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State of the Art Report - IA
3 2 BasicNotations eυ aao aL c d 刀 s u Wc l e n g t ho fc r a . c ko rc r a c kba . nd 一 . c kl e n g t h n i t i a l( p r e e x i s t i n g )n o t c ho rc ra ー i r. a c kl e n g t h 一 elasticallyequivalent(eff .tive)c も w o ・ d i m e n s i o n a l s p e c i m e n t h i c k n e s s o f 一 . c ke x t e n s i o n 一 elasticallyequivalentcra h a r a c t e r i s t i cd i m e n s i o n( o rs i z e )o fs t r u c t u r e( o rs p e c i m e n ) ー c ,飢 一 tensilestrength a c i n g 一 cracksp. d i s p l a . c e m e n t 一 w i d t ho fc r a c kba . nd G1c a . st i c i t y l 一 Young'smodulusofe . n e抗 r e s s , E'= E( 1-v2)f o rp l 一 E forpla e l e a s er a t e 一 energyr . t ef o rMode1 一 criticalenergyreleasera G 7 ー ー E E' G , G 姐 es t r a . i n f ra . c t u r ee n e r g y a . c t u r ee n e r gyo b t a i n e dbyt h eRILEMMe t h o d f r s t r e s s i n t e n s i t y f a . c t o r s f o r M o d e s 1 , I I , I I I , r e s p e c t i v e l y [ ( / ,K I I J < I I / , f r a c t u r e t o u g h n e s s c r i t i c a l s t r e s s i n t e n s i t y f a c t o r f o r M o d e 1 K/ c p l o a d maximum( u l t i m 乱t el o a d ) Pu 一r , α/ d e l a t i v ec r a c kl e n g t h α 一 brittlenessnumber F 一 r a c k o p e n i n gd i s p l ac e m e n t ー c 一 a le f f e c t iv ec r a c k t i pop e n i n gd i s p l a . c e ment dCTOD 一 critic. r. a ck-moutho p e n i n gd i s p l a c e m e n t dCMOD 一c も l o a d p o i n td i s p l a c e m e n hPD 正 s t r a i n r e s s σ 一 st o m i n a ls t r e s sa tu l t i m a t el o a d σN ー n 一 P o i s s o n ' s r a t i o U 一 sa 一 extensionofcrackorcrackband o t e n t i a len e r g y E ー p ι F r a c t u r eM e c h a n i c so fC o n c r e t e : C o n c e p t s, ModelsandD e t e r m i n a t i o n o fM a t e r i a lP r o p e r t i e s R e p o r tbyACIC o m m i t t e e4 4 6, F r a c t l i r eM e c h a n i c s . ー V e l l o r eS .Gopalaratnam 1•3 ( S e c r e t 町) Zd en 位 P .Baiant1 •2 (C 凶 m 切) 堕豆盟単語 1 O r a lBuyukozturk 1 L u i g iC ed o l i n 3 DavidDa . r w i n e lElices1 •3 M姐 u S h u J i nFang . l t e rG e r s t l e Wa N e i lM.Ha . w k i n s H i d e y u k iH o r i j l J e r e myI s e n b e r g V i c t o rC .L i1 Feng-BωL 泊1 S t e v e nL .McC a . be 3 S h e n g TaurMau 3 J a c k yMa . z a . r s S i d n e yM i n d e s s A n t o i n eE .Na . a m an1 C.De a . nNorman Ph i 1 l i pA .P f e i f f e r Gi U e sP i j a u d i e r C a b o t 3 1, a . 3 V i c t o rSaoum 1 aP.Sh a . h , 3 S u r e n d r. R ρ b e r t1 .S i e r a . k o w s k i 1 WimalSu a . r i s S t u a r tE.SWa.rtz1 •2 .Ts ub a . k i T a t s u ya C .V i p u l a n 姐 d 姐 1 1 M e t h iWe c ha . r a t a n a ∞gnizethe∞ntributionsofthefollowingnon-votingmembers: Thecommitteew i s h e st or e Fa . r hadA n s a r i1 3 Ra v i n d r a .G e t t u 1 AmeHi 1 l e r b o r g B.L .K a r i h a l ∞1 a . t l P e r eC .Pr i n h a . r dt l H姐 sW.Re 1M emberso fS u b c o m m i t t e e1( c h a i r e db yBa1組t )w h i c hp r e pa . r e dt h er e p o r t 2 P r i n c i p a lAu t h o r s 3 Con t r i b u t i n gA u t h o r s * T h i sr e p o r tw剖a. p p r o v edbyav o t eo ft h ef u l lACICommi t t e e4 4 6i nDecember1 9 8 9 .I t d o e sn o tr e f l . e c もr e s e a r c ha f t e rt h a td. a t e . 4 5 SYNOPSIS Introduction I nt h ef i r s to fi t 55 e r i e 5o ff o u rs t a t e o f t h e a r tr e p o r t su n d e rp r e p a r a t i o n,t h eC o m m i t t e ed e s c r i b e s t h eb a s i cc o n c e p t so ff r a c t u r em e c h a n i c so fc o n c r e t e , t h ee x i s t i n gt h e o r e t i c a lm o d e l s, a n dt h em e t h o d s 仔e r s自v er e a s o n sf o ri n t r o d u c i n gf r a c t u r e f o rd e t e r m i n i n gt h em a t e r i a lf r a c t u r ep a r a m e t e r s .C h a p t e r10 ,i n c l u d i n gsomec o d ep r o v i s i o n s :( 1 )a m e c h a n i c si n t oc e r t a i na s p e c t so fd e s i g no fc o n c r e t es t r u c t u r e s ,i . e .,e l i m i n a t e t h e o r e t i c a le n e r g ya r g u m e n t ;( 2 )t h en e e dt oa c h i e v eo b j e c t i v i t yo ff i n i t ee l e m e n ts o l u t i o n s 刷 r i o U 5meshs e n s i t i v i t y ;( 3 )t h ep r 叩 悶i v e( p r o p a g a t i n g )n a t u r eo ff a i l u r e ,i m p l i e dw h e n e v e rt h el o a d 刑l o n d e f l e c t i o nd i a g r a ml a c k say i e l dp l a t e a u ;( 4 )t h en e e dt or a t i o n a l l yp r e d i c td u c t i l i t ya n de n e r g ya b s o ,( 5 )t h ee f f e c to fs t r u c t u r es i z eo nt h en o m i n a ls t r e n g t h(iιnominal c a p a b i l i t y ;a n dm o s ti m p o r t a n t l y Sa tmaximumo ru l t i m a t el o a d )a sw e l la so nd u c t i l i t ya n de n e r g ya b s o r p t i o nc a p a b i l i t y .The s t問 5 仇c ti sd u et os t o r e de n e r g yr e l e a s ei n t ot h ef r a c t u r ef r o n t,a n di sn o tg o v e r n e db yWeibul 1 句p e s l z ee ' 回 r y .E xp e r i m e n t a le v i d e n c eo nt h ee x i s t e n c eo ft h es i z ee 仔' e c t,h i t h e r t oi g n o r e di nd e s i g n s t a t i s t i c a lt h p r a c t i c ea n dc o d ep r o v i s i o n s ,i 5d o c u m e n t e d . C h a p t e r2g i v e sab r i e fr e v i e wo ft h en e c e s s a r yb a s i cr e s u l t so fl i n e a re l a s t i cf r a c t u r em e c h a n i c s e p a r t u r e sf r o mt h i sc 1 a s 5 i c a lt h e o r ya r ec a u s e db yt h ee x i s t e n c eo fd i s t 巾Ijt e d ( l E F M ) .I nc o n c r e t e,d c r a c k i n g( o rd a m a g e )i nap r o g r e s s i v e l ys o f t e n i n gf r a c t u r ep r o c e s sz o n ew h i c hs u r r o u n d st h et i po fac o n t i n u o u sc r a c k.I nC h a p t e r3n o n l i n e a rf r a c t u r em o d e l sc h a r a c t e r i z i n gt h es o f t e n i n gs t r e s s d i s p l a c e m e n t s t 凶 nr e l a t i o n s( s u c ha st h o s eo fH i l l e r b o r g ' sf i c t i t i o u sc r a c km o d e l, c r a c kb a n dm o d e l,n o n l o c a l o rs t問 s s t r a i n s o f t e n i n gm o d e l s ,e t c . )a r ed e 5 c r i b e da n drandomp a r t i c l es i m u l a t i o no fa g g r e g a t em i c r o s t r u c t u r ei sd i s c u s s e d .Thep r i n c i p l e so fi m p l e m e n t a t i o no ft h e s em o d e l si nf i n i t ee l e m e n tp r o g r a m sa r ea l 5 0 o u t l i n e d .C h a p t e r4p r e s e n t ss i m p l e rn o n l i n e a rf r a c t u r em o d e l sw h i c hr e p r e s e n ta d a p t a t i o n so fl i n e a r e l a s t i cf r a c t u r em e c h a n i c s,s u c ha sJ e n qa n dS h a h ' sm o d e Ja n dt h eRc u r v e ,a l o n gw i t hd e t e r m i n a t i o n f T e c tl a wp r o p o s e db yB a f a n t .T h i sl a w ,d e s c r i b i n g o fg e o m e t r y d e p e n d e n tR c u r v e sf r o mt h es i z ee ,i sd i s c u s s e di n t h ea p p r o x i m a t ed e p e n d e n c eo ft h en o m i n a ls t r e s sa tmaximuml o a do ns t r u c t u r es i z e C h a p t e r5 ,a n ds t r u c t u r a lr e s p o n s ei 5c h a r a c t e r i z e db yt h eb r i t t l e n e s sn u m b e r . C h a p t e r6p r e s e n t si nc o n s i d e r a b l ed e t a i lt h ec u r r e n tm e t h o d sf o re x p e r i m e n t a la n da n a l y t i c a ld e t e r m i n a t i o no fm a t e r i a lf r a c t u r ep a r a m e t e r s, i n c l u d i n gt h eq u a s i l E F Mm e t h o d s, RllEM( w o r k o f f r a c t u 問) method,t h eJ e n q S h a ha n dK a r i h a l o o N a l l a t h a m b im e t h o d s ,a n dt h es i z e e f T e c tm e t h o d .E x p e r i m e n 白n i n g t a ld e t e r m i n a t i o no ft h ec h a r a c t e r i s t i cl e n g t hf o rn o n l o c a lc o n t i n u u mm o d e l sa n dt h es t r a i n 5 0 f n dm a t e r i a lp a r a m e t e r sf o rmodes1a n d1 1 1,s h e a rf r a c t u r e sa n dm i x e d p r o p e r t i e si st h e ne x a m i n e d,a modef r a c t u r ea r ea l s od i s c u s s e d .C h a p t e r7t h e np r o c e e d st od e 5 c r i b ev a r i o u si n f l u e n c i n gf a c t o r s ,s u c h a st h el o a d i n gr a t e,h u m i d i t ya n dt e m p e r a t u r e ,a sw e l la st h ee f T e c to fc y c l i cl o a d i n g .C h a p t e r8i s f T e c to fr e i n f o r c i n gb a r sa n dt h e i rb o n ds l i po nf r a c t u r ep r o p a g a t i o n,a n dt of r a c t u r e d e v o t e dt ot h ee トr e i n f o r c e dc o n c r e t e .C h a p t e r9d e a l sw i t hmoret h e o r e t i c a lp r o b l e m so fm o d e l i n gs y s t e m so f o ff i b e i n t e r a c t i n gc r a c k s .A t t e n t i o ni sf o c u s e do ns y s t e m so fp a r a l l e lg r o w i n gc r a c k s .T h e i rs t a b i l i t yd e c i d e s t h es p a c i n ga n dw i d t ho ft h ec r a c k sf r o mt h em e c h a n i c sv i e w p o i n t . f t e rad e c a d eo fr a p i dp r o g r e s si n問 s e a r c h,t h et i m ea p p e a r sr i p ef o ri n t r o d u c i n g I ti sc o n c l u d e dt h a t,a f r a c t u r em e c h a n i c 5i n t od e s i g np r a c t i c e .T h i ss h o u l dn o to n l yb r i n ga b o u tmoreu n i f o r ms a f e t ym a r g i n s , t h u si m p r o v i n gs a f e t ya n deconomyo fd e s i g n, b u ta l 5 0p a v et h ewayf o rs a f e ra n dmoree f f i c i e n tu s eo f h i g h p e r f o r m a n c ec o n c r e t e sa n dp e r m i td e s i g ne x t r a p o l a t i o n sb e y o n dt h er a n g eo fp r e v i o u se x p e r i m e n t s a n dd e s i g n . 且e s s, c o n c r e t e, c o n c r e t es t r u c t u r e s, c r a c ks p a c i n gandw i d t h, c r a c k i n g , d a . r n a g e KEYWORDS:B r i t t l e mech 姐 i ω, d e s i g nc o d e s, d u c t i l i t y ,f a i l u r e ,f i b e r r e i n C o r c e dc o n c r e t e ,nonlocalcontinuummodels, r e i n f o r c e dc o n c r e t e, s i z ee f f e c t, s t r a i ns o f t e n i n g, s t r u c t u r a ld e s i g n, t e s t i n gmethods, u l t i m a t el o a d s . Concretes t r u c t u r e sa r efuUo fc r a c k s .F a i l u r eo fωucretes t r u c t u r e st y p i c a l l yi n v o l v e s s t a b l egrowtho fl a r g ec r a ε k i n g z o n e 渇a ndt h ef o r m a t i o no fl a r g ef r a c t u r e sb e f o r et h emaximum 1 0 叫 I Sr e 叫 h e d .Yetd e s i g ni sn o tbasedonf r a c t u r emech a . n i c s, eventhought h eb a s i cf r a c t u r e mech a . n i c st h e o r yh a sbeena . v a i l a b l es i n c et h emiddleo ft h i sc e n t u r y .Sowhyhasn o tf r a c t u r e mecha . n i c sbeeni n t r o d u c e di n t o∞n c r e t ed e s i g n ? Have∞n c r e t ee n g i n e e r sbeeng u i l t yo f l Theformsoff r a c t u r emechanicswhichwerea v a i l a b l eu n t i lr e c e n t l y j g n o r a n c e ? Nota ta l. werea p p l i c a b l eo n l yt ohomogen 田 u sb r i t t l em a t e r i a l ssuch制 g l a s s, o rt ohomogeneous b r i t t l e -d u c t i l em e t a l s . Theq u e s t i o no fa p p l i c a b i J i t yo ft h e s ec l a s s i c a lt h e o r i 田 t oc o n c r e t e wase x p l o r e dl o n gago-thei d 伺 o fu s i n gt h es t r e s si n t e n s i t yf a . c t o rappeareda l r e a d yi nt h e 帥 r l y1 9 5 0 ' s( e . g ., B r e s l e r姐 dWollack, 1 9 5 2 )ands e r i o u si n v e s t i g a t i o n ss t a r 句 di nthe1 9 6 0 ' s ( e . g .,Kapl a . n , 1961, 姐dothers). Butthe叩swerw出 ,a . tt h a . ttime, n e g a t i v e( e . g K e s l e r , NausandLo t t, 1 9 7 1 ) .Asi snowunderstood, t h er e a s o nw剖 t h a ti nc o n c r e t es t r u c t u r e sone mustt a k ei n t oaccοunts t r a i n s o f t e n i n gduet od i s t r i b u t e dc r a c k i n g, l o c a l i z a t i o no fc r a c k i n g i n t ol a . r g e rf r a c t u r e sp r i o rt of a i l u r e , 組db r i d g i n gs t r e s s e sa tt h ef r a c t u r ef r o n t . A formo f f r a c t u r em民 h姐 iωthatc 組 bea p p J i e dt osuchs t r u c t u r e sh a sbeendevelopedo n J yd u r i n g t h el a s td e c a d e . g nh副 a . l r e a . d ys 田 nt wor e v o l u t i o n s .Thef i r s t, whichmadet h et e c h n o l o g y Concreted田 i o f∞n c r e t es t r u c t u r e 日p o s s i b l e, wast h edevelopmento ft h ee l a s t i cno ・t e n s i o n姐 a l y s i sd u r i n g 1 9 0 0・1 9 3 0 .Thesecondr e v o l u t i o n, b a s e donat h e o r yc o n c e i v e dc h i e s yd u r i n gthe1 9 3 0 ' s, was t h ei n t r o d u c t i o no fp l a s t i cl i m i t如 a l y s i s, whichoc 叩 r r e dd u r i n g1940-1970 .There釘 egood r e a s o n st ob e l i e v et h 抗 t h ei n t r o d u c t i o no ff r a c t u r emechanic8i n t ot h ed e s i g no fc o n c r e t e s t r u c t u r 倒 , bothreinforcedandunreiuforced,mightbethethirdmajorrevolution. The t h e o r y , f o r m u l a t e dmostlyd u r i n gt h eJ a s tdozeny e a r s, f i n a l l yappe a . r st ober i p e . Fr 】舵 a . c t 旬 ur ,問 er e s e a r c h e r sha . vea . tt h e戸 pr 伺 e 叩n tnodoubtt 凶 ha tu 凶 も hei I 叫 北 1 trod 町 u I 民 c “ ω も ぬ ∞ o i nof合 f r 悶a 拭c t 旬u r ,陀 ' eme 令ト c h a I 凶cs叫 m も ω ot h ed e s i g nc r i t e r i af o ra l lb r i 川 ef a i l u r e so fr e i n f o r c e dc o n c r e t es t r u c t u r e s( s u c h a sdiagon a . lshω,punchingshear,t o r s i o no rp u l lout,o rf o r∞n c r e t edams),c 組 b r i n g abouts i g n i f i c a n tb e n e f i t s .I twi 1 lmakei tp o s s i b l et oa c h i e v emoreuniforms a f e t ymargins, e s p e c i a l l yf o rs t r u c t u r e so fd i f f e r e n ts i z e s . This, i nturn, wi 1 limproveeconomya sw e l la s s t r u c t u r a lr e l ia . b i l i t y .1 もw i l lmakei tp o s s i b l et oi n t r o d u 回 目wd e s i g n sandu t i l i z enew∞n c r e t em a t e r i a l s .F r a c t u r em田 h a n i c sw i l lbep a r t i c u l a r l yimportantf o rhighs t r e n g t hc o n c r e t e s t r u c t u r e s, f i b e r r e i n f o r c e dc o n c r e t es t r u c t u r e s, c o n c r e t es t r u c t u r e so fu n u s u a l l yl a r g es i z e s, a . ndf o rp r e s t r e s s e ds t r u c t u r e s . Thea p p l i c a t i o no ff r a c t u r emechanicsi smostu r g e n tf o r s t r u c t u r 回 s uch槌 ∞n c r e t edamsandn u c l e a rr e a c t o rv e s s e l so rc o n t a iQ.IIle n t s, f o rwhicht h e s a f e t ycon 白 血s a . r ep a r t i c u l a r l yhigha . n d出 econsequenαso fap o t e n t i a ld i s a s t e renormous. S u r v e y so fc o n c r e t ef r a c t u r emechanicshaver e c e n t l ybeenpreparedbyv a r i o u scommitt e e s(Wittmann, 1983, 叩dElfgren,1989).However, duet ot h er a p i d l yadvancingr e s e a . rch, t h ec o n t e n t so ft h ep r e s e n ts t a t e of . 