An analysis of a non-planar crack under mixed mode I+III loading using infinitesimal dislocations with edge and screw average characters

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1 A lyss of o-pl k ude med mode IIII lodg usg ftesml dslotos wth edge d sew vege htes.n.b. ANONGBA U.F.R. Sees des Stutues de l tèe et de Tehologe Uvesté de oody B 58 Abdj ôte d Ivoe E-ml ddess: ogb@yhoo.f Abstt Ths ppe vestgtes the med mode IIII lodg of o-pl k tht flututes bout vege ftue ple ftely eteded sotop elst medum. The k s otuous y of ftely log o-stght dslotos wth ftesml Buges vetos. The dslotos e pepedul to the deto of ftue popgto hve bty peod smll shpe spedg the ple d the potos my be btly led wth espet to the deto. The dsloto dstbuto ots two types of dsloto: dslotos hve edge vege hte d espod to mode I lodg; dslotos hve sew vege hte d espod to mode III. Dsplemet d stess felds of the dslotos e fst gve. Epessos fo the stess bout the k tp stess testy ftos d k eteso foe G pe ut legth of the k fot e lso gve. The two spel ks hvg susodl d segmeted fot e osdeed: fomule fo vege k eteso foe < G > e povded d odtos ude whh < G > s mmum e estblshed. These odtos e the ofoted wth epemetl mesuemets of k-fot twst gle vesus ppled stess d foud to ofom to stutos whh o-pl ks e fvoubly obseved. eywods: k popgto d est; Eegy elese te; Dslotos; k mehs; Eegy methods. Itoduto I mtel otg tlly pl k subsequet ftue ude med mode IIII lodg geelly ous o o-pl sufe tht flututes bout the tl k ple. Ths obsevto s deved fom umeous epemets pefomed ude vous dffeet odtos. The boke sufe ehbts the followg fetues Fg. glss by Somme ] polymethyl methylte A by ooke d olld ] steel by Ytes d lle ] d Houle d eu 4] mog othes: Fo smll / 6% whee d e the ppled teso d she espetvely see Fg. fets B e lmost vetl φ π / d fets A hozotl φ A ; lto gle φ A of fets A eses ppetly wth the eteso of the k Somme ]; see lso Lw 5]. φ A eses fom to 45 ppomtely s eses fom zeo to fty - 4]. Fg. epodued fom ooke d olld ] shows ths ted. We stess tht the shpe of the k fot my dffe somewht fom the shemt oe show Fg.. my devte fom stght les fo ste. I the peset study we shll peset geel tetmet by epdg to Foue sees. The study of ftue med mode IIII solds eques o-pl k model tht povdes epessos fo physl quttes petet to dsuss k popgto. Fo ths pupose elevt qutty s the k eteso foe pe ut legth of the k fot o eegy elese te G. B

2 Se ftue poeeds though the moto of mosop legth of the k fot t ppes eessy to lulte vege < G > d look fo elto betwee φ A d φ B tht mmzes < G >. Ths elto my the be ofoted wth epemets s Fg.. A umbe of theoetl lyses devoted to o-pl ks hve bee publshed. Woks by Go 6] Xu et l. 7] d ovh et l. 8] efe to petubto methods whee effot s oetted upo povdg stess testy fto SIF fomule tht e gve to fst ode petubto. SIF epessos e the elted to G by the usul ple st elto. Ude suh odtos vegg G s pefomed the peset study Seto 4 smply edues the esult to tht of the pl k; ths evdetly does ot deped o the o-pl k elevt pmetes. Sml woks povdg fst ode SIF d volvg symptot lyss fo thee-dmesol elstty poblems hve lso bee publshed Bll d Llde 9]; Lzus et l. ]. Bll d Llde 9] wee oeed wth the stblty of mode I ks. ed mode lodg ws ot ttble. Lzus et l. ] ttempted to epl the ese of φ A wth k legth s mesued by Somme ]. They povded ppomte fomul fo G t pot A s defed Fg. to seod ode pmete deoted d γ / d by them tht s mesue of the devtve of φ A wth espet to k legth. The obted elto oveestmtes tul otto tes by ely odes of mgtude ]. Howeve they obted quet good vlue of the globl otto te by moe omple teo bsed o the mmzto of < G > d SIF epessos but oly fo thee o fou pot bedg epemets. I shot estg theoetl woks o o-pl k gowth e seously lmted whe ppled to med mode IIII lodg of mtels; possble lmttos e poted out the dsusso Seto 5. Sevel theoetl te hve bee poposed ttempt to epl the depedee of φ A o the to of the ppled stesses ]: these e bsed o the k-tp stess feld of pl k subjeted to med mode IIII lodg. They ll led to the sme equto fo modete ]; smple eplto s tht pet ks gow fom the pet k fot t twst gleφ A fom the ogl k ple fo whh they suffe o she stesses; φ t A. / edted gles of twst deped o osso s to d vy fom to 45 fo gg fom to fty. Ufotutely s be see fom Fg. mesued gles fll mkedly below theoetl pedto. The m of the peset study s to desbe model of o-pl k ude med mode IIII lodg povde epessos fo stess d k eteso foe G log the k fot pefom vege < G > of G estblsh elto lkg φ A d φ B tht mmzes < G > d flly ofot the popetes of the model wth epemet. The k ossts of otuous dstbuto of ftely log type d dslotos wth edge d sew vege htes. The dslotos e pepedul to the deto of ftue popgto. As fo the k fot Fg. the shpe of the dslotos the ple s peodl d flututes bout vege ftue ple. Elst felds of the two types of dsloto e gve Setos d. I Seto 4 lyss of the o-pl k s fst peseted; the ptul ks wth susodl d segmeted fots e osdeed. Seto 5 ofots the popetes of the model wth pevous woks d epemets d oludg emks e lsted Seto 6. A tes system thoughout.. Dsplemet d stess felds due to dsloto of edge vege hte.. lst dstoto s used

