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OPTICAL ANALYSIS OF STRESS ANDSTRAIN IN PHOTOELASTIC PARTICLEASSEMBLIES

H.G.B. ALLERSMATR diss)1560


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byHenderikus G.B. Allersma

Delft University of TechnologyThe Netherlands

Delft, 1987TR d ia1560


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to JannyBartAllard


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ACKNOWLEDGEMENTSThi s r esear ch was car r i ed out at t he Geot echni cal Labor at or y oft he Depar t ment of Ci vi l Engi neer i ng of t he Del f t Uni ver s i t y ofTechnol ogy, t he Net her l ands . I am gr at ef ul t o t he t echni calst af f of t he l abor at or y, A. Mensi nga, J . van Leeuwen andJ . J . de Vi ss er , who modi f i ed i deas i n r eal devi ces and t ook car ef or t he pr epar at i on of sever al t est model s .


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CONTENTS1 I NTRODUCTI ON 12 THEORY OF MEASUREMENT 72. 1 STRESS TENSOR 72. 1. 1 Cont act f or ces 72. 1. 2 St r ess t ensor i n gr anul ar mat er i al 82. 1. 3 St r es s di s t r i but i on i n a par t i c l e 102. 1. 4 Rel at i on of i nt er par t i c l e s t r ess and bul k s t r ess 122. 2 STRAI N TENSOR 152. 2. 1 Rel at i ve di spl acement of par t i c l es 152. 2. 2 Det er m nat i on of t he st r ai n t ensor 182. 3 OPTI CAL STRESS MEASUREMENT AT A MATERI AL POI NT 192. 3. 1 L i ght 192. 3. 2 Pol ar i sat i on 202. 3. 3 Doubl e r ef r act i on 222. 3. 4 Opt i cal f i l t er syst em 232. 3. 5 Par t i c l e as sensor s 312. 3. 6 Opt i cal aver agi ng i nhomogeneous st r ess 332. 4 DI SPLACEMENT MEASUREMENT 362. 4. 1 Det ect i on of mar ked par t i c l es 362. 4. 2 Det er m nat i on of cent r e mar k 382. 4. 3 El i m nat i on of noi s e 402. 5 DATA PROCESSI NG 412. 5. 1 Col l ect ed dat a 412. 5. 2 Pr i nc i pal s t r es s t r aj ec tor i es 422. 5. 3 St r es s di s t r i but i on 452. 5. 4 St r ai n t ensor 533 TEST SETUP 583. 1 OPTI CAL MEASURI NG DEVI CE 583. 1. 1 Mechani cal par t 583. 1. 2 Opt i cal syst em 603. 1. 3 El ec t r oni c ci r cui t 633. 1. 4 Sof t war e and measur i ng pr ocedur e 663. 2 MODELLI NG 733. 2. 1 Pr oduct i on of cr ushed gl ass 73


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3. 2. 2 Mechani cal pr oper t i es of cr ushed gl ass 763. 2. 3 Opt i cal pr oper t i es of crushed gl ass 803. 2. 4 Pr epar at i on of a model 813. 2. 5 Loadi ng sys t ems and sensor s 834 APPLI CATI ONS 854. 1 SHEAR 854. 1. 1 I nt r oduct i on 854. 1. 2 Theor y of shear 874. 1. 3 Shear devi ces used 954. 1. 4 Measur i ng r esul t s 984. 1. 5 Di scussi on 1154. 2 EXPERI MENTS W TH LABORATORY- SCALE PENETROMETERS 1214. 2. 1 I nt r oduct i on 1214. 2. 2 Model s used 1214. 2. 3 Measur i ng r esul t s 1244. 2. 4 Di scussi on 1414. 3 EXPERI MENTS W TH A LABORATORY- SCALE HOPPER 1484. 3. 1 Test set up 1494. 3. 2 Measur ed r esul t s 1504. 3. 3 Di scussi on 1505 DI SCUSSI ON 1555. 1 POSSI BI LI TI ES AND LI MI TATI ONS 1555. 2 ACCURACY 157

NOTATI ON 160REFERENCES 162SAMENVATTI NG 168


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1 I NTRODUCTI ONThe devel opment of mat er i al model s and cal cul at i on pr ocedur es,t o pr edi ct t he mechani cal behavi our of non- cohesi ve packedpar t i c l e as sembl i es i s s t i l l i n f ul l s wi ng. I n par t i c ul ar ,numer i cal cal cul at i ons based on t he f i ni t e el ement met hod havepr oved ver y power f ul . Cont r ar y t o t he cal cul at i on t echni ques,however , l i t t l e pr ogr ess has been made i n t he devel opment of newmeasur i ng pr i nc i pl es . Sensor i ng and dat a col l ect i on have i ndeed

been i mpr oved and mor e accur at e devi ces ar e devel oped t oi nvest i gat e t he behavi our of a sampl e of gr anul ar mat er i al . Themeasur i ng pr i nc i pl e, however , i s i n gener al s t i l l based on t hesame concept as many year s ago, namel y: measur ement of boundar yl oads or st r ess and di spl acement of boundar y segment s. Thest r esses and st r ai ns i n t he i nt er i or of a sampl e have t o bees t i mat ed f r om t he boundar y condi t i ons i n t hi s case, whi ch i spos s i bl e, o nl y, i f t he s t r es s di s t r i but i on and def or mat i on f i el dar e uni f or m The i nf or mat i on obt ai ned by convent i al measur i ngmet hods i s t her ef or e ver y r es t r i c t ed i f a gr anul ar mat er i al i ssubj ect ed t o compl ex boundar y condi t i ons. The boundar y condi t i onsi n model t es t s r el at i ng t o pr act i cal pr obl ems ar e i n gener al t oocompl ex t o yi el d usef ul i nf or mat i on about t he condi t i ons i n t hei nt er i or of t he gr anul ar mat er i al , so t hat i t i s not pos si bl e t oanal yse such pr obl ems i n det ai l .Because t he act ual mechani sms cannot be vi sual i sed i n mostt e s t s , i t i s not pr oper l y pos s i bl e t o ver i f y t he r es ul t s ofcal cul at i ons . Al t hough t he numer i cal l y cal cul at i on t echni quest hat have been devel oped ar e hi ghl y advanced, t he qual i t y of t her esul t s gr eat l y depends on t he s t r ess - s t r ai n r el at i ons used. Thest r essst r ai n r el at i ons ar e mai nl y based on t he behavi our ofsampl es of gr anul ar mat er i al whi ch ar e subj ect ed t o a knownuni f or m s t r es s condi t i on. A t ypi cal devi ce f or mat er i al t es t i ngi s t he t r i axi al appar at us, i n whi ch a sampl e can be subj ect ed t over y wel l def i ned boundar y st r esses. Thi s good cont r ol over t heboundar y st r ess es, however , r est r i ct s t he scope t o s i mul at econdi t i ons whi ch ar e mor e r eal i s t i c i n pr ac t i ce, such as s t r ess


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r ot at i on and cont i nuous def or mat i on.I t i s wel l known t hat t he behavi our of a packed par t i c l eassembl y depends gr eat l y on t he st r ess hi st or y and st r ess pat h i nt he mat er i al . I t i s t her ef or e not s uf f i ci ent t o i nves t i gat e asampl e under s i mpl e l oadi ng condi t i ons onl y; but i t i s al sonecess ar y t o know t he mechani cal behavi our of a s ampl e when i t i ssubj ect ed t o a s t r ess pat h s i m l ar t o t he one occur i ng i nr eal i t y. I n t he f i r s t i ns t ance, t hi s r equi r es knowl edge about t hest r ess pat h t o whi ch a gr anul ar mat er i al i s subj ect ed i n aspec i f i c pr ac t i cal s i t uat i on and, secondl y, a devi ce i s r equi r ed,whi ch can s i mul at e t hi s s t r ess condi t i on. Sever al devi ces havebeen devel oped i n whi ch mor e r eal i st i c st r ess condi t i ons can besi mul at ed. Exampl es ar e t he si mpl e- shear appar at us ( e. g.Roscoe, 1970) t or s i onal - shear appar at us ( e. g. Symes et a l . ,1982) , di r ect i onal - shear appar at us ( Ar t hur et a l . 1981) andmul t i axi al cubi c t es t cel l ( Moul d et a l . 1982) . For a par t i cul arpr obl em a choi ce has t o be made as t o whi ch mat er i al t est i ngdevi ce gi ves t he most usef ul r esul t s . Because c l ear i nf or mat i onabout t he expect ed st r ess pat h i s . of t en not avai l abl e, manyassumpt i ons have t o be made bef or e t he car ef ul l y measur edmat er i al pr oper t i es can be used i n cal cul at i ons.

To obt ai n mor e i nf or mat i on about t he st r essst r ai n behavi ouri n pr ac t i cal pr obl ems and t o ver i f y t he r esul t s of cal cul at i onprocedures, a mor e advanced measur i ng met hod i s r equi r ed whi chwoul d enabl e st r ess and st r ai n component s i n t he i nt er i or of t hegr anul ar mat er i al t o be vi sual i sed.Up t o now i t was poss i bl e onl y t o obt ai n det ai l ed i nf or mat i onabout t he s t r ai n i n t he i nt er i or of t he gr anul ar mat er i a l . Twomet hods ar e avai l abl e f or t he st r ai n measur ement s. The f i r stmet hod, devel oped by Rosc oe et al . ( 1963) , uses mar ked par t i c l eswhi ch ar e di s t r i but ed i n t he gr anul ar mat er i al , such as l eadshot . Because t he densi t y of l ead i s much mor e t han, f orexampl e, t hat of sand t he pos i t i on of t he l ead shot i n success i vest ages of a t est can be l ocat ed on a phot ogr aph by means of X-rays . Thi s met hod has been used by e. g. Br ansby et al . ( 1975)t o i nvest i gat e t he f l ow of gr anul ar mat er i al i n mode l t es t s of


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hopper s. The second met hod, whi ch was devel oped by But t er f i el det a l . , (1970) , uses st er eo- phot ogr ammet r y t o det er m ne t her el at i ve di spl acement s i n a gr anul ar mat er i al . I n t hi s met hodt wo successi ve phot ogr aphs of t he sur f ace of a sand sampl e ar eused. The t ext ur e of t he par t i c l e sur f ace i n combi nat i on wi t h anon- uni f or m def or mat i on show a mount ai nous ar ea i n a st er eov i ewer . The di f f er ence i n al t i t ude can be conver t ed i nt odi spl acement s by means of a st er eoscopi c pl ot t i ng machi ne. Thes t r ai n t ensor at a mat er i al poi nt can be der i ved f r om t her el at i ve di spl acement s of t hr ee poi nt s i n a r epr esent at i ver egi on.The measur ement of t he st r ess di st r i but i on i n t he gr anul armat er i a l , however , pr oved t o be mor e compl i cat ed. As cont r ast edwi t h el as t i c mat er i al s , t her e i s no uni que r el at i on bet weenst r ess and st r ai n whi ch can be used t o det er m ne t he st r essi ncr ement i n a r egi on wi t h a known def or mat i on. A mor e di r ectmet hod i s t her ef or e r equi r ed t o obt ai n i nf or mat i on about t hes t r es s di s t r i but i on. I t i s not pos s i bl e t o r eal i s e t hi s by meansof el ect r i cal s t r ai n gauges because t he l ar ge number of sensor s ,whi ch woul d have t o be di st r i but ed wi t hi n t he gr anul ar mat er i alt o per f or m syst emat i c measur ement s , woul d unaccept abl y di st ur bt he behavi our .At pr esent onl y t he opt i cal met hod of st r ess anal ysi s known asphot oel as t i c i t y i s avai l abl e f or i nves t i gat i ng i n det ai l t hest at e of s t r ess i n a gr anul ar mat er i al . The f undament al s of t hi st echni que wer e est abl i shed by Br ewst er , who i n 1811 di scover edt he l aw whi ch descr i bes t he phenomenon of pol ar i sat i on byr ef l ect i on. Anot her i mpor t ant st ep was achi eved i n 1852, whenHer apat h di scover ed t hat cr yst al s of a compl ex sal t cont ai ni ngqui ni ne, hydr i odi c ac i d and sul phur i c ac i d, possess t he abi l i t yt o absor b l i ght whi ch osci l l at es i n a spec i f i c pl ane. Theneedl e- shaped cr ys t al s , wi t h a gi r t h di amet er whi ch i scons i der abl y smal l er t han t he wave- l engt h of vi s i bl e l i ght , coul dbe used t o pr oduce synt het i c pol ar i sat i on sheet s . Thi s t echni quewas devel oped i n t he f i r st decades of 1900. The pr oduct i on ofl ar ge pol ar i sat i on sheet s has opened up t he poss i bi l i t y t o