叶l e a r tr e p o r t訂 eq u i t ed i f f e r e n t .A u n i f i e d , s y s t e m a t i c p r e s e n t a ' も i o n, r a t h e rthanac o m p i l a t i o no fa l lt h ec o n t r i b u t i o n sbyva . r i o u sa u t h o r s, i sa t temptedi nt h ep r e s e n ts t a t e -o f a r trepo 比 Ther e p o r ti saimedp r i m a r i l ya tr e s e a . r c h e r s, n o t ne 偲 s s a r i l ys p e c i a l i s t si nf r a c t u r emech 組 i c s .However, i ts h o u J da l s obeo fi n t e r e s tt od 田 i g n 吋 “ 7 6 e n g i n 田 r sb e c a u s ei td e s c r i b e sat h e o r yt h a . ti sl i k e l yt op r o f o u n d l yi n f l u e n c et h ed e s i g np r a . c t i c ei nt h ene a . rf u t u r e .S u b s e q u e n tr e p o r も sd e a l i n gw i t ha p p l i c も . ai o n si nd e s i g n , f i n i t ee l e m e n t a . c t u r e , anddynamicf r a . c t u r ean a . l y s i s, a r ei np r e p a . r a . t i o nbyACICommittee a n a l y s i so ff r (0) レ 4 4 6 . Chapter1 . W H YFRACTUREMECHANICS? ε F r a . c t u r emech 組 i c s, i nab r o a ds e n s e ,i sa .f a . i l u r et h e o r ywhich( 1 )u s e se n e r g yc r i t e , ap o s s i b l yi nc o n j u n c t i o nw i t hs t r e n g t hc r i t e r i a, 組d( 2 )w h i c ht a k e si n t o配 ∞un 色f a i l u r e r i p r o p a g a t i o nt h r o u g ht h es t r u c t u r e . . . E t?LU_ j _ t 1 .1F i v eReasonsf o rF r a c t u r eMechanicsApproach S i n 田 ∞n c r e t es t r u c t u r e shaveb 田 nd e s i g n e d姐 ds u c c e s s f u l l yb u i l ta c c o r d i n gt oc o d e s w h i c ht o t a l l yi g n o r ef r a c t u r em e c h a n i c st h 回 r y ,i tmight日eemunnec 閃 s a r yt och 姐 g et h e t h e r ea r ef i v ec o m p e l l i n gr e 出 o n sf o rd o i n gs o . c u r r e n tp r a ι t i c e .N e v e r t h e l e s s, U A B ~:I h Reason1 :EnergyRequiredf o rCrackFormation i tmustber e c o g n i 民 dt h a tw h i l ec r a ε ki n i t i a t i o n Fromt h es t r i c t l yp h y s i c a lv i e w p o i n t, maydependons t r e s s, t h ea c t u a lf o r m a t i o no fc r a c k sr e q u i r e sac e r t a i ne n e r g y t h ei r a c t u r e e n e r g y-whichr e p r e s e n t st h es u r f . a c ee n e r g yo fas o l i d . Hence ,energycriteriashouldbe u s e d .T h i s釘 gumentmights u 缶c et oap h y s i c i s tb u tn o t乱 d e s i g n e r . Butt h e r ea . r eo t h e r r e a s o n s . ← 園 ・ (d) ・ 嗣 . 自 ・ . ・ .・ .・ ・ 3 固圃. ( e ) l o a d Reason2 :ObjectivityofC a l c u l a t i o n s o r ymustbeo b j e c t i v ei nt h es e n s et h a tt h er e s u l to fc a l c u l a t i o n smade Anyp h y s i c a l山 e w i t hi tmustn o tdependons u b j e c t i v e ω p e c t ss u c h舗 t h ec h o i c eo fc o o r d i n a t e s, t h ec h o i c e o fmesh, e t c .I fathωryi sf o u n dt obeu n o b j e c t i v e , i tmustber e j e c 七e d .Therei snon e e d t oe v e n∞mparei tもoe x p e r i m e n t s .O b j e c t i v i t ycomesa h e a do fe x p e r i m e n t a lv e r i f i c a t i o n . l e m e n ta n a l y s i so fc o n c r e t ec r a c k i n gi s出 ec o n c e p to f Ap o w e r f u la p p r o a c ht of i n iもee sme a . r edc r a c k i n g, i n t r o d u c e dbyRa s h i d( 1 9 6 8 ) .Aαordingt ot h i sa p p r o a c h ,thes t r e s si na a t e r i a l, , anda f t e rr e a c h i n gt h i s f i n i t ee l e m e n ti sl i m i t e dbyt h et e n s i l es t r e n g t ho f出 em s t r e n g t hl i m i t,t h es t r e s si nt h ef i n i t ee l e m e n tmustd e c r e a s e .Asi n i t i a l l yp r a c t i 田 d ,出es t r e s s i nav e r t i c a ld r o p jb u t6 0 0 ni 七W剖 r e a l i z e dt h a t wasassumedt od e c r e a s es u d d e n l yt oz e r o, b e t t e randmorer e a l i s t i cr e s u l t sa . r eu s u a l l yo b t a i n e di ft h es t r e s si sr e d u c e dg r a d u a l l y , i . e ., t h em a t e r i a li sassumedt oe x h i b i ts t r a i n s o f t e n i n g( S c a n l o n, 1 9 7 1 ;L i nandS c o r d e l i s, 1 9 7 5 ) ; S田 F i g .1 .1 a .Thec o n c e p to fs t r a i n s o f t e n i n g , t h o u g h, p r o v e dt obeamixedb l e s s i n g .A f t e r a . i n s o f t e n i n gh吋 b e e ni m p l e m e n t e di nl a r g ef i n i t ee l e m e n tp r o g r a m . sandw i d e l ya p p l i e d, s t r i twasd i s c o v e r e dt h a tt h e∞n v e r g e n 偲 p r o p e r t i e sa . r ei n c o r r e c tandt h ec a l c u l a t i o nr e s u l t s . n a l y s t ' sc h o i c eo f山 emesh, i . e ., t h er e s u l t ss i g n i f i 白 川l y a r en o to b j e c t i v ew i t hr e g a r dt ot h ea c h a n g ei ft h emeshi sr e f i n e d(B 話 姐t ,1976,1982jBa . i a n tandC e d o l i n, 1 9 7 9, 1 9 8 0 ,1 9 8 3 ; B a i a n tandOh,1 9 8 3 a ;Darwin,1 9 8 5 jR o t s,Nauta ,KustersandBlaauwendraad,1 9 8 5 ) . . r ee n c o u n t e r e dwhenc r a c k i n gi smodeleda sd i s c r e t ei n t e r e l e m e n tc r a c k s, S i r n i l a rp r o b l e m sa ・~・ load A : f d e f L ( f ) ,~ crackI ength … energy i c h a n i c sor nonl o c a lf i n i t eelements { 山 刊 N o .o felements c r i t e r i o nand l o c α If i n i t eelements F i g .1 .1S p u r i o u sMeshS e n s i t i v i t y 9 8 ( α ) b剖 吋 ont h e山 e n g t hc o n c e p t( t m sa p p r o a . c hwasi n t r o d u 句 di n t of i n i t ee l e m e n ta n a l y s i sby 1 9 6 2, 組dbyNgoandScordelis, 1 9 6 7 ) . C l o u g h, e n s i も ivityωnbei l l u s t r a t e d, f o rexample , byt h er e c t a n Theproblemo fs p u r i o u sm包 hs g u l a rp a n e li nF i g .1 .1 bandc ,whichi ss u b j e c t e dt oau n i f o r mv e r t i c a ld i s p l a c e m e n ta tt h e t o pb o u n d a r y .As m a l lr e g i o nn e a rt h ec e n t e ro ft h el e f ts i d ei s掛 sumedt ohaveas l i g h t l y s m a l l e rs t r e n g t ht h 回 出er e s to ft h ep a n e l, andc o n s e q u e n t l ya .s mearedc r a . c kbands ta . r t s a . d i n gw i t htwof i n i t e g r o w i n gfroml e f tt or i g h t .Thes o l u t i o ni so b t a i n e dbyi n c r e m e n t a l l o e l e m e n tm e s h e so fv e r yd i f f e r e n tmeshs i z e sa sshown.Bys t a b i l i t yc h e c k si ti sf o u n dt h a t fs i n g l ee l e m e n tw i d t ha . tt h ec ra . c k i n gf r o n t t h ec r a c k i n gmusta l w a y sl o c a l i z ei n t o乱 bando ( F i g .l . lb, c ) . Thet y p i c a ln u m e r i c a lr e s u l t sf o r出 向 回 w e l l錨 v a r i o u so t h e rp r o b l e m sa r e a . t e di nF i g .1 .1 d, e, f .l nt h el o a d d e : f i e c t i o nd i 乱g ram( F i g .l . 1d ), i ti ss 伺 nt h a . tt h ep e a k i l l u s t r h ep o s t p e a ks o f t e n i n gi ss t r o n g l yd e p e n d e n tont h emeshs i z e , b e i n gr o u g h l y l o a da sw e l l剖 t p r o p o r t i o n a lt oh-1/2 wherehi st h ee l e m e n ts i z e .P l o t t i n gt h el o a d( r e a c t i o n )v e r s u st h e l e n g t ho ft h ec r a c kb組 d, l a r g ed i f f e r e n c e s制 a g a i nf o u n d( F i g .l . l e ) . r a c k i n gdeαe 出回 w i t ht h er e f i n e m e n to ft h ef i n i t e Thee n e r g ywhichi sd i s s i p a t e ddueもoc .l f )andc o n v e r g e st o0a sh→ O . e l e m e n tmesh( F i g.l Thef o r e g o i n gu n o b j e c t i v i t yi sp h y s i c a l l yu n a c c e p t a b l e .Theo n l ywa . yt oa . v o i di ti ssome formo ff r a c t u r emech a . ni c s .Bys p e c i f y i n gもhee n e r g yd i s s i p a t e dbyc r a c k i n gp e ru n i tl e n g t h o ft h ec r a c ko rt h ec r a c kband, t h eo v e r a l le n e r g yd i s s i p a t i o ni sf o r c e dt obei n d e p e n d e n t o ft h ee l e m e n ts u b d i v i s i o n( t h eh o r i z q n t a ld a s h e dl i n ei nF i g .l .l f ), a . n ds oi st h emaximum 1 0 0 0. platecu 他 ) 何 耐r i n g def. l F i g .1 . 2LoadD e s e c t i o nDiagramo fD u c t i l eandB r i t t l eS t r u c t u r e s Reason3 : LackofYieldPlateau B剖 edonl o a d d e f l e c t i o nd i a g r a . ms, onemayd i s t i n g u i s htwob a s i ct y p e so fs t r u c t u r a l f a i l u r e :p l a s t i candb r i t t l e . Thet y p i c a lc h a r a c t e r i s t i co fp l a s t i cf a i l u r ei st h a tt h es t r u c e d e g r e e o f f r e e d o mmechanisms u c ht h 抗 f a i l u r ei nv a r i o u sp a r t so f t u r ed e v e l o p sas i n g l t h es t r u c t u r ep r o c 田d ss i m u l t a n e o u s l y , i np r o p o r t i o nt oas i n g l ep a r a m e t e r .Suchf a i l u r e sa r e .2 a ) . m a n i f e s t e dbyt h ee x i s t e n c eo fal o n gy i e l dp l a t e a uont h el o a d d e f l e c t i o ndiagram( F i g .1 I ft h el o a d d e f l e c t i o ndiagramd o e sn o tha . v es u c hap l a t e a u, t h ef a i l u r ei sn o tp l a s t i cbu t b r i t t l e( o rb r i t t l e d u c t i l e )( F i g . 1 .2 b ) .I ft h e r ea r enos i g n i f i c a n tgωmetrice 仇c t ss u c h制 t h eP-se f f e c ti nb u c k l i n g, t h ea b s e n c eo fap la . t e a ui m p l i e st h ee x i s t e n c eo fs o f t en i n gi nt h e a . 1due七of r a c t u r e, c r a c k i n go ro t h e rda . m a g e ji ti m p l i e st h a t山 ef a . i l u r ep r o c e s sc a . n n o t m a t e r i d e v e l o pas i n g l ed e g r e e -o f ・ f r e e d o mmechanismb u tc o n s i s t so fp r o p a g a t i o no ft h ef a i l u r ez o n e t r u c t u r e .Soもhef a i l u r ei sn o n s i m u l t a n e o u sa . ndp r o p a g 乱t i n g . t b r o u g h o u t七hes Toi l l u s t r a t et h i sb e h a v i o r, c o n s i d e rt h ep u n c h i n gs h e a rf a . i l u r eo fas l a b( F i g .1 .3 ) .The t y p i c a l( a p p r o x i m a t e )d i s t r i b u t i o n so ft e n s i l es t r e s s σ a l o n gt h ef a . il u r es u r f a . c ea r edrawni n ft h em a t e r i a li sp l a s t i c, t h ec r o s ss e c t i o ng ra . d u a l l yp l a s t i c i z e su n t i la 1 1i t sp o i n t s t h ef i g u r e .I a r ea tt h ey i e l dl i m i t.However , i ft h em a t e r i a le x h i b i t ss o f t e n i n g, t h e nt h es も r e s speakmoves a c r o s st h ef a i l u r ez o n e, l e a v i n gar e d u c e ds t r e s s( s o f t e n i n g )i ni t swa . k e .Thes t r e s sr e d u c t i o n , i nw h i c h日 s e出 ep I 舗 t i cl i m i ta . n a l y s i si sn o t5 0f a ro f f . i sm i l do n l yi ft h es t r u c t u r ei ss m a l l I ft h es t r u c t u r ei sl a r g e, h o w e v e r, t h es t r e s sp r o f i l ed e v e l o p sa .s t e e ps t r e s sd r o pb e h i n dt h e p e a k s t r e s sp o i n t, andt h e r e f o r et h el i m i ta . n a l y s i ss o l u t i o n sg r o s s l yo v e r e s t i m a . t et h ef a . i l u r e (b) slze: plasticmoteri al (d) 平 large )L/ μt eigm ~-- 一、、/〆~~ ε " ) i L I/ 、 . . . .E ↑ ! ↑ ↑ ↑ F i g .1 .3P r o g r e s s i v eN a t u r eo fF a i l u r eI l l u 5 t ra . t e df o rPunchingS h e a ro f aS l a b 1 1 10 1 0 0 0 . Reason4 :EnergyAbsorptionC a p a b i l i t yandD u c t i l i t y Thea r e aundert h ee n t i r el o a dd e f l e c t i o nd i a g r a mr e p r e s 四 t st h ee n e r g ywhicht h es t r u c t u r ew i l la b s o r bd u r i n gf a i l u r eandmustt h e r e f o r ebeB u p p l i e dbyt h el o a d s .C o n s i d e r a t i o n o f出 i s四 位gyi simpor 同且te s p e c i a l l yf o rdynamicl o a d i n g ,anddeterminestheductilityof t h es t r u c t u r e .P l a s t i cl i m i ta n a l y s i sc 孤 g i v enoi n f o r m a t i o nont h ep o s t p e a kd e c l i n eo ft h e sp r o α別 . Someformo ff r a c t u r em e c h a n i c si sne 四 四 位y . l o a dandt h ee n e r g yd i s s i p a t e di n山 i 円1 0stlab tests Reason5 :S i z eE f f e c t Thes i z ee f f e c ti s, f o rd e s i g ne n g i n e e r s ,probablythemostcompellingre品目onf o ru s i n g ands oat h o r o u g hd i s c u s s i o ni si no r d e r . f r a c t u r em e c h a n i c s, Thes i z ee f f e c ti sd e f i n e dt h r o u g hacomp 釘 i s o no fg e o m e t r i c a l l ys i m i l a rs t r u c t u r e so fd i ι andi sc o n v e n i e n t l yc h a r a c t e r i z e di nt e r m so ft h en o m i n a ls t r e s s σN a tmaximum f e r e n ts i z e s, entheσN v a l u e sf o rg e o m e t r i c a l l ys i m i l a rs t r u c t u r 田 o fd i f f e r e nt ( u l t i m a t e )l o a d,九. Wh s i z e sa r et h es 晶me , wes a yt h a . tt h e r ei snos i z ee f f e c t .Ad e p e n d e n c eo fσN ont h es t r u c t u r e s i z e( d i m e n s i o n )i sc a l l e dt h es i 問 e f f e c t . o tr e p r e s 田 t姐 ya c t ua . ls t r e s si nt h es t r u c t u r eb u tma . ybe Then o m i n a ls t r e s sn田 dn u / b dwhent h es i m i l a r i t yi stw か d i m e n s i o n a l, o r出 凡/ c Fwhen d e f i n e ds i m p l y回 σN =P t h es i m i l a r i t yj st h r e e -d i m e n s i o n a l j b-t h i c k n e s so ft h etw 令 d i m e n s i o n a ls t r u c t u r e, 組dd a . ra c t e r i s t i cd i m e n s i o no ft h es t r u c t u r e, w h i c hma . ybec h o s e na . sa . nyd i m e n s i o n, e . g .パhe c h , o ri t ss p a n, s i n c eo n l yt h er e l a t i v ev a l u e so fσN m a t t e r . d e p t ho ft h ebeam A c c o r d i n gt ot h ec l a . s s i c a lt h e o r i e s, s u c ha se l a s t i ca n a l y s i sw i t ha l l o w a b l es t r e s s, p l a . si i c I im i tan a . l y s i s, a . sw e l l却 姐yo t h e rt h 田 r i e swhichu s esomet y p eo fs t r e n g t hl i m i to rf a i l u r e c r i t e r i o ni nt e r m so f耐 e s s e s( e . g .,v i s c o e l a . st i c i t y ,v i s c o p l a s t i c i t y ), σN i sc o n s t a n t,出品 i s, i n d e p e n d e n to ft h es t r u c t u r es i z e .T h i sma . ybei l l u s t r a t e d, e . g ., byc o n s i d e r i n gt h ee l a . s t i c 組 dp l a s t i cf o r m u l a . sf o rt h e柑 e n g t ho fbe a . msi nh e n d i n g , s h e a randt o r s i o n( r e g a . r d i n gt h e d e f i n i t i o n σN =凡/ b df o rt o r s i o n, n o t et h a . tonema . ys e t凡 =T u / rwhereTu = =u l t i m a t e t o r q u e,九==f o r c ea c t i n gonanarm , r , s u c ht h a もr / Ho rr / ai sc o n s t a n tf o rs i m i l a rs t r u c t u r e s o fd i f f e r e n ts i z e s jH =c r o s ss e c t i o nd e p t h, a=c r a c kl e n g t h ) .I ti ss e e nt h a t油 田ef o r m u l a s 釘 eo ft h esameforme x c e p tf o raf a . c t o r .Thus, i fwep l o もl o g σN v s .l o gd, t h ef a i l u r es t a t e s もr e n g t ho ry i e l dc r i t e r i o na r ea l w a . y sg i v e nbyah o r i z o n t a l l i n e( d a s h e dl i n e a c c o r d i n gt oas i nF i g .1 . 4 ) .Sof a i l u r e s乱c c o r d i n gt os t r e n g t ho ry i e l dc r i t e r i ae x : h i b i tnos i 田 e f f e c t . a . s t , f a i l u r e sg o v e r n 吋 b yl i n e a re l a s t i cf r a c t u r emech a . n i c se x h i b i tar a t h e rs t r o n g Byc o n t r s i z ee f f e c twhichi nF i g .1 . 4i sd 回 c r i b e dbyt h ei n c l i n e dd a s h e dl i n eo fs l o p e1 / 2 .Ther e a l i t y o ri l l u s t ra . t e dbyt h es o l i dc u r v ei nF i g .1. 