3 We osde dsloto t the og wth Buges veto b lyg deftely the deto d spedg the ple the fom of Foue sees s. os Hee s postve tege; wve umbe d d e mpltudes. We ssume to be smll d epess the plst dstoto β to fst ode ; ths gves j β b H b d the othe ompoets of β e zeo whee d H e the D delt futo d the j Hevsde step futo espetvely. Hee the fst tem s due to the stght edge dsloto. The oespodg dsplemet s kow ]. We shll theefoe oette o the seod tem deoted β. Its Foue fom my be wtte s β k k k β k k k β k k k e dk dk k 4 whee β π / k k k k k k e d e π π π / d β b s dd π / k k k k k k e d e π π π / k d k e el d equl to ± b / π d ged to ed 5 b os dd ; 6 k s tul umbe. d e o zeo oly whe k ± d β β b / π espetvely. The Foue fom of β my be β b π k k k k z e ze dk dk 7 whee z d z... Dsplemet d stess felds

4 The dsplemet u m m due to plst dstoto of the fom k β β k e whee k k k d hs bee obted by u ] to be u j m j l klj mk j k k k L k β k e. 8 Fo sotop mtel L mk k km λ µ k k µ λ µ k k k m 4 λ µ 9 whee k k k k d klj λ µ µ kl j kj l k lj j beg the oeke delt d λ d µ e Lmé ostts. Aodg to 8 k k u k L k k e f β s gve s β k e. I the peset se howeve m l kl mk β s gve by 7. The le theoy of elstty llows fo the supeposto of solutos so tht the oespodg soluto my be wtte s u m b π k k klkllmk k ze k kllmk k ze dk dk l β whh k k k k k k ge to ed d k k k k. Wth 9 d we my u b s os 4π d b u 4 os π whee I s I I k ] k e dk dk π k k 4 the subspt tkg the vlues o s lso 5 below; the tem s opeto tht ts o the fto wth I ;. ] s the th-ode modfed Bessel futo usully so deoted d j s the oeke delt. Flly the totl dsplemet tkes the fom: 4

5 5 b u u os s 4 π ] ] 5 ] s os 4 b u π. 6 u s the dsplemet due to stght edge dsloto: 4 l 8 b b u π π 4 t b b u π π. 7 The stess feld be obted by dffeettg the dsplemet. We fd: os s ] 4 ] 8 4 ] 4 os s 4 ] 8 4 j j j s os ] ] j j j j j j 8 whee subspt d j tke the vlues d d d espetvely / π µ b d j s the stess due to the stght edge dsloto: 4 d. 9 Note tht j j d.

6 6 We dte hee useful obsevto. Fom Eqs. 5 6 d 8 those due to dsloto wth the fom os s my be obted by emovg the symbol 5 6 d 8. ovesely fom the kowledge of the elst felds due to dsloto wth the smple fom s we ve t those oespodg to dsloto wth the moe geel fom smply by ddg to the felds d wtg os s sted of s d s os sted of os.. Dsplemet d stess felds due to dsloto of sew vege hte We osde dsloto wth Buges veto b lyg the deto d spedg the ple the Foue sees fom. The oly o-zeo ompoet of the plst dstoto s wtte to fst ode : b H b β tht s detl to. Hee the fst tem s due to the stght sew dsloto. The oespodg dsplemet s kow see fo ste u ]. The Foue fom of the seod tem s detl to tht 7 of β. To get the totl dsplemet feld we poeed etly s fo the dsloto of edge vege hte Seto. The esult s s os 4 b u π ] ] d os s 4 b u u π ] ] whee the subspt tkes the vlues d d u s the dsplemet due to stght sew dsloto ]: t b u π. The stess feld be obted by dffeettg the dsplemet. We get: s os ] ] s os

7 ] ] s os ] ] s os ] ] 4 whee the subspt tkes the vlues d d dsloto: µ b π Note tht d d. 4. Alyss of the o-pl k 4.. The model j s the stess due to stght sew d. 5 The dslotos wth edge Seto d sew Seto vege htes e ow osdeed to be otuously dstbuted ove the tevl to the ptul se whh the dslotos hve susodl shpe wll be vestgted lte Seto 4.6 d my seve hee to llustte Fg.. The medum s ssumed to be fte sotop d elst d subjeted to ufom ppled teso d she t fty. The dsloto dstbuto futo D fo the edges d fo the sews gves the umbe of dslotos smll tevl d bout s D d. Dslotos d hve Buges veto b d b espetvely d to yoe loted t ug pot beg gve by d oespodg le sese t / / / e ssoted. We e oeed wth the poblem of fdg the equlbum dstbutos of the dslotos ude the ombed to of the mutul epulsos d the foe eeted o them by d. We shll deve the equlbum eltos whh e gve by the odto tht eh of the ftesml dslotos s ted o by zeo totl foe log. We stess hee tht the peset lyss losely follows pevous woks epesetg k s otuous y of dslotos by Blby et l. 4] d Blby d Eshelby 5]. osde dsloto loted t Fg.. The foe df the deto eeted o t t bty pot by the dslotos d loted smll tevl d bout s obted fom the eh d oehle 6] fomul to be df t t D d b 6 7

8 whee d d e the stesses due to dsloto t d t j j d e the ompoets t of the dsloto le sese t. The dstbuto podues the foe t t D d b F. 7 The odto tht the totl foe t be zeo s evdetly t / t D d t / t. 8 Smlly the odto tht the totl foe the t / t D d deto be zeo t o dsloto s. 9 8 d 9 e tegl equtos tht povde the equlbum dstbutos D d D of the dslotos. 8 my be ged dffeet fom: fst epesso fo s etted fom 9; ths fomul s the todued 8 to obt ew elto. The p to solve my be wtte s t / t D d t / t D d. 4.. Dsloto dstbutos Assume fst tht the dslotos e stght pllel to the k the deto. We thus hve pl ple etedg fom to ude med mode IIII lodg. Ude suh odtos t / / see Setos d whee µ b / π d µ b / π ; beomes D d D d whee the uhy ppl vlues of the tegls e to be tke. The soluto s well kow 5]: D D π D D. π 8