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vi sual i se doubl e r ef r ac t i on i n model s of t r anspar ent mat er i al s .I n 1930 Col ter and Fi l on, bot h of t he Uni ver si t y of London,demonst r at ed t hat t he doubl y ref r act i ve pr oper t y coul d beut i l i z ed t o vi sual i se s t r es ses i n el as t i c mat er i al s . Si nc e t hen,phot oel ast i c i t y has i ncreas i ngl y become a usef ul t ool f or sol vi ngpr obl ems i n st r uct ur al engi neer i ng pr act i ce. The t heor y ofphot oel ast i c i t y and many appl i cat i ons t her eof ar e f or exampl edescr i bed i n Frocht ' s book (1946). I n t he l ast t en year s t he useof t he phot oel ast i c met hod has decl i ned dr amat i cal l y because manycompl ex boundar y val ue pr obl ems i n st r uct ur al engi neer i ng can nowbe sol ved wi t h power f ul comput er s and numer i cal cal cul at i onprocedures.

I n 1957 i t was demonst r at ed by Dant u as wel l as byWakabayashi , t hat phot oel ast i c i t y coul d be used to v i sual i se t het r ansm ssi on of f or ce i n a packed par t i c l e assembl y . The opt i calphenomenon i n a gr anul ar mat er i al , however , had t o be i nt er pr et edi n a qui t e di f f er ent way t han was usual i n el ast i c mat er i al s . I twas not poss i bl e t o cr eat e shar p i soc l i ni cs and i sochr omat i cs f oranal ys i ng t he s t r ess di s t r i but i on. I ns t ead of i sochr omat i c s apat t er n of cl ear st r i pes coul d be obser ved, whi ch wer e ass umed t or epr esent maj or pr i nc i pal s t r ess t r aj ec t or i es .The t r ansm ssi on of f or ce i n a gr anul ar mat er i al was anal ysedi n det ai l by De J ossel i n de J ong and Ver r ui j t (1969). Theyper f or med t est s wi t h cyl i ndr i cal d i sks of phot oel ast i c mat er i alt o si mul at e a t wo di mensi onal assembl y of a gr anul ar mat er i al .The i sochr omat i cs wer e used t o det er m ne t he magni t ude anddi r ec t i on of t he cont act f or ces bet ween t he di sks . Thi st echni que was l at er used by Dr escher and De J ossel i n de J ong( 1972) t o i nvest i gat e aspect s of a mat hemat i cal model f or t hef l ow of gr anul ar mat er i al . They descr i bed a met hod f ort r ans f or m ng t he di s t r i but i on of t he di scr et e cont ac t f or ces anddi spl acement s i n a r egi on i nt o a second- r ank t ensor , so t hat t het est r esul t s coul d be expr essed i n t er ms as usual be empl oyed i nsoi l mechani cs. They f ound exper i ment al l y t hat a t ensordesc r i bes t he di s t r i but i on of t he cont ac t f or ces i n a r egi onf ai r l y sat i s f ac t or i l y. Muc h r esear c h on t he di s t r i but i on of t he


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cont act f or ces and t he f abr i c st r uct ur e i n t wo- di mensi onalanal ogues of di sks was per f or med by Koni shi , Oda and Nemat - Nasser( 1982) . They used, f or exampl e, el l i pt i cal di s ks of phot oel as t i cmat er i al t o i nvest i gat e t he ani sot r opi c behavi our of a gr anul armat er i al . The t est t echni que wi t h di sks i s not sui t abl e f ori nves t i gat i ng t he st r ess di s t r i but i on i n model t es t s of pr ac t i calpr obl ems because an ass embl y of di sks has a ver y di scr et echar act er and i s r est r i c t ed t o t wo- di mensi onal assembl i es onl y.I n 1976 , Dr escher publ i shed an exper i ment al st udy onf l ow r ul es f or gr anul ar mat er i al i n whi ch he agai n used crushedgl ass as t he opt i cal l y sens i t i ve mat er i al . Because at t empt sunder t aken by Wakabayashi ( 1959) and De J ossel i n de J ong ( 1960)t o det er m ne the magni t ude of st r ess i n cr ushed gl ass , by meansof compensat or s , di d not yi el d sat i s f ac t or y r esul t s onl yi nf or mat i on about t he st r ess di r ect i on coul d be obt ai ned i nsampl es of cr ushed gl ass . I t was shown i n a qual i t at i ve way byWakabayas hi ( 1957, 1959) , Dr escher and De J ossel i n de J ong ( 1972)and Oda and Koni shi ( 1974) t hat t he aver age di r ect i on of t hel i ght s t r i pes vi s i bl e i n ci r cul ar l y pol ar i s ed l i ght appr oxi mat el ycoi nc i de wi t h t he l i nes of ac t i on of t he maj or pr i nc i pal s t r ess .Manual measur ement of t he di r ect i ons f r om phot ogr aphs, however ,was r at her subj ect i ve and ver y t i me consum ng, and t he di r ect i onof t he st r i pes est abl i shed i n phot ogr aphs appear ed t o bedependent on devi at i ons i n t he opt i cal f i l t er s , on t he wavel engt h di s t r i but i on of t he l i ght sour ce and on t he sens i t i vi t y oft he f i l m t o a par t i cul ar wave- l engt h. Becaus e ver y r eal i s t i cscal e model s can be pr epar ed wi t h cr ushed gl ass , i t woul d be ver y' usef ul i f mor e i nf or mat i on about st r esses and st r ai ns coul d beobt ai ned by us i ng the opt i cal pr oper t i es of g l ass .

The mai n pur pose of t hi s st udy was t o devel op opt i cal met hods,f or det er m ni ng s t r esses and s t r ai ns i n assembl i es ofphot oel ast i c gr anul ar mat er i al and t o desi gn a devi ce f orper f or m ng syst emat i c measur ement s i n scal e model s of pr act i calpr obl ems and mat er i al t est i ng devi ces . I n t he f i r s t i ns t ance aconnect i on was est abl i shed bet ween t he di st r i but i on of t he


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cont act f or ces at a mat er i al poi nt and t he opt i cal phenomenon i nt he gr anul ar mat er i al . Thi s r esul t ed i n D a met hod f ordet er m ni ng t wo st r ess component s i n a t hr ee- di mensi onal pl anest r ai n sampl e, 2) a cal cul at i on pr ocedur e f or det er m ni ngabsol ut e s t r esses , and 3) a pr ot ot ype of a devi ce f or per f or m ngt he measur ement s ( Al l er sma, 1982a) . At a l at er s t age acomput er - cont r ol l ed opt i cal devi ce was devel oped whi ch wasequi pped al so wi t h a di gi t al camer a. St r esses and di spl acement scan now be measur ed s i mul t aneousl y. The measur i ng met hod has beenappl i ed t o anal ysi ng t he mechani cal behavi our of gr anul armat er i al s i n sever al t es t s , whi ch s i mul at e pr act i cal pr obl ems andmat er i al t es t i ng dev i ces .

The t heor et i cal backgr ound of t he phot oel ast i c measur i ngmet hod, t he det er m nat i on of t he def or mat i on and t he dat apr ocess i ng ar e descr i bed i n Chapt er 2. I n Chapt er 3 t he l ayout ,cont r ol and oper at i on of t he aut omat ed opt i cal measur i ng devi cear e descr i bed, and t he pr oduct i on pr ocess of t he cr ushed gl assand t he pr epar at i on, l oadi ng and sensor i ng of a model ar eexpl ai ned. Some appl i cat i ons of t he opt i cal measur i ng met hod ar epr esent ed i n Chapt er 4. The st r essst r ai n behavi our under l ar geshear def or mat i ons has been i nvest i gat ed, and s ever almeasur ement s of penet r at i on t est s ar e pr esent ed and di scussed.F i nal l y i t i s shown how t he s t r ess di s t r i but i on changes i n ahopper i f some mat er i al f l ow has been t aken pl ace. I n Chapt er 5t he pos s i bi l i t i es , accur acy and l i m t at i ons of t he meas ur i ngmet hod ar e di scussed, and suggest i ons f or f ur t her devel opment oft he measur i ng met hod ar e pr oposed.


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2 THEORY OF MEASUREMENT2. 1 STRESS TENSOR2. 1. 1 Cont act f or ces

I f a l ar ge number of par t i c l es are br ought t oget her t her e wi l lbe cont act poi nt s bet ween t he par t i c l es. At each cont act poi ntt her e act s a f or ce due t o gr avi t y and/ or ext er nal l oads . Thedi r ect i on of t he f or ces and t he f r i ct i on and cohes i on bet ween t hepar t i c l es det er m ne whet her par t i c l es move wi t h r espect t o eachot her or not . At m cr o scal e t he di s t r i but i on of t he cont ac tf or ces i s ver y i nhomogeneous. Besi des body f or ces and ext er nall oad t her e ar e sever al ot her f act or s whi ch i nf l uence t hemagni t ude and di st r i but i on of t he cont act f or ces, such as ;- t he dens i t y of t he par t i c l e assembl y ,- t he s t r engt h of t he par t i c l es ,- t he shape of t he par t i c l es ,- t he el as t i c i t y of t he i ndi vi dual par t i c l es ,- t he par t i c l e si z e.The f i r s t f our i t ems i nf l uence t he number of cont act poi nt sbet ween t he par t i c l es. I f t he assembl y i s packed mor e densel y,mor e cont act poi nt s bet ween t he par t i c l es ar e to be expect ed. I ft he aver aged f or ce t r ansm ssi on i s t he same, t he magni t ude of t hei ndi vi dual cont ac t f or ces wi l l be smal l er . The s t r engt h of t he

par t i c l es i s r espons i bl e f or t he amount of c r ushi ng at t hecont ac t poi nt s , whi ch r esul t s i n a r edi s t r i but i on of t he cont ac tf o r ces . The number of cont act poi nt s bet ween t he par t i c l es i sdi r ect l y i nf l uenced by t he shape and t he e l as t i c i t y of t hepar t i c l es . The magni t ude of t he cont act f or ces i s mor e or l essi nver s el y pr opor t i onal t o t he par t i c l e s i z e. I f t he par t i cl es ar ever y smal l , t he magni t ude of t he cont act f or ces comes cl oser t ot hat of t he at om c f or ces . Thi s expl ai ns , f or exampl e, why ani ni t i al l y non- cohes i ve gr anul ar mat er i al shows a cohes i vebehavi our when i t i s moul ded t o a f i ne powder .I t i s usual i n soi l mechani cs t o descr i be t he di s t r i but i on of


