERXQGDU\ Γ  7KLV H[SUHVVLRQ LV VDWLVILHG D SULRUL E\ WKH XVH RI HLWKHU WUDFWLRQ RU GLVSODFHPHQW ERXQGDU\ FRQGLWLRQV +RZHYHU WKLV FKRLFH SURYLGHV GLIIHUHQW HVWLPD

WLRQV RI WKH KRPRJHQL]HG VWLIIQHVV WHQVRU DQG KDYH GLIIHUHQW LQWHUSUHWDWLRQV EDVHG RQ GLIIHUHQW DVVXPSWLRQV 2QH DVVXPSWLRQ LV WKDW WKH VWUDLQ ILHOG LQ WKH PLFUR DQG PDFUR

VFDOHV LV XQLIRUP ZKLFK LV NQRZQ DV WKH 9RLJW 7D\ORU DVVXPSWLRQ $QRWKHU DVVXPS

WLRQ LV WKDW WKH VWUHVV ILHOG LQ WKH PLFUR DQG PDFURVFDOHV LV XQLIRUP DOVR NQRZQ DV WKH 5HXVV +LOO DVVXPSWLRQ 7KHVH DVVXPSWLRQV SURYLGH XSSHU DQG ORZHU ERXQGV UH

VSHFWLYHO\ LQ WKH HVWLPDWLRQ RI WKH DFWXDO KRPRJHQL]HG VWLIIQHVV WHQVRU :KHUHDV WKH 9RLJW DVVXPSWLRQ FDQ EH LPSRVHG WKURXJK WKH XVHG RI XQLIRUP GLVSODFHPHQW ERXQGDU\

FRQGLWLRQV 8'%&V WKH 5HXVV DVVXPSWLRQ FDQ EH LPSRVHG E\ XVLQJ XQLIRUP WUDFWLRQ

ERXQGDU\ FRQGLWLRQV 87%&V $QRWKHU W\SH RI ERXQGDU\ FRQGLWLRQV ZKLFK RIWHQ

SURYLGH EHWWHU HVWLPDWLRQV RI WKH KRPRJHQL]HG SURSHUWLHV DUH SHULRGLF GLVSODFHPHQW

ERXQGDU\ FRQGLWLRQV 3%&V 7KHVH DUH EDVHG RQ WKH DVVXPSWLRQ RI SHULRGLFLW\ RQ WKH

GLVSODFHPHQW ILHOGV RQ RSSRVLWH ERXQGDULHV RI WKH 58& FI >@

 &RPSXWDWLRQDO GDPDJH DQG IUDFWXUH PHFKDQLFV

7KH VRFDOOHG FRQWLQXXP GDPDJH PHFKDQLFV &'0 LV DQ DSSURDFK GHDOLQJ ZLWK WKH GHJUDGDWLRQ RI WKH PHFKDQLFDO SHUIRUPDQFH RI VROLGV DV D FRQVHTXHQFH RI WKH SUHVHQFH RI PLFURFUDFNV DQG WKHLU HYROXWLRQ >  @ ,Q &'0 WKH FRQVWLWXWLYH EHKDYLRU RI VROLGV LV RIWHQ GHVFULEHG LQ D SKHQRPHQRORJLFDO PDQQHU ZLWK WKH DLG RI GDPDJH YDUL

DEOHV DVVRFLDWHG ZLWK WKH UHGXFWLRQ RI WKH ORDGEHDULQJ FDSDELOLWLHV RI VROLGV ,Q WKH IL

QLWH HOHPHQW FRQWH[W WKLV LV LPSOHPHQWHG DV D SKHQRPHQRORJLFDO FRQVWLWXWLYH EHKDYLRU DW DQ LQWHJUDWLRQ SRLQW OHYHO 7KH VRFDOOHG ORFDO DSSURDFKHV WR GDPDJH PRGHOLQJ FDQ KDYH SDWKRORJLFDO PHVK GHSHQGHQF\ VLQFH WKH GHJUDGDWLRQ RI WKH VWUDLQ HQHUJ\ GHQVLW\

LV GLUHFWO\ UHODWHG WR WKH HOHPHQW VL]H 7R DOOHYLDWH WKLV UHJXODUL]DWLRQ VFKHPHV VXFK DV QRQORFDO FRQWLQXXP GDPDJH PRGHOV RU JUDGLHQWHQKDQFHG GDPDJH PRGHOV DUH ZLGHO\

XVHG >@

&RPSXWDWLRQDO IUDFWXUH PHFKDQLFV FDQ EH FRQFHLYHG DV DQ DSSURDFK GHDOLQJ ZLWK FUDFNV ZLWK D ILQLWH OHQJWK WR PRGHO IUDFWXUH LQ VROLGV 6HYHUDO WHFKQLTXHV FDQ EH IRXQG LQ WKH OLWHUDWXUH IRU WKLV SXUSRVH XVLQJ WKH ILQLWH HOHPHQW PHWKRG DOWKRXJK QRW H[

FOXVLYHO\ 7R WKLV H[WHQW WZR ELJ JURXSV FDQ EH FDWHJRUL]HG LQ GLVFRQWLQXRXV RIWHQ UHIHUUHG WR DV GLVFUHWH DQG LQ FRQWLQXRXV RIWHQ UHIHUUHG WR DV VPHDUHG DSSURDFKHV

'LVFRQWLQXRXV DSSURDFKHV DOORZ GLVFRQWLQXLW\ RI WKH GLVSODFHPHQW ILHOG DFURVV IUDF

WXUH VXUIDFHV 'LVFRQWLQXLWLHV FDQ EH FRQVLGHUHG E\ GXSOLFDWLQJ QRGHV DW WKH ERXQGDULHV RI WKH HOHPHQWV UHPHVKLQJ > @ LQWHUIDFH HOHPHQWV XVLQJ FRKHVLYH ]RQH PRGHOV

>    @ XVLQJ HPEHGGHG VWURQJ GLVFRQWLQXLWLHV DW DQ LQWUDHOHPHQW OHYHO RU WKH H[WHQGHG ILQLWH HOHPHQW PHWKRG >      @ WR PHQWLRQ D IHZ

0RVW RI WKHVH WHFKQLTXHV UHTXLUH DGGLWLRQDO DGKRF FULWHULD EDVHG RQ FLUFXPIHUHQWLDO VWUHVVHV VWUDLQ HQHUJ\ GHQVLWLHV HQHUJ\ UHOHDVH UDWHV >@ RU YLUWXDO FUDFN FORVXUH WHFK

QLTXHV >@ WR GHWHUPLQH WKH GLUHFWLRQ RI WKH FUDFN SDWKV 0RUHRYHU WUDFNLQJ RI WKH GLVFRQWLQXLWLHV LV QHHGHG ZKLFK FDQ EHFRPH D FKDOOHQJLQJ WDVN IRU DUELWUDU\ FUDFN SDWKV LQYROYLQJ EUDQFKLQJ DQG PHUJLQJ

2Q WKH RWKHU KDQG FRQWLQXRXV DSSURDFKHV WR PRGHO IUDFWXUH FRQVLGHU FRQWLQXLW\ RI WKH GLVSODFHPHQW ILHOG 3HUKDSV WKH PRVW SRSXODU WHFKQLTXHV DUH SHULG\QDPLF PRGHOV DQG WKH SKDVHILHOG 3) DSSURDFK WR EULWWOH IUDFWXUH 7KH ODWWHU DGRSWHG LQ WKH SUHVHQW WKHVLV WUDFHV EDFN RQ WKH UHIRUPXODWLRQ RI WKH *ULIILWK¶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