9 9 D oespods to the dstbuto of stght edges ude pue mode I lodg d D to stght sews ude pue mode III. The oespodg eltve dsplemets of the k fes the d detos e: / / b π φ / / b π φ. Thus t ppes tht D s ubouded t ± d the φ uve s vetl t these ed pots. We ow tu to the dslotos wth shpe Eq.. To solve we hoose fo oveee o the vege ftue ple. Ths mes tht whee stsfes the equto. osequetly wth the help of the dsloto stess felds Seto d we my wte the vous stesses ppeg s ] sg s os / R ] s os / R s os / ] ] sg R 4 whee / sg d ] R s the bouded pt of the modfed Bessel futo of fst ode ] : ] / ] R. beomes / / / d D ; s os d D f ] sg s os / d D R ; s os / d D g 5 whee ] sg / ; R f ] /

10 sg R ] ] g ; /. 6 Note tht f d g e bouded whe teds to. 5 s system of two tegl equtos wth uhy type sgul keels. The umel esoluto of ths system wth two ukows D d D my be pefomed the sme wy s fo sgle equto wth oe ukow Aogb 7]; Aogb d Vtek 8]. The soluto eds D D N N D α T / whee D α T / < ; 7 / D π s the soluto of the homogeeous equto / D d T e the hebyshev polyomls of fst kd N d the oeffets α umelly usg 5 s desbed by Aogb 7]. oespodg eltve dsplemets d of the fes of the k the d detos e smlly os / b N α s φ π b N α s os π / d e obted φ φ Stesses bout the k fot Whe the equlbum dstbutos D d D of the dslotos hve bee foud Seto 4. the totl stess due to the o-pl k t y pot s j j j j 9 whee j j j D d d ; 4 hee j o s the stess feld podued by dsloto t the og wth Buges veto b o b. j the ppled stess s equl to zeo eept d. We ow desbe the vous stesses the eghbouhood of the k fot t. Fst we shll j detl the lulto fo d the se the poedue s ommo to ll gve the othe stesses. We ssume to be smll d osde the Lu sees epso of 8 to fst ode wth espet to :

11 o 4 whee o s the omplemety pt of the sees. 4 eds s os os ] ] o s R o 4 whee R desgtes omplemety tems tht e bouded. We fd 4 to hve the fom os R o s. 4 We shll detfy below to ts ubouded tems s teds to ; oly the ubouded tems 4 s teds to zeo my otbute. Ths dsds the tem wth R. O the le ppomto of the peset study we lso dsd the tem wth the sgulty / beuse t s popotol to whh we my sy to be smll qutty of ode se we ssume to be smll; ddto ths sgulty does ot led to well-defed vlue of the k eteso foe s lulted Seto 4.5 dvegg tems oug the esult. s gve by the followg epesso oespodg to the tem wth / 4 oly D d. 44 We tegte 44 tkg fo D the stght edge dsloto dstbuto D oespodg to pl k; the wtg the esult of the tegto s << s > d etg tems wth / s oly we ve t I s 45 π s whee s d e bty d The poedue to obt π s the SIF fo the pl k mode I lodg. I s sml to tht desbed bove fo. The esult s

12 whee III s 46 π s eds III π s the SIF fo the pl k mode III lodg. Usg 9 45 d 46 s I III. 47 π s oeedg sml me wth the othe stess ompoets we get s I III π s s I III π s d 48 whee the subspt tkes the vlues d ; g s d e bty s > d s smll. A sgulty of the type s << / s volved the lulto of d / s the fl the step oespodg to 4-44 to whh thee s o sgulty of the type esult; osequetly these stesses e set equl to zeo. Whe gvg the stesses 48 we hve estted ouselves to sgul tems s s teds to zeo oly; ths s beuse y othe otbuto beomes eglgble fo suffetly smll vlue of s. oe pesely we estt ouselves to sgultes of the type s ; ths s the sgulty tht omes to ply the study of pl ks d gves well-defed vlue to the k eteso foe Stess testy ftos We efe to the k fot loted t Fg.. Let e e e be othooml bss ssoted wth the tes oodte system wth og t the ete of ou k system. At bty pot of the k fot osde othooml bss e e e wth the followg defto: e e ; e d e e the oml deted towds the postve -deto d tget deted towds the postve -deto vetos to the k fot. Ths lol oodte system s used below to defe stess testy ftos. The followg elto holds: e e e / e osθ sθ e 49 s os θ θ e whee θ s the gle betwee e d e. We hve

13 / os θ / / s θ. 5 The stess ompoets tsfom fom oe oodte system to the othe s follows: j j j j B B 5 whee subspts d j efe to the lol oodte system tthed to d the mt j B s gve 49. I the geel study of thee dmesol pl ks thee e two wys fo defg SIF by the study of ethe the eltve dsplemet of the fes of the k o the stesses. We shll use the SIF defto bsed o the study of the stesses Bu 9]: π lm / s I π lm / s II π lm / s III 5 whee j e the ompoets of the stess t bty pot o the k sufe hed of the k fot eltvely to e e e. s desgtes the dste betwee d mesued the deto. Net we lulte I II d III suessvely. Wth the help of 5 t follows os s s os θ θ θ θ s os θ θ os s os θ θ θ. 5 The lulto of s j hs bee pefomed Seto 4.. The eltos 5 d 48 ssoted wth the defto 5 the led to / I I III / III III I d II 54