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t he cont act f or ces wi t h a second- r ank t ensor , because i t i s t henpossi bl e t o appl y t he f undament al s of cont i nuum mechani cs. I t hast o be r eal i sed, however , t hat a second- r ank t ensor gi ves as i mpl i f i ed i mage of r eal i t y. The di scr et e char act er of gr anul armat er i al and i nf or mat i on on, f or exampl e, t he number of c ont actf or ces and t he magni t udes of t he i ndi vi dual cont act f or ces i s noti ncl uded i n a t ensor . At pr esent , however t her e i s no bet t er wayt o symbol i se the f or ce di s t r i but i on. Ther ef or e t he opt i calmeasur i ng met hod has been so desi gned, t hat par amet er s ar eobt ai ned whi ch can be used t o der i ve t he component s of a t ensor ,so t hat t he di s t r i but i on of t he cont act f or ces i s descr i bed i n as i m l ar way , as i s usual i n soi l mechani cs .2. 1. 2 St r ess t ensor i n gr anul ar mat er i al

I n cont i nuum mechani cs , s t r ess i s def i ned as t he r esul t antf or ce ac t i ng on a uni t ar ea. Thi s def i ni t i on i s based onhomogeneous mat er i al s, whi ch means t hat t he el ement ar y par t i cl es,i . e. at oms, ar e ver y smal l i n compar i son wi t h t he st r essgr adi ent s . I f cont i nuum mechani cs i s appl i ed t o gr anul armat er i al , ar eas have t o be cons i der ed whi ch cont ai n suf f i c i entpar t i c l es . On t he ot her hand, i t i s not per m t t ed t o cons i dert oo l ar ge ar ea uni t s , because i n t hat case i nf or mat i on on t hes t r es s gr adi ent i s l os t i n s ampl es wi t h a non- uni f or m s t r es sdi st r i but i on. A uni t ar ea can be assumed t o be r epr esent at i ve i fa smal l devi at i on i n sur f ace does not s i gni f i cant l y change t heaver age s t r ess . However , i t i s not al ways sui t abl y poss i bl e t odef i ne a r epr esent at i ve ar ea i n a gr anul ar mat er i al , becausel ar ge gr adi ent s i n st r ess and st r ai n may occur . The t hi ckness ofa shear band, f or exampl e, i s about t en par t i c l e di amet er s , sot hat t he cont act f or ces of onl y a f ew par t i c l es ar er epr es ent at i ve of t he s t r es s s t a t e. T hi s i s of cour s e ver y l i t t l ei n compar i son wi t h homogeneous mat er i al s and i t i s not qui t ecl ear how such phenomena have t o be i nt er pr et ed.I f t he macr o st r ess i n a r egi on i s not ver y i nhomogeneous, t heaver agi ng pr ocedur e, whi ch i s descr i bed by Dr escher and


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De J os s el i n de J ong ( 1972) can be used to t r ansf or m di scr et econt act f or ces i nt o st r ess compcAaent s. I f a r egi on wi t h vol ume Vi s cons i der ed wi t h a non- uni f or m s t r es s s t at e o. ., whi ch i s i nequi l i br i um t hen the average stress o. . i s def i ned by ij = fv ij d v ( 2 : l )

whi ch, by usi ng Gauss' s di ver gence t heor em can be wr i t t en as

* ij =hi x i ( m ) T j ( m ) ( 2 : 2 )m=li n whi chu =number of di sc ret e f or ces i nt er sec t ed byt he boundar y of Vx. = i - co- or di nat e of the i nt er sec t i on poi nt oft he mt h contact fo rce (i =l , 2, 3)T ( m) = j _ component of the mt h f o r c e ( j =l , 2, 3)Eq. 2: 2 was used by Dr escher and De J ossel i n de J ong todet er m ne the stress tensor i n a r egi on of a t wo di mensi onalass embl y of phot oel as t i c di s k s . The cont act f or ces wer edet er m ned f r om the i s o c l i n i c s i n the di s ks , us i ng a proceduredevel oped by De J o s s e l i n de J ong and Ver r ui j t (1969). Contrary tousual phot oel ast i c measur i ng t echni ques, i t was pos s i bl e i n t h i sway t o det er m ne t he compl et e st r ess t ensor i n a r epr esent at i ve

r egi on. Si nce the di s t r i but i on of the cont act f or ces was known,i t was pos s i bl e to pl ot a Maxwel l di agr am of the f or ces whi chi nt er sec t a hypot het i cal c i r cul ar boundar y . An exampl e of such adi agr am i s pr esent ed i n Fi g. 2: 1. The mor e cl osel y t he di agr amappr oaches an el l i ps e, the better the f or c e di s t r i but i on can bedescr i bed by a t ensor . Taken i nt o account t hat a r el at i vel y smal lnumber of par t i c l es wer e cons i der ed, f ai r l y good agr eement wi t han el l i ps e i s observed.The anal ys i s of Dr escher and De J ossel i n de J ong hasdemonst r at ed t hat the di s t r i but i on of t he cont act f or ces can bedes cr i bed f ai r l y wel l wi t h a second- r ank t ensor , and agr eementcan be expect ed to be better i f the r egi on under cons i der at i on


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o 3 10 kg

Fi g. 2: 1 Maxwel l di agr am of a ci r cul ar r egi on i n a t wo-di mensi onal assembl y of 150 di sks ( f r om Dr escher and De J ossel i nde J ong, 1972) .cont ai ns mor e par t i c l es .Because t he cont act f or ces creat e s t r esses i n t he di sks , t hest r ess t ensor whi ch descr i bes t he di s t r i but i on of t he cont actf or ces i n a r egi on can al so be der i ved by aver agi ng a l l t hei nt er nal s t r es ses of al l t he di s ks i n a r egi on cons i der ed, as i sshown by eq. 2: l . I t was f ound poss i bl e t o det er m ne t wocomponent s of t he aver age st r ess t ensor i n a body of phot oel ast i cmat er i al by opt i cal means . Because t he whol e vol ume of t he bodyi s consi der ed i n t hi s met hod, t he body may be of any shape.Si nce t he opt i cal aver agi ng pr ocedur e i s al so val i d over t het hi ckness of a sampl e, t he aver age st r ess component s can al so beder i ved i n t hr ee- di mens i onal assembl i es of ar bi t r ar i l y shapedpar t i c l es of phot oel as t i c mat er i al . Because t he opt i cal aver agi ngcan be per f or med aut omat i cal l y, s yst emat i c measur ement s can beper f or med i n scal e model s wi t h mor e r eal i s t i c par t i c l es .2. 1. 3 St r es s di s t r i but i on i n a par t i c l e

The par t i c l es pl ay an i mpor t ant r ol e i n t he opt i cal measur i ngt echni que. I n f act , t hey ar e used as gauges whi ch t r ans l at e t hedi scr et e cont act f or ces i nt o t ensor component s and t hey maket hese component s measur abl e. I n t he opt i cal measur i ng met hod


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empl oyed t he st r ess component s ar e accumul at ed vect or i al l y overt he sur f ace of a r egi on. The s t r ess di s t r i but i on i n t he r egi onconsi der ed t her ef or e need not t o be homogeneous. However ,because t he magni t ude of t he pr i nc i pal s t r ess di f f er ence (a - a ) ,whi ch can be measur ed i s l i m t ed, i t i s necessar y t o know how t hes t r esses var y at m cr o scal e. I nf or mat i on on t he s t r essdi s t r i but i on at m cr o scal e can be obt ai ned by cal cul at i ng t hes t r ess di s t r i but i on i n a body whi ch i s l oaded by di sc r et e f or ces .I f a par t i c l e i s s chemat i sed as a c i r cul ar di sk t he s t r essdi s t r i but i on can be cal cul at ed wi t h an anal yt i cal sol ut i on basedon el ast i c t heor y ( Ti moshenko and Goodi er , 1951) . I n t he case ofa di amet r i cal l oad t he di s t r i but i on of t he pr i nc i pal s t r es sdi f f er ence and t he pr i nci pal s t r es s di r ec t i on, t y i s gi ven by4P ( a2- x z - y 2)(a -a ) = ( 2: 3)1 2 ira Cx2+( a - y ) z ] Cx 2+( a+y) 23

2xy* - . - r - r - 7 ( 2 : 4 )x - y +ai n whi chP = t he appl i ed di amet r i cal l oada = t he r adi us of t he di skx, y = t he co- or di nat es of a poi nt i n t he di sk (-a


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M

Fi g. 2: 2 The di s t r i but i on of t he pr i nc i pal st r es s di f f er enceand t he maj or pr i nc i pal s t r ess di r ect i on i n a di amet r i cal l y l oaded di sk.t hi s case. Fur t her mor e, i t i s not t o be expect ed t hat t he shapeof t he boundar y has much i nf l uence on t he var i at i on i n t hepr i nc i pal s t r es s di f f er ence, so t hat i t i s not neces s ar y t oi nvest i gat e a gr eat var i et y of par t i c l e shapes .2. 1. 4 Rel at i on of i nt er par t i c l e s t r ess and bul k s t r ess

A mat er i al poi nt i n a par t i c l e assembl y i s a vol ume whi chcont ai ns s uf f i c i ent par t i c l es f or obt ai ni ng r epr es ent at i veaver aged val ues of t he st r ess component s. I n t he case of at hr ee- di mensi onal pl ane st r ai n sampl e, of whi ch t he sur f ace i sl ar ge i n compar i son wi t h t he t hi ckness , a mat er i al poi nt i s , f orexampl e, a cyl i nder wi t h a l engt h equal t o t he t hi ckness of t he


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F i g. 2: 3 Di s t r i but i on of t he cont act f or ces at t he boundar y ofa cyl i ndr i c al mat er i al poi nt .sampl e. Such a cyl i ndr i cal mat er i al poi nt cont ai ns a number ofpar t i c l es over t he t hi ckness as wel l as over t he sur f ace. Eachpar t i c l e i s l oaded by a number of cont act f or ces . I n F i g. 2: 3 i ti s demonst r at ed schemat i cal l y how t he cont act f or ces ar edi s t r i but ed over t he hypot het i cal cyl i ndr i cal boundar y. As wi l lbe t he case i n pr act i ce, t he i ndi vi dual cont act f or ces ar e notal l pl ot t ed per pendi cul ar l y t o t he axi s of t he cyl i nder .Fur t her mor e, a cohesi onl ess gr anul ar s ampl e has t o be suppor t edat t he ent i r e boundar y , so t hat i n r eal i t y t he s t r ess condi t i oni s t hr ee- di mens i onal . However , i f t he t hi ckness of t he sampl e i ssuf f i c i ent l y l ar ge, t he f or ce component s per pendi cul ar t o t hepl ane of t he sampl e ar e assumed t o have no s i gni f i cant i nf l uenceon t he pl ane def or mat i on.The aver aged val ue of t he component s of t he t ensor , whi chdescr i be t he aver aged s t r ess i n t he cyl i ndr i cal mat er i al poi nt ,


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F i g. 2: 4 Repr esent at i on of t he cont r i but i on of one cont actf or ce t o the aver aged s t r ess i n a cyl i ndr i cal r egi on.can be cal cul at ed wi t h eq. 2: 2. I n Fi g. 2: 4 i t i s shown, how a m t hcont act f or ce T has t o be r esol ved i nt o di f f er ent component s.I t i s assumed t hat a sampl e can be l oaded i n such a way t hatone of t he pr i nc i pal s t r esses , e. g. a , i s per pendi cul ar t o t heboundar y of t he sampl e and t her ef or e a and a act i n t he pl aneof t he sampl e. The st at e of st r ess at a poi nt of a pl ane sampl ei s r epr esent ed i n a Mohr di agr am i n F i g. 2: 5. I t i s t o be not edt hat compr essi ve st r ess has a negat i ve si gn. The aver agi ngpr ocedur e i s based on t he whol e vol ume of t he mat er i al poi nt . I nr eal i t y , however , onl y a par t of t he vol ume i s occupi ed by t hegr anul ar mat er i al , whi l e t he por es ar e f i l l ed wi t h ai r or al i qui d. The aver age s t r ess i n t he par t i c l e bodi es i s t her ef or el ar ger t han cal cul at ed wi t h t he aver agi ng pr ocedur e. Al t hought hi s poi nt i s debat abl e i t i s negl ec t ed, because a di scuss i onl eads t o new det ai l s such as t he phenomenon t hat s ome par t i cl es