4 .T h i s f o rc o n c r e t es t r u c t u r e si sat r a n s i t i o n a lbeh前 i c u r v ea p p r o a c h e st h eh o r i z o n t a l l i n ef o rt h es t r e n g t hc r i t e r i o ni ft h es t r u c t u r ei sv e r ys m a l l, 組 dt b ei n c l i n e ds t r a i g h tl i n ef o rl i n e a re l a s t i cf r a . c t u r em e c h a n i c si ft h es t r u c t u r ei sv e r yl a r g e v e r ys m a l l "姐 d“ v e 可 l a r g e "w i l lbec l a r i f i e dbyE q .5 . 1 1 ) .T h i s ( t h ep r e c i s emeaningo f“ s i z ee f f e c t, w h i c hi sg e n e r a l l yi g n o r e dbyc u r r e n tc o d e s( w i t haf e we x c e p t i o n s ), i so b v i o u s l y i m p o r t a n ti nd e s i g n . i 1 i t y . Anothers i z ee f f e c twhichc a l l sf o rt h eu s eo ff r a c t u r em e c h a n i c si se f f e c to fs i z eond u c t larger moststructures ・ s ; z e ) log ( F i g . 1 . 4 F T a c t u r eM?chmesSizemeetf o rGeometriqllySimilustmct u r e so fDi J f e r e n tS i z e s nom. stress O N re. lde f l . F i g .1 .5L o a d D e f l e c t i o nD i a g r a m so fG e o m e t r i c a l l yS i m i l a rS t r u c t u r e s 四 o fD i f f e r e n tS i z (b) v . ~~三 F i g . 1 . 6 P o s s i b l eF i n a lF r a c t u r eL o c a t i o n sCausedbyRandomnesso f S t r e n g t h 円・園田・ーー 幽 12 13 D u c t i l i ho fas t r u e s u r emaybec h a r a C 4 e r i z e dbyt h ed e f o r m a t i o na twhich出 es t r u c t u r ef a i l s i ng e n e r a l, f a i l u r eo c c u r sr i g h ta . tt h ei n i t i a t i o no ft h em a c r o s c o pi cc r a c kg r o w t h .F o rt h o s e s 山 ctures, f u n c t i o n σ(P , x)i sk n o w n ; σ(P , x )=P S e { と )where(=x/ d=relativeωrdinate d= c ha . r a . c t e r i s t i cd i m e n s i o no ft h es t r u c t u r e, ands e ( { )i st h ee l a . s t i cs t r 回 sd i s t r i v e c t o r s, b u t i o nduet ou n i t1 0 0 0(P= 1 ) .Thes 3 0 m ea s s u m p t i o n sha . v eo f t e nb e e ni m p l i e di nv a r i o u s W e i b u l l t y p es t u d i e so fc o n c r e t es t r u c t u r e s . C o n c r e t es t r u c t u r e s, h o w e v e r , b e h 3 0 v ed i f f e r e n t l y . Duet or e i n f o r 四m enta sw e l la st h e 3 o i n s o f もe n i n gi naI a . r g ez o n eo fm i c r o c r a c k i n ga h e a do f出 et i po fao o n t i n u o u s e x i s t e n c eo fs t r 低k ,concretestruct町 田 donotf a i la tc r a c ki n i t i a t i o n .I nf a c t, d e s i g nc o d e sr e q u i r et h e c r f a i l u r e10Mt obes i g n i f i c 叩 t l yh i g h e rt h a nt h ec r a c ki n i t i a t i o nl o a d ;f o rb e n d i n g, . a tl e a . s t1 .2 5 e s s e db e a . m sand1 .2t i m e sh i g h e rf o rp r e s t r e s s e dbe a . m s( a c c o r d i n g t i m e sh i g h e rf o ru印 刷r もoA CI3 1 8 ), b u ti np r a . c t i c et h i sr a t i oi su s u a l l ymuchh i g h e r .C o n s e q u e n t l y , 品 ∞ncrete a c r o s c o p i cc r a c kgrow t hp r i o rt o s t r u c t u r eu n d e r g o e sp r o n o u n c e di n e l a s t i cd e f o r m a t i o n岨 dm s u c ht h a tt h es t r e s sd i s t r i b u t i o n r e a c h i n gt h ef a i l u r el o a d .T h i sc a u s e ss t r e s sr e d i st r i b u t i o n s, σ(P , x)a . tf a i l u r ei sv e r yd i f f e r e n tf r o mt h ee l a s t i cs t r e s sd i s t r i b u t i o ns e( C ). T h i sd i s t r i b u t i o ni sd i f f i c u l tt od e t e r m i n e, b u tt h en e a r t i pasympto t i ce l a s t i cs t r e s sf i e l d , PS ( { ) , m a . y b e u s e d a s a n 叩 p r 似 i m a t i o n3 0 も d i s t a n c e sn o t t o o f a . r f r o m a n d n o t t o o c 1 0 s e e a c r o c r a c ka . tt h emomento ff a i l u r e .Now , duet os i n g u l a r i t yo ft h i sf i e l d, t ot h et i po f出 em t h es t r e s sv a . l u e sf ,紅白e rawayfromt h et i po ft h em a c r o c r a c ka r er e l a t i v e l ys m a l la . ndmake ' N ( E q .1 .2 )∞m p 3 0 r e dt o出e山 間 偲 i nt h evolume竹 o ft h e an e g l i g i b l ec o n t r i b u t i o nt oO 台a c t u r ep r o c e s sz o n e砿 oundt h et i p .T h i so fc o u r s er e f l e c t st h ef a . c tt h a tt h evolumei nw h i c h t h em a c r o c r a c kt i pa tf a i l u r em i g h tbel o c a t e d( a sd i c t a t e dbyt h el a w so fm e c h 3 0 n i α )i sv e r y s m a l l( e . g .t h ed i a g o n a ls h 悶 c r a c ki nF i g .L6bc a . n notgrowt o w a r dt h el o w e rmidspan , r e g a r d l e s so ft h es t r e n g t hv a l u e st h e r e ) . regiωort o w a r dt h eu p p e rl e f tc o r n e ro ft h ebeam C o n s e q u e n t l y ,thestatistica . ls i z ee f f e c tmustbes m a l l e rt h a ni nau n i f o r m l ys t r e s s e d wheref a . il u r ei sp r e c i p i t a t e dbyam3 o c r o c r a c k3 0 n y w h e r ew i thi nt h evolumeo ft h e t e n s i l eb a r, a sp o i n t e do u tbyBa . zant(1986,1987a), i fo n ec a l i b r a t e sWe i b u l lpa . r ame t e r 目 b a r . Thus, m 姐 dσ ont h eb a s i so fu n i a x i a . ls t r e n g t ht e s t sa . n dt h e nu s e st h esamep3 0 r a m e t e r st o o rdi 乱g o n a ls h e a rf a i l u r e , o n emustf i n dt h es t a t i s t i c 3 0 ls i z ee f f e c tt obe p r e d i c tPJandO'N f e v e nt h o u g hl i m i t e dt e s tr e s u l t sf o rd i a g o na . ls h e a rf 3 0 i l u r e sha . veb e e n r a t h e rs m a l l . Thus, s u c c e s s f u l l yf i t t e dbyf o r m u l a sb a s e dont h eW e i b u l ld i s t r i b u t i o n, t hes i z ee f f e c td 3 0 t af o rb o t h t h ed i a g o n a . ls h e a . rf a i l u r e sa . ndt h eu n i a x i a lf a i l u r e sc a n n o tbes u c c e s s f u l l yf i t t e du s i n gt h e samem a t e r i a . lp a r a . m e t e r s( u n l e s s姐 i 且∞ r r e c もe l a s t i cs t r e s sf i e l dP s . ( ( )i su s e d ) . Thus , t h ep r i n c i pa . lf a u l to ft h eW e i b u l l t y p es t3 o t i s t i c a . le x p l a na . t i o n so ft h es i z ee f f e c ti n 民 e f f e c tc a u s e dbyt her e d i s t r i b u t i o no fs もr e s s c o n c r e t es t r u c t u r e si sthωtheyi g n o r et h es i σ( P , x )priort ω of . 仏 悩 a 必i l u r e .T h i ss i z ee f f e c t, w h i c hi so ft h ef r a . c t u r em配 ha . n i c st y p e , i s踊 S 民 i a . t 吋 w i t ht h ee n e r g yr e l e a s ei n t ot h ef r o n to fa .l a r g ec r a c ka . n dwo ul de x i s te v e ni ft h em a t e r i a l b e h a v i o rw e r ed e t e r m i n i s t i c . t h ep r o p e ra p p r o a c hi st of i tt h es i z ee f f e c td a t a .f o rac O D e r e t es t r u c t ur ef i r s t T h e r e f o r e, .f r a c t u r em e c h a n i c st h e o r y, . a ndo n l yi fsomep a r to ft h eo b s e r v e ds i z ee f f e c trem a . i n s bya u n a c c o u n t e df o ri tm晶ybea t t r i b u t e dt oW e i b u l l t y p es ta . t i s t i c a lphe no me na( Ba . z a n t, 1 9 8 6, 1 9 8 7 a ) . I lt h e o r yt h e nd e s c r i b e sas i z ee f f e c to ft h evolumeo ft hes t r u c t u r e, expr e s si n gt h e Weibu m お む ニ; r r r ニむ と2 2 J ぷ立 エ 7 1 J i ozrJ;i;:::; :eu; A i s pl a 捌 l e n tc o n k o l( ie .i 叫 叫 m n 叩 l p O ω s 吋 d i s が p μ I a . 蹴 c 回e 悶醐 m 1 吋 i c e 偲s f a 芯 i li 泊 n色 出 h 隠 凶 e i rp o s t p e a k, s t r 3 o i n s o f t e n i n gr 却炉・ I na .p l o to fσN v e r s u sd e s e c t i o n, t h e d e v 3 o r a . c t e r i z e dbyaもmgent(dashdl i n ei nFig.15)dac e r t a i neon 伽 n も f a i l u r ep o i n ti sch な丘公立政ぷh ; 2 2 e ; 出品治誌とぷねじ ; Z d t t ; : 沈 t y p es t 1 O L ni nFig.15.Asi l l u s t r晶,t e d, f a i l u r eo c c u r sc l o s e rt ot h epeak回 S l z emcre 国 ω . T h i se f T e di s a g a i ng e n e r a l l yp r e d i c t e dbyf r a c t u r emechanichduet ot h ef a c tt h a ti na l 3 0 r g e rs t r u c t u r emo 陀 s t r 3 0 i ne n e r g yi s肝 包l 3 0 b l eもod r i v et h ep r o pa . g a t i o no ft h ef a il u r ez o n e . Ad e c z e a s eo fd u c t i l i t yo fas U 1 1 c t u r er e p r e s e n t sani n c r e a s ei ni t sb r i t t l e n e s s Thew e l l k n o w ne f f e c to fs t r u c t u r es i z eo rm四 l b e rs i z eonc r 3 0 c ks p 3 0 c i n ga . n dc r a c kw i d t h a . n i c 渇 ・ T hes p u 巾 u se f f e c to fmeshs i 田(Re剖on2 )c 組 i sa l s oe x p l i C 3 0 b l ebyf r 3 0 c t u r emech ber e g a . r d e da s0 3∞n s e q u e n 田 o ft h es t r u c t u 凶 s i z ee f f e c t( t h i sc 姐 b eshownby∞n s i d e r i n g s t r u c h u m so fd i f e r e n ts i z ebuwitht h esamemeshsim , 組dt h e ns c a l i n ga l ls t r u c t u r e st o o l o n gw i t ht h em e s h e s ) . t h esames i z e3 1 .2I sWeibull'sS t a t i s t i c a lTheoryofS i z eE f f e c tApplicable? T r a d i t i o n a l l y , t h es i z ee 仔e c th a sb e e ne x p l a i n e ds t a t i s t i c a l l y , byr a . n domnesso f出 es t r e n g t h value.Thef a i l u r el o a do fac h & i ni sdderminedbyt h eminimumv a l u eo ft h es t r e n g t ho f もt h el o n g e rt h e t h el i n k si nt h echain-andt h esutMicalBime t r e e ti sduetobhef a c ttha c h a i n , t h esm a . l l e ri sふ耐e n g 出 v a l u et h a ti sl i k e l yt obee n c o u n t e r e di nt h ec h a i n .T h i s e x p l a n a t i o n ,whichi sc e r t a i n l yc o r r e c tf o rt h esimebdo b s e r v e di nt h ef a i l u r eo fal o p g i sd e s c r i b e dbyWeibd'sw 哨 e 北 a k 悶 e 剖s 凶 も l l n k f o r m I vs h r e s s e dc o n c r e t ebMi nh e n S i o n( F i g .l a ), s “ t a “ t 蜘 i s “ t d 3 0eq u e s t i o n a b l ef o rt h ef o l l o w i n gr e a ω s o n . 凶 抗 胤 凶 紅 . r 問 , 1 9 3 9 ; Z a i b e v姐 dWitt 中ann, 1 9 7 3 ; A c c o r d i n gt oW e i b u l l w p et h e o r yo f f a i l u r e ( W e i b u l l M i l 邸 h ia . ndZ a i t s e v, 1 9 8 1 ;組 dC a r p i n t e r i, 1 9 8 6 ), t h ep r o b a b i l i t yo ff a i l u r eo fas t r u c t u r e u n d e r1 0 0 0Pωdt h emeannomin a . ls t r e s s3 0 tf a i l u r ea . r e : 日 r Lr但 豆 号斗 ( P )=1-e xp{- I JVIσ'0 O ' N I Vr =主=土 f ∞[ 1-P帥 ( P ) ] d P J ( 1 . 1 ) ( 1 . 2 ) b d b d1 0 WI l e z eP=meanload.x=coordinatev e c t o r s , V=volumeo fs b mctuEe , E 4=r e p r e s e n も 3 0 t i v e volumeo fm a h e r i a l{ t h es m a l l e s tvolumef o rwhicham a t e r i a lw i t hd i s c r e t emicrostrmure t i n u u m ), m =W e i b u l lmoduluso ft h ema も e r i a . l , σ。 =sca . l ep a r a m e t e r . anbet r e 3 0 t e d回 a n sf u n c t i o nσ(P , x ), r e p r e s e n t i n gt h es t r e s s Thekeyt oa p p l i c a t i o n so f U 1 eW e i b u l l山 田 町 i s e dbyl o 3 o dP 3 0 tp o i n tx( f o r伽 t h 並r 砕 e e 吋d im 叩 e n 凶 s B 叩 i 叩 1mp 戸 ふ ふ i k h T n m 1 im u ∞ , 。 1 4 1 5 f a c も もh a tt h el a r g e rt h evolume , t h eg r 帥 t e ri st h ec h a n c eo fen ∞unteringacriticalmicroscopic 乱r加 t h el o a . da ti n i t i a t i o no fc r a ε k i n g( o rdamage)i s f l a wt h a tt r i g g e r sf a i l u r e . Thus, 回 f ∞n伺 med, W e i b u l l t y p e油 田r yi s, ofωurse, a p p l i c a b l e .S a o u r i d i s( 1 9 8 9 )demons t I 叫edt h a t 田 e f f e c ti nu n n o t c h e db e a m s .I n d 田 d ,出 l o n ga . st h e r e bya n a l y s i so fL' H e r m i t e ' st e s t so fs i a r enos 紅 白8 r e d i s t r i b u t i o n s, i . e . ,a l ls t r 側 部 a r ef i x e di nt h e i rp r o p o r t i o nt ot h e1 0 0 0 ,the i n t e r a c t i o no ft h ee l e m 四 t so ft h es t r u c t u r ei sm a t h e m a t i c a l l ye q u i v a l e n tt oas e r i e sc o u p l i n g o fe l e m e n t s , t h esamea . si nau n i a x i a l l ys t r 邸 s e db a ro fv a r i a b l ec r o s ss e c t i o n .When出 e r ea r e h o w e v e r, t h es t r u c t u r eb e h a v e sa sa∞m b i n a t i o no fs e r i e sandp a r a l l e l s t r e s sr e d i s t r i b u t i o n s, c o u p l i n g s . ButW e i b u l l ' st h e o r yi sva . l i do n l yf o rs e r i e sc o u p l i n g, 舗 i nac h a i no fe l e m e n 旬 ( h e n c et h eterm“ w e a k e s t l i n k山 t i s t i c s ") . 吋 . . - +-lz 1 b (D) hosuh︿ド・ H ・凶広 { ロ) whereB = (2G b/h) 1/ 2 / f ;, d o= (h/2k)(d/α )=r e f e r e n c es i z ew h i c hd e p e n d sont h es h a p e jE o ft h es t r u c t u r eb u ti si n d e p e n d e n to fs t r u c t u r es i z ei ft h es t r u c t u r e sa r eg e o m e t r i c a l l ys i m i l a r / α=c o n s t a n t ) ;刀 =t e n s i l es t 問 n g t h, i n t r o d u c e df o rc o n v e n i e n c e jandh= w i d t h ( b e c a u s ed o ft h ec r a c kbandf r o n t, whichmaybet r e a t e da p p r o x i m a t e l y出 c o n s t a n t, i n d e p e n d e n t C t u r es i z e . E m p i r i c a l 1 y, do竺 nd hereda = maximuma g g r e g a t es i z eandn i s o fs t r u aw sv a r i e d, b u td e p e n d sons t r u c t u r eg e o m e t r y .E q .1 .4 a p p r o x i m a t e l yc o n s t a n twhend01' d ai w i l lbed i s c u s s e dmorei nC h a p t e r5andd e r i v e di nAppendix1 . ++zb ( 1 .4 ) ︻ J 'i s=d / ゐ ~ 4ト 。d , . 、r、・ y whereI I=卯 po t 総e 凶 n t i 凶 叫e 叩n e 釘r g y “ 自 t 町 o r 吋 i 泊 nt 出 hes 蜘 t む r 町 uc t 旬u r 凧 e , b=t h i c k n e s sand匂 Gf=f r a c t u r ee 叩n ぽ e r g y 2) ( d i m e n s i o n刀 J 1 ' mη =en 巴r 喝g yn e e d e dt ω o c α r 白 t 旬e晶 f 仕 r 配 a c t 削u r 児e 叫 o rc . α r a c khando fu山 n 1 t h et h i r dd i r e c t i o n )u n i tw i d t h . Thev a l u eo fGj i sa p p r o x i m a t e l y∞n s t a n tandr e p r e s e n t s .2f o rt h en o m i n a ls t r e s s, oneo b t a i n st h es i z ee f f e c 七 am a t e r i a lp r o p e r t y .S o l v i n gfromE q .1 lawp r o p o s e dhyB a z a n t( 1 9 8 4 a ) : σN= B f:(I+β)ー1/2, ‘ - ug UU仏∞ω切﹄咽阿古母音目的包ヨHEmEugu 刷局む回同 Ud ~ Thef r . a c t u r em e c h a n i c st y p es i z ee f f e c t, w h i c hi sduet oe n e r g yr e l e a s e ,canbesimply a . l e x p l a i n e dbyc o n s i d e r i n gt h eu n i f o r m l ys t r e s s e dp l a n e lw i t hac r a c ko rc r a c kbando fi n i t i l e n g t ha : 0 ao, showni nF i g .1 .7 .I tmaybeimaginedt h a tt h ef o r m a t i o no f. ac r a c kband o ft h i c k n e s shr e d u c e st h es t r a i ne n e r g yd e n s i t y σゐ /2Ei nt h ec r o s s h a t c h e da r 伺 t oz e r o(E :0 e l a s t i cmoduluso fc o n c r e t e ) . Whent h ec ra . c kba . nde x t e n d sbya, t h ea d d i t i o n a ls t r a i n e n e r g yt h a ti sr e l e a . se dcomesfromt h ed e n s e l yc r o s s h a t c h e ds t r i po fh o r i z o n t a ld i m e n s i o n .7 a ) .I ft h ef a i l u r emodesa r egωmetric a . l l y5 i 凶 l a r, 拙i su s u a l l y由 ec 剖 e , t h e n Aα(Fig.1 t h el a r g e rt h ep a n e l, t h el o n g e ri st h ec r a c kbanda tf a i l u r e .C o n s e q u e n t l y , t h ea r e ao ft h e b e i n gg i v e nbyh6a+ 2ka6awherek=e m p i r i c a l d e n s e l yc r o s s h a t c h e ds t r i pi sa l s ol a r g e r, c o n s t a n td e p e n d i n gont h es ha . p eo ft h es t r u c t u r e .T h i si l l u s t r a t e st h a . t , i na .l a . r g e rs t r u c t u r e , moree n e r g yi sr e l e a . s e dbyt h esamee x t e n s i o no ft h ec r a c kb a n d .Thee n e r g yr e l e a s e dfrom t h es t r i pi s6 I T / 6 α : 0 Gjb , i .e ., 土 (h6a+2ka6a) 生 , b 6a、 ' 2E=G -J 回同 1 .3SimpleEnergyExplanationo fS i z eE f f e c t z y コ 17 16 n em i g h tg e t出 ei m p r e s s i o nt h a t h h i se x p l a n a t i o no ft b es i z ee f e c tw o r k so n l yf o r L回 to p d b andb u tn o tforashrpl i n ec r a c k, c o n s i d e rt h es a g E ep a n e lwithalinec r a ι ko f 山 F i g .1 .7bf o r5凶 叫a n e l so f制 e r e n ts 問 Iωn 伺 いherei sa l w a y s lengthωhow h e a dofthetipofacontinuouscmkof回 mef i n i t el e n g t h i z e a b l e合acurepm-zonea , i nt h ec r u d e s 七a p p r o x i m a t i o n , bec o n s i d e r e dcoastanb.Overb h el e n g t ho ft h h i c hmay . Becauseo f出 i sz o n e, t h e zn t h et r a n s v e r s enormalstmsg r a d u a l l yd r o p sfromれ oO 部 t i c a l l y叩 l V 伽 tc r a c kl e n g t hwhichc a u 附 t h e附 a ω s s 鈴e 刊0 o fs t r 日 a 叩 , t 旬e r i 凶 a l i 抱 sl o n g e r出 t h釦 a nt h ec o n t i n u o u sc α r a 配 . c k1 佃 g t 出 h, a, bya .d i s t 組 問 c whichi sa p p r o x i m a t e l y ,cvazies, b u tmuchl e s sbhani np r o p o r t i o ntoUIe am a t e r i a lC O I l s h a r i b . 