FDQ EH DGGUHVVHG

,Q WKH SUHVHQW WKHVLV ERWK D &'0 DQG D 3) DSSURDFK WR IUDFWXUH DUH LPSOHPHQWHG WR LQYHVWLJDWH VRPH HIIHFWV DW WKH PHVRVFRSLF VFDOH 7KH HIIHFW RI UHODWLYH OD\HU VKLIWLQJ LQ WKH GDPDJH QXFOHDWLRQ DQG FUDFN JURZWK LQ ZRYHQ FRPSRVLWH ODPLQDWHV LV DGGUHVVHG

$ PXOWLVFDOH &'0 DSSURDFK LV ILUVW DGRSWHG LQ 3DSHU ,, 7KLV LV XVHG WR LQYHVWLJDWH

WKH ORFDWLRQ DQG UHGLVWULEXWLRQ RI GDPDJH GHYHORSPHQW EHWZHHQ WKH ILEHU EXQGOHV DQG

PDWUL[ GXH WR WKH UHODWLYH OD\HU VKLIWLQJ EHWZHHQ OD\HUV )RU D PRUH GHWDLOHG GHVFULS

WLRQ RI GDPDJH GHYHORSPHQW EDVHG RQ WKH GHVFULSWLRQ RI WKH HYROXWLRQ RI FUDFNV 3) PRGHOLQJ RI EULWWOH IUDFWXUH LV XVHG DV D KLJKILGHOLW\ DSSURDFK GHVFULELQJ FUDFNV DQG WR PRGHO FUDFN SURSDJDWLRQ )LUVW LQ 3DSHU ,,, D 3) DSSURDFK LV IRUPXODWHG DQG LWV FD

SDELOLWLHV DUH H[SORUHG E\ VWXG\LQJ IUDFWXUH RI FRPSRVLWHV LQ WKH PLFURVFRSLF VFDOH DQG DW WKH FRXSRQ VFDOH EHORZ DQG DERYH WKH PHVRVFRSLF VFDOH 7KHQ LQ 3DSHU ,9 FUDFN GHYHORSPHQW DW DQ HDUO\ VWDJH LQ ZRYHQ FRPSRVLWHV LV LQYHVWLJDWHG DW WKH PHVRVFRSLF VFDOH

 $ PXOWLVFDOH FRQWLQXXP GDPDJH PHFKDQLFV DSSURDFK

$ JHQHUDO DQLVRWURSLF FRQWLQXXP GDPDJH PRGHO 0DWUL[ DQG ILEHU EXQGOHV LQ WKH WUDQVYHUVH DQG VKHDU ILEHU GLUHFWLRQV

7KH FRQWLQXXP GDPDJH PRGHO IRU WKH SRO\PHU PDWUL[ DQG WUDQVYHUVHVKHDU ILEHU GLUHF

WLRQV DVVXPHV WKDW WKH HODVWLF VWUDLQ HQHUJLHV LQ WKH GDPDJHG DQG XQGDPDJHG PDWHULDOV IROORZ D VLPLODU IRUP )RU VPDOO GHIRUPDWLRQV DQG DGLDEDWLF SURFHVVHV WKH *LEEV IUHH HQHUJ\ GHQVLW\ LV JLYHQ E\ > @

χ = (2ρ) ⁻¹ σ :  = (2ρ) ⁻¹ σ : D : ˜ C ⁻¹ : D : σ (2.4) ZKHUH ρ LV WKH GHQVLW\ σ LV WKH &DXFK\ VWUHVV WHQVRU  LV WKH LQILQLWHVLPDO VWUDLQ WHQVRU

C LV WKH VWLIIQHVV PDWUL[ DQG WKH GDPDJH WHQVRU D = (1 − D ˜ i j ) ⁻¹ FRQWDLQV WKH GDPDJH SDUDPHWHUV D _{i j}  ZLWK i j    7KH LQGH[  UHIHUV WR WKH ILEHU GLUHFWLRQ DQG 

WR WKH WUDQVYHUVH SODQHV 7KH YDOXHV RI D _{i j} UDQJH IURP  XQGDPDJHG PDWHULDO WR  FRPSOHWHO\ GDPDJHG PDWHULDO ,Q XQDEULGJHG QRWDWLRQ χ FDQ EH ZULWWHQ DV

χ = 1 2 ρ

 σ ² ₁₁

(1 − D 11 ) ² E ˜ ₁₁ + σ ² ₂₂

(1 − D 22 ) ² E ˜ ₂₂ + σ ² ₃₃ (1 − D 33 ) ² E ˜ ₃₃

−

 ˜ ν 21

E ˜ ₂₂ + ˜ ν 12

E ˜ ₁₁

 σ 11 σ 22

(1 − D 11 )(1 − D 22 ) −

 ν ˜ 32

E ˜ ₃₃ + ν ˜ 23

E ˜ ₂₂

 σ 22 σ 33

(1 − D 22 )(1 − D 33 )

−

 ν ˜ 31

E ˜ ₃₃ + ν ˜ 13

E ˜ ₁₁

 σ 11 σ 33

(1 − D 11 )(1 − D 33 ) + σ ² ₁₂

2 (1 − D 12 ) ² G ˜ ₁₂ + σ ² ₂₃ 2 (1 − D 23 ) ² G ˜ ₂₃ + σ ² ₁₃

2 (1 − D 13 ) ² G ˜ ₁₃



(2.5)

ZKHUH ˜ ν i j DUH WKH 3RLVVRQ¶V UDWLRV ˜ E _{i j} DQG ˜ G _{i j} DUH WKH QRUPDO DQG WKH VKHDU HODVWLF PRGXOL DQG WKH WLOGH UHIHUV WR WKH XQGDPDJHG SURSHUW\ 1RZ OHW C GHQRWH WKH GDPDJHG VWLIIQHVV WHQVRU LH C ⁻¹ = D : ˜ C ⁻¹ : D 7KHQ RQH REWDLQ IURP (T 

C ⁻¹ = ρ ∂ ² χ

∂ σ ²  (2.6)

0RUHRYHU LW LV DVVXPHG WKDW GDPDJH LV GHYHORSHG ZKHQ D GDPDJH VXUIDFH RI WKH IRUP > @

f = 

Y : H : Y

 

ˆ g

−(c   1 (e

− 1)

γ

+γ 0 ) ≥ 0 (2.7)

FRQWUROV WKH GDPDJH HYROXWLRQ ,Q  H LV GHILQHG DV D WHQVRU LQ 9RLJW QRWDWLRQ ZLWK FRPSRQHQWV WKDW DUH FKDUDFWHULVWLF RI WKH PDWHULDO 7KHVH FRPSRQHQWV UHODWH WKH LQWHUDFWLRQ DPRQJ WKH GLIIHUHQW GDPDJH YDULDEOHV D _{i j} WKURXJK WKH GDPDJH GULYLQJ IRUFH Y 7KXV Y : H : Y ˆg JLYHV D VFDODU YDOXH FORVHO\ UHODWHG WR WKH *ULIILWK HQHUJ\ 7KH GDPDJH GULYLQJ IRUFH Y LV HQHUJ\ FRQMXJDWH WR D