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"lf r "

\ ^"Vx

" T

> \

" " / /

,' - " p o l e

r*

x* " Y Y

Fi g. 2: 5agram Repr esent at i on of t he st r ess component s i n a Mohr di -

ar e not l oaded at al l or ver y l i t t l e.An opt i cal met hod can be appl i ed t o det er m ne aver aged st r esscomponent s whi ch r ef er t o a mat er i al poi nt i n a pl ane st r ai nsampl e. I t can be demonst r at ed wi t h t he t heor y, whi ch descr i best wo di mens i onal phot oel as t i c i t y, t hat t he opt i cal aver agi ng overt he t hi ckness and sur f ace of a mat er i al poi nt i s al most t he sameas t he aver agi ng pr ocedur e wi t h eq. 2: l .2. 2 STRAI N TENSOR2. 2. 1 Rel at i ve di spl acement of par t i c l es

I f a gr anul ar mat er i al i s l oaded, t he par t i c l es wi l l move i nr el at i on t o each ot her . The d i spl acement s of t he i ndi vi dualpar t i cl es cause some macr o def or mat i on of a sampl e. Toi nves t i gat e t he s t r ai n behavi our of a s ampl e i n det ai l , i t i snecessar y t o know t he di spl acement s i n a r epr esent at i ve r egi on.Because i t i s not conveni ent t o use t he di spl acement s of al l t hei ndi vi dual par t i c l es i n f or mul as , a met hod i s r equi r ed t ochar act er i se t he di spl acement s at a mat er i al po i nt . I n t he sameway as t he s t r esses i t i s usual t o desc r i be t he r el at i ve


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e 2 Ux.xUy.y

Fi g. 2: 6 Repr esent at i on of t he st r ai n component s i n a Mohr di agr am ( r el at i ve di spl acement di agr am) .di spl acement s bet ween t he par t i cl es at a mat er i al poi nt by meansof a second- r ank t ensor . The component s of t he st r ai n t ensor at amat er i al poi nt ar e repr esent ed i n a Mohr di agr am i n F i g. 2: 6. Thest r ai n t ensor i s i n gener al not symmet r i cal , due t o t her ot at i on, u of t he mat er i al .I n Fi g. 2: 7 t he di s t r i but i on of t he di r ec t i ons of t hedi spl acement s of a t ensor i al def or mat i on f i el d i s pl ot t ed. I t i sassumed t hat UJ =0 and t hat t he vol ume r emai ns const ant : hence t hepr i nci pal st r ai ns E and E have t he same absol ut e val ue. Todemonst r at e t hat t he di spl acement s i n a gr anul ar mat er i al can bedescr i bed r easonabl y wel l wi t h a t ensor , a s i m l ar pat t er n hasbeen cr eat ed by means of t wo negat i ves on whi ch t he par t i cl est r uct ur e of t wo successi ve st ages of a pl ane sampl e ar e f i xed( Fi g. 2: 8) . I f t he f i el d i s subj ect ed t o a homogeneousdef or mat i on t he maj or di r ec t i ons of s t r ai n become v i s i bl e. Thi st echni que was used by De J ossel i n de J ong, 1959, t o anal yse t hedef or mat i on i n a t wo di mensi onal assembl y of rods. Al t houghgood aver agi ng i s obt ai ned over a l ar ge amount of par t i c l es, t hi st echni que i s not al ways sui t abl e f or obt ai ni ng det ai l edi nf or mat i on on t he def or mat i on, because a l ar ge r egi on wi t h ahomogeneous def or mat i on i s r equi r ed.


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. .' * l t t \ \ \ \

Fi g. 2: 7 Cal cul at ed di s t r i but i on of t he di r ec t i ons of t he di s pl acement s of a t ensor i al def or mat i on f i el d.

Fi g. 2: 8 Di s t r i but i on of t he di r ec t i ons of d i s pl acement s ,vi sual i sed i n a pl ane s t r ai n gr anul ar mat er i al .


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2. 2. 2 Det er m nat i on of t he s t r ai n t ensorCont r ar y t o t he opt i cal s t r ess measur ement , i t i s notpr act i cabl e t o obt ai n st r ai n component s whi ch ar e based oncont i nuous aver agi ng of t he di spl acement s over t he vol ume of amat er i al poi nt . Ther ef or e t he det er m nat i on of t he s t r ai n t ensorhas t o be based on t he di spl acement s of some di scr et e poi nt s. Thedi spl acement s of onl y t hr ee poi nt s ar e r equi r ed t o cal cul at e t hest r ai n component s at a mat er i al poi nt . At pr esent t her e ar e t wo

measur i ng pr i nc i pl es avai l abl e f or det er m ni ng t he di spl acementof a di scr et e poi nt . I n t he most advanced t echni ques t wosuccess i ve exposur es of t he par t i c l e st r uct ur e at t he sur f ace ofa sampl e ar e used t o det er m ne t he di spl acement of a poi nt bymeans of t he st er eo- phot ogr ammet r i c met hod ( But t er f i el d etal . , 1970) . The advant age of t hi s met hod i s t hat t he di spl acementat t he sur f ace can be measur ed mor e or l ess cont i nuousl y, so t hatl ar ge st r ai n gr adi ent s can be obser ved. A di sadvant age i s ,however , t hat t he di gi t i sat i on of t he di spl acement s r equi r esexpensi ve pr eci si on devi ces and i s t i me consum ng. Because t hemeasur i ng pr ocedur e cannot be aut omat ed i n a si mpl e way, i t i snot pr act i cabl e t o appl y t hi s met hod t o t he i nt er pr et at i on of al ar ge number of phot ogr aphs . Fur t her , i t i s not s i mpl y poss i bl et o f ol l ow t he pat t er n of a s i ngl e po i nt , whi ch i s f or exampl eusef ul i n t est s wi t h l ar ge def or mat i ons .

A si mpl er t echni que f or det er m ni ng t he di spl acement s ofdi sc r et e poi nt s i s t o measur e t he co- or di nat es of l abel l edpar t i c l es i n success i ve s t ages of a t es t . I f a pl ane sampl e i snot t r anspar ent , par t i c l es wi t h a hi gher dens i t y e. g. l ead shot ,can be di st r i but ed i n a pl ane i n t he sampl e and X- r ays can beused t o f i x t he act ual pos i t i on of l ead bal l s on a phot ogr aph.Thi s met hod was used, f or exampl e, by Br ansby et al . , 1973 t oi nvest i gat e t he def or mat i on i n pl ane hopper s. Thi s t echni que i sver y at t r ac t i ve i n t est s wi t h c r ushed gl ass . Because t heassembl y of gl ass par t i c l es i s made t r anspar ent by sat ur at i ng thepor es wi t h a l i qui d whi ch has t he same r ef r act i on i ndex as g l ass ,bl ack mar ks can be made vi si bl e si mpl y by nor mal l i ght .


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Fur t her mor e par t i cl es, of t he same densi t y can be used such asbl ack col our ed gl ass . Because, due t o smal l devi at i ons of t her ef r ac t i on i ndex of gl as s , t he par t i c l e s t r uct ur e i s vi s i bl e i nan assembl y t oo, a st er eo obser vat i on of successi ve phot ogr aphscan be used t o suppor t t he i nt er pr et at i on of t he di spl acement s ofmar ked par t i c l es .Sever al t ypes of di gi t i ser s ar e avai l abl e t o det er m ne t heco- or di nat es of t he mar ks f r om phot ogr aphs . I n gener al , however ,t he di gi t i sat i on of t he co- or di nat es i s r at her t i me- consum ng,and i f c r oss hai r s ar e used i t i s not al ways easy t o def i ne apar t i cul ar poi nt on s i m l ar mar ks of success i ve t es t s .Fur t her mor e, t he phot ogr aphi c st ep i s not conveni ent . To pr eventer r or s due t o s t r engt h of t he f i l m mat er i al , i t i s neces sar y t ouse gl ass pl at es wi t h a l i ght - sens i t i ve l ayer . However , adi sadvant age i s t hat a camer a whi ch handl es gl ass pl at es i s notsui t abl e f or aut omat i on.To el i m nat e t he phot ogr aphi c s t ep, a di gi t al camer a i sappl i ed i n t hi s r esear ch t o det er m ne t he pos i t i on of mar ksdi r ect l y i n a t est model . The advant age of t hi s met hod i s t hatt he pr ocedur e can be compl et el y aut omat ed and t he r esul t s ar edi r ec t l y avai l abl e, so t hat t he pr ogr ess of a t es t can beobser ved i n r eal t i me.2. 3 OPTI CAL STRESS MEASUREMENT AT A MATERI AL POI NT2. 3. 1 L i ght

L i ght i s used as t he medi um f or det er m ni ng t he s t r essdi st r i but i on and def or mat i on i n a t r anspar ent assembl y ofopt i cal l y sens i t i ve gl ass par t i c l es . The maj or advant age ofl i ght i s t hat i t has no ef f ect on t he mechani cal behavi our of t hepar t i c l es . Onl y secondar y ef f ect s , such as heat pr oduct i on of al i ght sour ce, can i nf l uence t he measur ement , because t emper at ur egr adi ent s caus es addi t i onal i nt er nal s t r es s es i n gl as s . For t hi sr eason a l aser sour ce appear ed t o be ver y sui t abl e, because ani nt ens i ve l i ght beam i s pr oduced wi t h m ni mum pr oduct i on of heat


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r adi at i on. Fur t her mor e, l i ght wi t h a ver y smal l r ange i n wavel engt h i s obt ai ned, so t hat t he mat hemat i cal descr i pt i on ofphot oel as t i c i t y i s i n ver y cl ose agr eement wi t h t he r eal i t y. Fort hi s pur pose a He- Ne l aser i s used, whi ch pr oduces l i ght wi t h awave- l engt h of X=633nm Li ght i s a compl ex phenomenon andsever al t heor et i cal model s have been devel oped i n or der t odescr i be t hi s phenomenon compl et el y. For t hi s pur pose, however ,t he c l assi cal et her - wave t heor y of Huygens i s qui t e adequat e.Accor di ng t o t hi s t heor y, space i s f i l l ed wi t h a hypot het i calper f ect el ast i c medi um and l i ght i s a wave phenomenon caused bya di st ur bance i n t hi s medi um The di st ur bance consi st s ofvi br at i ons of t he el ement ar y par t i c l es . I t i s assumed t hat t hevi br at i ons t ake pl ace i n a di r ec t i on per pendi cul ar t o thedi r ec t i on of pr opagat i on of t he wave. I n ord i nar y l i ght t heel ement ar y par t i cl es ar e subj ect ed t o many har moni c waves , whi chvi br at e i n r andom di r ect i ons and whi ch di f f er i n ampl i t ude andphase. The mot i on of a par t i c l e i s t her ef or e ent i r el y r andom asi s s chemat i sed i n F i g. 2: 9. I t i s not poss i bl e t o desc r i beor di nar y l i ght wi t h s i mpl e mat hemat i cal f or mul as. To per f or mphot oel ast i c measur ement s or di nar y l i ght has t o be modi f i ed i nt oa wel l - def i ned phenomenon whi ch can be r eal i sed by means of apol ar i sat i on f i l t er .2. 3. 2 Pol ar i sat i on

The r andom behavi our of or di nar y l i ght can be modi f i ed i nt o awel l - def i ned wave mot i on by means of pol ar i sat i on. Pol ar i sat i oncan be achi eved by di chr oi c c r yst al s , such as her apat i t e( di scover ed by Her apat h i n 1852) . The physi cal model ass umest hat or di nar y l i ght i s conver t ed i nt o t wo mut ual l y per pendi cul arpl ane- pol ar i sed beams when i t ent er s t he cr yst al . Because t heabsor pt i ve capac i t y of di chr oi c cr yst al s i s much gr eat er f or oneof t hese beams, onl y one pl ane- pol ar i sed beam i s t r ansm t t ed( Fi g. 2: 10) . I n pl ane- pol ar i sed l i ght t he l i ght vec t o r vi br at eshar moni ous l y i n par al l el pl anes , as i s shown schemat i cal l y f or


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F i g. 2: 9 Di agr am of t he mot i on of or di nar y l i ght , pr oj ec t ed ona pl ane per pendi cul ar t o t he l i ght beam

Unpolaiad Inddint U(ht

CourUty M. Orabau and Polanid Corporation

Fi g. 2: 10 Schemat i c r epr esent at i on of t he pr oduct i on of pol ar i s ed l i ght by a di c hr oi c cr y st al ( f r om Fr oc ht , 1946) .