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Z a : 組 . n t組 dCa ω . 0, 1 9 8 筋6 吋 a ) . Comparisonswiththebulko ft e s td a t aa . v a i l a bl ei n出 e l i t e r a . t u r eweremadef o rd i a g o n a ls h e a rf a i l u r e so fb o t hu n p r e s t r e s s e dandp r e s t r e s s e dbeams, 姐 db eamsw i t h o u tandw i t hs t i r r u p s( B a z a n tandKim, 1 9 8 4 ;Ba . Z 叩ta ndSun, 1 9 8 7 ;Ba . Z a n t 姐 dCω, 1 9 8 6 b ), a sw e l la st o r s i o n a lf a i l u r e s( B a z a : ntandS e n e r1 9 8 7 ) .S t a t i s t i c a la n a l y s i s W剖 i n c l u d e di nt h e s es t u d i e s . 23 p P Series Model P Uc Uc r Uel 1 .5E xplanationo fS i z eE f f e c tonD u c t i l i t y S t r u c t u r a la c t i o ni sn o r m a l l yac o m b i n a t i o no fs e r i e sandp a r a l l e lc o u p l i n g so ft h ec r a . c k i n g 却 l a i n e dbyt h es e r i ω z o n e sandt h eu n c r a c k e d( e l a s t i c )z o n e s .Thes i z ee f f e c tond u d i l i t yi se c o u p l i n ga s p e c t .C o n s i d e rac r a c k i n ge l e m e n tc o u p l e dw i t hane l a s t i ce l e m e n ta sshowni n F i g .1 .1 3 .Thel o a d d i s p l a c e m e n td ia . gramso ft h e s ee l e m e n t sa r ea l s os h o w n .S i n 白 山ef o r c e t h er e s p o n s eo fb o t h i nb o t he l e m e n t si st h esameandt h ed e f o r m a t i o n sa r es u p e r i m p o s e d, e l e m e n t scombinedi so b t a i n e dbyp a s s i n gah o r i z o n t a l l i n ea te a c hl e v e lP andsummingt h e , 舗 s howni nF i g .1 .1 3 .I ft h ee l a s t i ce l e m e n ti ss u f f i c i e n t l y c o r r e s p o n d i n gd e f o r m a t i o n sa回 db s o f t, t h i sc 組 o b v i o u s l yp r o d u c eal o a d d i自p l a c e m e n td i a g r a mw h i c he x h i b i t st h es o c a 且e d s n a p b a c k, i nw h i c ht h ed i s p l a c e m e n td i . a g r . a mt u r n sb a c k. a tap o s i t i v es l o p e .Thes n . a pback b e h a v i o ri su n s t a b l ee v e nu n d e rd i s p l a c e m e n t∞n t r o l, andt h es t r u c t u r ef a i l sa tt h emaximum d i s p l a . c ement, l a . b e l e da sU cr • T h i sd i s p l a c e m e n tr e p r e s e n t sa .d u c t i l i も yl i m i tf o rt h es y s t e m . S i n c et h ea d d i t i o no fa : ne l a . s t i ce l e m e n ti se q u i v . a l e n tt oi n c r e a s i n gt h es i z eo ft h es t r u c t u r e , i ti sc l e a rt h a t姐 i n c r e a s eo fs i z et e n d sもod 田 r e a s ed u c t i l i t y .D u c t i l i t yi sn o tama . t e r i a l a n i c sandd e p e n d s p r o p e r t yb u tas t r u d u r ep r o p e r t yw h i c hi sg o v e r n e dbyf r a c t u r em配 h ons t r u c t u r es i z e( a sw e l la st h et y p eo f∞n c r e t e ) .I ti sworthn o t i 碍 t h . a tHawkins( 1 9 8 4 ) i d e n t i f i e dt w e n t y n i n ep r o v i s i o n si nt h eACICodew h i c hseemt obee m p i r i c a lb u tc o u l d . r i o u sd u c t i l i t yl i m i泊 t i o n s, p r o b a b l ybee x p l a i n e dbyf r a c t u r em e c h a n i c s . Theyi n c l u d eva c o n d i t i o n sf o rminimumr e i n f o r c e m e n t, c r a c ks p . a c i n gandc r a c kw i d t h, e t c . h ee x p e r i m e n t a l l yo b s e r v e ds t r u d u r a ls i z ee 宜e c t, 国 wella . st h er e l a t e d Tosumup,t , p r e s e n t st h emo s tp o t e nta r 伊 m entf o rt h ea p p l i c a t i o no f s p u r i o u se f f e c to ft h emeshs i z e a n i c st oc o n c r e t es t r u c t u r e s . f r a c t u r em民 h F i g .1 . 13Ch 臼 g ei nP o s t P e a kLoad-De B . e c t i o nDiagramDuet oSe r i e s C o u p l i n gw i t hE l a s t i cE l e m e n t Mode I 江 E F i g 2 .1ModesI , IImdI I I( O p e n i n g , PMeshmAnti-PlaneSh 問 Fl ' a c t u r e s ) U 2 4 25 0, C h a p t e r 2.ESSENTIA_L !!-~S!!_LTS FROMLINEAR ELASTICFRACTUREMECHANICS I nl i n e a re l a s t i cf r a c t u r em e c h a n i c s(LEFM)i ti sassumedt h a ta l lo ft h ef r a c t u r ep r o c 倒 happensa t出 ec r a c kt i pandt h a ももh ee n t i r evolumeo ft h ebodyr e m a i n se l a s t i c . Under t h eq u e s t i o n so fc r 拭 kp r o p a g a t i o nands t r u c t u r a lf a i l u r ec a nbes o l v e dby t h i sa s s u m p t i o n, methodso fl i n e a re l a s t i c i t y . i s t i n g u i s ht h r e ee l e m e n t a r y{ r a c t u r emod 民 M odes1 , I I姐 dI I I, a 1s o I ti sc o n v e n i e n ttod t h ep l a n a rs h e a rmodeandt h ea n t i p l a n es h e a rmode;s e eF i g .2 .1 . c a l l e dt h eo p e n i n gmode, Modes1andI Ia r ep l a n a rs y m m e t r i ca . nda n t i s y m m e t r i c, w h i l eModeI I Ii st h r e e -d i m e n s i o n a l . . a durei sal i n e a rc o m b i n a t i o no ft h e s et h r e em o d e s . G e n e r a lf r stress r e l i e f zone 2 . 1S t r e s sS i n g u l a r i t y I n t r o d u c t i o no fa .c r a . c ki n も oa .l i n e a re l a s t i cbodyp r o d u c e ss t r e s sc o n c e n t r a . t i o n sn e a rt h e a . t e dbyt h ep e r t u r b a t i o no ft h et r a j 民 t o r i 田 o ft h emaximum c r a c kt i p s .T h i smaybei l l u s t r wi t ha 1 1t h e p r i n c i p a ls t r e s sshowni nF i g .2 . 2 . Thes t r e s sf i e l di ss i n g u l a ra tt h ec r a c kt i p, 伺 S∞ mponentsa p p r o a c h i n gi n f i n i t ya st h er a . di a ld i s t a n c erfromt h ec r a c kt i p D o n z e r os t r t h es 紅 白B t e n d st oz e r o( F i g .2 . 3 ) .l nas u f f i c i e n t l yc l o s en e i g h b o r h o o do ft h es h a r pc r a c kt i p, components町 j a r et h esamer e g a r d l e s so ft h es h a p eo ft h ebodyandt h emannero fl o a d i n g , 姐 dm aybee x p r 田s e d部: σL=I ufl( l } ) ( 2 付 t ,σu=J(llfi~[(())(2附)ー1/2 , σUI = kmLYI(6)(2πrt /2 1 12 / F i g .2. 2P r i n c i p a lSt r e s sT r a j e c t o r i e si naC r a c k e dSpecimen ( 2 . 1 ) H e r et h es u b s c r i p t sands u p e r s c r i p t s1 , 1 1and1 1 1r e f e rt ot h ee l e m e n t a r ymodes, 8i st h ep o l a r J( I, /(IIa nd1(111a r ep a r a m e t e r sc a l l e d七hes t r e s si n t e n s i t yf a c t o r s, andf u n c t i o n sん a n g l e, a 削 h esamer e g a r d l ω so ft h ebodygeom 向 a ndt h emannero fl o a d i n g .F o rexample , f [ l( 0 ) =0ωα(1-s i n α s i n 3 α ), f( 0 )=∞sα(1+sinαsin3α), f l 2 ( 0 )= Cοsαsin2αcos3α, where / 2 ;s e ee . g ., Kn o t t( 1 9 7 3 ), B r o e k( 1 9 7 4 ), OwenandFawkes( 1 9 8 3 ), H e l l a n( 1 9 8 4 ), 回d α=0 o p e l a r( 1 9 8 5 ) . Kanninen姐 dP ム 2 . 2EnergyC r i t e r i o n Thef a c tt h a t,a c c o r d i n gt ot h et h 田 r yo fe l a s もi c i t y , もh es t r e s sn e a rt h ec r 晶c kt i pa p p r o a ε h e s . n n i t y , nom a t t e rhows m a l lt h el o a d , wasn o t e dbyG r i f f i t h( 1 9 2 1, 1 9 2 4 )ont h eb a s i so ft h e i n i fl i n e a re l a s t i c i t y p r e v i o u ss o l u t i o nf o re l l i p t i c a lh o l e sbyI n g l i s( 1 9 1 3 ).Hec o n c l u d e dt h a t, onec a n n o ti n t r o d u c eas t r e n g t hc r i t e r i o na sac o n d i t i o no ff a i l u r e , b凶 musti n s t e a d i su s e d, n e r g y d e c i d ef a i l u r eont h eb a s i so fane n e r g yc r i t e r i o n . Ast h ec r a c kt i pp r o p a g a t e s,e n e r g yf i o wi s f l o w si n t ot h ec r a c kt i pw h e r ei ti sd i s s i p a t e dbyt h ef r a c t u r ep r o印 5S. Thee c h a r a c t e r i z e dbyt h ee n e r g yr e l e a s er a t ew h i c hi se x p r e s s e d出: θ l l ( α )_ . 1r T T I ,Lla, TTI . d .aJ l H ( a +す )-I I ( α ーτ ) J b=ーすf 2E ( 2 . 2 ) a . rC r a c kT i p F i g . 2 . 3S t r e s sD i s t r i b u t i o nNe 26 27 i nwhichI I= U- W = p o t e n t i a len 釘g yo ft h es t r u c t u r e ,W = workofloads, andU = s t r a i ne n e r g yo ft h es t r u c t u r ea saf u n c t i o no ft h ec r a c kl e n g t ha .E q .2 . 2a l s og i v e saf i n i t e d i f f e r e n c ea . pproxim 叫i o nwhichma . ybeu s 吋 t oc a . l c u l a も eG byt h ef i n i t eelem 叩 t method. F o rt h a tp u r p o s eonema . ymodelt h ec r a c k剖 al i n egapb e t w . 四na d j a 伺 n te l e m e n t sa . nd c a l c u l a . t et h es t r a i ne n e r g ys t o r e di nt h emeshf o rt h ec r a c kt i pd i s p l a c e dbye i もh e r-Aα/2 a / 2 .I n s t e a do fu s i n g姐 i n t e r e l e m e n tl i n ec r 拭 k , onemay , f o rt h es a k eo fc o n v e n i e n c e , o rA modelt h ec r a c kbyωnsideringabando fe l e m e n t st oh a v ez ぽ os t i f f n e s s( i foneu s e samesh 姐a. b o u t0 . 1o ft h ec r o s ss e c t i o nd i m e n s i o n, t h e o fs q u a r ee l e m e n t swhoses i z ei sn o tmoret h 閃 B a Z a . n t e r r o ro fc o n s i d e r i n gac r a c kbandi n s t e a do fal i n ec r a c ki su s u a l l yl e s s山 拍 1%;s 姐 dC e d o l i n, 1 9 7 9 ) . 伍t h, t h ec o n d i t i o no fc r a c kp r o p a g a t i o n( c r i t i c a ls t a t e )i s A c c o r d i n gt oG r i s i z 回, t h es t r e s si n t e n s i t y{ a c t o ri sp r o p o r t i o n a lt o出 es q u a r er o o to fもhes i 田, a ndt h ee n e r g y r e l e a s er a t ei sp r o p o r t i o n a lt ot h es i z eo ft h es t r u c t u r e . t h ec o n d i t i o no fmode1c r a c kp r o p乱 . g a t i o n( c r i t i c a ls t a t e )c a nbe I n s t e a do fE q .2 . 3, e x p r e s s e di nt e r m so ft h es t r e s si n t e n s i t yf a c t o ra s : , Gi st h ef r a c t u r ee n e r g y , whichh a sもhed i m e n s i o no fJ/ が ( o rN/m)andr e p r e s e n t saba . si c m a t e r i a lp r o p e r t y .F o rG く G , t h ec r a c kc a n n o tp r o p a g a t e ,andforthecaseG >G e q u i l i b r i u mi si m p o s s i b l e .I fG =G anddG / θa>0, w h i c hi sn o r m a l l yt h ec a s e , t h ec r a c k 吋 c o n t r o l( i . e ., t h es t r u c t u r ef a i l s ) .Theree x i s t, ' howev e r , somec a s e si n i su n s t a b l eu n d e r1 0 andt h e nt h ec r a c kc a ngrowu n d e rl o a dc o n t r o li nas t a b l e whichG =GfanddG/θα<0, m a n n e r . , I Ia nd1 1 1maybee x p r e s s e dont h eb a s i so ft h es t r e s s Thee n e r g yr e l e a s er a t ef o rmodes1 i n t e n s i t yf a c t o r sa sf o I I o w s : , , , /E ', GII=K l d E ', Gm = =K l μ GI= Ki I I/ ( 2 . 4 ) i 立w h i c hμ =e l a s t i cs h e a rmodulus, f o rt h ec a s eo fp l a n es t r e s sE'= E = Y o u n g ' se l a s t i c 民 f o rt h ec 剖 eo fp l a n es t r a i nE'=E/(l-v2), andv= P o i s s o n ' sr a t i o .F o rg e n e r a l modul l o a d i n g, t h et o ta . le n e r g yr e l e a s er a . t ei s : G=GI+G I I+G m ( 2 . 5 ) a . lt ot h ea p p l i e dl o a d , andmayg e n e r a l l ybe Thes t r e s si n t e n s i t yf a c t o r sa r ep r o p o r t i o n e x p r e s s e di nt h ef o r m : α = α/d ( 2 . 6 ) i nw h i c hfi sac e r ta . i nn o n d i m e n s i o n a lf u n c t i o no ft h er e l . a t i v ec r a c kl e n g t hα( d=c h a r a c t e r and~(α) =f (α) . j 石 : : : : :a . no 山e rn o n -d i m e n s i o n a lf u n c t i o n .F o r i s t i cs t r u c t u r ed i m e n s i o n ), v a r i o u ss i m p l eg e o m e t r i e so fn o t c h e df r a c t u r es p e c i m e n s ,accurateexpr田 sionsf o rf u n c t i o n fareavailablei nもe x t b o o k sandh a n d b o o k s( e . g ., Tadae ta l ., 1 9 8 5, andMur a . k a m i , 1 9 8 7 ) . t h ef u n c t i o nfc a na l w a y sbec a l c u l a t e dbyl i n e a re l a s t i ca n a l y s i s je . g ., F o ro t h e rg e o m e t r i e s, t h r o u g haf i n i t ee l e m e n tp r o g r a m .F o rt h es p e c i a lc a s eo fas i n g l el i n ec r a c ko fl e n g t hai nan もi n f i n i t yt on o m i n a ls t r 回 SσN i nt h ed i r 配 t i o nnormalt ot h ec r a c k i n f i n i t es o l i ds u b j e c t e da on eh硝 f ( α)= 1 .E q .2 . 6s h o w s由民 f o rg 加 m e t r i c a l l ys i m i l a . rs t r u c t u r e so fd i f f e r e n t p l a n e, ( 2 . 7 ) r i t i c a lv a l u eo fK[, w h i c hi sa l s oc a l l e df r a c t u r et o u g h n e s sandr e p r e s e n t s i nwhichKlc=c r i a lp r o p e r t y ;/(Ic= G f E ' .HE q .2 . 7i ss u b s t i t u t e di n t oE q .2 . 6, t h en o m i n a l抑 制 a回 以e a tf a i l u r e( c r a c kp r o p a g a t i o n )i so b t a i n e da s : Kf c σN =百 ( 2 . 3 ) G=G 石 f ( α )= b : :v'd~(α), I f I=E b dVJ .• -. ¥ , d KI=K1c Kl c ( 2 . 8 ) 雨 =万 ね I tmayben o t e d出 a t, a c c o r d i n gt oE q .2 . 8, 十ogd+∞M ( 2 . 9 ) l o g σN = ・ 3 T h i sr e l a t i o nshowst h a . tt h es i z ee f f e c tp l o もa c c o r d i n gt ol i n e a . re l a s t i cf r a c t u r em e c h a n i c si s 佃 i 吋 i n e ds t r a i g h tl i n eo fs l o p e-1/2( F i g .1 刈. 2 . 3LimitsofA p p l i c a b i l i t y I nr e a l i t y , t h ef r a c t u r ep r o c e s sc a n n o tt a k ep l a c ea tap o i n t . Thef 旨a c t u r ep r o c e s sz o n e musth a v esomef i n i t es i 民.A c c o r d i n gt oI r w i n( 1 9 5 8 ), ac r u d ee s t i m a t eo ft h el e n g t hr o f 臼i n e dbys e t t i n gt h et r a n s v e r s en o r m a ls t r e s si nE q .2. 1 t h ef r a c t u r ep r o c e s sz o n emaybeob .Thisy i e l d s ωbee q u a lt ot h et e n s i l es t r e n g t h刀 , 0 l o ,1/(f1:( . =2 r =2 1 1 ' 1 1 " G _ ' . P n -= -K~c _ E' U - f ? -f f ( 2 . 