Y = ρ ∂ χ

∂ D  (2.8)

7KH VHFRQG WHUP LQ  γ FRUUHVSRQGV WR D KDUGHQLQJVRIWHQLQJ IXQFWLRQ GHSHQGLQJ RQ WKH PDWHULDO SDUDPHWHUV c ₁  c ₂  WKH RQVHW RI GDPDJH JURZWK γ 0  DQG D KDUGHQLQJ YDULDEOH δ 7KH GDPDJH JURZWK LV JLYHQ E\

D = ˙λ ˙ ∂ f

∂ Y  (2.9)

ZKHUH ˙ λ LV D VRFDOOHG GDPDJH PXOWLSOLHU DQG WKDW UHSUHVHQWV WKH PDJQLWXGH RI WKH GDPDJH LQFUHPHQW DQG ∂ f /∂ Y LWV GLUHFWLRQ ,W LV KHUH DVVXPHG WKDW KDUGHQLQJ UDWH ˙δ LV UHODWHG WR ˙ λ DV >@

δ = ˙λ ˙ ∂ f

∂ γ  (2.10)

.DUXVK.XKQ7XFNHUW\SH ORDGLQJXQORDGLQJ FRQGLWLRQV DUH DSSOLHG WR UHVWULFW WKH DG

PLVVLEOH YDOXHV RI ˙ λ DQG ˙f

λ ≥ 0; f ≤ 0; ˙λf = 0 ˙ (2.11)

7KH ILUVW FRQGLWLRQ HQVXUHV LUUHYHUVLELOLW\ RI WKH GDPDJH SURFHVV WKH VHFRQG FRQGLWLRQ LV UHODWHG WR WKH GDPDJH VXUIDFH DQG WKH WKLUG FRQGLWLRQ LV D EDODQFH ODZ IRU WKH GDPDJH HYROXWLRQ 7KHUHIRUH WKH YDOXH RI ˙ λ LV GHWHUPLQHG SURYLGHG WKDW GDPDJH LV HYROYLQJ  f = ˙f = 0 LH

˙ f = ∂ f

∂ Y : ˙ Y + ∂ f

∂ γ γ = 0, f = 0. ˙ (2.12)

2QH REVHUYHV WKDW (TV  DUH LQVSLUHG IURP FODVVLFDO WKHRULHV RI DVVRFLDWHG SODVWLFLW\ 6LQFH Y Y(, D DQG γ γ(δ) WKHLU FRUUHVSRQGLQJ UDWHV FDQ EH REWDLQHG

7KXV XVLQJ WKH UDWHV WRJHWKHU ZLWK (TV  DQG E\ FRQVLGHULQJ LQ (T 

WKDW ∂ f /∂ γ = −1 RQH FDQ H[SUHVV ˙λ XVLQJ (T  LH

λ = − ˙

∂ Y ∂ f : ^{∂ Y} _

∂ f ∂ Y : ^{∂ Y} _{∂ D} : ^{∂ f} _{∂ Y} + ^{∂ γ} _{∂ δ} : ˙  (2.13) 7KHUHIRUH WRJHWKHU ZLWK (T  WKH KDUGHQLQJ HYROXWLRQ LV

δ = −˙λ ˙ (2.14)

7KH VWUHVV WHQVRU DQG WKH FRQVLVWHQW WDQJHQW VWLIIQHVV WHQVRU DUH FRPSXWHG DV

σ = ∂ χ()

∂    = ∂ σ

∂  = ∂ ² χ()

∂  ²  (2.15)

UHVSHFWLYHO\ ,PSOHPHQWDWLRQ LV GRQH LQ WKH FRQVWLWXWLYH EHKDYLRU RI D VWDQGDUG ILQLWH HOHPHQW URXWLQH XVLQJ D VWDQGDUG UHWXUQ PDSSLQJ DOJRULWKP ZLWK DQ HODVWLF SUHGLFWRU

GDPDJH FRUUHFWRU

2EVHUYH WKDW WKLV PRGHO LV UDWKHU JHQHUDO DQG WKURXJK WKH GHILQLWLRQ RI WKH FRPSR

QHQWV RI H RQH FDQ GHILQH HLWKHU DQ LVRWURSLF RU DQ DQLVRWURSLF GDPDJH PRGHO ZKHUH WKH GDPDJH YDULDEOHV FDQ LQWHUDFW ZLWK HDFK RWKHU )RU WKH ILEHU EXQGOHV WKHVH SDUDPH

WHUV ZHUH FDOLEUDWHG IURP PLFURPHFKDQLFDO PRGHOV DQG WKH WHFKQLTXH ZLOO EH H[SODLQHG LQ WKH QH[W VHFWLRQ

&RQWLQXXP GDPDJH PRGHO RI WKH ILEHU EXQGOHV IRU WKH ILEHU GLUHFWLRQ

7R REWDLQ WKH GDPDJH GHYHORSPHQW LQ WKH ILEHU EXQGOHV LQ WKH ILEHU GLUHFWLRQ D GLIIHU

HQW DSSURDFK WKDQ DQ DEUXSW UHGXFWLRQ RI WKH PRGXOXV RI HODVWLFLW\ LQ WKH ILEHU GLUHFWLRQ LV IROORZHG ,QVWHDG WKH FRQVWLWXWLYH EHKDYLRU RI WKH ILEHU EXQGOH LV DSSUR[LPDWHG E\

FRQVLGHULQJ WKH QRQOLQHDU EHKDYLRU RI WKH FRQVWLWXHQWV 7KH SRO\PHU PDWUL[ LV FRQVLG

HUHG WR EH ZLWK D GU\ ILEHU EXQGOH LQ DQ LVRVWUDLQ 9RLJW FRQILJXUDWLRQ UHIHUUHG WR DV WKH PLFURPRGHO 7KLV PLFURPRGHO FRQVLGHUV GDPDJH LQ ERWK WKH GU\ ILEHU EXQGOHV DQG WKH PDWUL[ DQG LV XVHG WR FDOLEUDWH D QHZ GDPDJH HYROXWLRQ IRU WKH ILEHU EXQGOH RU D PHVRPRGHO

,W LV DVVXPHG WKDW WKH GDPDJH EHKDYLRU RI WKH ILEHU EXQGOH PHVRPRGHO FDQ EH PRGHOHG LQ D VLPLODU ZD\ DV IRU WKH GDPDJH GHYHORSPHQW RI D GU\ ILEHU EXQGOH >@

7KHUHIRUH D FRQWLQXXP GDPDJH PRGHO RI WKH GU\ ILEHU EXQGOHV LV ILUVW GHVFULEHG

$V D WHQVLOH VWUDLQ LV DSSOLHG WR D GU\ EXQGOH RI ILEHUV VRPH RI WKH ILEHUV ZLWK IODZV EHJLQ WR IDLO DQG WKH VWUHVV RQ WKH UHPDLQLQJ ILEHUV LV UHGLVWULEXWHG DQG LQFUHDVHV

7R DFFRXQW IRU WKH VWRFKDVWLF QDWXUH RI WKH WHQVLOH VWUHQJWK RI WKH ILEHUV D :HLEXOO GLVWULEXWLRQ >@ LV RIWHQ DGRSWHG >  @