14 t pot log the fot of the o-pl k wth hlf legth. SIF epessos 54 be used to povde fomul fo the k eteso foe see Seto 5. below Eegy osdetos I the followg epesso fo the devtve G of the eegy of the system wth espet to k e s deved. Ths seves to dsuss the tto of k moto. We follow Aogb 7] d the poedue s dpted fom Blby d Eshelby 5]. Allow the ght-hd fot of the o-pl k wth shpe use Fg. to llustte to dve sy gdly fo smplty fom to but pply foes to the feshly fomed sufes to pevet eltve dsplemet; the eegy of the system s ulteed. Now llow these foes to el to zeo so tht the k eteds effetvely fom to. The wok doe by these foes oespods to deese of the eegy of the system whh we shll estmte the eegy of the system ossts of the elst eegy of the medum d the eegy of the lodg mehsm. The elemet ds dld l us pllel to the k fot of the sufe hed of the k fot t pot my be defed by ds γds whee γ s the ut veto pepedul to ds potg to the postve deto. We obt ds / / / ddl. The elevt ompoets of the foe tg o ds e j ds j the deto d j ds j the deto the summto oveto o epeted subspts pples whee j e stesses hed of the shote k; thus the eegy hge j j j j u ssoted wth ds s ds u / ds / whee d e the dffeees dsplemet oss the legtheed k just behd ts tp the d detos. It beomes ow le tht whe the k dves fom to the eegy deese ssoted wth sufe elemet dl s E j j ds u / ds u j j u / u / u d dl. 55 Let G be the devtve of the eegy of the system wth espet to k e. G oespods to the lmtg vlue tke by E / dl s deeses to zeo. Stesses geelly osst of tems tht e ethe bouded o ubouded s teds to ; oly those stess tems tht e sgul my otbute o-zeo vlue to G; the bouded tems ll otbute othg. Usg 48 d defg s d s I III s 56 π s we ve t G lm E / dl / G j 4

15 G 57 whee G d lm /. u d d 58 Epesso 57 gves the vlue of G t bty pot log the fot of the o-pl k wth hlf legth. The lulto of u depeds o the wy the eteso of the ght-hd fot of the k fom to s pefomed. Whe u s obted fom dstbuto of dslotos pepedul to the -deto we mpltly ssume gd k-fot dsplemet. I tht se u my be obted fom the soluto of 5 modfed to llow fo the ft tht the k eteds fom to sted of fom to. Appomte epessos fo G d oespod to pl dstbuto of stght edge d sew dslotos. I tht se Blby d Eshelby 5] hve show tht I G I / E G III G III / E G 59 whee E s Youg s modulus. Adoptg ppomto 59 d defg G s I III G G / G G we obt G / 5. 6 Fo the pl k wth stght fot the deese of the eegy of the system E dvded by the sufe elemet dl s defed s the k eteso foe pe ut edge legth of the k fot see fo emple Blby d Eshelby 5]. I the peset study we shll efe to G 57 s the k eteso foe pe ut legth of the k fot. I Seto 4.6 d 4.7 we gve moe detled despto of G fo two spel ks Susodl k fot Hee ou k model s ssumed to hve susodl fot Fg. gve s / s π 6 whee s / λ d λ π / s the wvelegth. I the followg we epess the edued k eteso foe 6 deoted G fo the susodl k the sptl vege < G > of G defed s 5

16 λ < G > / λ G d d flly odtos ude whh G > hs etemum. Fst we povde the ssoted fomule d the wth the help of plots desbe oespodg behvous d popetes of the model. The edued vlue G 6 of the k eteso foe depeds o the podut tφ ; φ the k-fot lto gle s the ute gle mesued the ple pepedul to the k popgto deto betwee the k fot d the vege ftue ple; G < t φ t os φ t φ os π 7 t φ s π t φ os π s π 5 π. 6 Fo gve vlue d k pofleφ 6 gves the edued k eteso foe t bty pot s o the k fot. I the lulto of < > G the vege vlues of tems wth t φ d t φ odd powes the bkets 6 otbute othg; the two othe volved tegls e gve espetvely by the omplete ellpt tegl of the fst kd d fomule.67 the tble of tegls by Gdshtey d Ryzhk ]. We get < G osφ > t φ F ;;s φ 7 t φ ;;s 4 F φ 6 whee F s Guss s hypegeomet futo. The odto fo etemum of < G > φ s < G > / φ. We hve < G φ whee > osφ u u u u tφ F ;;s φ s φ F ;;s φ 4 u tφ t φ F ;;s φ s φ tφf ;;s φ 4 5 u tφ t φ F ;;s φ s φ tφf ;;s φ < G > / φ oespods to fdg the oots of polyoml of ode ; ths gves 4u 4 u u u 8u 7 u 6

17 Fo gve vlue < G > 6 s futo of φ tht my ot etemums mmum d mmum the ssoted φ vlues of whh e gve by 66. Fgues 4 5 d 6 e plots of G 6 < G > 6 d pots φ 66 espetvely. Guss s hypegeomet futos e estted to the e fst tems of the ssoted sees. Oe dstgush two types of behvou depedg o. Whe >. 67 fo / ote tht s obted fom 66 s φ teds to zeo G > pesets mmum lge th Fg. 5. Fo gve G > eses otuously wth < φ fom vlue oespodg toφ up to mmum bout.7 t φ 4 fo d the deeses deftely fo lge φ. The vlue of φ t whh < G > s mmum s gve by 66 d epoted s futo of Fg. 6. φ vlues e loted betwee zeo d vlue deoted L L φ 4. /. φ els the deomto 66 d the ssoted teds to fty. Fom φ d Fg. 6 eses fst slowly wth φ d the s φ ppohes L vey pdly to fty. φ oespods to vetl symptote. The behvou of G fot s llustted Fg. 4. G < L φ log the k s mmum o the vege ftue ple wth slght symmety: pots t / p / G slghtly lge th those fo whh p s eve; G eses wth φ o the vege ftue ple. G s mmum bout the wve peks oespodg to / λ j / / j tul umbe Fg. 4 d deeses thee s φ eses. Whe < G > pesets mmum smlle th Fgs. 5 d 6. Fo gve G > < deeses wth φ fom φ to mmum lose to.94 t φ 47. fo. d the eses otuously fo lge φ. The φ vlues t whh < G > s mmum e gve by 66 d epoted s futo of o Fg. 6. These e lge th mmum φ 47. umeto 66.e.. Fomφ H log the k fot s gve Fg. 4b. H < H φ els the Fg. 6 eses slowly wth φ. The behvou of G G tkes lge vlues bout the wve peks d eses thee wth φ. G tkes smlle vlues bout the me ftue ple d deeses thee wth φ. Assumg tht o-pl k popgto ous whe < G > s lge th the odto fo k moto oespods to > ; the stes fo the tto of k moto e loted o the vege ftue ple. Whe o-pl k popgto s uepeted Segmeted k fot Hee the k fot s segmeted s llustted Fg. d B s tke s og. s the odd d λ λ A λb peodl wth espet to whee λ A d λ B Fg. e the pojeted legth log of fet A d B espetvely. s gve ove wvelegth s tφ B λb / tφ A λ λ B / λb / λa]. 67 As the se of susodl k fot Seto 4.6 we et osde suessvely the omlzed k eteso foe ow deoted G the vege < G > / λ G d d ultmtely the odto fo etemum of < G v >. v v λ v 7