1 0 ) a . lp r o p e r t i e s, 組dt h e r e f o r ei s Notet h a tt h i sl e n g t hi se x p r e s s e do n l yi nt e r m so fm a t e r i am a t e r i a . lp r o p e r t y , t o o .Ana l t e r n a t i v ee s t i m a t eo ft h es i z eo ft h ef r a c t u r ep r o c e s sz o n eo f ∞ncretecanbeb出 edonthemaximum. a g g r e g a t es i z eda • B話相も andOh( 1 9 8 3 a )c o n c l u d e d もt h el e n g t hande f f e c t i v ew i d t ho ft h ef r a c t u r ep r o c e s sz o n eo fc o n c r e t ei nt h r e e p o i n t t h a r e s p e c t i v e l y . b e n ds p e c i m e n sa r er o u g h l y1 2 d 組 d3da, Ji smuchs m a l l e rt h a nt h ec r o s ss e c L i n e a re l a s t i cf r a c t u r em e c h a n i c si sa p p l i c a b l ewhenr a . t i s f I edf o rmostc o n c r e t es t r u c t u r e s, t i o nd i m e n s i o no ft h es t r u c t u r e .T h i sc o n d i t i o ni sn o ts w i t ht h ep o s s i b l ee x c e p もi o no fsomev e r yl a . r g es t r u c t u r e ss u c ha sc o n c r e t ed a m s . However, amorep r e c i s ec r i t e r i o nf o rt h ea p p l i c a b i l i t yo fl i n e a re l a s t i c企配t u r em e c h a n i c s, whicha l s o t a k e si n t oa c c o u n tt h es t r u c t u r es h a . peandt h em組 問ro fl o a d i n g, c 組 beg i v e ni nt e r m so f t h e回 ・ 臼 且edb r i t t l e n e s snumber, β, w h i c hwi I Jb ee x p l a i n e di nS e c .5 . 2 . Thei n a p p l i c a b i l i t yo fl i n e a re l a s t i cf r a c t u r em e c h a n i c st ob r i t t l ef a i l u r e so ft y p i c a l∞E c r e t es t r u c t u r e si sc l e a r l ya p p a r e n tfromt h et e s tr e s u l t sshowni nF i g .1 . 8 -1 .11 . Thed a t a . p o i n t si n d i c a t eam i l d e rs i z ee f f e c tt h 回 t h es t r a i g h tl i n eo fs l o p e1 / 2 . Q 28 29 (0) Lineor Frocture Chapter3 .NONLIN~AR_ ~~AC_TP"~ß MODELS WITHSOFTENINGZONE Ther e a s o nf o rd e v i a t i o n so fc o n c r e t eb e h a v i o rf r o ml i n e a re l a s t i cf r a c t u r em e c h a n i c s i st h ed e v e l o p m e n to far e l a t i v e l yl a r g e企a c t u r ep r o c e s sz o n ewhichu n d e r g o e sp r o g r e s s i v e s o f t e n i n gdamageduet om i c r o c r a c k i n g .Thee f f e c to ft h i sm i c r o c r a c k i n gi s :( 1 )t or e d u c et h e a 四 o fe n e 曙 yt h a tc a nber e l e a s e di n t o仕l ec r a c kt i p iand( 2 )a tt h esameもimet oi n c r e a s e t h ec o m b i n e ds u r f a c ea r e ao fc r a c k i n g, andt h u se n h a n c et h ee n e r g ya b s o r p t i o nc a p a . b i l i t yo f t h ef r 拭 t u r ep r o c e s sz o n e . , ar e l a t i o nd e s c r i b i n gt h es o f t e n i n gdamagen e e d st obei n c l u d e di nt h ef r 配 t u r e T h e r e f o r e m o d e l .T h i sc a nbedoneb a s i c a l l yi ntwoways :( 1 )i nt h eformo fa .s t r e s s d i s p l a c e m e n t ,o r( 2 )a .s t r e s s s t r a i nr e l a t i o nf o rt h es t r a i n r e l a t i o nf o rt h ef r o n t a lz o n eo fal i n ec r a c k s o f t e n i n g( m i c r o c r a c k i n g )r e g i o ni nf r o n to ft h emainc r a c k . Wew i l lf i r s td e s c r i b et h e s e s 田 e . g .P l a n 舗 a p p r o a c h e sandt h e nd i s c u s st h e i rr e l a t i v em e r i t s( F o rf u r t h e rd i s c u s s i o n s, andE l i c e s , 1 9 8 8 a ;恥 t s, 1 9 郎). I ng e n e r a l ,onem可 distinguishtwotypeso fn o n l i n e a rf r a c t u r em e c h a n i c s :( 1 )d u c t i l e ( m e t a l s ), and( 2 )n o n d u c t i l e( c o n c r e t e, a sw e l la s田 r a m i c s ) ;s e eF i g .3. 1 .I nc o n t r 却 t t ol i n e a r e l a s t i cf r a c t u r em e c h a n i c s, t h en o n l i n e a rz o n ei sl a r g ef o rb o t ht y p 邸 o fn o n l i n e a rf r a c t u r e m e c h a n i c s .F o rd u c t i l ef r a c t u r em e c h a n i c s, mosto ft h en o n l i n e a rz o n eu n d e r g o e sp l a . s t i c ,民t p l 制 t i c i t y ,andt h ef r a c t u r ep r o c e s sz o n ei ss t i l 1a .v e r ys m a l lp a r to f h a r d e n i n go rp e r f t h en o n l i n e a rz o n e . Byc o n t r a s t, f o rn o n d u c t i l ef r 邸 t u r em e c h a n i c s, w h i c hi st h ec a s ef o r , t h ef r a c t u r ep r o c e s sz o n ei sl a r g eando c c u p i e sn e a r l yt h ee n t i r en o n l i n e a rz o n e . c o n c r e t e Thus, a l t h o u g hmanyr e s u l t so ft h ef r a c t u r et h e o r yo fm e t a l swhiche v o l v e de a r l i e ra r eu s e f u l , mosto fthemc a n n o tbed i r 民 t l yt r a . n s p l a n t e d . Re m a r k s :Thef r a c t u r ep r o c e s sz o n ei sd e f i n e da st h ez o n ei nwhicht h em a t e r i a lu n d e r ,i . e ., t h es t r e s sn o r m a lt ot h ec r a c k ・p l a n ed e c r e a s e sw i t hi n c r e a s i n g g o e ss t r a i n s o f t e n i n g 凶r a i n .Thes t r e s sc a nbeu n d e r s t o o da st h em a c r o s t r e s so ra v e r a g es t r e s sσ ; ;t h a ti sc a l c u l a t e d回 ( l / f lt ) t i m e st h ef o r c er e s u l t a n tt r a . n s m i t t e da c r o s sa r e af l nw h i c ht h eh e t e r o g e 1o n岡 山 m a t e r i a lw i t ha 邸r e g a t e s回 dm i c r o c r a c k sc a nbea p p r o x i m a t e d出 acontinuum( o n t h em a c r o s c a l e ) .A l t e r n a t i v e l y , ( a n dc u s t o m a r i l y ), σ匂=ん σ' ( ] d V jv .whereσ arethemi, m a t r i xandi n t e r f a . c e s, whichshowh i g hrandom c r o s t r e s s e s( a c t u a ls t r e s s e si nt h ea g g r e g a t e s l o c a ls c a t t e r ), a . ndv , .i sf o rt h er e p r e s e n t a t i v evolumeo ft h em a t e r i a l , d e f i n e db e l o wE q .1 .2 回 i st h ec h a r a c t e r i s t i cl e n g t hi ) . Thesec o n c e p t s乱 開 d e v e l o p e dmorep r e c i s e l yi nt h e ( i t ss i s t a t i s t i c a lt h e o r yo fh e t e r o g e n e o u sm a t e r i a l s( e . g .K r ' るn e r , 1 9 6 7 ;Krumhansl, 1 9 6 8 ) . (b) Metols (c) Concrete ; r 3 . 1SofteningStress-DisplacementRelations T h i sa p p r o a c hd e v e l o p e d回 am o d i f i c a t i o no fas i m i l a ra p p r o a c hp r e v i o u s l yf o r m u l a t e d f o rm e t a l s . Dugdale( 1 9 6 0 )組 dB a r e n b l a t t( 1 9 5 9, 1 9 6 2 )p r o p o s e dt h a tap l a s t i c( y i e l d i n g c o h e s i v e )z o n eo fac e r t a i nf i n i t el e n g t hmuste x i s ta tt h ef r o n to fac r a c k( F i g .3 . 2 a ) . The もt h es t r e s s e sfromt h ef r a c t u r ep r o c e s sz o n ec a n c e l l e n g t ho ft h . i sz o n emustbes u c ht h a t h es t r e s ss i n g u l a r i t yc a u s e da tt h et i po ft h ee q u i v a l e n te l a s t i cc r a c kbyt h ea p p l i e dl o a d F i g . 3 . 1 L i n e a rZone(L), N o n L i n e a rZone( N )andF r a c t u r eP r o c e s s Zo ne( S )i nF r a c t u r eo fD i f f e r e n tMa . t e r i a l s 31 30 ( i . e ., ](1 = 0 )・ Thec r a c ko p e n i n ga tt h eb e g i n n i n go ft h ep la . st i cz o n e ,wherethes も r e s s 回 l yd r o p st oz e r o, mayber e g a r d e da sam a t e r i a lp r o p e r t ywhichc o n t r o l sp r o p a g a t i o n s u d d ( F i g .3 . 2 b ) . F o rsomem e t a l sando t h e rm a t e r i a l si twωlatern o t e dt h a tt h ec o h e s i v ez o n es h o u l d 邸 h i b i tag r a d u a lr a t h e rt h 阻 s u d d e ns t r e s sd r o p,c h a r a c t e r i z e dbyas o f t e n i n gr e l a t i o no f t h en o r m a ls t r e s sσa . c r o s st h ec r a . c kv s .t h ec r a c k o p e n i n gd i s p l a c e m e n t ι ( F i g .3 . 2 d ) js e e K f o u r iandMi I l e r( 1 9 7 4 ), 姐dWnuk(1974). Forconcrete, t h i st y p eo f e . g .Knauss( 1 9 7 4 ), modelwasp r o p o s e dbyH i l l e r b o r g, Modeer姐 dP 悦 r s s o n( 1 9 7 6 )u n d e rt h enameo ff i .c t i t i o u s c r 8 心km o d e l js e ea l s oPet 位 e , r s “ 旬 f i c ω 叫 色 “i t “ 1 叩 0 u s ♂ "r e f e r s色 ωothef.畠配 c t もt 出 ha tt h e卯 po r 尚 “ t 10 叩, no fac r a c kwhicht r a n s m i t st e n s i l es t r e s sc a n n o t 鈴e p 鉱 a r 叫 a もt i o no ft h es u r ( , 仏 a α渇 町iも 吋 山 her e a lc r a c ke n d sa tt h ep o ω i 叫 n beac o n t i n u o u sc r a c kw i t hf u l ls t w h e r et h es t r e 団 i sr e d u c e dt oz e r O jF i g .3 . 2 d . )I nt h i smodel , whichha 日b e e nw i d e l ya p p l i e d i nf i n i もee l e m e n ta n a l y s i so fc o n c r e t ef r a c t u r e, t h em a t e r i a lf r a c t u r ep r o p e r t i e sa r ed e f i n e d byt h es o f t e n i n gs t r 田s -d i s p l a c e m e n tr e l a . t i o n : “ (0) E 仰M C0 .d ( 3 . 2 ) 叩 Oc (e) ε F i g . 3 . 2S t r e s sD i s t r i b u t i o n s1 nandN e a rt h eF r a c t u r eP r o c e s sZ o nei n D i f f e r e n tModels σ J 九 ∞ fIM 一 -G, ( s e eF i g .3 . 2 d ), where円 i st h e抗r e s si nt h ed i r e c t i o nn o r m a lt ot h ec r a c k .The紅 e au n d e r a . c t u r ee n e r g yo ft h em a t e r i a l, i . e . : t h ec u r v er e p r e s e n t st h ef r (d) ( c ) 噌 σ首 =f (t 5 c) qd dc 、 ‘ . , , E-- 担 , , 、 ・ 1nnn~ Thec r a c kb e g i llS t oopenwhent h es t r e s sa tt h et i pr e a . c h e st h et e n s i l es t r e n g t hl i m i tf : . I ft h es h a p eo ft h es o f t e n i n gc u r v ei sf i x e d, t h e nt h ef r a c t u r epropertie~ a r ec o m p l e t e l y . T hep r e c i s es h a p eo ft h es o f t e n i n gs t r e s s c h a r a c t e r i z e dbytwop a r a m e t e r s : 刀 andGJ d i s p l a . c ementdiagramh a sac o n s i d e r 功l ei n f i u e n c eont h ec a l c u l a t i o nr e s u l 比 I nv a r i o u s i nwhichc a s et h es t r e s si sr e d u c e d w o r k st h i ss h a p ei sc o n s i d e r e dt r i a n g u l a ro rb i l i n e a r, も oz e r oa ta五n i t ed i s p l a c e m e n t6 x p o n e n t i a ls h a p eh a sa l s ob e e nu s e d . Whent h e 0・ Ane 七h eb e h a v i o rI s制 s k e t c h e di nF i g .3 . 2 d . s o f t e n i n gz o n ei su n l o a d e dandr e l o a d e d( F i g .3 . 2 d ), 色i t i o u sc r a . c kmodelwasv e r i f i e d岨 dc a l i b r a 七e dbyv a r i o u sc o m p a r i s o n sw i t h H i l l e r b o r g ' sf i c t e s td a t a . .However, i ts e e m st h a . ta . ne x h a u s t i v ec o m p a r i 回 nw i t ha l lt h ei m p o r t a . n tc o n c r e t e f r a c t u r eda . t . afromt h el i t e r a t u r eh a sn o ty e tbe 四 p r e s e n t e d .Butduet oe q u i v a l e n c ew i t h , もh ee x t e n s i v ec o m p a r i s o n so ft h el a t t e rw i t ht e s tda 凶 ( B a z a n tand t h ec r a c kb叩 dmodel Oh, 1 9 8 3a . )i n d i r e c t l yv a l i d a t et h ef i c t i t i o u sc r a c km o d e l . Thes h a p eo ft h es t r e s s d i s p l a c e m e n tc u r v ewass t u d i e dont h eb a s i so fm i c r o m e c h a n i c so f . a c k sbyH o r i i( 1 9 8 8 ), H o r i ie ta l .( 1 9 8 7, 1 9 8 8 ), a . n dBa . i a n t( 1 9 8 7 b ), andonc e r a m i c s m i c r o c r w i t hs i m i l . a rb e h a v i o rbyO r t i z( 1 9 8 8 ) .Fromt h e s em o d e l si tt r a n s p i r e dt h a tt h em i c r o c r a c k a . b i l i t yo ft h es t r e s s d i s p l a c e m e n tc u r v e . c o a l e s c e n c eg e n e r a l l yt e n d st op r o d u c es n a p b a c ki n s t t h ee x i s t i n gme a . s u r e m e n t s, a l b e i tl i m i t e di ns c o p e , don o tshowa . nys n a p b a c k .1 t However, m. a y w e l lbet h a . tsomeo t h e rmechanism, s u c ha . sf r i c t i o n a lp u l l o u to fa g g r e g a t e. a ndf r a g m e n t s 32 成 8 , e l i m i n a t e st h es n a p b a c k . fromt h ec r a c kf a 3 . 2S o f t e n i n gS t r e s s S t r a i nR e l a t i o n s S i n 田 t h ec r a c k si nc o n c r e t ea r en o ts t r a i g h tb u tt o r t u o u s, andt h em i c r o c r a c k i n gz o n ei n f r o n to f出 ec o n t i n u o u sf r a c t u r ei sn o tl i k e l yt od e v e l o pa l o n gas t r a i g h tl i n e, t h eb e h a v i o r r a c t u r ep r o c 伺 sz o n ec 岨 e q u a l l yw e l lbed e s c r i b e dbys t r e s s -s t r a i nr e l a t i o n sw i t h o f出 ef 品司s o f t e n i n g, i . e ., d e c l i n i n gs t r e s s抗 i n c r e a s i n gs t r a i n .T h i sa p p r o a c hi sq u i t ec o n v e n i e n t s t r a a . t i o no ft h en o d e so ftwoa d j a c e n te l e m e n t sn e e r l s f o rc o m p u t e rprogrammings i n c enos e p a r t ob ei n t r o d u c e dandf r a c t u r ei sh a n d l e dbya d j u s t m e n t so ft h ei n c r e m e n t a ls t i f f n e s so ff i n i t e b a s i c a l l yi nt h esameway回 anyi n e l a s t i cb e h a v i o r . e l e m e n t s, n t r o d u c e di nf i n i t ee l e m e n t S t r a i n s o f t e n i n gi nt h eformo fas u d d e nv e r t i c a ld r o pw踊 i h i d( 1 9 6 8 ) .Then e e dt oc o n s i d e rp r o g r e s s i v es t r a i n s o f t e n i n gi n加 l s i o nwas a n a l y s i sbyRas ∞gni田 dbySc晶 . n l a n( 1 9 7 1 ), whoi n t r o d u c e das e q u 担問 o fs m a l ls t r e s sd r o p s . Thef a c t r e x p e r i m e n t a l l yo b s e r v e dbyL'H e r m i t e thatωncretee x h i b i t ss t r a i n s o f t e n i n gi nt e n s i o nw槌 e ( 1 9 5 9 ) ,RuschandH i l s d o r f( 1 9 6 3 ),Hughes釦 dCh a . pman( 1 9 6 6 ),andEvans乱 . n dMarathe ) .Extensivea . ndc a r e f u l l yc o n t r o l l e dm e a s u r e m e n t sw e r er e c e n t l yr e p o r t e dbyP e t 聞 s o n ( 1 9回 ( 1 9 8 1 ),Re i n h a . r d tandC o r n e l l i s e n( 1 9 8 4 ),GopalaratnamandShah( 1 9 8 5 ),ando t h e r s . Fromt h econtinuumm e c h a n i c sv i e w p o i n t, t h e∞n c e p to fs もr a i n s o f t e n i n gi n v o l v e sc e r t a i n 回 m a t h e m a t i c a ld i 伍c u l t i e s, s u c ha . si m a g i n a r ywa 問 s p e e d( o rc h a n g eo ft y p eo ft h e s e v e p a r t i a ld i f f e r e n t i a le q u a t i o no fm o t j o nf r o mh y p e r b o l i ct oe l l i p t i c )andi l l p o s e d n e s so ft h e , w h i c hw e r ep o i n t e do u tand叩 a l y z e dbyHadamard( 1 9 0 3 ), Thom 剖 b o u n d a r yv a l u ep r o b l e m ( 1 9 6 1 ),B a z a n t( 1 9 7 6 ),S姐 d l e r( 1 9 8 4 ), ReadandH号: g e m i e r( 1 9 8 4 ), WuandF r e u n d( 1 9 8 4 ), Ba . i a n t, B e l y t s c h k o組 dChang( 1 9 8 4 ), ando t h e r s . Thec h i e fproblemi st h a tt h ez o n eo f e n e r g yd i s s i p a t i o nt e n d st ol o c a l i z et oaz o n eo fz e r ovolume( as u r f a c e )s ot h a tt h et o t a l ぽ g yd i s s i p a t i o na tf a i l u r ei si n c o r r e c t l yi n d i c a t e dt ob ez e r o( c f .r e v i e wbyBa . i a n t, 1 9 8 6 ) . en 侃c u l t i e sa r ec i r c u m v e n t e di fs もr a i n s o f t e n i n gi nf i n i t ee l e m e n tm o d e l si si n t r o d u c e d T h e s ed i t b r o u g hf r a c t u r ec o n c e p t si n0 問 。f t w o f o r m s :( 1 )山 ec r a c kbandmodeland( 2 )t h en o n l o c a l mode l .Thel a t t e rr e q u i r e saf i n e rmeshb u ta l l o w sb e t t e rr e s o l u t i o no fl o c a l i z e ds t r a i nf i e l d s . 33 o fag r a d u a ls o f t e n i n g .L a t e r∞mp 釘 i s o n sw i t hnum ぽ o u st e s tr e s u l t s,howe v e r .i n d i c a t e d h u sc r e a t i n g a r e l a s i v e b i o gf r a e b u r e t h a tt h er e d u c t i o no fs t r e s st oz e r omustbegradud,t 聞 c p r o z o n ea h e a do ft h ef r o n to f山 f r a c t u r e( F i g .3 . 