F ( ˜σ) = 1 − exp

 −l f

L ₀  σ ˜

σ ˜ 0

_β 

 (2.16)

WR GHVFULEH WKH SUREDELOLW\ RI D ILEHU RI OHQJWK l _f WR IDLO DW RU EHORZ D VWUHVV ˜ σ 7KH SDUDPHWHUV ˜ σ 0 UHSUHVHQWV WKH VFDOH SDUDPHWHU RI WKH GLVWULEXWLRQ DQG β LV UHODWHG WR WKH VFDWWHU RI WKH GDWD DQG FDQ EH GHWHUPLQHG H[SHULPHQWDOO\ IURP ILEHU VWUHQJWK H[SHUL

PHQWV FDUULHG RXW ZLWK D JDXJH OHQJWK L ₀  (T  FDQ EH UHZULWWHQ DV

F ( ˜σ) = 1 − exp 

−α ˜σ ^β 

 (2.17)

ZLWK

α = l _f L ₀ σ ˜ ^β 0



7KH SHUFHQWDJH RI XQEURNHQ ILEHUV LV JLYHQ E\ 1 − F( ˜σ) DQG WKHQ WKH DSSDUHQW VWUHVV LQ WKH ILEHU EXQGOH σ LV HTXDO WR WKH SURGXFW EHWZHHQ WKH VWUHVV LQ XQEURNHQ ILEHUV DQG WKH SHUFHQWDJH RI XQEURNHQ ILEHUV

σ = ˜σ (1 − F( ˜σ)) = ˜σ exp 

−α ˜σ ^β 

 (2.18)

7KH VWUHVV PD[LPL]LQJ  LV JLYHQ E\

σ ˜ ^PD[ = (αβ) ^−1/β  (2.19)

DQG E\ VXEVWLWXWLQJ  LQ  WKH FULWLFDO VWUHVV RI WKH GU\ ILEHU EXQGOH LV DFKLHYHG

σ ^{c r} _b = (αβe) ^−1/β  (2.20)

7KH GDPDJH HYROXWLRQ RI WKH ILEHU EXQGOHV LQ WKH ILEHU GLUHFWLRQ D ₁₁ ^{F I B}  LV GLUHFWO\ GH

VFULEHG E\ F ( ˜σ) DV

D ₁₁ ^{F I B} = F( ˜σ) = 1 − exp(−α ˜σ ^β ) (2.21)

E\ DVVXPLQJ D VLPLODU :HLEXOO SDUDPHWHU β DV IRU D ZHOO NQRZQ GU\ ILEHU EXQGOH >

@ 7KH ILEHU EXQGOH FULWLFDO VWUHVV LV WKHQ GHWHUPLQHG XVLQJ D ILQLWH HOHPHQW PRGHO FRQVLGHULQJ WKH GU\ ILEHU EXQGOH DQG WKH PDWUL[ XQGHU LVRVWUDLQ FRQGLWLRQV $ GDPDJH SURJUHVVLRQ PRGHO IRU WKH PDWUL[ PRGHOHG DV GHVFULEHG LQ WKH SUHYLRXV VHFWLRQ DQG D SURJUHVVLYH GDPDJH PRGHO IRU WKH GU\ ILEHU EXQGOHV ZLWK :HLEXOO SDUDPHWHUV NQRZQ

DQG GHVFULEHG LQ >@ LV XVHG 7KHQ E\ XVLQJ (T  WKH UHPDLQLQJ :HLEXOO SDUDPHWHU α FDQ EH HVWLPDWHG DQG DQ HTXLYDOHQW SURJUHVVLYH GDPDJH PRGHO IRU WKH ILEHU EXQGOHV LV GHVFULEHG 2EVHUYH WKDW WKLV PRGHO LV VWUHVVEDVHG FRPSXWHG DW DQ LQWHJUDWLRQ SRLQW OHYHO 0RUHRYHU LUUHYHUVLELOLW\ RI WKH GDPDJH YDULDEOH FDQ EH GLUHFWO\

HQIRUFHG WKURXJK WKH ORDGLQJ KLVWRU\

0HVRVFDOH GDPDJH PRGHOLQJ IURP UHVSRQVHV DW WKH PLFURVFDOH

)DLOXUH DQDO\VLV RI WDSH ODPLQDWHV LV W\SLFDOO\ SHUIRUPHG DW D SO\ OHYHO ZLWK SDUDPH

WHUV WR EH GHWHUPLQHG H[SHULPHQWDOO\ 'LIIHUHQW IDLOXUH FULWHULD VXFK DV 3XFN 7VDL:X

&XQW]H DPRQJ RWKHUV KDYH EHHQ WHVWHG LQ WKH ZRUOGZLGH IDLOXUH H[HUFLVH >@ 7KHVH FULWHULD DUH SK\VLFDOO\ EDVHG RU SRO\QRPLDO DSSURDFKHV DQG GHVSLWH PDQ\ RI WKHP DUH ZHOO DFFHSWHG E\ WKH FRPPXQLW\ VRPH GR QRW DOZD\V SURYLGH VLPLODU HVWLPDWLRQV 7KH FRPSOH[LW\ LQ WKH SUHGLFWLRQV LQFUHDVHV ZKHQ IDLOXUH KDV WR EH FRQVLGHUHG LQ D PXOWL

VFDOH IDVKLRQ LH WKH PLFURVFRSLF IDLOXUH DW WKH FRQVWLWXHQW OHYHO LQFOXGLQJ WKH ILEHU

PDWUL[ DQG WKH LQWHUIDFH KDV WR EH WUDQVIHUUHG DW WKH SO\ OHYHO ZKLFK LQ WXUQ VKRXOG EH UHIOHFWHG LQ WKH ILQDO IDLOXUH RI WKH ODPLQDWH RU WKH VWUXFWXUDO SDUW $ PXOWLVFDOH DS

Isotropic damage model (matrix)

a)

Isotropic damage model (matrix)

Anisotropic damage model (fiber bundles)

b)

Figure 2.2. Approaches for damage modeling at different scales. a) Microscopic scale, b) mesoscopic scale (anisotropic damage model calibrated from the micro- scopic scale).

SURDFK IRU PRGHOLQJ WKH VWUXFWXUDO LQWHJULW\ RI WKH FRPSRVLWH FDQ EH SHUIRUPHG ZLWK WKH DLG RI WKH VRFDOOHG JHQHUDOL]HG PHWKRG RI FHOOV *0& :LWK WKLV WHFKQLTXH WKH *0&

LV XVHG WR PRGHO IDLOXUH DW WKH PLFURVFRSLF VFDOH 7KH *0& LV EDVHG RQ PLFURPHFKDQ

LFV DQG LW ZDV LQLWLDOO\ GHYHORSHG E\ 3DOH\ DQG $ERXGL >@ DQG ODWHU RSWLPL]HG E\

3LQGHUD DQG %HGQDUF\N >@ ,Q WKH ILQLWH HOHPHQW FRQWH[W WKH DQDO\VLV RI FRPSRVLWHV

LV SHUIRUPHG E\ VROYLQJ WZR QHVWHG ERXQGDU\ YDOXH SUREOHPV 7KH VWUXFWXUDO VFDOH

FRPSXWHV WKH FRQVWLWXWLYH UHVSRQVH DW DQ LQWHJUDWLRQ SRLQW OHYHO EDVHG RQ WKH VROXWLRQ

DW WKH PLFURVFDOH REWDLQHG ZLWK WKH *0& 7KH PLFURPHFKDQLFDO UHVSRQVH LV REWDLQHG

E\ DQDO\]LQJ IDLOXUH DW WKH FRQVWLWXHQW OHYHO ZKLFK FDQ EH EDVHG RQ WKH ILEHU PDWUL[

DQG LQWHUIDFH LQWHUDFWLRQV 7KH WKHRUHWLFDO EDVLV RI WKH *0& LPSOHPHQWDWLRQDO GHWDLOV DQG VRPH DSSOLFDWLRQV VWXG\LQJ IDLOXUH LQ FRPSRVLWHV FDQ EH IRXQG LQ WKH OLWHUDWXUH