18 Usg 6 G v o peod eds G v osφb tφ B t φ B < λb / os φ A tφa t φa ] λ B / λb / λa. 68 Fo gve d k pofle φ 68 povdes the edued k eteso foe t bty A φb pot o the segmeted k fot. G v tkes ostt vlues o fet A d B. The vege < G v > my be wtte s < G v > v v v 69 whee v / ε osφ ε osφ A B v / ε sφ ε sφ A B v / ε tφ sφ sφ 7 A A B d ε λ / λ tφ / tφ. B A A B We estt ouselves to the odto fo etemum fo < G v > wth espet to φ A by ellg < G > / φ. We hve fom 69 v A < G v > / φ A v v v 7 whee v / φa fε / φ AosφA osφb fε sφa v / φ A fε / φ AsφA sφb ε A v / φa fε / φ A tφasφ A sφb fε sφa sφb s φa / os v v f osφ v d φ 7 f ε / ε fε / φ A fε / tφb os φa. 7 < G > / φ leds to fdg the oots of polyoml of ode ; ths gves v A v v v v v v. 74 v v Equto 74 s the equed soluto. It leds to postve vlues d gees wth etemums obseved o < G v > s futo ofφ A. A 8

19 Fgues 7 8 d 9 e plots of G v 68 < G v > 69 d pots φ A 74 fo φ B ostt. Fo gveφ B oe dstgush dffeet behvous s futo of. Ths despto pples to yφ B. osde φ B π /.57 7 to llustte. Fg. 9 gves the tevl..99]. Fst eses wth φ A fom. φ A to.99 φ A 4 ; these φ A oespod to mmum lge th fo < G v > whe plotted gstφ A ; ths s the se fo the p φ A.5.8 Fg. 8. The fom mmum. 99 φ A 4 Fg. 9 deeses wthφ A ; these φ A oespod to mmum fo < G v > ; ths s the se fo the pot φ Fg. 8. Outsde..99] G > deeses otuously wth A φ A fo <.. Fg. 8 fo ste d eses deftely wth φ A fo > Fg. 8. Fg. 7 shows G v log the k fot fo dffeet..8 d.5 oespodg to vlues loted below sde d bove..99] wth φ A π / 7.85 d φ B π / G v tkes ostt dffeet vlues gve by Equto 68 tevl ] λ B / λb / d ] λ B / λb / λa d eses wth Fg. 7. I summy fo gve φb thee ests vlue sy t gve by 74 whe φ A suh tht: Whe > t thee e φa vlues lmted by zeo s dsussed bove oespodg to < G v > > ; ude suh odtos o-pl k moto s possble. Whe t < G v > fo y φ A ; o-pl k popgto s uepeted. 5. Dsusso 5.. O the hypotheses of the peset study Epessos fo the dsplemet d stess felds of dsloto of edge vege hte Seto e estblshed fom le fom wth espet to of plst dstoto β ote tht the et elto s β b H. Relto s vld ude the odto smll oly the shpe of beg bty wth the meg tht thee s o estto o ts sptl devtves / d / ptully. No ddtol hypothess s todued see Seto dtg tht dsplemet d stess felds 5 6 d 8 e lso vld ude smll oly. Ths odto pples to oespodg elst felds of dsloto of sew vege hte Seto se sme le fom wth espet to of plst dstoto β s used. Fom stesses due to dslotos we obt suessvely stesses < v j 48 bout the k fot d k eteso foe G 57 pe ut legth of the k fot. The lulto of these quttes volves tegls see 44 d 58 tht e eeuted ppomtely by usg elst felds due to pl dstbuto of stght dslotos. Howeve these ddtol hypotheses hve o det fluee o the geometl ftos.e. those wth d ts sptl devtves / d / peset esults d 6 d epessos 54 fo stess testy ftos deved detly fom 48. I shot the esults of the peset study e obted ude the odto smll oly. Thee s o estto o the sptl devtves of. 5.. k eteso foe We hve deved epesso 6 fo the k eteso foe G pe ut legth of the k fot. Thee ests the ltetue o o-pl ks see Xu et l. 7] d Lzus et l. ] 9

20 mog othes poedue to povde othe fomul fo G. Ths ossts lultg stess testy ftos see 54 fo emple ssoted wth the o-pl k d devg G by usg the usul ple st elto tht eds E E G III I / / the peset k geomety d ppled lodgs. To ompe both eltos of the k eteso foe we shll ssume fo smplty / d / to be smll d develop the vous epessos fo G o equvletly G to seod ode / d /. Relto 6 beomes fst estmto of G : G. 75 Relto 54 beomes III I I I III III d II. 76 The ple st elto the gves seod estmto of G : G. 77 The fst two tems G 75 d G 77 e detl dtg tht both epessos gee to fst ode. Howeve the tems of seod ode.e. the two lst tems 75 d 77 e dffeet. To udestd the og of ths dsepy let us deve the usul ple st elto epltly. We efe to deftos d ottos Setos 4.4 d 4.5. The eegy deese see Seto 4.5 ssoted wth sufe elemet dl loted t pot hed of the k fot eds u dld u dld E 78 whee u d e the dffeees dsplemet oss the legtheed k just behd ts tp the e d e detos fo e see Seto 4.4; j e defed Seto 4.4 d e stesses bout the k fot mesued eltvely to e e e. Fom the defto of the stess testy ftos 5 we my wte