2 e ) .T h i s{ . お or m 町m 叫晶机叫 4 l 似 o ∞ n,吋 wh i 凶 c hw剖 炉 g i 討 v 佃, e n .1 泊 nd e t a i l匂 byB 加 a . i a 副n 凶t( 仰 1 凶9 鈎 8 幻 2 ) , 制dBa . i a n t凶 Oh(1983a )h a sb e e nshownhagmwith cf r a c t u r et 回t d a t aa v a i l a b l ei nt h eH t e r 畠t u r e , p a r t i c u l a r J yt h o s eont h e e s s e n t i a l l ya l lt h eb担 i , t h eR c u r v e( s e eChap 附 4 ), andt h ed i f f e r e n c e s e f f e c to fs p e c i m e ns i z eont h emaximuml o a d 色w eenv a r i o u ss p e c i m e ng 印 m e t r i e s .Att h esamet i m ei tw回 s hownt h a tt h ec r a c kband be exceP4when modelaadm i l e r b O I r g ' sa c t i t i 0 1 1 8c r a c kmodelg i v ee s s e n t i a l l yt h es a m e r e s u l t s, c l o s e l ys p a c e dp a r a l l e lc r a c k soccur-(Thus, t h ee x t e n s i v ee x p e r i m e n t a lj u s t i n c a t i o no fb h e t i t i o u sc r a c kmode . l) c r a c kbandmodela l s oi n d i r e c t l yp r o v i d e dj u s t i f i c a t i o nf o rt h e貧c S o f t e n i n gi sc a u s e dbyf r a c t u r i n g抗 日h, t f, w h i c hi ss u p e r i m p o s e donS h ee l a s t l citmn. h o o s i n . l ! : Assuminga 1 1t h ec r a c k st ob ep a r a l l e lands m e a r e d( c o n t i n u o u s l yd i s t r i b u t e d ),andc a x i sX2t oben o r m a lもot h ec r a c k s, weh a v ei ntwod i m e n s i o n s( x )-X2p l 如 4 j : ロ “ 引 十 日 ) (i;)= [?;:?;;:i212/Fl( The , ∞lumnmatriCE泡 onthel e f thand姐 dr i g h tbands i d e sc o n s i s to ft h es t r a i n阻 ds t r 側 c o m p o n e n t s(γ=2 f 1 2= s h e a ra n g l e ), C1111, C1122 = C2211, C2222 姐 dC1212 釘 et h ee l a s t i c m p l i 姐 C 民 組dβis姐 e m p i r i c a lf a c t o r ,( 0く F壬 l ), c a l l e dt h es } m r ! ? e t e n t i o nfadf ( i n t r o d u c e dbyS u i d a nandS c h n o b r i c h, 1 9 7 2, 叩dY u z u g u l l u組 dS c h n o b r i c h, 1 9 7 3, P h i l l i p s, lm ,岨dP h i l l i p s組 dZ i e n k i e w i e z ,1976).Ift h em a t e r i a li si s o k o p i c ,C1111=C2m= 22l 1= りE', Cll~12 : : : :C 〆/ E', C1212 = 2 { i nw h i c hE'= E, v ': :1f o rp l a n e d E '=E/(l-1 1 ), 〆=ν/(1-v)forplanestrain-ThefracturingMmincanbei nc o r p o r a t d i n t ot h e∞m p l i a n c e , i nw h i c ht h ee x p r e s s i o nf=,[ωα222/(1ω ) J σ 2 2h部 b田 ni n t r o d u c e d : 姐 b er e g a r d e da . sdamageandc 叩 b ec o n s i d e r e dt obeaf u n c t i o no ft hes t r a i nnorm i . l ωC t ot h ec r a c k, f 叫 o rt h emaximump r i n c i p a ls t r a i n j ω / ( 1ー ω)=ゆ( f 2 2 ) .I n i t i a l l yweha . v e ω : : : :0( n od叩 a g e ), andc o m p l e t edamage( c o n t i n u o u sf r a c t u 吋 occursf o rω =1 . Always r a c t u r ee n e r g yi so b t a i n e da s 0 : : ; ω 三1.Thef ∞ 叫 べ 山口 , 3 . 2 . 1CrackBandModel Theb a s i ci d e ao f出 ec r a c kbandmodel , w h i c hwasp r o p o s 吋 b yBa . ia n t( 1 9 7 6 ), i s : ( 1 )t oc h a r a c t e r i 配 t h em a t e r i a lb e h a v i o ri nt h ef r a c t u r ep r o c e s sz o n ei nas r 問 a r e dmanner .s t r a i n s o f t e n i n gc o n s t i t u t i v er e l a . t i o n, a . nd t h r o u g ha ft h ef r o n to ft h es t r a i n s o f t e n i n gz o n e( c r a c kb a n d ), ( 2 )t oi m p o s eaf i x e dw i d t hW c o r e p r e s 叩 t i n gam a t e r i a lp r o p e r t y . Thei m p o s i t i o no fc o n s t a n t叫 i sr e q u i r e di no r d e rt oa v o i ds p u r i o u smeshs e n s i t i v i t y 姐 da c h i e v eo b j e c t i v i t y , a s s u r i n gt h a tt h ee n e r g yd i s s i p a t i o nduet of r a c t u r ep e ru n i tl e n g t h ( a n du n i tw i d t h )i sac o n s t a n t, e q u a lt ot h ef r a c t u r ee n e r g yo ft h em a t e r i a l, GJ.I nk 伺 p i n g w i t ht h ec l a . s s i c a la p p r o a c ht os m e a r e dc r a c k i n g, t h ed e t a i l e df o r m u l a t i o no ft h ec r a c kband model( Ba . i a n tandC e d o l i n,1 9 7 9,1 9 8 0,1 9 8 3 )f i r s te m p l o y e das u d d e ns t r e s sd r o pi n s 旬a d ( 3 . 3 ) J GJ::::Wcf句 d f ( 3. 4 ) Thef u l l yc r a c k e ds t. a t ei so b t a i n e df o rω →1. l tw剖 shown( B a z a n tandOh, 1 9 8 3a . ) t h 前 i fE q .3. 4i si n v e r t e dandt h el i m i to ft h ee n s u i n gs t i 汗n e s sm a t r i xf o rω → Oise a l c , f l a 叫 ther田 ult∞incideswiththewel 1 k r 附 ns t i f f n 倒 m前 市 f o ramater 凶 t h a tiscMEd 伽 n a l l y姐 dc o 凶 削o u s l y , a . si n t r o d u c e dbyR a . s h i d( 19 6 8) ・ u n i d i r e E q . 3 . 4 r e g e c t so n l yav e r ys p e c i a lformo fdamage.Ing e n e r a l,damagen e e d st ob e i n t r o d u c e dt h r o u g ht h ef o r m u l a t i o nO ft h ei n t e r n a lv a r i a b l et h e o r y( f o rC O Z I e r e b e , 間 e . g. P i j a u d i e r C a b o tandM a z a r s, 1 9 8 9, 姐dPijaudier-Ca . b o tandBa . i a n t .1 9 8 8 ) 蜘 a i nr e 1 a t i o nu n d e r l y i n gt h ec α r 舵 a ckh n dm m 吋 O d 由似 e li お sd Theu n i a x i a ls o f t e n i n gs t r e s s 凶凶i 砲刷 db 匂 ' y f 加 叩 un c t ω t 1 叩 0 n 吋 ゆ 州 仲 仇 ( 恥 仇 正 叫 匂 幻 2 叫 , ) 仏 2 )w 耐h i c hd e t 白 i悶 d制晶略郁 g 酔e ω The削 1 1 t ω t s a r 陀' e , 却 a ss 回叩 e 伽 n 邸 伽 凶 s 凶 s 討 i i 川 t も u 伽 i 片 v 刊e 叫川 t 吋 o ぱft h es o f t e n i n gs t r 色伺 s 目 仔 s 凶 古 仕 r 問a 吋 J nr ,閃 e l 晶 a “ も t i o n卸 a r 陀et h o s ef o rt h ef 貧 i c t i 比 t i o 田 u 1 悶s c r a c kmode l .1 ta p p e a r s “ω ω “ λ い ; 34 35 t h a tt h es i m p l ef o r m u l aゆ( f )=(Ej 刀)expa ( f-f w i t he m p i r i c a lc o n s t a n t sα姐 dfp, i s p), i . e ., at r i a n g u l a rs t r e s s s t r a i nd i a g r a m ,h副 g e n e r a l l ya d e q u a t e . As t r a i g h tl i n es o f t e n i n g, a l s ob e e nu s e ds u c c e s s f u l l y . Therea r etwo討mplev a r i a n t st ot h ec 阻止 b andmode . lTheo r i g i n a lone(Ba . Z a n tandOh , 1 9 8 3 a )p r e s u m e st h a tt h es m e a r e dp a r a l l e lc r a c k s伽 r tt of o r mi nt h ed i r e c t i o nn o r m a lt o t h emaximump r i n c i p a ls t r e s sb u ts u b s e q u e n t l yt h ec r a c ko r i e n t a t i o ni sf i x e di nt h em a t e r i a l 白川 r e s e a r c h(eιGuptaandAkbar , e v e ni ft h ep r i n c i p a ls t r e s sd i r e c t i o nr o t a t e s .Morer e 1 9 8 4 ;Cope ,1984)附 mstoindicatethati ti sb e t t e rt oassumet h a tt h ec r a c ko r i e n t 抗1 0 n r o t a t e sw i t ht h ed i r e c t i o no ft h emaximump r i n c i p a ls t r e s 8, w h i c hmeanst h a ts h e a rs もr e s s e s sv a r i a n t, t h eg e n e r a lt r i a x i a ls t r e s s s t r a i nr e l a t i o n ont h ec r a c kp l a n ec a nn e v e ra r i s e .F o r山 i f o rt h em i c r o c r a c k e dm a t e r i a l叩 bew r i t t e ni nt h ef o r m(Ba.Z叫叫 L i n, 1 9 8 8 b ) : f;; =l Il " ;; ;1 - 十ーーーーー一ー_'l7' T 1 _.nr'rl__ l I'Tr ( 3 . 5 ) 叶 i nw h i c hni= =d i r e c t i o nc o s i n e soft h ec u r r e 凶 ma 杭. x 1 m 宜m ω もe n s o 討叫 s 削u b s c α n 中 p t 匂si m p l ysummation)andC仰 ,n i st h ee l 嗣 t i c(undamaged)c o m p l i a n c e t e n s o r . Whent h ep r i n c i p a ls t r e s sd i r e c t i o n sr o 同t es i g n i f i c a n t l y , t h en o n r o t a もi n gc r a c kmethod mustbeg e n e r a l i z e dt oa l l o wf o rt h ef o r m a t i o no fs e c o n d a r yandt e r t i a r yc r a c k i n go fd i f f e r e n t o r i e n t a t i o n s .Suchm u l t i d i r e c t i o n a ls m e a r e dc r a c k i n gm o d e l sw e r ee s p e c i a l l yp e r f e c t e dbyde B o r s t( 1 9 8 4 ) . 1 nr e a l i t y , t h em i c r o c r a c k sp r i o rt o出 . ef i n a lc o n t i n u o u sf r a c t u r ea r ed i s t r i b u t e do v e ra l l o r i e n t a t i o n s, w i t hd i f f e r e n tf r e q u e n c i e sf o rv a r i o u so r i e n t a t i o n s .T h i sf e a t u r ei sc a p t u r e dby . Z a n tandOh, 1 9 8 5 )w h i c hs 田 m st obep h y s i c a l l yt h emostr e a l i s t i c t h em i c r o p l a n emodel(Ba o n c e p t u a l l ys i m p l e s もmod e lo fdamageduet oc r a c k i n gb u ti sv e r ydemanding a sw e l l邸 c f o r∞mputerもime.1 nt h i smodel( w h i c hwi l Jb ec o n s i d e r e df u r t h e ri nS e c t i o n3 . 2 . 5 ), t h e h y p o t h e s i st h a tt h ec r a c k i n gs t r a i nt e n s o ri sa d d i t i v et ot h ee l a s t i cs t r a i nt e n s o ri sabandon 吋 叫 c r a c k i n gI sm o d e l l e da ss t r a i n s o f t e ni ngs e p a r a t 均 onp l a n 伺 o fa . lo r i e n t a t i o n s( c a l l e d m i c r o p l a n e s ), s u b j e c tt o出 eh y p o t h e s i st h a tt h es t r a i ncomponentsone a c hm i c r o p l a n ea r e t h er e s o l v e dcomponentso ft h e( m a c r o s c o p i c )s t r a i n .Thes t r e 蹴 s a . f t e rc r a c k i n gont h ep l a n e s , b u to v e r a l le q u i l i b r i u mi se n f o r c e dby o fv a r i o u so r i e n t a t i o n sa r en o te x a c t l yi ne q u i l i b r i u m u s i n gt h ep r i n c i p l eo fv i r t u a lw o r k . I tmaybep o i n t e do u tt h a tl i n e a . re l a s t i cs t a . b i l i t ya . n a l y s i si nwhicht h em i c r o c r a c k sa r e 剖 s umedt ogrowi nahomogen ∞uselasticcontinuumindicatesthatthesofteningstressd i s p l a c 即 l e n to rs t r e s ss t r a i nr e l a t i o ns h o u l de x h i b i tamaximumd i s p l a c e m e n to rs t r a i n a f t e rwhicht h es t r e s ss u d d e n l yd r o p st oz e r o(B a . Z 姐t ,1987b). Buti ti sno も ( s n a p b a c k ), y e tknownw h e t h e rt h ep r e d i c t i o no fs n a p b a c ka l s or e s u l t sfrommodel st a k i n gi n t oa c c o u n t 田, i n e l a s t i cb e h a v i o r, andf r i c t i o n . inhomog e n e i t i . p p r o x i m a t e l yWc 竺 3 d d Thew i d t ho ft h ec r a c kbandf r o n tc a nbeassumedt obea a( a = maximumaggregates i z e ) .T h i sc o n c l u s i o nw踊 drawnon出eb a s i so foptimumf i t t i n g . Z a n tandOh, 1 9 8 3a . ) .However , t h eoptimumwasn o ts h a r p ;叫 o fnumeroust e s td a t a(Ba v a l u e sr a . n g i n gfromda t o6 d "g a v ealmo s te q ua l 1 ygoodr e s u l t s, p r o v i d e d, o f∞u r s e ,the α . p o s t p e a kf u n c t i o nO ((22)w出 a d j us t e df o re a c hd i f f e r e n tv a l u eo f叫 Oncet h es h a p eo ft h es o f t e n i n gs t r e s s s もr a i nr e l a t i o ni sf i x e d, t h ec r a c kba . n dmodeli s f u l l yc h a r a . c t e r i z e dbyt h r e em a t e r i a . lp a r a m e t e r s : , Ga . n dWc ( a l t h o u g ht h ei n f l u e n c eo f a l u ei sr a t h e rweakf o rs i t u a -もi o n sw i t hi s o l a t e dc r a c k s ) .Byc o n t r a s t, t h ef i c t i t i o u s t h e叫 -v 晶c kmodelh a so n l ytwob a s i cp a r a m e t e r s,刀姐dGf ・ Whyt h ee . 対r a .parameteri nt h e c r c r a c kb姐 dmodel?Thee x t r ap a r a m e t e r , ω ' c, i si m p o r t a n to n l yi ns i t u a t i o n swhent h e r ea r e p叫 l e lc r a c k s( 句 ・ ,i nt h epresωoft e n s i o nr e i n f o r c e m e n t ) ;t h e nw b a s i c a l l yd e t e r m i n e s a sam a t e r i a lp r o p e r t y .I ts h o u l dbep o i n t e do u t出品t t h eminimump o s s i b l ec r a c ks p a c i n g, t h ef i c t i t i o u sc r a c kmodel , b e c a u s eo fi t sl a c ko ft h ee x t r aparameterωc, c 組 g i v er 田u l t s 4hatuen o bo b j e c t i v e i ns i 4 1 1 a t i o n sw i t hp a r a l l e lc l o s e l ys p a c e dc r a c k s -(Ac e r t a i nl e n g t hh a s d d i t i o n a lma . t e r i a lp a r a m e t e ri n出 ef i c t i t i o u sc r a ε km o d e l ;h o w e v e r , a l s ob e e nd e f i n e d組 組 a i nc o n t r a s ttoωc, i ti sn o tani n d e p e n d e n tp a r a m e t e r . ) e q u i r e dbyt h ec r a c kb岨 dmodel, maybet ∞smallin Thef i n i t ee l e m e n ts i z eh=叫, r t h ec a s eo fv e r yl a r g es t r u c t u r e s .I nt h a t日 s e, i ti sp o s s i b l et oe n l a r g et h ee l e m e n ts i 配 ( i . e . u s eh>叫 )p r o v i d e dt h a tt h es o f も 匂 e 刷凶 n 1 自 帥 品mee n e r 略 gyd i s s i p a t i o n ,GI- Thisi si l l u s t r a t e di nF i g .3 . 3 .I nv i e wo ft h es e r i e sc o u p l i n g model , a l r e a d yd i s c u s s e di nc o n n e c t i o nw i t hF i g .1 .8 ,thegivens もr e s s s t r a i ndiagramOPA f o rt h es t r a i n s o f t e n i n gc r a c kbandn 田 d st ober e p l a 臼 d ,atincreasingelementsizeh,by 必a g r a m sOPB, OPC, OPD, e t c . ,suchthatwhenthe紅 側 underanyofthesediagamh esamef r a 山 r ee n e r g yv a l u eGJ i so b t a i n e d( F i g .3 . 3 c ) .I nt e r m so f i sm u l t i p l i e dbyht t h es t r e s s d i s p l a c e m e n tdiagramf o rlmghhho f S n i t ee l e m e n t , ac h a a g ei nhr e q u i r e st h a t 閃 m entd i a g r . a m0 1 2i nF i g.3 . 3 bber e p l a c e dbyd i a g r a m s0 3 2, 0 4 2, t h ea c t u a ls t r e s s d i s p l a 0 5 2, e t c .Theses t r e 時 d isplacemeMdiagramhavethepropedyt h a tt h ea mu n d e rb h e r ni c o n s t a n t, t h u s回 s u r i n gc o n s t a . n tf r a c t u r ee n e r g yG/ . Onec a n, o fc o u r s e, a l s ou s ee l e m e n t sw i t hhく ωc( s a y0 . 1 ωふ p r o v i d e dt h ep o s t p e a k i a g r a mOPFi nF i g .3.3cωthatt h eσ-8dia~ram s l o p ei sd e c r e a s e da sshownbyt h eσ-fd 0 8 2( F i g .3 品)woulda g a i nhav et h esamea r e a鎚 0 1 2 .I nt h i sc 槌 e .therowd c r i k e d e l e m e n t sc a nben a r r o w 町七h ant h ea d j a c e n te l e m e n t sandt h emeshmayl o o ka sshowni n r e p l a c e dbyh .O b v i o u s l y , whenh→ 0, t h ec r a c kb姐 dmodeli nt h e F i g .3. 4ab u tw i t hω。 l i m i tbecomesi 判 icaltoHi I l e r b o r g' smodel( w h i c hi nもh iss e n s ei sas p e c i a lc 蹴 o ft h e c r a c kbandmoden. o f t e n i n gs l o p eg e t ss t e e p e r,u n t i lf o rac e r t a i n Ast h ee l e ments i z ei si n c r e a s e d,もhes e l e m e n ts i z ehav e r t i c a ls t r e s sd r o pOPCor42iso b t a i n e d . F o ras t i l ll a r g e re l e m e n ts i z e, whichwouldcausenmericaldi 血 叫t i 偲 J thediagrmOPDor052woulde x h i b i ts n a p b a c k l nt h a . tc a s e , onema . yr e p l a . c et h es n a p b a c ksegment5 2o ft h ee l e ment鈴 阿 佐d i s p l a 臼m ent 67inFig.3.3b.ThepoiMofV e d i c a id r o pi sddermined d l a g r a mbyaverbids t r e s sd r o p, a g a i nfrom出 ec o n d i t i o nth4thea r e au n d e rt h ediagram067mustbet h esame槌 t h e a r e au n d e rt h ed i a g r a m012o r0 4 2.T h i sc o n s i d e r a t i o ni n d i c a t e s出 a tt h ee q u i v a l e n tt e n s i l e s t r e n g t hj んofthelargef i n i t ee l e m e n to fs i z eh>ωci sg i v e nby : 1 , : 1 、 吾 ん= h n=竺生 一紘 一 一 切 1 [ 2 一 刀2 ( f o r h三 h o ) ( 3. 