>         @ 7KHVH SDUWLFXODULWLHV DUH RPLWWHG KHUH IRU WKH VDNH RI EUHYLW\ &XUUHQWO\ WKH *0& LV LPSOHPHQWHG LQ WKH FRGH PLFURPH

FKDQLFV DQDO\VLV FRGH ZLWK WKH JHQHUDOL]HG PHWKRG RI FHOOV 0$&*0& GHYHORSHG DW 1$6$ *OHQQ 5HVHDUFK &HQWHU >@ DQG ZKLFK KDV EHHQ DOVR FRXSOHG ZLWK WKH ILQLWH HOHPHQW PHWKRG LQ WKH VRFDOOHG )($0$& IUDPHZRUN >@

Micro-scale approaches Meso-scale approaches

a)

b)

Figure 2.3. A comparison of the micro-scale approach using the GMC at an integra- tion point level and meso-scale approach using the calibrated continuum damage model. a) Fractional reduction of the C ₂₂ component of the stiffness tensor under transverse tension, b) fractional reduction of the C ₁₂ component of the stiffness tensor under in-plane shear. After [22].

7KH DERYHPHQWLRQHG DSSURDFK LV XVHG WR REWDLQ WKH QRQOLQHDU UHVSRQVH RI WKH ILEHU EXQGOHV GXH WR WKH GDPDJH HYROXWLRQ 7KH DVVXPSWLRQ LV WKDW WKH PDLQ VRXUFH RI QRQ

OLQHDULW\ LV GXH WR WKH SURJUHVVLYH GDPDJH RI WKH PDWUL[ DW WKH PLFURVFRSLF VFDOH )LJ

D ZKLFK KDV EHHQ SUHYLRXVO\ VKRZQ WR EH D UHDVRQDEOH DSSUR[LPDWLRQ >@ ,W LV ZRUWK QRWLFLQJ WKDW ZLWK WKLV DSSURDFK RQO\ H[SHULPHQWDO FKDUDFWHUL]DWLRQ RI WKH PD

WUL[ ZDV QHFHVVDU\ IRU WKH SKHQRPHQRORJLFDO FKDUDFWHUL]DWLRQ RI DQ LVRWURSLF GDPDJH PRGHO 7UDQVYHUVH WHQVLRQ ORQJLWXGLQDO VKHDU DQG WUDQVYHUVH VKHDU UHVSRQVHV DUH RE

WDLQHG 6XFK UHVSRQVHV DUH XVHG WR FDOLEUDWH WKH PHVRVFDOH FRQWLQXXP GDPDJH PRGHO SUHVHQWHG LQ WKH SUHYLRXV VHFWLRQ ZKLFK JLYHV DV D UHVXOW DQ DQLVRWURSLF GDPDJH EH

KDYLRU RI WKH KRPRJHQHRXV ILEHU EXQGOHV )LJ E 7KH DQLVRWURSLF GDPDJH PRGHO ZDV FDOLEUDWHG LQ D SUHYLRXV ZRUN >@ ZKHUH LW ZDV IRXQG WKDW WKH PHVRVFDOH GDPDJH PRGHO ZDV DEOH WR FDSWXUH GDPDJH GHYHORSPHQW LQ VLPLODU ORFDWLRQV DV LQ WKH PXOWL

VFDOH PRGHOLQJ DSSURDFK XVLQJ WKH *0& )LJ 

 $ SKDVHILHOG DSSURDFK WR EULWWOH IUDFWXUH 9DULDWLRQDO IRUPXODWLRQ

Figure 2.4. Schematic representation of a sharp crack and its regularized approxi- mation in a solid body with domain Ω.

3KDVHILHOG WHFKQLTXHV IRU IUDFWXUH DUH EDVHG RQ WKH YDULDWLRQDO IRUP RI WKH *ULIILWK HQHUJ\ >  @

Π(m, d) =



Ω

[(1 − d)   ²

g(d)

Ψ + () + Ψ − ()] dΩ

 

VWUDLQ HQHUJ\

+ G c



Γ

d Γ 

 

IUDFWXUH HQHUJ\

(2.22)

UHSUHVHQWLQJ WKH WRWDO LQWHUQDO HQHUJ\ Π LQ D GRPDLQ RI DQ HODVWLF ERG\ Ω DQG HYROY

LQJ LQWHUQDO GLVFRQWLQXLWLHV Γ  )LJ  7KH ILUVW WHUP LQ  FRUUHVSRQGV WR D GHFRPSRVLWLRQ RI WKH HODVWLF VWUDLQ HQHUJ\ GHQVLW\ Ψ() = Ψ ₊ () + Ψ ₋ () WR EH GH

WDLOHG ODWHU 7KH &DXFK\¶V LQILQLWHVLPDO VWUDLQ WHQVRU (m) ò(∇m + ∇ ^T m) LV DV

VXPHG ZKHUH m LV WKH GLVSODFHPHQW ILHOG DQG ∇ WKH JUDGLHQW RSHUDWRU 7KH VWUDLQ HQHUJ\ GHQVLW\ LV GHJUDGHG E\ D VRFDOOHG GHJUDGDWLRQ IXQFWLRQ g (d) GHSHQGLQJ RQ WKH SKDVHILHOG SDUDPHWHU d ZKLFK UDQJHV IURP d  XQEURNHQ WR d  EURNHQ PDWHULDO SKDVHV 7KH VHFRQG LQWHJUDO LQ  FRUUHVSRQGV WR WKH FRQWULEXWLRQ RI WKH IUDFWXUH HQHUJ\ GHSHQGLQJ RQ WKH IUDFWXUH WRXJKQHVV RI WKH PDWHULDO G _c  DQG WKH IUDF

WXUH VXUIDFH 7KH GLVFUHWH FUDFN SDWK Γ  LV UHJXODUL]HG XVLQJ D VRFDOOHG FUDFN GHQVLW\

IXQFWLRQDO >  @ γ(d, ∇d, ) GHSHQGLQJ RQ d LWV JUDGLHQW ∇d DQG D OHQJWK

VFDOH SDUDPHWHU  7KLV ODVW SDUDPHWHU JRYHUQV WKH UHJXODUL]HG FUDFN WRSRORJ\ DQG GHWHUPLQHV WKH ZLGWK RI WKH FUDFN DSSUR[LPDWLRQ 7KH ODVW LQWHJUDO LQ  LV WKHQ DSSUR[LPDWHG E\ D YROXPH LQWHJUDO > @

G _c



Γ

d Γ ≈ G c



Ω

γ(d, ∇d, ) dΩ (2.23)