21 I s I III s. 79 Hee d I III III e gve by 54 d d by 56; s <<. To get the usul ple st elto we wte fo u sml wy s fo stesses 79 I I III III u u u u 8 whee u d e oespodg eltve dsplemets of the fes of pl k wth stght fot. Fom the wok by Blby d Eshelby 5] use the elto 86 we hve u b I III / s / π π ; 8 hee the dsplemets e mesued t just behd the tp of the legtheed k so tht s s postve. Usg 78 to 8 we ve t G lm E / dl I III / / E E ; 8 we ote tht followg losely the pevous wok o pl k by Blby d Eshelby 5] see the developmet fom the eltos to 5 my be helpful the ouse of the lulto. The dsepy obseved fo seod ode tems G 75 d G 77 ogtes fom oet fom 8 fo u. The et eltos e u u osθ u sθ u sθ u osθ u 8 u whee d e defed Seto 4.5. Whe 79 d 8 e todued 78 d u 8 s ppomted by fo G. Hee sted of 8 the oet ppomto would be u u θ u III I os sθ u I I I III s osθ u III III u 8 we eove seod ode tems luded epesso 6 θ u 84

22 whee I I osθ III sθ III I sθ III osθ 85 d /. The k eteso foe G pe ut legth of the k fot the eds I I III III / G / E E. 86 We would lke to pot out tht stesses bout the k fot 48 by the wy they hve bee estblshed see Seto 4. ot y stess tem wth sgulty of the type s. It thus ppes tht SIF epessos 54 deved detly fom 48 odg to defto 5 vlble the ltetue Bu 9] fo emple effetvely be used to lulte the k eteso foe; howeve the oveet fomul fo G would be 86 d ot the usul elto ompso wth pevous theoetl lyses Oe my dstgush btly two types of studes volvg the o-pl hte of loded ks. O the oe hd thee e woks povdg fomule fo elst quttes stess stess testy ftos dsplemet bout the fot of k tht s etely o-pl s the peset study does: Go 6] Xu et l. 7] Bll d Llde 9] Aogb 7] ovh et l. 8] mog othes. O the othe hd thee e woks teested elst quttes bout the fot of tlly pl k tht dopts ove shot dste of futhe popgto o-pl ofguto: the eet wok of Lzus et l. ] fo ste. osde fst type of studes by Go 6] Xu et l. 7] ovh et l. 8] d Bll d Llde 9]. These uthos hve osdeed fte sold wth o-pl k whose sufe / ou otto depeds o both d d my be vewed s slght petubto of pl semfte stght edge k; we stess tht the wok by Bll d Llde 9] depeds o oly. I ll these studes both d ts sptl devtves / d / e tke smll d le epessos of SIF e gve. These esults hve ow pplto. Atully these pply to k tht popgtes essetlly ude mode I lodg but whose fot fo vous possble esos suffes slght petubto. Ideed whe mode II lodg ddto to mode I s ppled to pl k loted the subsequet ftue popgto pth depts fom Edog d Sh ] Rdo et l. ] d odto smll s volted; fo ppled mode III d s metoed ele Seto s geelly smll but / mesued the k fot deto my be lge o fets B fo ste Fgue eve ude smll. Let us ssume s these ele woks both d ts sptl devtves to be smll. Ude suh odtos oly le tems the edued k eteso foe e osdeed to suvve d we my wte G. 87 Ths my be tke fom 75 o 77. I Setos 4.6 d 4.7 we hve pefomed vege < G > of G usg 6. The sme poedue usg 87 edues the esult to tht of the pl k s obvous whe the k fot hs susodl shpe Seto 4.6; fo the segmeted k fot Seto 4.7 se t s ssumed tht / s smll ths mes tht t φa φa d t φb φb ; beuse φ A d φ B hve sml smll mgtude < G > usg 87 s vtully oe. Hee to dsuss

23 mosop k popetes t s eessy to use moe elbote epessos suh s 6 vld ude odto smll oly see Seto 5.. Oe my justly be teested detled ompso betwee stess o SIF fomule obted the peset study d those deved ele woks Go 6]; Xu et l. 7]; ovh et l. 8]; Bll d Llde 9]; f oe gees wth both SIF defto 5 d stess eltos 48 bout the opl k fot oe must lso gee wth the pot of vew tht SIF eltos 54 e omplete.e. thee s o mssg tem whe omped wth SIF epessos vlble these ele studes. Atully suh ompso s ot stghtfowd beuse ou k geomety Seto 4. s vey dffeet fom the oe metoed bove dopted these pevous studes. Theefoe ou esults e epeted to be dffeet fom thes. Futhemoe we hve obted fomule fo the stess bout the k fot fom the stess felds due to dslotos whees ele woks used petubto methods ssoted wth bttle ftue mehs. Ths leds to dffeet deftos d pesettos of esults; ptul the wok of ovh et l. 8] tht mkes etesve use of the stess feld due to the pl k the lulto of SIF epessos volves lge umbe of tegtos pefomed ove the ete k fot o k sufe see the developmet fom.5 to.6 fo emple. Ths otsts wth the peset method of lyss whh g s fully osstet whee SIF eltos 54 e deved detly fom the lytl foms of the stess bout the o-pl k fot 48; o use s mde of the stess feld due to the pl k. Howeve geel ompso befly be mde log the followg les. We ote fst tht stesses d 48 e zeo bout ou k fot ledg to II 54 whees II tkes o-zeo vlue fo the opl sem-fte k see 6 Bll d Llde 9] d.4 ovh et l. 8]. Futhemoe ude pue mode I d III lodgs the III d I dued by the shpe of the opl k e to fst ode / I d / / III espetvely use 76. These gee fom wth oespodg esults peseted by Bll d Llde 9] see the eltos 6 d 7 d ovh et l. 8] see the.4; by gee fom we me tht esults gve by these uthos e popotol to the devtve / of the petubto s the peset wok. We wt to stess hee tht Lzus et l. ] sml ompso hve foud geemet betwee some of the esults the elto 5 d pevous esults of Bll d Llde 9] the 6 d ovh et l. 8] the.4. We ow tu to Lzus et l. ] whh osde tlly pl sem-fte stght edge k k fot the -deto tht dopts o-pl ofguto fte ftue popgto ove shot dste the otto the ew k sufe ou otto beg d depedet. The oly smll pmete s o equvletly ; thee s o estto o the sptl devtves of s the peset study. A questo tht be posed s: ude whh odtos the ppoh of these uthos be used to epess the stess felds bout the fot of mosop k etely o-pl? It ppes evdet tht t s eessy tht be suffetly lge. They povde stess testy ftos to fst ode d use the usul ple st elto see 8 fo ste to estmte the k eteso foe G. As we hve show Seto 5. G deved ude suh odtos s ute to fst ode oly both petubto o d ts sptl devtves. Howeve they hve ppled to epemets esults fo the k eteso foe epessed to seod ode d s suh pesumbly ute sptl devtves of. To ed hee we would lke to stess tht fomul 6 fo edued k eteso foe Seto 4.5 ppes to be dequte whe ppled to epemets pefomed ude med mode IIII lodg ou k geomety ompso wth epemets We ow ompe ou lyss wth epemets d fst fous o the stuto whee the vege vlue of the edued k eteso foe s lge th d the k-fot lto gle φ A oφ tkes modete vlues.e. gg fom zeo to 5 ppomtely. Fom Seto 4 t s see tht ths ous whe the to she/teso of the ppled stesses s lge th. 67