6 ) 37 36 ( e . g ., BaZa n t阻 dC e d o l i n, 1 9 7 9, 1 9 8 0 )i nw h i c hh oi st h ee l e m e n ts i z ef o rw h i c hav e r t i c a l s t r e s sd r o p( d i a g r a m0 4 2 )i so b t a i n e d .N o t et h a tt h ee xp r e s s i o nf o rh oi ss i m i l a rt oI r w i r 内 E q .2 . 1 0f o rt h es i z eo ft h ey i e l d i n gz o n e . も ss u b j e c t e dt o Thef o r e g o i n ga d j u s t m e n t sa r ei d e a l l yd e f i n e df o rs q u a r ef i n i t ee l e m e n h es i d e s ) .However , e x t e n s i o n st on o n s q u a r ee l e m e n t s t e n s i o no rs h e a r( n o r m a lo rp a r a l l e lもot a r ep o s s i b l e .I nt h i sc 制 e , i ft h ed i m e n s i o no ft h ee l e m e n ti ne a ε hd i r e c t i o ni sa b o u tt h esam e , onemaya g a i nu s eE q .3 . 7i nw h i c hh=ゾ 互 , whereA =elementarea( i n2 D ) . e l a t i o nhf;q/2E= hof?/2E= G/wherehr e p r e s e n t se i t h e r E q .3 . 7f o l l o w sfromもher t h ea c t u a . lw i d t ho ft h ec r a ε k i n ge l e m e n ti nt h ed i r 配 t i o nn o r m a lt ot h ec r a c k s, p r o v i d e dt h e elem 回 t i ss q u a r eo rr e c t a n g u l a r , o rt h ee f f e c t i v ew i d t ho ft h ee l e m e n td e f i n e da . sh=叫 A . , f A 6 w h e r eA . /, Aba r e出 e紅 e a . so ft h ee l e m e n t姐 do ft h ec r a c kbandw i t h i nt h ee l e m e n t .F o r e l e m e n 七w i d t bh<h o, f o rwhicht h eposιpeaks l o p ei sr e d u c e db u tt h es t r e n g t hl i m i もi sk e p t , 出e向 e時蜘a.i nd i a g r a mmaybee x p r e s s e d, a c ∞rdingtothes e r i e smodel( F i g .1 . 8 ), a 8刀 a . sf o l l o w s : E s o f t+(1一生 ) f u n h= 竺 f o rhく 1 1 0 ) ( 3 . 7 ) f=竺Z nl L, . -ω ~c竺 Ab ( 作 ト' -+ V -,~ð !σ パゴ~ b/ ー し/' ¥ 工6 ( b ) 4 3 5 (c)σ 汗 f f ; eQ= ( d ) w~~国 TIN-上 f . o / t, f u n l= t h es t r a i n sc o r r 田p o n d i n gt ot h esames t r e s sa c c o r d i n gt ot h es t r a i n s o f t e n i n g s t r e s s s t r a . i nd i a g r a mand叩 u n l o a d i n gs t r e s s s t ra . Ind i a g r a ms t a r t i n gf r o mt h epe 北 s t r e s s p o i n t . I fl a r g e re l e m e n t sn e e dt obeu s e d, a n o t h e rp o s s i b i l i t yi sもok e e pt h es i z eo fもh ee l e m e n t s ont h el i n eo fc r a c kbandc o n s t a n t, e q u a lt os i z ewc, ande n l a r g ea l lt h er e m a i n i n ge l e m e n t s 4 a.I ft h ec r a c kbandi smucht h i n n e rt h 姐 t h ea d j a c e n t血l i t ee l e m e n t s, a sshowni nF i g .3. 山 em odel ,o f∞u r s e, be ∞mespracticallyi d e n t i c a lt ot h ef i c t i t i o u sc r a c kmode . l ,thecrackbandmodela sw e l l I nt h ec a s eo fc r a c kp r o p a g a t i o ni nana r b i t r a r yd i r e c t i o n 拙 t h ef i c t i t i o u sc r a c kmodelr e q u i r e sr e m e s h i n gs o出 a tt h eboundaryl i n e so ft h ec r a c k r a c kl i n ewouldc o n f o r mt ot h ei n t e r e l e m e 叫 b o u n d a r i e s( 知 t h ef i c t i t i o u s b姐 d白 色hec c r a c kmodel , t h e s et e c h n i q t 瑚 w e r et op e r f e c t e dbyI n g r a f f e a( 1 9 8 5 ), I n g r a f f e a叩 dG e r s t l e 組dIngraffeaandSaouma(1985). I ft h er e m e s h i n gi sn o tdone ,i ti ss t i l Jp o 回b l e ( 1 9 8 5 ), t oa p p r o x i m a t e l yr e p r e s e n tf r a c t u r er u n n i n gi nana r b i t r a r yd i r e c t i o nbya l l o w i n gt h ec r a c k bandt oh a v eaz i g z a gformωshowni nF i g.3. 4 b .Somea d j u s t m e n t so ft h ef r a c t u r ee n e r g y a r et h e nne 偲 s s a r y七ot a k ei n t oa ι c o u n tt h e品 刊r a g ew i d t ho fs u c haz i g z a gband( B a z a n t, 1 9 部品) Evenw i t hs u c ha d j u s t m e n ω, h o w e v e r, t h e r ei sac e r t a i nb i a simposedbyt h emesho r i i fs h e a rs t r e s s e sp a r a l l e lt ot h eo v e r a l lc r a c kb叫 dd i r e c t i o na r i s e ,the e n t a t i o n .M o r e o v e r, z i g z a gbandc a ni n t r o d u ω s p u r i o u ss h e a . rl o c k i n go ft h eOPPωitef a c e s .Thesep r o b l emsc a n ∞mee i t h e rbyi n もr odudngan o n l o c a lv e r s i o no ft h emodel, d e s c r i b e dl a 句r , o rby beo v e r 自∞n t i n u o u sa l o n gt h eb o u n d e n r i c h i n gt b ef i n i t ee l e m e n te i t h e rw i t has t r a i nf i e l dt h a ti sd i a r i 回 o f姐 a r b i t r a r i l yo r i e n t e db姐 d, 槌 proposedbyDroz(1987)f o rt e n s i o n, ( s i m i l a rt ot h e m o d e l sr e p o r t e dbyO r t i ze ta l ., 1 9 8 7 ;LeroyandO r t i z, 1 9 8 9, 加dBelytschkoe ta l ., 1 9 8 8f o r s h e a rb姐 d s )o rw i t hd i s p l a c e m e n tf i e l dd i s c o n t i n u o u sa l o n gal i n e, a sp r o p o s e dbyD v o r k i n e ta 1 .( 1 9 8 9 ) . Thec r a c kbandmodelo f f e r st h ep o s s i b i l i t yo fi n t r o d u c i n gt h ei n s u e n c eo fno n s i n g u l a r q) ε 4 σ F i g. 3 .3E f f e c to fS i z eont h eS t e e p n e s so fPos t P e a kS o f t e n i n g ( α ) (b) F i g . 3. 4M eshest oR e p r e s e n taVeryNarrowCrackBanda n daZ i g Z a g Band 39 38 t h r e e d i m e n s i o n a ls t r e s s e sonf r a c t u r e .F o rexample , i ti sknownt h a tac o m p r e s s i v es t r e s s σ z p a r a l l e lt ot h ec r a c kp l a n ep r o m o t e sc r a c k i n g .Based0 0t h eknowns h a p eo fもheb ia x : i a lf a i l u r e , t h i si n f l u e n c 怠 ( f o rむ く 0 )maybet a k e ni n t oa c c o u o tbyr e p l a . c i n g e n v e l o p eo fc o n c r e t e w i t h ( 3 . 8 ) 万=f t ( l+σ ,,/ f D : f (B a . i a n t, 1 9 8 5 a )wheref~ = compr 白 s i o ns t r e n g t h .F o rσ= 一f 疋~ ( c o m p r e s s i o nf ' 必 a a 泊 i l l 町 u l 開 r e ), . on whicha g r 問 w i t hF i g .3 . 9 . Note油 ot h a t, i nc o n t r a . s t, σ"h 剖 n oe f f e c t musth a v e斤 =0, i fl i n e a re l a s t i cf r a c t u r em民 h a n i c si su s e d ;n o rwhenl i n ec r a c km o d e l sw i t hs o f t e n i n gz o n e s a r eu s e d . a r g eg e n e r a Jp u r p o s ef i n i t ee l e m e n t Thec r a c kbandmodelh a sb e e ni m p l e m e n t e d泊 somel ∞des( e . g .DIANA, TEMP-STRESS, NONSAP). 3 . 2 . 2CrackLayerModel Av a r i a n to ft h ec r a c kbandmod 巴1 i sr e p r e s e n t e dbyC h u d n o v s k y ' sc r a c kl a y e rmodel (B 飽 s e n d o r f fandChudnovsky , 1 9 8 4 ;Chudnovsky , 1 9 8 6 ) .T h i smodeli smoreg e n e r a lt h 叩 t h e c r a c kba . ndmodeli nt h a tt h ewid もho f t h ec r a ε kbandf r o n ti sn o tc o n s i d e r e dt obeac o n s t a n t . Ra t h e ri ti sa l l o w e dt oexpando rs h r i n kb a s e donam a t e r i a lp r o p e r t y .T h i sp r o p e r t yl i n k sa c e r t a i np a t h i n d e p e n d e n ti n t e g r a lt h a . tc h a r a c t e r i z . 田 t h ee n e r g yc h a n g eduet ot h ee x p a n s i o n r a c kf r o n twhichc h a r a c t e r i z e st h ee n e r g y t oa n o t h e rp a t h i n d e p e n d e n ti n t e g r a la r o u n d出 ec f l o wi n t ot h ec r a c kf r o n tduet oc r a c kp r o p a g a . t i o n( 町c e ' sJ i n t e g r a l ) . T h i smodel, how e v e r, h a sn o tb e e nd e v e l o p e df o ra p p l i c a t i o ni n∞n c r e t eandh a sn o tb伺 n comp a r e dt ot y p i c a lt e s td a t af o rc o n c r e t e . Thep r o p e r t i e sg o v e r n i n gt h ebande x p a n s i o n n u g h tbeu n i d e n t i f i a b l efrom出 ee x p e r i m e n t a ld a t aa tt h ec u r r e n td e g r e eo fs o p h i s t i c a t i o n o ft e s t i n gr o e t h o d s . 3 . 2 . 3CompositeFr actureModel Thec o m p o s i t ef r a c t u r emodeli se s s e n t i a l l yana d a p t a t i o no ft h ec r a c kb 組 dm odelt o f i n i t ee J e m e n t so fal a r g e rs i 配 t h a nt h ec r a c kbandw i d t hW c, t h u sp e r m i t t i n gl a r g es t r u c t u r e s t obea n a l y z e dw i t hac r u d e rm e s h . Themodela s s u m e st h a tac r a c kba . n do fw i d t hωc, w h i c hi sam a t e r i a lp r o p e r t y( 町 ane q u i v a l e n tl i n ec r a c k )i sembeddedw i t h i na1 釘 g e rf i n i t e a p p r o x i m a t e l y酪 i l l u s t r a t e di nF i g .3 . 3 d .T h i si d 伺 , whichi ss i n u l a rt oE q .3 . 8, wω e l e m e n t, p r o p o s e df i r s tf o rp l 拙 t i cs h e a rba . nd sbyP i e t r u s z c z a kandMroz( 1 9 7 9 )andl a t e rd e v e l o p e d f o rt e n s i l ec r a c k i n go fc o n c r e t ebyWillame ta . l( 1 9 8 5,1 9 8 6 ) .I ft h ee l e m e n ts i z es h r i n k s t h i smodelbecomese s s e n t i a l l yi d e n t i c a It ot h ec r a c kbandmode l . Thec r a c kband toωc, w i t h i nt h ef i n i t ee l e m e nもi si n t r o d u c e dbye n r i c h i n gt h esmoothd i s t r i b u t i o nf u n c t i o nw i t ha . d i s c o n t i n u o u ss t r a i nf i e J df u n c t i o nw h i c hc h a r a . c t e r i z e sc r a ε k i n g .The乱p p r o a c hi ss i n u l a rt o t h o s ebyDroz( 1 9 8 7 ), O r t i ze ta J .( 1 9 8 7 ), B e l y t s c h k oe ta l .( 1 9 8 8 ), L e r o yandO r t i z( 1 9 8 9 ) 叫 dD v o r k i ne ta l .( 1 9 8 9 )a l r e a d ym e n t i o n e da tt h eendo fS e c .3 . 2 .1 . 3 . 2 . 4NoniocalContinuumwithLocalS t r a i n(NonlocalDamage) An o n l o c a lcontinuumi sac o n t i n u u mi nw h i c hsomes t a t ev a r i a b l ed e p e n d sn o to n l yon t h es t r e s s 田 o rs t r a i n sa tt h esamep o i n tb u ta l s oont h es t r e s so rs 七r a i n五e l di nt h en e i g h b o r hoodd t h epoint.Asshownbyk r i s n e r ( 1 9 6 7 ), kmmhand(1968), kmin( 1 9 6 8 ), E r i n g e n 姐 dE d e l e n ( 1 9 7 2 ), ando t h e r s(cf.Bdan t ,1986) ,thenonlocalconcepti sa p p r o p r i a t ef o r s t a t i d i c a l l yh e t e r o g e n e o u sm a t e r i a l sandf o l l o w sl o g i c a l l yfroms t a b i s t i c a lm i c r o m e c h a n i c a l ∞nsidera . t i o n s . R怠 切n t l y ,thenonloca . la p p r o a c hh a sb e e nshownt obev e r ye f f e c t i v ef o r 企a c t u r er o e c h 姐 i c so fd i s t r i b u t e dc r a c k i n g . Then o n l o c a lc o n c e p tr e p r e s e n t sag e n e r a la p p r o a c hw h i c hmakesi tp o s s i b l et ou s es t r e s s a . i nr e l a t i o n sw i t hs t r a i n s o f t e n i n g . Asm e n t i o n e db e f o r e ,i ft h e5もr a i n s o f t e n i n g∞n c e p t s t r , c a u s i n R i si n t r o d u c e di nt h el o c a lc o n s u l 1 1 1 1 I R s o f t e n i n gz o n e sl o c a l i z et oav a n i s h i n gvolume i m p r o p e r∞n v e r g e n 明 o nmeshr e f i r 悶 n e n s, s p u r i o u smdEseMtivityandp h y s i c a l l y i n c o r r e c t p r e d i c t i o n ss u c ha sf a i l u r ew i t hv a n i s h i n ge n e r g yd i s s i p a t i o n舗 もhemeshs i z ei sr e f i n e dt o z e r o . Toa v o i ds u c hb e h a v i o r, t h ecomput 抗i o n a lmodelmusti n c l u d esomema . thema t i c a l d e v i 田 t h a . tI i m i t st h el o c a l i z a t i o n( Ba . z a n t, Be l y t s c h k oandC 1 加 1 9 ,1984). Thesimplest, a l b e i tc r u d 田 t , wayt ol i n u tl o c a l i z a t i o ni st oi m p o s eaf i x e ds i z e, ωc,ont hef r o n t a lf i n i t e e l e m e n t, a sdonei nt h ec r a c kba . ndmodel( u n l e s st h ep o s t p e a ks t r e s s s t r a i nr e l a t i o ni s 宣c i a l l ym o d i f i e d ) .However, t h i sa p p r o a c hmakesi ti m p o s s i b J et or e s o l v et hef i e l di nt h e a r t i f r a c t u r ep r o α出 z o n ei t s e l f ,o rd e t e r n u n ehowt hef r a c t u r ep r o c e s sz o n ew i d t hc o u l dv a r y c t u r eg r o w t h( n om o d i f i c a t i o no fs t r e 時 町a i nr e l a t i o n sc o u l dbep e r m i も tdf O } d u r i n g山 e企a 告 白 ) . t h o s epurpo A moreg e n e r a lwayt ol i m i tl o c a l i z a t i o n ,i no r d e rt oc o p ew i s hs u c hp r o b l e m s, i st h e n o n l o c a lc o n t i n u u m .Thec o n c e p tmaybei n t r o d u c e di nv a . r i o u sf o r m s, s u c h剖 a v e r a g i n go f s t r a i n so rs t r a i n r e l a t e dqu 姐 t i t i 田 o v e ran e i g h b o r h o o do fap o i n t( Bazant, 1 9 8 4 b ;Ba . i a n t, B e l y t s c h k o叩 dChang , 1 9 8 4 ), o ri 山 吋u c t i o no f白 川 姐ds e c o n d s p a t i a lf l e d U t i v e s O f町 ams o ro t h e rqu 岨 t i t i e si n t ot h ec o n s t i t u t i v er e l a t i o n(eιSchreyer岨 dChen,1 9 8 6;andf o r 机 m ediumf o r m u l a t i o nf o rr o c k sbySuJem組 dVardo 由, k i s, 1988)Thes p a t i a l t h eC o s s e r も i o nc a nbeo b t a i n e dbye x p a n d i n g出 es p a t i a Ja . v e r a g i n gi nt e g r a li nT a y l o r d e r i v a t i v ef o r m u l a s e r i e s . Onlyt h es p a t i a l. a v e r a g i n ga p p r o a c hh倒 的 f a rb 四 nw o r k e do u ti nd e t a i lf o r凶e c r e t e . m u l t i d i m e n s i o n a la n a l y s i so ff r a c t u r eo fm a t e r i a l ss u c h出∞n s p a t i a la v e r a g i n gwasa p p l i e dt h r o ught h et o t a ls t r a i n sus e di n 1 nt h ee a r l yf o r m u l a t i o n, t h a もa p p r o a c h, w h i c hr e s u l t e di n S h es t r a i n s o f t e n i n gc o n s t i t u t i v er e l a t i o n . H o w e v e r, 叫 a l 1 凶 e 吋di 帥 m b 伽 凶 r i c 臼a t 旬e c ∞ o 凶 n 凶 t i 泊 n削 u 1 1 ha 刊 v es 叩o m ロ ' l e∞ c omp叫 u l 比 凶 t d 叫 a 品 t i o n a l l yi n c o n v e n i e n tp 戸r 叩 o pe 釘r 凶 t l 郎 e 8 .R e c e n t l y( Baian tandP i j a u d i e r -Cabot, 1 9 8 7,1 9 8 8 a ;P i j a u d i e r C a b o t組 dBdmt,1987;BatantandL i n ,1988a, 邸 川 ,i tw f o u n dt h a tamoreeRectiveapproachi st oc o n s i d e r b i l e e l a s t i cs t r a i n sωlocal( n o r a g i n g ), a 干 da p p l yt h en o n l o 叫a. v e r 叫ngonlytothesofteningpartoftheは 凶nswhichrepmejt d l s t r i b u t e dm i c r o c r a z k i n g T h I Snewa p p r o a c hh a st h eadvan t a g et ha tt h econtinuumhid o n d i t i o n sr e m a i nt h esame酪 f o rt h e e q u a t i o n so fe q u i l i b r i u ma sw e l la s出 eboundaryc c “ “ h …ム 40 41 c l a s s i c a ll o c a lcontinuum , andt h a tt h e r ei snop o s s i b i l i t yo fz e r o e n e r g yi n s t a b i l i t ymodes 均 shownt oe x i s tf o rf u l l yn o n l o c a lm o d e l s ) .I th a sa l s ob e e nmathema もi c a l l yp r o v e n ( p r e v i o u 出品 t h ea v e r a g i n go ft h ef r a c t u r i n gp a r to ft h es t r a i n si ss u 伍c i e n tt op r e v e n tl o c a l i z a t i o n o fs t r a i n s o f t e n i n gdamagei n t oaz o n eo fz e r ovolume.Theu n l o a d i n g r e l o a d i n gb e h a v i o ri s a l w a y sl o c a li nt h i sa . p p r o 邸 h . . a t i a . la v e r a g i n go p e r . a t o r , d e n o t e dby姐 o v e r b a r, maybed e f i n e di nt e r m so fs t r a . i n Thes p e l la sf o l l o w s : l t l l ( X )= Y T T~__\ ( s-x ) { f l l ( s ) } dV L ' ( x-S ) ( E 1 1 ( S ) ) d V r(x)Jvα ¥ , , . . ¥ , , .=J vα ( 3 . 9 ) I _ ' 1 _ _ ¥ . . I t I " _ I _ _¥ _α ( s-x V r(x)= l _0 ' ( 8-x)dV ,α (x-s )一一 一 一一) V r( X ) ( 3 . 1 0 ) ( f 1 1 )=ε11i fE l l>0, ( 1 : 1 1 )=0i ff 1 1壬O ( 3 . 1 1 ) i nw h i c h Jv-¥ - --r" , -, - 1 . 