ZKHUH WKH UHJXODUL]DWLRQ PRGHO >  @

γ(d, ∇d, ) = 3

8 (d + ² ∇d · ∇d) (2.24)

LV XVHG 7KH FKRLFH RI  UHIHUUHG WR DV D ILUVWRUGHU FUDFN GHQVLW\ IXQFWLRQDO LV

UHODWHG WR D GDPDJH GLVVLSDWLRQ IXQFWLRQ LQVSLUHG E\ JUDGLHQW GDPDJH PRGHOV >

@ ,W KDV WKH DGYDQWDJH RI SURPRWLQJ PRUH ORFDOL]DWLRQ WKDQ VRPH KLJKHURUGHU IRUPXODWLRQV ZKHUH GDPDJH VWDUWV WR HYROYH DV VRRQ DV WKH PDWHULDO LV ORDGHG FI >

 @ IRU IXUWKHU GHWDLOV

7KH SKDVHILHOG RU FUDFN SURILOH d (x) DFWLQJ LQ D GLVWDQFH x LV REWDLQHG IURP WKH YDULDWLRQDO PLQLPL]DWLRQ RI  OHDGLQJ WR WKH FKDUDFWHULVWLF HTXDWLRQ 1 −2 ² d ^ (x) = 0 ZLWK VROXWLRQ d (x) = (1 − ^−|x| ₂ ) ²  ,W UHSUHVHQWV D GLIIXVLYH DSSUR[LPDWLRQ WR D VKDUS FUDFN ZKHUH WKH FUDFN IURQW LV LQ D SHUSHQGLFXODU GLUHFWLRQ WR x

6WUDLQ HQHUJ\ GHFRPSRVLWLRQ

7KH VWUDLQ HQHUJ\ GHQVLW\ LV GHFRPSRVHG LQWR WHQVLOH Ψ ₊ () DQG FRPSUHVVLYH Ψ ₋ ()

FRQWULEXWLRQV RUGHU WR GLVWLQJXLVK EHWZHHQ IUDFWXUH EHKDYLRU LQ WHQVLRQ DQG FRPSUHV

VLRQ >  @ 7ZR GHFRPSRVLWLRQV EDVHG RQ D K\GURVWDWLFGHYLDWRULF DQG D VSHF

WUDO GHFRPSRVLWLRQ RI WKH VWUDLQ WHQVRU DUH FRQVLGHUHG

7KH K\GURVWDWLFGHYLDWRULF GHFRPSRVLWLRQ VXJJHVWHG E\ $PRU HW DO >@ DVVXPHV WKDW FUDFN JURZWK WDNHV SODFH GXH WR SRVLWLYH GLODWDWLRQDO DQG GLVWRUWLRQDO FKDQJHV

>@

Ψ + () = 1 2  n

 (WU()) ² 

+ + μWU (e : e)  Ψ ₋ () = 1

2  n

 (WU()) ² 

− 

(2.25)

ZKHUH  n = λ + 2μ/n GHQRWHV WKH EXON PRGXOXV DQG n GHQRWHV WKH GLPHQVLRQDOLW\

7KH SDUDPHWHUV λ DQG μ DUH WKH /DPp FRQVWDQWV e WKH GHYLDWRULF VWUDLQ WHQVRU DQG WKH EUDFNHW RSHUDWRU 〈x〉 ± = (x ± |x|)/2

7KH VSHFWUDO GHFRPSRVLWLRQ VXJJHVWHG E\ 0LHKH HW DO >@ DVVXPHV WKDW FUDFN JURZWK LV LQGXFHG E\ WKH SRVLWLYH SULQFLSDO VWUDLQV

Ψ ± () = 1

2 λ 〈WU()〉 ² _± + μWU 

 ² _± 

 (2.26)

ZKHUH  =  + +  − DQG  ± =  _n

i=1 〈 i 〉 ± ± a i ⊗ a i  ZKHUH  i DUH WKH SULQFLSDO YDOXHV DQG a i WKH SULQFLSDO GLUHFWLRQV

7KH VWUHVV WHQVRU DQG WKH FRQVLVWHQW WDQJHQW VWLIIQHVV WHQVRU DUH FRPSXWHG DV

σ = g(d) ∂ Ψ ₊ ()

∂  + ∂ Ψ ₋ ()

∂    = ∂ σ

∂  = g(d) ∂ ² Ψ ₊ ()

∂  ² + ∂ ² Ψ ₋ ()

∂  ²  (2.27) UHVSHFWLYHO\ ZKHUH Ψ + DQG Ψ − GHSHQG RQ HDFK GHFRPSRVLWLRQ FRQVLGHUHG

/HQJWKVFDOH SDUDPHWHU HVWLPDWLRQ

7KH OHQJWK VFDOH SDUDPHWHU FDQ EH UHODWHG WR RWKHU PDWHULDO SDUDPHWHUV E\ FRQVLGHULQJ WKH RQVHW RI GDPDJH :H ILUVW GHILQH WKH WRWDO HQHUJ\ GHQVLW\ W IURP (TV 

W = 1

2  :  :  + 3

8 G _c (d + ² ∇d · ∇d). (2.28)

)URP WKHUPRG\QDPLF DUJXPHQWV D FKDQJH LQ WKH WRWDO HQHUJ\ GHQVLW\ LV UHTXLUHG WR EH QRQQHJDWLYH ZKHQ HQHUJ\ GLVVLSDWHV RXW RI WKH V\VWHP >   @

−∂ W

∂ d ≤ 0. (2.29)

$ KRPRJHQHRXV VROXWLRQ FDQ EH REWDLQHG XQGHU XQLD[LDO WHQVLOH VWUHVV VWDWH LQ WKH HODVWLF OLPLW ZKHUH WKH JUDGLHQW RI WKH GDPDJH YDULDEOH YDQLVKHV &RQVLGHU DQ LQLWLDOO\

XQLD[LDO VWUHVV VWDWH σ x XQGDPDJHG 7KHQ (T  EHFRPHV

− ∂ W

∂ d | d=0 = σ ² _x E ^ − 3

8 G _c ≤ 0 ⇒ σ x ≤ σ c =

 3G _c E ^

8 , (2.30)

ZKHUH E ^ = E IRU SODQH VWUHVV DQG E ^ = E/(1 − ν ² ) IRU SODQH VWUDLQ FRQGLWLRQV E LV WKH

<RXQJ¶V PRGXOXV DQG ν WKH 3RLVVRQ¶V UDWLR (T  DOORZV IRU WKH GHILQLWLRQ RI D FULWLFDO PD[LPXP FRKHVLYH VWUHVV σ c  )LQDOO\ IURP  RQH JHWV WKH UHODWLRQ

= 3 8

G _c E ^

σ ² _c , (2.31)

ZKLFK LV XVHG WR HVWLPDWH WKH OHQJWK VFDOH SDUDPHWHU

)XUWKHUPRUH WR HQIRUFH WKH LUUHYHUVLELOLW\ RI WKH IUDFWXUH SURFHVV Ψ ₊ LQ (T  LV UHSODFHG ZLWK LWV PD[LPXP YDOXH GXULQJ ORDGLQJ KLVWRU\ XVXDOO\ UHIHUUHG WR DV  LQ OLW