24 / the se of k wth susodl fot Fgs. 5 d 6; howeve whe the k hs segmeted fot s Fg. Seto 4.7 the oespodg s otbly smlle. 8 fo φ B π / d. foφ B π /. 57 Fgs. 8 d 9 fo emple. Assumg tht k moto my ou whe the vege vlue of the omlzed k eteso foe s lge th ou lyss suggests tht o-pl ks wth segmeted fot wth φb π / e fvoued t smll. Ths s full geemet wth epemets ]. We lso meto ppetly teestg popety of the model. osde k wth segmeted fot d tke φ B π /. 57 to llustte Fg. 8; we hve dted Seto 4.7 tht fo suffetly lge. 4 the vege < G v > eses otuously wth φ A fom the vlue φ A. Ths suggests tht φ A my ese gdully s the k eteds. Howeve t s le tht the k model of the peset study by ostuto s uble to desbe qutttvely y hge of φ A wth k legth s depted by Somme ] fo ste. We hve epoted o Fg. uve the pots φ 66 fo whh G > 6 s mmum d lge th > We hve lso plotted o Fg. the pots φ φ ostt Equto 74 fo whh G > 69 s mmum d lge th : uve B < v oespods to φ B π / d uve 4 toφ B π / 4. These ddtol uves povded by the peset study fll sde epemetl pots fo eltvely smll. It s hzdous to go futhe to ompso beuse fo the epemetl pots gve Fg. we hve o fomto bout oespodgφ B. A detled ompso wth epemets wth the m of mpovg estg models should tke full out of the whole obseved shpe of the k fot. 6. oludg emks A model of o-pl k of fte legth ude med mode IIII lodg flututg bout vege ftue ple hs bee vestgted. The k fot hs bty < peodl shpe spedg ple pepedul to the k popgto deto. We hve estblshed epessos fo both the stess bout the k fot 48 d the k eteso foe G pe ut legth of the k fot We hve veged G ove fo two spel ks susodl 6 d segmeted 69 k fot d estblshed odtos lkg ppled stess to k-fot pofle ude whh the vege k eteso foe < G > s mmum d lge th the vlue oespodg to tht of the pl k ude med mode IIII. These odtos ofom to epemetl stutos whh o-pl ks e fvoubly obseved. Othe epessos fo the stess bout the k fot d the k eteso foe my be obted fom 7 d 8. Itodug 7 to 44 d tegtg yeld ew epessos dffeet fom 48 fo the stess bout the k fot sees foms whose tems volve hebyshev polyomls of seod kd. These eltos fo the stess ssoted wth 8 llow oe by usg 55 to gve epesso fo the k eteso foe dffeet fom 57. A omptve lyss volvg the vous dffeet fomule d epemetl fdgs ludg ew detled obsevtos of the whole shpe of the k fot s of ptul teest. The k s mosop ou modellg Seto 4. se t us deftely the deto. Ths s well-fomed k wth legth oughly of the ode of mm ody lbotoy epemets. I the ple the k s suouded by fte medum ude med mode IIII wth lodgs ppled t fty. I ths geomety thee s ppetly o eso fo the o-pl k to deped o. Thus the edued k eteso foe 6 ems ptlly uhged whe ddto oe ssumes the k fot to deped o k legth. Ths s beuse the ddtol stesses tht ome to ply the lulto of the k eteso foe d 48 e zeo both ses k A 4

25 fot beg oly o both d depedet; lso the othe stesses d 48 em uhged fom. Oly whe ppled mode II lodg ddto s ssumed o ou k model wll the k legth ply le ole the ftue popetes of the ked sold. The peset modellg ssumes the shpe of the k fot to be depedet of k legth. It my be gued tht ths s seous lmtto. Howeve the peset wok deseves osdeto fo umbe of esos. Fst the method of lyss s uommo d eveled tself s pble of opg wth the moe elbote poblem of the geel lodg mode IIIIII of o-pl ks solds. Seod ou esults desbe deeply the depedee of ude med mode IIII lodg. Addtol studes e equed to vestgte the geel lodg of o-pl ks usg the method of lyss dopted the peset study. Ths poblem eques tht be both d depedet. 5