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Fi g. 2: 11 Gr aphi c r epr esent at i on of t he wave mot i on of pol ar i sed l i ght .one pl ane i n F i g. 2: 11. Accor di ng t o t he wave t heor y pol ar i sedl i ght ' can be descr i bed by

Asi n( ) t ) w = 2i r- r ad/ s (2 :5 )wer e a = t he act ual ampl i t ude at t i me tA = t he maxi mum ampl i t udeid = t he angul ar vel oci t yc = t he pr opagat i on speed of l i ghtx = t he wave- l engt h of t he l i ght usedThe i nt ens i t y I of a pl ane- pol ar i sed l i ght beam i s def i ned by

I = 2A* (2 :6 )

2. 3. 3 Doubl e r ef r act i onI sot r opi c t r anspar ent mat er i al s , such as gl ass , becomet empor ar i l y doubl e- r ef r act i ve when subj ect ed t o shear st r esses.The physi cal backgr ound of t hi s phenomenon i s t hat t he vel oci t yof pr opagat i on of l i ght whi ch vi br at es i n t he pr i nc i pal s t r ess

di r ec t i ons i s di f f er ent . The vel oc i t y of pr opagat i on decr easesi f t he s t r es s , ac t i ng on t he pl ane of vi br at i on, i nc re as es . Todescr i be t he doubl e- r ef r act i ve pr oper t y t he f ol l owi ng phys i calmodel i s assumed ( Fi g. 2: 12)


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POLARIZER1/4- \ PLHTE

'. CIRCULRR"LIGHT

ELLIPTICAL LIG HTROTATING FILTER

LIGHT SENSOR (2Fi g. 2: 13 Lay- out of t he opt i cal f i l t er s ys t emI n opt i cs per manent doubl e- r ef r act i ve f i l t er s ar e used t opr oduce l i ght wi t h par t i cul ar pr oper t i es . A t ypi cal exampl e ofs uch a f i l t er i s a quar t er - wave pl at e. I n t hi s f i l t er t he

r et ar dat i on bet ween t he waves whi ch vi br at e i n t he di r ect i on oft he pr i nc i pal axes i s exact l y a quar t er of t he wave- l engt h of t hel i ght used.2. 3. 4 Opt i cal f i l t er s ys tem

The appl i cat i on of s t r ess opt i cs t o par t i c l e assembl i esr equi r es some pr oper t i es of t he opt i cal f i l t er sys t em whi ch ar eusual l y not of i mpor t ance f or s t r ess anal ys i s i n bodi esconsi st i ng of a homogeneous mat er i al .Al t hough t he assembl y of gl ass par t i c l es i s made t r anspar entwi t h a l i qui d, t he absor pt i on of l i ght due t o r ef r ac t i on andr ef l ect i on i s not homogeneous over t he sur f ace of t he pl anesampl e and depends on t he def or mat i on. I t i s t her ef or e necessar yt hat t he i nt ens i t y of t he i nc i dent l i ght , i nc l udi ng l osses due t oabsor pt i on by t he opt i cal f i l t er s, can be measur ed when t he modeli s s t r essed.The st r ess t ensor at a mat er i al poi nt has t o be der i ved f r om


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t he m cro s t r esses i n t he par t i c l es of a smal l r egi on. Theaccumul at i on of t he opt i cal ef f ect s over t he sur f ace andt hi ckness of t he l i ght beam shoul d t her ef or e be i n agr eement wi t ht he vec t or i al addi t i on of t he m c ro s t r es s es .The l ayout of t he opt i cal sys t em whi ch best meet s t her equi r ement s i s shown schemat i cal l y i n Fi g. 2: 13. Monochr omat i c,l i near l y pol ar i sed l i ght , pr oduced by a He- Ne l aser , i s modi f i edi nt o c i r cul ar l y pol ar i sed l i ght by means of a so cal l ed quar t er -wave pl at e. I f t he ci r c ul ar - pol ar i s ed l i ght beam has beent r ansm t t ed t hr ough a model of doubl e- r ef r ac t i ve mat er i al , anaddi t i onal phase di f f er ence S causes an el l i pt i cal mot i on of t hel i ght vec t or . A r ot at i ng pol ar i s at i on f i l t er ( anal ys er ) and al i ght s ens or ar e us ed t o anal ys e t he el l i pt i cal l y pol ar i s edl i g h t .The i ni t i al pol ar i sed l i ght beam whi ch can be desc r i bed wi t heq. 2: 5, ent er s t he quar t er - wave pl at e i n such a way t hat t hepl ane of pol ar i s at i on i s i nc l i ne at 45 wi t h r es pec t t o t heopt i cal axes of t he pl at e. The l i ght wave i s r esol ved i n t hedi r ect i ons of t he opt i cal axes q and q and can be descr i bed byqx = i AJ ~2cos( wt ) ( 2: 7a)q2 = | Af 2cos ( ut ) ( 2: 7b)

When t he l i ght l eaves t he f i l t er , t he r et ar dat i on has caused aphase di f f er ence of TT/ 2 bet ween t he per pendi cul ar waves. Theequat i ons desc r i bi ng t he vi br at i on i n the di r ec t i on of t heopt i cal axes now becomeq = | Af 2cos(uj t ) ( 2: 8a)q2 = | A4" 2C os ( ut - r = | Af 2si n( J t ) ( 2: 8b)

Compoundi ng t hese t wo per pendi cul ar waves, a ci r cul ar mot i on of


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t he l i ght i s obt ai ned whi ch can be wr i t t en i n t wo di mensi ons wi t hr = | AJ ~2 ( 2 : 9a)()> = u i t (2:9b)

wer e r and $ ar e the l engt h and t he di r ect i on of t he l i ght vect orat t i me t , r es pec t i vel y. The i nt ens i t y of t hi s c i r cul arpol ar i sed l i ght , known as t he i ni t i al l i ght i nt ens i t y, I , i sI = 2r 2= A2 ( 2: 10)o

When t he c i r cul ar - pol ar i sed l i ght ent er s t he t empor ar y doubl e-r ef r ac t i ve mat er i al of t he t es t model , t he c i r cul ar mot i on i sr esol ved i nt o t wo per pendi cul ar waves agai n, whi ch vi br at e i n t hedi r ec t i ons of t he pr i nc i pal s t r esses , as r epr esent ed bya = i AJ Ts i n( wt ) ( 2: l l a)2 2a i = | AJ ~2cos( wt ) ( 2: l l b)

wer e a and a ar e t he ampl i t udes of t he l i ght wave i n t he aand a di r ec t i on, r espec t i vel y. The pr opagat i on vel oc i t y of t het wo waves ar e not equal , and the vel oc i t y di f f er ence e, whi ch i spr opor t i onal t o t he di f f er ence of t he pr i nc i pal s t r es s es , i sgi ven by

E = K( a - a ) ( 2: 12)wer e K i s an opt i cal mat er i al const ant .The vel oci t y di f f er ence causes a phase r et ar dat i on S bet weent he waves whi ch i s pr opor t i onal t o t he t hi ckness s of t he model ,t he wave- l enght X of t he l i ght used and t he aver age vel oci t y c'of t he l i ght i n t he doubl e- r ef r ac t i ve mat er i al . The expr ess i on


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f or S becomesS = ^r ( a - a ) ( 2: 13)

XC 1 2I n i n a t es t X, c ' , K and s ar e cons t ant s ; t hi s equat i on cant her ef or e be wr i t t en as( a -a ) = MS ( 2: 14)

wer e M i s a f act or t ak i ng account of al l par amet er s i nf l uenci ngt he r et ar dat i on S.The phase di f f er ence causes an el l i pt i cal mot i on of t he l i ghtwhen i t l eaves t he model , as i s r epr esent ed schemat i cal l y i nFi g. 2: 14. Thi s mot i on can be descr i bed wi t ha 2 = i AJ ~2si n( 0J t +S) ( 2: 15a)8 L I = AJ Tcosdj Ut ) ( 2: 15b)

I f t he l i ght ent er s t he s econd pol ar i s at i on f i l t er ( anal ys er ) t hel i ght vec t or r ( F i g. 2: 14) wi l l be resol ved i nt o t wo per pendi cul arcomponent s, but onl y one component , whi ch coi nci des wi t h t hepol ar i sat i on pl ane wi l l be t r ansm t t ed. The ampl i t ude of t het r ansm t t ed wave depends on t he angl e y bet ween t he pol ar i sat i onpl ane and t he a - axi s and can be descr i bed bya = A. T2Csi n( ut+S) si nY+cos( wt ) cosyD ( 2: 16)

The val ue of wt at whi ch a i s a maxi mum A , can be f ound byy ydi f f er ent i at i ng eq. 2: 16. Af t er some r ear r angement t hi s gi ves

wt = a r c t g f . c o ^S . A = $ ( 2: 17)^ (. cotgy+si nSj Tand t hus

A = ?AJ TCsi n( 0+6) si n>f +cos( t ) cosY3 ( 2: 18)y *


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anal yzer

F i g. 2: 14 Gr aphi c al r epr es ent at i on of t he l i ght mo t i o n whe n i th a s be en t r ans m t t ed t hr o ugh t he t es t mo de l .

max

0 Pmax. Pmin.rotation analyzer

Fi g. 2: 15 Light i nt e ns i t y I as aang le p of the an al ys er .