HUDWXUH > @ 7KH IUDFWXUH HYROXWLRQ LV JRYHUQHG E\ WKH PLQLPL]DWLRQ RI WKH HQHUJ\

IXQFWLRQDO (T  'LVFUHWL]DWLRQ LV GRQH E\ LQWHUSRODWLQJ WKH GLVSODFHPHQW DQG SKDVHILHOG YDULDEOHV ZLWK VWDQGDUG ILQLWH HOHPHQW VKDSH IXQFWLRQV >@ $ W\SLFDO GLV

FUHWL]DWLRQ H[DPSOH FDQ EH IRXQG LQ >@ 7KH VROXWLRQ LV EDVHG RQ D 1HZWRQ5DSKVRQ VROYHUW\SH LQ D VHTXHQWLDOO\ FRXSOHG VRFDOOHG VWDJJHUHG VFKHPH >@ ZKHUH WKH GLVSODFHPHQW m DQG GDPDJH d DUH LQFUHPHQWHG VHTXHQWLDOO\ ,W LV NQRZQ KRZHYHU WKDW ODUJHGLVSODFHPHQW LQFUHPHQWV FRXOG XQGHUHVWLPDWH WKH UDWH RI FUDFN SURSDJDWLRQ DQG RYHUHVWLPDWH WKH ORDGFDUU\LQJ FDSDELOLW\ >@ 7KHUHIRUH SVHXGRWLPHVWHS LV UHGXFHG XQWLO WKH VROXWLRQ GRHV QRW FKDQJH VLJQLILFDQWO\ $GGLWLRQDOO\ WR DYRLG XQGHUHVWLPDW

LQJ FUDFN JURZWK WKH m DQG d VROXWLRQV ZLWKLQ D SVHXGRWLPH VWHS DUH LQFUHPHQWHG XQWLO WKH PD[LPXP GDPDJH LQFUHPHQW LV ZLWKLQ D WROHUDQFH WRO d  IROORZLQJ >@

 'HODPLQDWLRQ RQVHW LQ FRPSRVLWH ODPLQDWHV

'HODPLQDWLRQ LV DQ LQWHUODPLQDU PRGH RI IUDFWXUH LQ FRPSRVLWH PDWHULDOV ,W ZDV SHU

KDSV DIWHU WKH SLRQHHULQJ ZRUN RI 3LSHV DQG 3DJDQR >@ ZKHQ VLJQLILFDQW DWWHQWLRQ ZDV SDLG WR VWXG\ WKLV IDLOXUH PHFKDQLVP LQ FRPSRVLWH ODPLQDWHV GXULQJ VHUYLFH ORDGV

>@ ,Q W\SLFDO WDSH ODPLQDWHV GHODPLQDWLRQ FDQ DULVH GXH WR WKH H[LVWHQFH RI VWUHVVHV DW WKH LQWHUVHFWLRQ RI WKH IUHHHGJH DQG WKH LQWHUODPLQDU SODQH RI ODPLQDWH VXEMHFWHG WR D XQLIRUP D[LDO H[WHQVLRQ 7KLV ZDV H[WHQVLYHO\ VWXGLHG LQ WKH SDVW GHFDGHV IRU W\SL

FDO WDSH ODPLQDWHV VKRZLQJ WKDW WKH IUHH HGJH HIIHFWV DSSHDU DW D FHUWDLQ GLVWDQFH IURP

WKH IUHH HGJH DQG WKDW WKH VWDFNLQJ VHTXHQFH KDG D VLJQLILFDQW LPSDFW RQ LQWHUODPLQDU

VWUHVVHV FDXVLQJ SUHPDWXUH IDLOXUH >   @ 7R LOOXVWUDWH WKLV FRQVLGHU DV DQ

H[DPSOH D >@ _s ODPLQDWH )LJ D ZLWK OHQJWK L ZLGWK b < L DQG WRWDO ODPLQDWH

WKLFNQHVV t _l << b VXEMHFWHG WR D PDFURVFRSLF XQLD[LDO H[WHQVLRQ LQ WKH xGLUHFWLRQ

a) b)

Figure 2.5. An example of a typical free-edge effect in a unidirectional tape [0/90] s

laminate under uniaxial extension. a) Schematic representation, b) finite element solutions compared to the solutions of Wang et al. [122] and Nguyen et al. [123]

$W WKH IUHHHGJHV y = 0 DQG y = b WKH PDFURVFRSLF VXUIDFH WUDFWLRQ LV ]HUR t = 0 3ODQHVWUHVV FRQGLWLRQV SUHYDLO DW WKH LQWHULRU RI WKH ODPLQDWH ZKHUH LQSODQH VWUHVVHV PLJKW GLIIHU EHWZHHQ OD\HUV GXH WR WKH GLIIHUHQW SO\ RULHQWDWLRQV DQG VWDFNLQJ RUGHU

7R VDWLVI\ HTXLOLEULXP DQG WKH PDFURVFRSLF WUDFWLRQIUHH FRQGLWLRQV DW WKH IUHHHGJHV

LQWHUODPLQDU VWUHVVHV DULVH DW WKHVH ]RQHV YDQLVKLQJ DW D GLVWDQFH RI DSSUR[LPDWHO\ t _l DFFRUGLQJ WR WKH 6DLQW9HQDQW SULQFLSOH >@ VKRZQ LQ )LJ E

,QWHUODPLQDU VKHDU WHVWV DUH RIWHQ XVHG WR GHWHUPLQH WKH LQWHUODPLQDU VKHDU VWUHQJWK RI FRPSRVLWH ODPLQDWHV RIWHQ UHODWHG WR WKH GHODPLQDWLRQ RQVHW 7KHVH WHVWV W\SLFDOO\

LQGXFH D VWDWH RI VKHDU VWUHVV RIWHQ OHDGLQJ WR GHODPLQDWLRQ

,Q WKH SUHVHQW WKHVLV GHODPLQDWLRQ RQVHW DULVLQJ DV D FRQVHTXHQFH RI IUHHHGJH HI

IHFWV GXULQJ XQLIRUP XQLD[LDO H[WHQVLRQ DQG DV D FRQVHTXHQFH RI D VWDWH RI VKHDU VWUHVV

DUH VWXGLHG 7KH HIIHFW RI OD\HU VKLIWLQJ LV LQYHVWLJDWHG LQ ERWK DQDO\VHV DQG FRQFOXVLRQV

DUH GUDZQ WRJHWKHU ZLWK H[SHULPHQWDO REVHUYDWLRQV

 1XPHULFDO DQG H[SHULPHQWDO SURFHGXUHV

 &RPSXWDWLRQDO KRPRJHQL]DWLRQ RI KHWHURJHQHRXV VWUXFWXUHV ZLWK SUHIHUHQWLDO SHULRGLFLW\

:RYHQ FRPSRVLWH ODPLQDWHV FDQ EH LGHDOL]HG ZLWK DQ 58& WR DQDO\]H WKHP 7KH XVH RI FRPSXWDWLRQDO KRPRJHQL]DWLRQ LV D FRPPRQ SUDFWLFH WR REWDLQ WKH KRPRJHQL]HG SURS