26 Refeees ] E. Somme Fomto of ftue les glss Egeeg Ftue ehs ].L. ooke D.D. olld Ftue popgto pths ude med mode lodg wth etgul bloks of polymethyl methylte Joul of Geophysl Reseh ] J.R. Ytes.J. lle ed mode I-III ftgue thesholds fogg steel Ftgue Ft. Eg. te. Stut ] F. Houle A. eu Fssuto p ftgue sous sollttos polymodles mode I odulé mode III pemet d u e pou otos 6NDV4 ém. S. Rev. étll ] B.R. Lw Ftue of bttle solds mbdge Uvesty ess mbdge 99. 6] H. Go Thee-dmesol slghtly o-pl ks ASE Joul of Appled ehs ] G. Xu A.F. Bowe. Otz A lyss of o-pl k gowth ude med mode lodg Itetol Joul of Solds d Stutues ] A.B. ovh H. Go J.R. Wlls O petubtos of ple ks Itetol Joul of Solds d Stutues ] R.. Bll H. Llde Thee-dmesol stblty lyss of pl stght ks popgtg qussttlly ude type I lodg Itetol Joul of Ftue ] V. Lzus J.-B. Leblod S.-E. ouhf k fot otto d segmetto med mode I III o I II III. -t I: lulto of stess testy ftos Joul of the ehs d hyss of Solds t II: ompso wth epemets Ibd ] D.D. olld. Segll.T. Deley Fomto d tepetto of dltt ehelo ks Geol. So. Am. Bull ].N.B. Aogb The dsplemet d stess felds due to susodl edge dsloto hys Sttus Sold B ] T. u The otuum theoy of dslotos Adves tels Reseh ] B.A. Blby A.H. ottell.h. Swde The sped of plst yeld fom oth oeedgs Royl Soety Lodo A ] B.A. Blby J.D. Eshelby Dslotos d the theoy of ftue I: Ftue Vol. Lebowtz H. Ed. Adem ess New Yok 968 pp ].O. eh J.S. oehle The foe eeted o dslotos d the stess feld podued by them hysl Revew ].N.B. Aogb A o-pl k lyss usg otuously dstbuted susodl edge dslotos d le elstty hys Sttus Sold B

27 8].N.B. Aogb V. Vtek Sgfe of the devtos of the k fot to the ple pepedul to the k popgto deto- I. k-fot dsloto geeto Itetol Joul of Ftue ] H.D. Bu éque de l uptue fgle sso s 978. ] I.S. Gdshtey I.. Ryzhk Tble of tegls sees d poduts Adem ess Lodo 994. ] F. Edog G.. Sh O the k eteso pltes ude ple lodg d tsvese she Ts. Am. So. eh. Egg ].. Rdo.S. Leve L.E. ulve Ftue toughess of A ude bl stess I: Ftue Vol. UW ess 977 pp

28 Fgue ptos Fg.. Shemt shpe of k fot med mode IIII lodg of sold ple pepedul to the deto of ftue popgto. The k flututes bout the ple d ossts of pl fets wth lto gle φ A d B φ t pot A d B of the vege ftue ple espetvely. I ths geomety the k s subjeted to ppled teso deto d she the deto. the Fg.. k-fot lto gle φ A degees s futo of the to she/teso of the ppled stesses polymethyl methylte A o legls: symbol d steel o s epodued fom ooke d olld ]. uve oespods to Eq. the tet uve to Eq. 66 wth > ; uve d 4 oespod to Eq. 74 wth φ B π / d π / 4 espetvely plots beg estted to φ fo whh G > s mmum d lge th.. 8 A. A < v Fg.. A wvy k wth susodl fot etedg fom to d subjeted to ufom d t fty. The k flututes bout me ple whh s llustted wth smll mpltude sy oughly of the ode of µ m o smlle. Fg. 4. Nomlzed k eteso foe edued by the wvelegth φ π / 8.97 π / 4. 9 d /. 7 λ. G 6 fo the susodl k fot s futo of : the uves e ssoted wth k pofles π. b. : φ π /. 85 π /. 8 d π / /. Fg. 5. Redued k eteso foe veged ove < G > 6 s futo of k pofle φ susodl k fot. uves wth mmum e ssoted wth d > those wth mmum wth. d.4 < ; /. The vlues of φ t whh G > s mmum o mmum e gve by 66 d plotted Fg. 6. < Fg. 6. ots φ 66 fo whh G > 6 s mmum d lge th >. 67 d < mmum d smlle th < whe plotted gst φ s Fgue 5; / ; susodl k fot. Fg. 7. Nomlzed k eteso foe G v 68 fo the segmeted k fot s futo of edued by λb defed Fg.. The uves oespod to..8 d.5; they tke dffeet ostt vlues tevls ] / / d ] / / / ε wheeε λb / λa tφ A / tφb. φ A π / φ B π /. 57 d /. Fg. 8. Redued k eteso foe veged ove < G v > 69 s futo of the lto gle φ A of k-fot segmet A segmeted k fot; φ B π /.57 7 ; /.. : the oespodg uve deeses otuously s φ A eses.. 8 : the ssoted uve dsplys lol mmum bove d lol mmum below t φ A vlues gve by 74 see lso Fg. 9 uve π / : the oespodg uve eses otuously wth φ A. 8

29 Fg. 9. ots φ A φ B ostt Eq. 74 fo whh < G v > 69 s etemum whe plotted gst φ A s Fg. 8 wth. 8 ; / ; segmeted k fot. The uves oespod to φ B π / π /. 57 d π / 6 : y of these uves pots befoe fom the left the mmum oespod to < G v > mmum d lge th d pots fte the mmum to < G v > mmum d smlle th. 9

30 Fgue

31 Fgue

32 Fgue

33 Fgue 4 4 G π/8.97 π/ π/ / λ

34 Fgue 4b.5 G π / π /.8.5 π / / λ 4

35 Fgue 5 < G.5 > φ degees 5

36 Fgue 6 6

37 Fgue 7 G v / λ B 7

38 Fgue 8 < G v > φ Adegees 8

39 Fgue π / 6.5. π / π / φ Adegees 9