180

f unc t i on of t he r o t at i on


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The l i ght i nt ens i t y as a f unc t i on of y i s nowI = 2A2 ( 2: 19)Y Y

Because t he maxi mum ampl i t udes i n eq. 2: 15 ar e equal , i t can beconcl uded t hat t he pos i t i on of t he el l i pse i s al ways symmet r i calwi t h r espect t o t he pr i nc i pal s t r esses ; so i n t he case ofFi g. 2: 14, I i s a maxi mum i f y= 45 and a m ni mum i f -y= 135.Subst i t ut i ng t hese ext r eme val ues i nt o eq. 2: 17 gi ves r espect i vel y


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F i g. 2: 16 Di f f er ence of t he maxi mum and t he m ni mum l i ght i nt ens i t y as a f unct i on of t he phase r et ar dat i on.r et ar dat i on, f or exampl e, wi t h

, 1 - Imax rE +1 . ,max mm', 1 - l . .o . _ f max m n^

S = ar cs i nl j = 1( 2: 23)

I f t he i ni t i al pos i t i on of t he anal ys er i s known, t he di r ec t i onof t he pr i nc i pal s t r esses can be deduced f r om t he r ot at i onangl e B of t he anal yser a t maxi mum l i ght i nt ens i t y. I n t he caseof Fi g. 2: 15 t he pr i nc i pal s t r es s di r ec t i ons ar et|) = B - 45l " max ( 2 : 2 4a)\\> = B +452 max ( 2: 24b)

A s i mpl i f i ed equat i on c an be der i v ed t o c al c ul at e I , bec aus eoI i s appar ent l y a har mo ni c f unc t i o n of y wi t h a per i o d of 180 .An equi v al ent equat i on i s t hen

( I - I ) s i n2Y+Imax ( 2: 25)Subs t i t ut i ng t he expr es s i ons f or t he i nt ens i t i es gi ves

1 = 1 ( s i nSs i n2Y+l )Y 0 ' ( 2: 26)


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wer e y may al so be wr i t t en as y = P-i p The r ange of S i s l i m t ed because I - I . dec reases i f Smax mmbecomes l ar ger t han I T r adi ans ( Fi g. 2: 16) . However , a f avour abl ec i r c ums t anc e i s t hat t he pr i nc i pal s t r es s di f f er enc e i n par t i cl eassembl i es i s usual l y smal l i n compar i son wi t h t hat f ound i nt est s on homogeneous cohesi ve mat er i al s. I f a sampl e of gl asspar t i c l es wi t h a t hi ckness of 40 mm i s used i n a model t est , amaxi mum pr i nc i pal s t r ess di f f er ence of about 700 kN/ m can bemeasur ed. I n t he t est s descr i bed i n t hi s paper t he maxi mumpr i nci pal st r ess was i n gener al not mor e t han 250 kN/ m ( 8 and i|) i s basedon a pl ane s t r ess model of phot oel ast i c mat er i al , wi t h ahomogeneous st r ess di st r i but i on at t he mat er i al poi nt consi der ed.I n t he case of a pl ane model of cr ushed gl ass t he i ndi vi dualpar t i c l es ar e used as an opt i cal sensor whi ch t r ans l at e t hedi s t r i but i on of t he cont ac t f or ces i nt o opt i cal l y measur abl es t r ess component s . Apar t f r om opt i cal i mper f ec t i ons of t hegr anul ar mat er i al , sever al ot her phenomena ar e not i n agr eementwi t h t he assumpt i ons on whi ch t he mat hemat i cal descr i pt i on of t hebehavi our of l i ght i n doubl e- r ef r ac t i ve mat er i al i s based. Thef ol l owi ng devi at i ons can be not ed;- t he model i s not homogeneous;- t he s t r es s di s t r i but i on i n t he par t i c l es i s not pl ane;- t he st r ess di st r i but i on i s not homogeneous over t he sur f aceand t hi ckness of a mat er i al poi nt .I t i s not t o be expect ed t hat t he i nhomogeneous char act er of agr anul ar sampl e has a s i gni f i cant i nf l uence on t he val i di t y oft he mat hemat i cal descr i pt i on of t he opt i cal measur ement . I ndeed,

t he s t r es ses i n t he opt i c al l y sens i t i ve mat er i al of a par t i cl eassembl y ar e l ar ger t han i f t he same space wer e f i l l ed wi t h ahomogeneous opt i cal l y sensi t i ve mat er i al . However , because t het hi ckness of t he model i ncl udes al so t he por e vol ume, t he


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F i g. 2: 17 Vec t or i al addi t i on of t wo Mohr c i r c l es .r el at i on bet ween t he r et ar dat i on S and (a -a ) i n eq. 2: 13 i s i n

1 2 ^agr eement wi t h t he def i ni t i on of s t r ess i n a gr anul ar mat er i al .Mor eover , t he l i near r el at i on bet ween t he par amet er s i n eq. 2: 13demonst r at es t hat t he opt i cal const ant f or an assembl y of cr ushedmat er i al i s t heor et i cal l y t he same as f or a homogeneous mat er i al .I n t est s i t was at t empt ed t o appr oach t he pl ane st r esscondi t i on as much as poss i bl e. However , t he m cr o st r esses i nt he par t i c l es cannot be pr event ed f r om devi at i ng f r om t he pl anest at e of s t r ess . Thi s i s not i n agr eement wi t h t he t wo-di mensi onal s t r ess opt i cs . Al t hough t her e i s no mat hemat i cal orexper i ment al ev i dence, i t i s pr esumed t hat t he opt i cal l ydet er m ned st r ess component s descr i be a t wo- di mensi onal st r essst at e i n t he pl ane of t he sampl e, whi ch i s i n agr eement wi t h t heaveraged stress.Cont r ar y t o t he assumpt i ons made f or t he t heor et i cal anal ysi sof t he opt i cal f i l t er s ys t em t he s t r es s di s t r i but i on i s nothomogeneous over t he sur f ace and t hi ckness of a mat er i al poi ntconsi der ed. Si nce t he vol umet r i c aver aged st r ess i s obt ai ned byvect or i al addi t i on of t he component s of t he m cr o s t r esses , t heopt i cal measur ement i s r equi r ed to pr oduce a s i m l ar r esul t . Duet o t hi s r equi r ement sever al combi nat i ons of opt i cal f i l t er s ar enot s ui t abl e f or t hi s appl i c at i on. A c i r cul ar pol ar i s cope(F roch t , 1946) , f or exampl e, adds t he pr i nc i pal s t r ess


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di f f er enc e s ov er t he s ur f ac e a s s c a l a r s , whi l e i n t he c a s e ofpo l a r i s e r s on b ot h s i des of t he model r ot at i ng s y nc hr onous l y( e. g. Re dn er , 1976) , . equal s t r es s e s i nc l i ned at 45 a r es uppr es s e d. I t i s demons t r at ed be l o w t hat t he f i l t er s y s t em us edi n t hi s r es e ar c h a dds t he s t r es s c o mpo nent s al mos t v ec t o r i a l l y .An addi t i onal adv ant age of t hi s f i l t er s y s t em over ot her s y s t e msi s t hat a s e pa r at i o n i s obt ai ned be t we en t he di r ec t i o ns of aand a . I n par t i c ul ar , t hi s i s c onveni ent i n s ampl es wi t h ac ompl ex st r es s di s t r i but i on.2. 3. 6 Opt i c a l a v er a gi n g i nho mo ge ne ous s t r e s s

T he o pt i c a l l y me as u r e d c o mp on en t s of t he s t r e s s t ens o r a r e t hel engt h S of a v ec t o r and i t s o r i e nt a t i o n 2tJ ). I n a Mo hr d i a gr a mt hi s v ec t o r i s t he l i ne s e gment be t we en t he po l e a nd t he c e nt r eof t he Mo hr c i r c l e . T he addi t i on of s t r es s t ens o r s c an beper f or med by v ec t o r i a l a ddi t i o n of al l t he l i ne s egment s of t heMo hr d i a gr a ms . An e xa mp l e of a t wo - d i me ns i o na l s t a t e of s t r es si s s hown i n F i g . 2: 17. T he o pt i c a l l y me as u r a bl e pa r a met er s S , ijja nd S' , i j )' r epr es ent t wo r egi ons wi t h a d i f f er ent s t at e ofs t r e s s , and S" , i ))" i s t he r es ul t of t he vec t or i a l addi t i on. Themat hemat i c al r el at i on i s

6" = J t S2+S' Z+2SS' c os 2( i j j - U> ) D ( 2: 27a)\|> = i - arcos[i j - rrcos2i j ) --| 7rcos2tJ )' ] ( 2: 27 b)

T he o pt i c a l a dd i t i o n of t he t e ns o r c o mp on ent s c a n be s e pa r a t e di nt o an a ddi t i o n o ver t he t hi c k ne s s of a t es t mo del a nd ana ddi t i on ov er t he cr os s - s ec t i o n of t he l i ght bea m.

The opt i c al addi t i o n over t he s ur f ac e c an be i nv es t i gat ed wi t he q. 2: 26. I f t he s ur f ac e of a mat er i a l poi nt i s di v i ded i nt o t woequal par t s wi t h di f f er ent s t r es s e s eq. 2: 26 bec o mes

I _ = I c i s i nSs i n2( p- i | ) ) +| - si nS ' s i n2( p- \ |> ) +13 ( 2 : 28)


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The maxi mum and t he m ni mum val ue of I and t he val ues of 6 t omat ch can be f ound numer i cal l y. The val ues f or S" and t|j" can nowbe cal cul at ed f r om eq. 2: 23 and eq. 2: 24 and ver i f i ed wi t h t her esul t s of eq. 2: 27. Some numer i cal exampl es of t he vect or i al andopt i cal addi t i on ar e gi ven i n Tabl e. 2: 1.

t wo r egi ons wi t hdi f f er ent s t r es s* i000000

s1020252020

6'10401252020

* ;01045454530

opt i c al addi t i onsur faceS"2058. 112. 17. 128. 034. 5

* ;06. 740232315

t hi cknessS"2059. 112. 37. 128. 034. 5

< ) > ; 06. 040232114

vect or i aladdi t i onS"2059. 212. 27. 128. 334. 6

* ;06. 740. 322. 522. 515Tabl e 2: 1. Compar i son of opt i cal and vect or i al addi t i on ofopt i cal l y measur abl e st r ess component s.

I t appear s t hat t he opt i cal addi t i on i s i n r easonabl e agr eementwi t h t he vect or i al addi t i on. The er r or i s a maxi mumi f l ^ i j r i = 45 and i ncr eases i f S+S' or | S- S' | i ncr eases .Al s o, i t can be deduced f r om eq. 2: 28 t hat t he aver age l i ghti nt ens i t y dur i ng t he r ot at i on of t he anal yser i s i ndependent oft he s t r ess di s t r i but i on i n t he measur i ng r egi on. The sum of t hel i ght i nt ens i t y, over a r ot at i on of 180 of t he anal yser , can beder i ved wi t h

r i >E I Q = < I s i n S s i n 2 ( 8 - i | ) )+I s i n S ' s i n 2 ( B - i | ) ' ) + I ) d B ( 2 : 2 9 )P 0 J 2 o r ^ l 2 o ^ T i o ^Si nce 2P r anges f r om 0 t o 2 I T , t he f i r s t t wo t er ms do notcont r i but e t o t he summat i on. The cont r i but i on of t he l ast t er mcor r esponds wi t h 1801 . The aver aged l i ght i nt ensi t y dur i ngr ot at i on i s = I , so t hat I can be der i ved f r om t he aver agedl i ght i nt ens i t y al so i f t he s t r ess di s t r i but i on i s i nhomogeneous


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2 \ d 2 HNHLYSER

F i g. 2: 18 Behavi our of l i ght t r ansm t t ed t hr ough t wo l ayer s ofdoubl e- r ef r ac t i ve mat er i al wi t h di f f er ent s t r es s es .over t he cr oss sect i on of t he l i ght beam