HUWLHV RI WKH HQWLUH ODPLQDWH ,Q WKH OLWHUDWXUH RQH FDQ ILQG HLWKHU WKH XVH RI 3%&V RU 8'%&V > @ IRU WKLV SXUSRVH ,I 3%&V DUH XVHG RQH DSSURDFK LV WR FRQVLGHU DQ 58& SHULRGLF LQ WKH WKUHH GLUHFWLRQV >   @ ZKLFK UHSUHVHQWV D ODPLQDWH ZLWK SHUIHFWO\ DOLJQHG OD\HUV RQ WRS RI HDFK RWKHU $QRWKHU DSSURDFK LV WR FRQVLGHU WKH H[DFW QXPEHU RI OD\HUV RI WKH ODPLQDWH DQG FRQVLGHU LQSODQH SHULRGLFLW\ >  @

:LWK WKLV ODVW DSSURDFK LW LV QRW FOHDU WKHQ KRZ WR HVWLPDWH WKH RXWRISODQH SURSHUWLHV DQG KRZ VHQVLWLYH WKHVH DUH ZLWK UHVSHFW WR WKH W\SH RI ERXQGDU\ FRQGLWLRQV LPSRVHG

0RUHRYHU LW LV XQNQRZQ KRZ WKH VWUHVV DQG VWUDLQ ILHOGV EHFRPH DIIHFWHG $OO WKHVH LV

VXHV DUH FUXFLDO IRU WKH SURSHU PRGHOLQJ RI WKH PHVRVFDOH UHVSRQVH RI ZRYHQ FRPSRVLWH ODPLQDWHV DQG ZHUH DGGUHVVHG LQ 3DSHU ,

a)

x z y

b)

Figure 3.1. Application of computational homogenization in periodic structures.

a) Heterogeneous structure with randomly distributed inclusions, b) woven com- posite laminate.

7KH HIIHFW RI %&V LQ WKH HVWLPDWHG SURSHUWLHV RI VWUXFWXUHV ZLWK SUHIHUHQWLDO SHULRG

LFLW\ LV ILUVW VWXGLHG $ JHQHUDO KHWHURJHQHRXV 58& )LJ D LV FRQVLGHUHG $UUD\V RI VXFK D JHQHUDO 58& DUH FUHDWHG WR UHSUHVHQW SUHIHUHQWLDO SHULRGLFLW\ LQ RQH WZR DQG WKUHHGLPHQVLRQV (ODVWLF SURSHUWLHV DUH LQYHVWLJDWHG XVLQJ 8'%&V 87%&V 3%&V

LQSODQH 3%&V DQG D PL[ RI %&V $IWHU VXFK %&V DUH DSSOLHG WR D ZRYHQ FRPSRVLWH

ODPLQDWH $ VLQJOH ODPLQD DQG XS WR D WKUHHOD\HUHG ZRYHQ FRPSRVLWH ODPLQDWH ZLWK

WKH OD\HUV SHUIHFWO\ DOLJQHG DUH FRQVLGHUHG )LJ E 7KH VHQVLWLYLW\ RI WKH HVWLPDWHG

SURSHUWLHV ZLWK UHVSHFW WR WKH %&V LV VKRZQ DQG GLIIHUHQFHV EHWZHHQ D VLQJOH ODPLQD

DQG D ODPLQDWH DUH LOOXVWUDWHG DQG FRPSDUHG ZLWK H[SHULPHQWV 0RUHRYHU VXUIDFH VWUDLQ ILHOGV SURGXFHG ZLWK WKH GLIIHUHQW %&V DUH FRPSDUHG ZLWK H[SHULPHQWDO GDWD UHSRUWHG LQ WKH OLWHUDWXUH

 /D\HU VKLIWLQJ HIIHFWV LQ WKH HODVWLF UHVSRQVH DQG GDPDJH QXFOHDWLRQ DQG JURZWK

x

z y λ

W y

t l φ a)

hy z^A0(x)

z^Au(x) z^Al(x) (x, φ) (x, φ) (x, φ)

x

z

θ^A θ^B

h^At h^Bt

h^Bt

h^Ay h^By

dx h^Am

h^Bm

h^Bm h^Am

h^Am h^Am

φ

x = 0

A B

z^Bl z^B0 z^Bu

b)

Figure 3.2. Woven composite models devoted to investigating the layer shifting effect. a) Three dimensional model, b) two-dimensional (analytical) approach.

7KH UHODWLYH OD\HU VKLIWLQJ EHWZHHQ OD\HUV φ KDV EHHQ UHSRUWHG WR EH DQ LPSRUWDQW PHVRVWUXFWXUDO LPSHUIHFWLRQ LQ ZRYHQ FRPSRVLWH ODPLQDWHV >  @ VLQFH VWUDLQ DQG VWUHVV ILHOGV > @ DV ZHOO DV WKH PDFURVFRSLF UHVSRQVH DUH VLJQLILFDQWO\ DI

IHFWHG > @ +HUH WKH HIIHFW RI WKH UHODWLYH OD\HU VKLIWLQJ EHWZHHQ OD\HUV RI ZRYHQ FRPSRVLWH ODPLQDWHV LV FRQVLGHUHG WR LQYHVWLJDWH ILUVW WKH PHVRVWUXFWXUDO HIIHFWV LQ WKH JOREDO HODVWLF UHVSRQVH 0RGHOV ZHUH FRQVLGHUHG ZLWK D ILQLWH QXPEHU RI OD\HUV LH

WKURXJK WKH WKLFNQHVV GLUHFWLRQ ZDV FRQVLGHUHG WR EH QRQSHULRGLF 7KLV LV VWXGLHG QXPHULFDOO\ ZLWK WKH DLG RI IXOO\ WKUHHGLPHQVLRQDO ILQLWH HOHPHQW PRGHOV )LJ D

XVLQJ FRPSXWDWLRQDO KRPRJHQL]DWLRQ DQG ZLWK D VLPSOLILHG WZRGLPHQVLRQDO DQDO\WLFDO PRGHO )LJ E EDVHG RQ FODVVLFDO ODPLQDWLRQ WKHRU\ WR EHWWHU XQGHUVWDQG WKH XQGHU

O\LQJ PHFKDQLVPV 6XFK DQ DQDO\WLFDO PRGHO LV GHVFULEHG LQ WKH DSSHQGL[ RI 3DSHU ,, 7KHUHDIWHU D FRQWLQXXP GDPDJH PRGHO LV LPSOHPHQWHG LQ QXPHULFDO PRGHOV ZLWK GLIIHUHQW VKLIWLQJ FRQILJXUDWLRQV WR EHWWHU XQGHUVWDQG WKH UROH RI WKH GDPDJH QXFOHDWLRQ DQG JURZWK DV D FRQVHTXHQFH RI OD\HU VKLIWLQJ 7KHVH SURFHGXUHV DUH DGGUHVVHG LQ PRUH GHWDLO DV D SDUW RI 3DSHU ,,

 3KDVHILHOG PRGHOLQJ RI EULWWOH IUDFWXUH LQ FRPSRVLWH VWUXFWXUHV

0RGHOLQJ IDLOXUH LQ ILEHUUHLQIRUFHG FRPSRVLWH PDWHULDOV LV D FKDOOHQJLQJ WDVN GXH WR

WKH FRPSOLFDWHG IDLOXUH PHFKDQLVPV LQYROYHG KDSSHQLQJ DW GLIIHUHQW OHQJWK VFDOHV $W

FRXSRQ OHYHO WHVWLQJ VWUHQJWK SUHGLFWLRQ RI QRWFKHG ODPLQDWHV LV DQ H[DPSOH )RU WKLV