The opt i cal aver agi ng of t he st r ess component s over t het hi ckness i s i ndependent of t he f i l t er sys tem used. Toi nvest i gat e t he agr eement bet ween t he vect or i al and opt i caladdi t i on, i t i s assumed t hat t he l i ght descr i bed by eq. 2: 15 i st r ansm t t ed t hr ough a second l ayer of doubl e- r ef r act i ve mat er i alwi t h a di f f er ent r et ar dat i on and pr i nc i pal s t r ess di r ec t i on. Thet wo per pendi cul ar waves of eq. 2: 15 ar e t r ansf or med i nt oa' = a cosot - a si na2 2 1a.' = a cosa+a s i na1 1 2

wer e a' and a' ar e t he vi br at i ons i n t he a' and a' di r ec t i on of1 2 1 2t he second l ayer , r espect i vel y, and a. i s t he i ncl i nat i onbet ween o and a' ( Fi g. 2: 18) . When t he l i ght l eaves t he secondl ayer an addi t i onal r et ar dat i on S' has t aken pl ace. Subs t i t ut i ngt he f unct i ons of a and a , eq. 2: 30 becomes .1 2 ^a = 2A 2Ccosasi n( wt +S+S' ) - si nacosCut +S' ) D ( 2: 31a)

( 2: 30 )( 2 : 3 0b)


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a' = ~A4~2Ccosotcos(( j t )+si n


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LI GHT SOURCELENSI MRGE 1 1>

MRRK-CRUSHEDG LHSS

\

F i g. 2: 19 Di agr am of t he opt i cal sys t em t o det ec t a bl ack mar ki n an assembl y of cr ushed gl ass.par t i c l es whi ch ar e s i t uat ed i n a pl ane hal f way t hr ough t het hi ckness of a sampl e. To el i m nat e t he phot ogr aphi c st ep adi gi t al camer a i s used t o det er m ne t he co- or di nat es of t he mar ksdi r ect l y i n t he pl ane sampl e. S i nce t he ar ea cons i der ed and t henumber of mar ks ar e r at her l ar ge ( e. g. 900 cm wi t h 40 t o 100mar ks ) i t i s not poss i bl e t o anal yse t he t ot al a r ea accur at el yenough al l at once. Ther ef or e t he pos i t i on of t he di gi t al camer ai s cont r ol l ed by an accur at e x- y scanner , so t hat t he ar ea of t hesampl e can be di vi ded i nt o smal l squar e el ement s whi ch can beanal ysed consecut i vel y .The most i mpor t ant par t of t he camer a i s an i nt egr at ed ci r cui twi t h a wi ndow of about 6 mm squar e whi ch cont ai ns a mat r i x of128x128 l i ght - sens i t i ve pi xel s ( di odes ) . El ec t r oni c s i n t hecamer a conver t s t he l i ght i nt ens i t y, r ecei ved by each pi xel , i nt oa vol t age whi ch i s connect ed t o an out put l i ne f or a shor t t i me.The vol t age of t he pi xel s can be r ead by a comput er . Asynchr oni sat i on s i gnal , whi ch i s al so pr oduced by t he camer a i sused t o separ at e t he s i gnal s of t he pi xel s . I f t he i mage of ar egi on wi t h a bl ack mar k, whi ch has a di amet er of about 2 mm i sf ocused one- t o- one on t he wi ndow of t he sensor ( Fi g. 2: 19) , anumber of pi xel s wi l l r ecei ve a l ow l i ght l evel . The out putvol t age of a pi xel i s conver t ed i nt o bl ack ( 1) or whi t e ( 0 ) ,dependi ng on whet her t he vol t age i s r espect i vel y l ower or hi ghert han a r ef er ence val ue. The s t at us of each pi xel i s r ead l i ne byl i ne and i s st or ed sequent i al l y i n t he comput er memor y. The


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COLUMN

127

ROM

127

o0o128

o10129

o2130

o127255

o16383

Fi g. 2: 20 Rel at i on bet ween t he memor y l ocat i ons and t he r ow andcol umn number of a pi ct ur e el ement .r el at i on bet ween t he r ow and t he col umn number and t he act ualmemor y l ocat i on i s r epr esent ed i n Fi g. 2: 20.2. 4. 2 Det er m nat i on of cent r e mar k

The pos i t i on of t he cent r e of t he i mage of a bl ack par t i c l e ,wi t h r espect t o t he upper l ef t cor ner of t he pi xel mat r i x can beder i ved f r om t he r ow and t he col umn number of t he dar k pi xel s.I f t he number of t he memor y l ocat i on of t he f i r s t pi xel i sassumed t o be zer o, t he co- or di nat es of a pi xel ( xcor r esponds wi t h t he memor y l ocat i on n can be cal cul at ed f r omy ) w h i c hxP = i n t S ) sy = n s - k y ' p J P

( 2 : 3 4 a )( 2 : 3 4 b )

wer e k i s t he number of p i xel s i n a r ow and s i s t he di st ancebet ween t he pi xel s. The cent r e of t he i mage of a bl ack mar k( x , y ) can be f ound by addi ng t oget her t he x and y co-


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or di nat es of al l t he bl ack pi xel s and di vi di ng t he r esul t s by t het ot al amount t as f or mul at ed byx = ^Z x ( 2 : 3 5a)m t py = i ry ( 2: 35b)-'m t Jp

The absol ut e posi t i on of a mar k can be cal cul at ed f r om t he knownco- or di nat es of t he upper l ef t - hand cor ner of t he pi xel mat r i xwi t h r espect t o t he or i gi n of t he scanner .To pr event er r or s i n the i nt er pr et at i on of t he i mage of a mar ksever al cont r ol par amet er s ar e used, such as- t he number of dar k pi xel s of t he i mage of a mar k;- t he number of dar k pi xel s at t he boundar y of t he pi xel mat r i x;- t he measur ed di spl acement of a mar k.The number of dar k pi xel s whi ch i s cover ed by a mar k can beused t o det er m ne t he sur f ace of t he bl ack i mage, so t hat t he

comput er can di s t i ngui sh a mar k f r om a smal l acci dent al di r t yspot. The number of bl ack pi xel s at t he boundar y of t he pi xelmat r i x i ndi cat es i f a mar k i s c l ose t o a boundar y segment of t hemodel . I f t he measur ed di spl acement of a mar k af t er adef or mat i on st ep i s unr easonabl y l ar ge, e. g. i n compar i son wi t hmeasur ed di spl acement s of boundar y segment s , i t can be concl udedt hat t her e i s somet hi ng wr ong.Because t he whol e i mage of a mar k i s used t o det er m ne t he

cent re, , t he accur acy i s not s i gni f i cant l y dependent on t he shapeof t he mar k. However , t he assembl y of cr ushed gl ass i s notper f ec t l y t r anspar ent due t o smal l devi at i ons i n t he r ef r ac t i oni ndex and di r t , so t hat t he i mage of a mar k i s of t en accompani edby mor e or l ess r andom y di st r i but ed bl ack spot s ( noi s e) . Toi ncr ease t he accur acy a pr ocedur e has been devel oped t o el i m nat et he noi se bef or e t he co- or di nat es of a mar k ar e det er m neddef i ni t i vel y.


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b

O m~

OFi g. 2: 21 Exampl e of an i mage of a mar k and noi se on t he pi xelmat r i x of t he i mage sensor .

2. 4. 3 El i m nat i on of noi seA machi ne l anguage r out i ne has been devel oped t o el i m nat enoi se i n an i mage of a bl ack mar k. An exampl e of an i mage i sr epr esent ed i n F i g. 2: 21. The mar k i s l ocat ed by a c i r c l e, whi l esome i r r egul ar i ndi cat ed r egi ons ar e supposed t o be bl ack f i el dsdue t o di r t i n t he assembl y. Because t he sur f ace of t he mar kdom nat es and t he noi se i s di st r i but ed mor e or l ess at r andomt he mai n poi nt of t he bl ack pi xel s i s al ways s i t uat ed wi t hi n t heboundar y of t he i mage of t he mar k. I n Fi g. 2: 21 t he pr ovi si onalcent r e of t he mar k i s i ndi cat ed by t he i nt er sect i on poi nt of t hel i nes aa' and bb' . To c l ean t he i mage t he pr ogr am st ar t s wi t h api xel i n t he f i r s t r ow whi ch i s l ocat ed on l i ne aa' . The pi xelnumber i s r educed unt i l t he l ef t - hand par t of t he boundar y of t hemat r i x i s r eached. The cont ent s of t he cor r espondi ng memor yl ocat i ons ar e checked f or bl ack or whi t e. I f a whi t e pi xel hasbeen obser ved once, al l t he f ol l owi ng pi xel s i n t hat segment of

t he r ow ar e al so set t o whi t e. Next , t he r i ght - hand segment oft he r ow i s pr ocessed on t he same way. Thi s pr ocedur e i s r epeat ed


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f or al l t he r ows , and t he col umns ar e si m l ar l y pr ocessed. Ther esul t i s t hat al l t he bl ack spot s whi ch ar e not connect ed wi t ht he bl ack r egi on of t he mar k ar e el i m nat ed, so t hat t he cent r eof t he mar k can now be det er m ned def i ni t i vel y.2. 5 DATA PROCESSI NG2. 5. 1 Col l ec t ed dat a

Dur i ng t he measur ement of t he par amet er s whi ch descr i be t hecondi t i on i n t he gr anul ar mat er i al of a t est model a cons i der abl eamount of dat a i s pr oduced. Si nce t he di gi t i sed condi t i ons ofsever al t est s ar e r equi r ed t o anal yse t he s t r ess s t r ai n behavi ourof t he gr anul ar mat er i al , i t i s necessar y that t he dat a of eachst age can be r ead easi l y by a comput er . Ther ef or e t he measur eddat a of t he di f f er ent s t ages i s s t or ed i n f i l es on a di s k. Thef i l e name i ncl udes t he name of t he t est and a number whi chr epr esent s t he sequence. The f ol l owi ng dat a ar e s t or ed i n a f i l e- name, r ef er ence number of t he measur ement , dat e, t i me, numberof s t r ai n gauges , t hi ckness of t he sampl e;- co- or di nat es of t he angul ar poi nt s of t he boundar y , t he l ocat i on of s t r ai n gauges ; .- number of r ows and col umns of t he nodal poi nt s at whi ch t hest r ess component s ar e measur ed, t he di st ance bet ween t he nodalpoi nt s , and t he co- or di nat es of t he f i r s t nodal poi nt of t hemes h;- number of r ows and col umns of bl ack mar ks i n t he f i el d;- number of mar ks at t he boundar y;- a mat r i x c ont ai ni ng t he r el at i ve pr i nc i pal s t r es s di f f er enc es ;- a mat r i x cont ai ni ng t he pr i nc i pal s t r es s di r ec t i ons ;- a mat r i x cont ai ni ng t he co- or di nat es of t he bl ack mar ks ; ,- t he out put of t he s t r ai n gauges .

The dat a i s col l ect ed by a s i ngl e- boar d m cr ocomput er whi chcont r ol s t he measur i ng devi ce ( Sect i on 3 . 4 . 1 ) . The di gi t s ar et r ansm t t ed t o a mor e power f ul comput er , whi ch st or es t he dat aand gr aphi cal l y di spl ays t he el ement ar y par amet er s of t he


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Fi g. 2: 22 a) Gr aphi cal r epr esent at i on of t he di s t r i but i on oft he pr i nci pal st r ess di r ect i ons, based on measur ement s at 100di s cr et e poi nt s , b ) Vi s i bl e pat t er n of l i ght s t r i pes , v i ewed wi t ha ci r cul ar pol ar i scope.s t resses , di spl acement s and st r ai n gauges gr aphi cal l y. The cour seof a t est can be conveni ent l y moni t or ed i n t hi s way. I n a l at erst age t he dat a of one or mor e t est s ar e subj ect ed t o a mor edet ai l ed anal yses .2. 5. 2 Pr i nci pal s t r es s t r aj ect or i es

Si nc e t he di gi t al , poi nt i nf or mat i on of t he pr i nc i pal s t r es sdi r ec t i ons i s not ver y conveni ent f or vi sual i nspec t i on, acomput er pr ogr am has been devel oped whi ch conver t s t he di scr et emeasur ement s i nt o a cont i nuous r egul ar pat t er n. The pat t er n i sf or med by cur ved l i nes , pr i nc i pal s t r ess t r aj ec t or i es , of whi cht he t angent at any par t i cul ar poi nt r epr esent s t he pr i nc i palst r ess di r ect i on at t hat poi nt of t he sampl e. A comput er pl ot ofsuch a pat t er n i s shown i n F i g. 2: 22a. The pl ot t i ng pr ocedur e i s


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s t ar t ed i n poi nt s at ' t he boundar y. The di st ance bet ween t hes t ar t i ng poi nt s det er m nes t he concent r at i on of t he t r aj ec t or i es .The poi nt s i n t he f i el d of a t r aj ect or y ar e t he end poi nt s ofsmal l l i ne segment s, whi ch ar e t he chor ds of ci r cl e segment sdescr i bi ng t he l ocal cur ve. The di r ect i on of t he chor d i s f oundby means of an i t er at i on pr ocedur e. Due t o t hi s pr ocedur e t heshape of a t r aj ect or y i s not dependent of t he pl ot t i ng di r ect i on.A t r aj ect or y i s assumed t o have been compl et ed when t he boundar yi s r eached agai n. To pr event a heavy l ocal concent r at i on oft r aj ec t or i es due t o conver gence, t he co- or di nat es of t he s tar t i ngpoi nt and end poi nt of each t r aj ect or y ar e r emember ed. A newt r aj ec t or y i s pl ot t ed onl y i f i t s end i s not t oo c l ose t o one oft he pr evi ous st ar t i ng or end po i nt s . When a pi ct ur e i s al mostcompl et e, al l t he di s t ances bet ween t he i nt er sect i on poi nt s oft he t r aj ect or i es and boundar y ar e checked. I f a t oo l ar ge adi st ance i s f ound, some concessi ons ar e made wi t h r egar d t o t hem ni mum di st ance. The choi ce wet her a concessi on i s made or noti s dependent on a f ew par amet er s. The val ue of t he par amet er s,whi ch depends on t he shape of t he model and t he l oadi ng pr ogr amhave t o be changed manual l y.I n t he f i r s t i ns tance t he maj or pr i nc i pal s t r es s t r aj ec tor i es ar epl ot t ed. The s t ar t i ng poi nt of t he l onges t t r aj ect or y i sr emember ed, because t he poi nt s of t hi s t r aj ect or y ar e used ass t ar t i ng poi nt s f or t he m nor pr i nc i pal s t r es s t r aj ec t or i es . Theadvant age of t hi s pr ocedur e i s t hat a ver y r egul ar di s t r i but i onof t r aj ect or i es i s obt ai ned. Si nce not a l l boundar y segment s ar er eached i n al l cases i n t hi s way, a r out i ne i s used t o f i nd andf i l l l a r g e gaps. I f a pi c t ur e i s not accept abl e, an opt i on can beac t i vat ed t o def i ne s t ar t poi nt s of ext r a t r aj ec t or i es manual l y.Manual hel p , however , i s i n gener al not necessar y.Because t he pr i nci pal st r ess di r ect i ons wer e measur ed atdi scr et e po i nt s , an i nt er pol at i on pr ocedur e had t o be used t odet er m ne t he di r ec t i on at an ar bi t r ar y poi nt . A s i mpl e l i nearpr ocedur e i s used, whi ch pr oceeds as f ol l ows ( F i g. 2: 23)


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x i-y i * " x 2.y2) v ( 2 : 3 8a)xx 2 2 l z x Ta = 4( 0 +o- +( o - o )cos2i | )) ( 2: 38 )yy 2 2 l 2 l ^a = a =- ( or - o )s i n2t| ) ( 2 : 3 8c ) )xy yx 2 2 l T

Subs t i t ut i ng eq. 2: 38 i nt o eq. 2: 37 gi ves| ~( ( a +a - ( a - a ) cos2i | ) ) ~f - ( i ( a - o )si n2i j ) ) = 0 ( 2 : 39a)oX Z 2 1 2 1 dy Z 2 1| - ( i ( cr +a +( a -a )cos2x| ) ) - f - ( ( a - a ) si n2i | ) ) = 0 ( 2: 39b)dy Z 2 1 2 1 dX Z 2 1Di f f er ent i at i ng eq. 2: 39 (a l l t er ms ar e f unc t i ons of x and y ) and

r ot at i ng t he x- axi s i n t he di r ec t i on of a ( i | )=0) t he equi l i br i umequat i ons r educe t o-Ji-(a -a )|f- = 0 (2:40a)dt 2 1 dt1 2

2 1wer e t and t ar e or t hogonal cur v i l i near co- o r di nat es whi chc oi nc i de wi t h t he pr i nc i pal s t r es s t r aj ec t or i es . The equat i onsder i ved, whi ch ar e known as t he Lam- Maxwel l equat i ons ofequi l i br i um can be used t o ca l cul a t e t he i nc r ement of a or aal ong a pr i nc i pal s t r es s t r aj ec t or y, i f ( o' 2" 0r1) a n d ^ a r e knownat each poi nt of a t es t mode l . Si nce t he pr i nc i pa l s t r essdi f f er ence and di r ec t i on have t o be appr oxi mat ed by i nt er po l a t i on( f ol l owi ng t he pr ocedur e of eq. 2: 36) , t he i nt egr at i on of eq. 2: 40has t o be per f or med numer i ca l l y . An i nt egr at i on s t ep a l onga - c o - o r di na t e i s s hown gr aphi c al l y i n F i g. 2: 24. Thet r apez oi dal i nt egr at i on r ul e gi v es


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c ^ -0

" , )F i g. 2: 24 I nt egr at i on s t ep al ong a a - t r aj ec t or y.

F i g . 2 : 2 5 S i g n c o n v e n t i o n f o r dij>/dt ( = 1 / R ) .Aa, X{K-%>r)A*K-vU7)B} ( 2: 41)

I f At i s al ways t aken as pos i t i ve, t he s i gn of di |>dt det er m nest he s i gn of Aa ( a i s negat i ve f or pressu re ) . The s i gnof di j i /dt depends on t he cur ve of t he t r aj ect or y, whi ch i s showngr aphi cal l y i n F i g. 2: 25. The i ncrement of a al ong a t - co or di nat e can be s i m l ar l y cal cul at ed.For t he cal cul at i on of t he st r ess t ensor at an ar bi t r ar y poi ntof a model , t he r eal val ue f or a or a at some poi nt and a


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Fi g. 2: 26 Mohr di agr am of dat a measur ed at a mat er i al poi ntcl ose at t he boundar y.mul t i pl i cat i on f act or M ( i n eq. 2: 14) have t o be known. Thesepar amet er s can be der i ved i f , bes i des t he opt i cal l y measur edpar amet er s 8 and i|), t wo nor mal st r esses on a pl ane ar e al soknown. I n F i g. 2: 26 a poi nt on a boundar y i s shown at whi ch t henor mal st r ess a , i t s di r ect i on p, t he pr i nci pal s t r es sdi r ect i on ij) and t he r el at i ve pr i nc i pal s t r ess di f f er ence S ar eknown. I t i s shown gr aphi cal l y t hat t he pol e P and t he or i gi n oft he Mohr di agr am can be det er m ned f r om t hese dat a. Thepr i nc i pal st r esses at t he poi nt consi der ed can be expr essed i nM, 5, a , 4) and 0 wi t h

o = a - MS( cos2( B- i | ) ) +l ) = or -l n 2 Ti n MT ( 2 : 4 2a)a = or +MS( cos2( B- i l ) ) - l ) = a +MT2 n & l n 2 ( 2 : 4 2b)

To det er m ne t he mul t i pl i cat i on f ac t or M, t he nor mal s t r ess at asecond poi nt i s used ( Fi g. 2: 27) . The r el at i ve i ncr ement of af r om A t o B can be cal cul at ed usi ng eq. 2: 41 and i s ass umed t o beAa = MA ; t hus ( o )_= ( a, ) . +MAl 1 XJ 1 A ( 2: 43)

W t h eq. 2: 42 t hi s gi vesM = ( V B-


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Fi g. 2: 27 Dat a used t o cal cul at e M, a and a .t ensor at an ar bi t r ar y mat er i al poi nt of t he model , ei t her poi ntA or poi nt B can be used as st ar t i ng poi nt f or t he numer i cali nt egr at i on wi t h eq. 2: 41.The cal cul at i on pr ocedur e i s used i n a comput er pr ogr am t oder i ve st r ess maps f r om t he measur ed dat a. Si nce t he boundar y ofa model i s i n gener al not compl et el y cover ed by t he mesh of t hemeasur i ng poi nt s , t he f i r s t ac t i on i n t he pr ogr am i s t o expandt he mat r i x whi ch cont ai ns t he f i el d i nf or mat i on. The dat a i n t he 'new nodal poi nt s ar e set t o t he same val ue as t he near est poi nt sof t he or i gi nal mat r i x. Ever y t i me a pai r of r ows and col umns i sadded, i t i s checked i f t he boundar y of t he model i s l ocat edwi t hi n t he new boundar y of t he mesh. I f t he mat r i x i s expandeds uf f i c i ent l y, t he val ue of t he mul t i pl i c at i on f ac t or M i sdet er m ned, us i ng al l t he poss i bl e combi nat i ons of poi nt s at t heboundar y wher e t he nor mal st r ess i s measur ed. I f t oo l ar gedi f f er ences ar e f ound, e. g. due t o a f aul t y l oad c e l l , t hepr ogr am as ks f or manual hel p. I f t he mul t i pl i cat i on f act or hasbeen det er m ned, t he absol ut e val ues of t he pr i nc i pal s t r essdi f f er ences can be cal cul at ed. An exampl e of t he di s t r i but i on


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Fi g. 2: 28 Di s t r i but i on of t he pr i nc i pal s t r es s di f f er enc e,based on 100 measur i ng poi nt s .

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Fi g. 2: 29 Cal cul at ed s t r ess di s t r i but i on i n a sampl e, us i ngopt i cal l y measur ed dat a (1 scal e = 100 kN/ m or 1 cm) .


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of {a -a ) i n a t es t i s s hown gr aphi cal l y i n F i g. 2: 28.Next , t he st at e of st r ess at t he cent r e of t he sampl e can bedet er m ned by cal cul at i ng t he i ncr ement of a or a f r om t hepoi nt s at t he boundar y, wi t h known nor mal s t r ess , t o t he cent r e.I f t he cal cul at ed st r ess component s ar e i n good agr eement wi t heach ot her , aver aged val ues ar e used t o set t he par amet er s whi chdescr i be t he s t r ess s t at e at t he cent r e. For f ur t hercal cul at i ons t he cent r e of t he model i s used as t he s t ar t i ngpoi nt f or t he numer i cal i nt egr at i on pr ocedur e. I n t he f i r s ti nst ance t he di st r i but i on of t he nor mal s t r esses at t he boundar yar e cal cul at ed. Some of t hese st r esses can be compar ed wi t h t hemeasur ed nor mal s t r esses t o i l l ust r at e t he accur acy of t he st r essmap obt ai ned. I n Fi g. 2: 29 t he measur ed nor mal st r esses ar esymbol i sed by ar r ows. Sever al ot her par amet er s can be cal cul at edt o vi sual i se t he s t r ess condi t i on of t he model . I n F i g. 2: 29 t het wo- di mensi onal s t at e of s t r ess at some mat er i al poi nt s i sder i ved, and t he di s t r i but i on of t he mobi l i sed f r i c t i on angl e d>

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