1 8-xl¥2 x )= 1一 寸e ~I) 時- x l<k l ,otherwiseα(8-x)=0 t z 』 s-x ! 0 x 1 i 1k2-'_kl_ ft h es t r u c t u r e , x=c o o r d i n a t ev e c t o r, s=c o o r d i n a t ev e c t o r so fa d j a c e n t H e r eV=volumeo p o i n t s, α =g i v e nw e i g h t i n gf u n c t i o no ft h ed i s t 組 c eI s-x !, c o n s i d e r e dもob eam a t e r i a l , .hasapproximately, b u tn o te x a c t l y , t h esamem田 n i n g槌 t h er e p r 争 p r o p e r t y( F i g .3 . 5 ) .v s e n t a t i v evolumei nt h es t a t i s t i c a lt h e o r yo fh e t e r o g e n e o u sm a t e r i a l s . n i f o r m(α=1 )o v e rvolumev , .r e p r e s e n t i n g Thew e i g h t i n gf u n c t i o nc o u l db ed e f i n e d回 u a .c i r c l ei ntwod i m e n s i o n s, ab a rs e g m e n ti noned i m e n s i o n , o ras p h e r ei nt h r e ed i m e n s i o n s, w i t haz e r ov a l u eo u t s i d eVr • However, i th a sb e e ne x p e r i e n c e dt h a tt h ec a l c u l a t i o n sc o n v e r g e 自s i b l ec h o i c e( B a z a n t組 dP i j a u di e r ・C abot, b e t t e ri ft h ew e i g h t i n gf u n c t i o ni ss m o o t h .Onepo 1 9 8 7, 1 9 8 8 a ;Baz 組 t andL i n , 1 9 8 8 b )i saG a u s s i a nd i s t r i b u t i o nf u n c t i o n, which , h o w e v e r, h a s ts e e m sp r e f e r a b l et ou s eap o l y n o m i a lb e l l s h a p e d n o n z e r ov a l u e so v e rt h ee n t i r es t r u c t u r e .I e . g .( F i g .3. 5 ) f u n c t i o nw h i c hi se x a c t l yz e r obeyondac e r t a i nd i s t a n c e, α ( ト (0) ( 3 . 1 2 ) sanorm a . l i z i n gc o n s t a . n td e t e r u u n e d50t h a tt h ei n t e g r a lo fαover品 l i n es e g m e n t, H e r eki o ra .c i r c l e, o ras p h e r eo fs i z ek lwouldbet h esame舗 もh ei n t e g r a lo fα =1o v e rt h esame r e g i o n jk= 1 5 / 1 6= 0 . 9 3 7 5f o rl D ,( 3 / 4 ) 1 / 2= 0 . 9 0 8 6f o r2D,k= ( 1 0 5 / 1 9 2 ) 1 / 3= 0 . 8 1 7 8 . r a . m e t e rli st h es o c a l 1 e dc h a . r a c t e r i s t i cl e n g t ho ft h en o n l o c a lc o n t i n u 町n , f o r3 D .Thepa w h i c hi sam a t e r i a lp r o p e r t yt h a td e f i n e st h es i z eo f出 ea v e r a g i n gvolumeandi sd e t e r m i n e d byt h es i z eo ft h ei n h o m o g e n e i t i e si nt h em i c r o s t r u c t u r e .F o ro n eq u i t et y p i c a lc o n c r e t e, m伺 s u r e m e n t s( B a i a n tandP i j a u d i e r ・C abot, 1 9 8 8 b )haveshownt h a t{=2 . 7 d . .whered . .= maximuma g g r e g . a t es i z e( S e c .6 . 9 ) . 叩 o ft h ebody ,p紅色 o ft h ea . v e r a g i n gz o n ep r o t r u d e s I fp o i n txi st o oc l o s et ot h es u r f a beyondt h es u r f a 偲. T h i si sh a n d l e dbyd e l e t i n gt h ep r o t r u d i n gp a r tfromt h ei n t e g r a t i o n もi o nc a u s e sv , .( x )t od e p e n donl o c a t i o nx . r e g i o n .T h i sd e l e t h ei 凶e g r a l si nE q .3 . 1 0阻 d3 . 1 1a .a r ea p p r o x i m a t e l y I nf i n i t ee l e m e n tc a l c u l a t i o n s, e v a l u a t e d舗 f i n i t esumso v e ra l lt h ei n t e g r a . t i o np o i n も so fa l lもhee l e m e n t si nt h es t r u c t u r e . F i g . 3 . 5W e i g h t i n gFunc t i onf o rN o n -Loc a lCo ntinuumTbeoryandRe p r e s e n t a t i v eVolumeo ft h eM a t e r i a l 、‘、 、 、. -、 0 . 3七 A ー 。c o n c r e t e 0- m o r t ar ト-ーご竺丸、 ユ ー 0 . 5 、 、 、 E E・0.7J 出 0 . 0 . 4 0 . 8 1 .2 1 .6 l o g { d / da) F i g . 3 . 6 S i z eE R e c tC a l c u l a t e dbyN o n L o c a lContinuumT h e o r y( B a t a n t ,1988b)forExponential( d a s h e dcurve)andL i n e a r( d a s hd o t andL i n , Comparedw i t hS i z eE R e c tLaw v e ) T e n s i l eS t r a i nS o f t e n i n gC u r v e s s o l i dcurve)andw i t ht e s tr e s u l t so fBat a n t組 d o fBasant(Eq.1.4, a . t ap o i n t s ) P f e i f f e r( 1 9 8 7 )( d 42 43 , t h ei n t e g r a t i o np o i n t swhosed i s t 日 c e sf r o mp o i n txe x c e e dk lmaybeo m i t t e d However fromt h esums i n c ef o rthemα=0. I d e a l l y ,出ef ormo ff u n c t i o n α ( s-x )s h o u l dbed e r i v a b l efrommicromecha 且i c a . l m o d e l i n g o ft h em i c r o s t r u C h u r ew i t ha g g r e g a t e sandm i c m c r a c k s, samea st h eformo ff u n c t i o n O ( ε )i n , t h i sg o a li st o odimcuuatp r e s e n t, w h i l ea tt h esamet i m et h e E q .3.4姐d3.6.However D r e c I s ef o r mo ff u n c t i o n α ( s-x )h a so n l yl i t t l ei n f l . u e n c eonωmput a . t i o nr e s u l t s( 出i sf . a ι b 即 tionαfromt 制 d a . t . ap帥 lem a . t i c )・ Whati si m p o r t a n ti st h e LakesideMiaωtiono ff a l u eo fi( p r o v i d e dt h ebodyi sn o tも∞ l a r g ecomparedt ol ) . ThesimpledEtonlocalmlSUMMErlodel, appropmheforu n i d i r ω i o n a lc r a c k i n g, i so b t a i n e dbyg e n e r a l i z i n gt h eC E a c kbandmodel(B 話 相ta ndL i n , 1 9 8 8 b )・ Onea . l t e m a t i v ei st o , t h e M m et h ec r a c kd i r e c t i o nt obeaxeda tt h et i m et h ec r a c k ss t a r tt oform.Int h i scue mnlodgeneralizationi so b t a i n e dbyreplacingωinE q .3. 4w i t hn o n l o 叫 d amageρ, s u c h Ani n t e r e s t i n g , 阻dp o t e n t i a l l yu s e f u lp r o p e r t yo ft h en o n l o c a lf o r m u l a t i o ni st h a tt h e r s eo fl o a d i n g . The w i d t ho ft h ec r a c k i n gz o n ei sn o tf i x e db u tv a r i e sd u r i n gt h e∞u c o n s e q u e n 田 i st h a tt h en o n l o c a lf r a c t u r emodeld o e sn o tc o r r e s p o n dt oau n i q u es t r e s s d i s p l a c e m e n td i a g r a mf o rt h el i n ec r a c kmodel五 (c t i t i o u sc r a c k ) .T h i sf a c tmighte 却l a i n whyi th a sp r o v e dr a t h e rd i f f i c u l tt oo b t. a i nu n i q u ev a l u e so ft h ef r a c t u r ee n e r g yont h eb 8 0 s i s o ft h ef i c t i t i o u sc r a c kmode . l 自o ben o t e dt h a tal a r g ef i n i t ee l e m e n tp r o g r a . mf o rt h en o n l o c a lf r 8 0 c t u r emodel I ts h o u l da l I l yr u n sf a s t e rt h a nt h ec o r r e s p o n d i n gl o c a lp r o g r a . m ,d e s p i t et b ef a c tt h 8 0 t8od d i t i o na . l u s u a ∞mputertimei sn e e d e dもoc a l c u l a t et h es p a t i a l晶. v e r 8 o g 回 .T her 同 s ona p p a r e n t l yi st h a も もh e a . la . v e r a g i n gs t a . b i l i z e st h er e s p o n s e ,t h u sc 晶u s i n gt h ei t e r a . t i o n si nt h el o a d i n gs t e p s n o n l o c t oc o n v e r g ef a s t e r. t h a t 3.2.5 MultidirectionalCrackingandNontensileFr acture w t1=cp(fI) ( 3 . 1 3 ) ( 1- . l 山a i n臼 l c u l 8 0 t e dfromt h en o n l o c a l( a v e r a g e d ) wheref Ii st h emaximumn o n l o c a lp r i n c i pa s t r a . i nt e n s o rf i ; 氏∞ c ∞ oI n J 1 Thes e s 叩u medt 句or o t a . t ew 叩i t 出 ht h emaximump r i n c i p a ls t r e s s, i so b t a i n e dbyr e p l a c i n gωwithwi n Eq.3.6 I th i sb e e nverihd畑 山n tandL i n,1 9 8 8 b )t h a tt h i sn o n l o c a lg e n e 叫 i z a t i o no ft h e c r a c kbandmodelロ閃e r m 首 i i t 日 s伽 t h eu s 問 巴 eo fh 脳 伝 f i n 泊 1 吋 i 向 s 叩 pu 巾 u 凶 sr r 附 hs 鈴e n E 比 t i v 吋 i i t 句 yandi m p r o p e rc o n v e r g e n 侃 F o re l e m e nts i z e sno自m a l l e rt h a nl , t h en o n l o c a l m o d e lbecomesi d e n t i c a lt ob i l ec r a £kb andmodel , aadt h eskesE4rainE e l a t i o n f o rt h ef i n i t ee l e m e n twhichu n d e r g o 田 c r a c k i n gmustbea d j u s t e da c ∞rdingtotheelem 田も s i z e ー 却 a . l r e a d ye x p l a i n e d . i th a sa l s ob e e nshownt h a tt h enonlodmmedcmkingm o d e l p r o v i d e s祖 e x c e l l e n t 批 o fv a r i o u sf r a d u r et e s tdata-Seee . g ., Fig.3.6, i nwhichdIet e s tr e s u l t sfromBdaab andP f e i f f e r( 1 9 8 7 )a r ecomparedw i 出 n o n l o ca . lf i n i t ee l e m e n tr e s u l t so fBaZa n tandL i n 回 h ed組 dd a s h d o tc u r v e 唱 ) 叫dw i t ht h es i 田 e f f e c tlaw( E q .1 ムsolidcurve)(the ( 1 9 8 8 b )( d d a s h e dc u r v ec o r r e s p o n d sもoas o f t e n i n gt e n s i l e山 倍 付 も 問nr e l a t i o ni n. a ne x p o n e n t i心 form , whichf i t sもheda . t ab e t t e rt h. a n出 eda . s hd o tc u r v ew h i c h∞r r 凶 p o n d st oa .l i n e a rs t r e s s s t r a . i n r e l a t i o n ; s t i l lb e t t e ra t sc o u l d p r o b a b l y b e o b t a i n e dw i t ho b h e rs h a p e so f t h i sr e l a t i o n )・Fr om Fig.35its h o u l dbee s p e c i a l l yn o t e dもh 8 0 tan o n l o c a lf i n i t ee l e m e n tc o d ee x h i b i t st h ec o r r e c t 剖 a 句p 戸 pr 似 o X l 凶 r m 町】 胤 a 凶 t . e l yd e s c r i b e dbyt h e t r a n s i t i o n a . ls i z ee f f e c to fn o n l i n e a rf r a c t u r em e c h a n i c s, s 討i z ee f 晶 T 島 ' e c tlawp 戸r 叩 o p伺 o s e 吋 : db 句yBa おE 叫 も 1 北 ι .By∞ c o n t r a 泌 . s t, も 山 h el o c a lf 貧 i 凶 n i も 旬 ee l e m 阻 e nも乱 n 凶a 乱 . 々s i se x h i b i t s nos 討 i z 民ee 釘e c d t, a so b s e r v e dong m m e t r i c a l l ys i m i l a zs p e c i m e n so fd i f e r e n ts i z e s , w i t hs i m i l a r m e s h e s . Onep r a c t i c a la . d v 姐 t a g eo ft h en o n l o c a lf o r m u l a t i o ni st h a ti ft h ehitee l e m e n ts i z e s u e l e s st h a na b o u tl/3oft h ec h a r a c t e r i s t i cl e n g t ht , t h e nもh e r ei snod i r e c t i o n a lb i a so ft h e mdlwithREardLoc r a c kpropagaM.ThisMbeen附 i f i e d(B a . Z a n tandL i n, 1 9 働 )by i t h胡a.l i g n e d叩 a r emesh組 da . mlyzingthefmureofbhesamer e c t a n g u l a rs pぬ menw 訂 em esh.Thus, t h en O B l o c a l m o d e lcmbeu s e di ng e n e r a lf E a z u r es i t u a b i o n s s l. a n t e ds q u ゐ i “ F o r∞n c r e t es t r u c t u r e ss u b j e c t e dt o∞m p l i c a t e dl o a . d i n gh i s t o r i e s, i tmaybene 伺 s s a r y 切組a.l y z ef r a c t u r et a . k i n gi n t oa c ∞unt出 eeXIstenceofmultidir庇 tionalcracking(BaZ祖t 1 9 8 3 ;deB o r s t,1 9 8 4,1 9 8 7 , . a1 9 8 7 b ;deB o r s tandNa . u t a ,1 9 8 5 ;O t t o s e n阻 dDahlblom ,1 9 8 6 ; Willame t仏 1 9 8 7 ;Ro t s,1 9 8 8 ;Ro t s組 dB l 品uwendr a . a d,1 9 8 9 ;C r i s f i e l dandW i l l s,1 9 8 9 ) . ,whichwas Ane f f e c t i v ea p p r o a c ht os u c hb e h a v i o ra p p e a r st obet h em i c r o p l a n emodel 山 由edam a . g el a w .I n s t e a . do fam a c r c ト a l r e a d ym e n t i o n e db e l o wE q .3 . 6i nc o n n e c t i o nw i s c o p i cs t r e s s s t r a i nr e l a t i o n, t h em a t e r i a lb e h a v i o ri nt h em i c r o p l a n emodeli sc h a r a c t e r i z e d i n d e p e n d e n t l yonp l a n e so fv a r i o u so r i e n t a t i o n s, c a l l e dt h em i c r o p l a n e s .I np r a c t i , 田 only0 8 吋 , . a c c o r d i n gt oan u m e r i ca . li n t e g r a t i o n c e r t a i nnumbero fd i s c r e t em i c r o p l a n e si sc o n s i d e r f o r m u l af o rs p 8 0 t i a ld i r e c t i o n s . Them a c r o s c o p i cr e s p o n s ei sd e t e r m i n e da sac e r t a i na v e r 田 p o n s e sfrom出 ei n d i v i d u a lm i c r o p l a n e s, o b t a i n e da c c o r d i n gt ot h ep r i n c i p l e a g eo ft h er o fv i r t u a lw o r k .T h i smodel , whoseb a s i cc o n c e p ti ss i m i l a rt ot h es l i pt h e o r yo fp l a s t i c i t y ( T a y l o r,1 9 3 8 ;B a t d o r f姐 dB u d i a n s k i,1 9 4 9 ),wasf i r s td e v e l o p e di na .l o c a lf o r m(Ba 加 l t a n dOh , 1 9 8 5 ) .I twasa . l s oshownt h a tt h i smodelc a nr e p r e s e n tw e l ln o to n l yt h ee x i s t i n g , i n c l u d i n gt h ee f f e c t so f出 et r a n s t e s td a t aont e n s i l ef r a c t u r eb u ta . l s ot h o s eonc r a c ks h e a r v e r s en o r m a ls t r e s sands h e a r i n d u c e de x p a n s i o n晶c r o s st h ec r a c k(Ba おn t叩 dGambarov , . a 1 9 8 4 ) . Re c e n t l y(BaZa 叫 a ndO z b o l t, 1 9 8 9 ), t h em i c r o p l a n emodelh a sbeeng e n e r a l i z e dt o an o n l o c a lf o r m . T h e r ea r eo t h e rt y p e so ff r a c t u r ew h i c hn e e dt obemodeledf o rc o n c r e t es t r u c t u r e s .A newt h 即時 w i l lhavet obed e v e l o p e df o rB r a z i l i a nt e n s i l es p l i t t i n gf r a c t u r eo fv e r yl a r g e whichc a n n o tb ea d e q u a t e l y阻 a l y z e dw i t ht h ee x i s t i n gもe n s i l ef r 叫 t u r em o d e l s . c y l i n d e r s, oa n a l y s i sbyt h e Ont h eo t h e rhand,s h e a rf r a c t u r es e e m s,a tl e a s tp a r t i a l l y ,am~nable t e x i s t i n gt e n s i l ec r a c k i n gm o d e l s .I nt h o s em o d e l s, t h es h e a rf r a c t u r ei sd e s c r i b e d祖 aband o fi n c l i n e dm i c r o c r a ι k s, g o v e r n e dbyas o f t e n i n gs t r e s s s t r a . i nr e l8 o t i o n . 3.3Stress-Displacementvs. Stress-StrainSofteningRelations T h e r eh a sb e e nani n c e s s a . n td e b a t eont h er e l a t i v em e r i t sandd e f i c i e n c i e so ft h el i n e 44 4 5 c r a c kandc r a . c kbandr e p r 白 e n t a t i o n so ff r a c t u r e .Therea r et h r e ev i e w p o i n t sもom e n t i o n : 3 . 3 . 1I s o l a t e dCracks:MootPointComputationally F町民 o n emustr e a l i z et h a tt h el i n ec r a . c kmodel( i . e . t h ef i c t i t i o u sc ra . c kmodelo f 1 l e r b o r g )andt h ec ra . c kb組 dmodely i e l da b o u tt h esamer e s u l t s( w i t hdぽe r e n 側 o fa b o u t Hi 1 %)i ft h es t r e s sd i s p l a . c e m e n tr e l a t i o ni nt h ef i r s tmodeland出 es もr e s s 叫 r a i nr e l a t i o ni nt h e s e ∞ndmodelareca . l i b r a t e di ns u c hawayt h a t ι = ωc . f ' ( 3 . 1 4 ) i . e ., t h ec r a . c ko p e n i n gd i s p l a . c ementi stakenωthef r a c t u r es t ra . i n, f . f, a . c c u m u l a t e do v e rt h e .T h i se q u i v a l e n c ei sa l r e a d ys u g g e s t e dbyt h ef a . ctt h a . t , i nl i n e a r w i d t hωco ft h ec r a c kb岨 d e l a . s t i cf ra . c t u r em e c h a n i c s, ani n t e r e l e m e n tl i n ec r a c k組 das i n g l e -e l e m e n tw i d ec r a . c kband w i t hasymmetryl i n e出 品 c o i n c i d e sw i t ht h 邑l i n ec r a c kg i v ee s s e n t i a l l yt h esamer e s u l t si n a . 8 er a t e, w i t hd i f f e r e n c e so fo n l ya b o u t1%, p r o v i d e dt h e t h ec a l c u l a t i o no ft h ee n e r g yr e l e e l e m e n ts i z e sn e a rt h ef r a c t u r ef r o n tdon o te x c e e da b o u t1 / 1 0o ft h ec r o s ss e c t i o nd i m e n s i o n ( B a i 姐 t . a ndC e d o l i n, 1 9 7 9 ) .I nf a . c t , t h ed i f f e r e n 伺 b e t w e e nt h e s etwomethodso fc a l c u l a t i o n もl a r g e rt h a nt h ee r r o ro f o ft h ee n e r g yr e l e a s er a t ef o rl i n e a re l a s t i cf r a c t u r em e c h a n i c si sno a p p r o x i m a t i n gt h econtinuumbyf i n i t ee l e m e n t si nt h ef i r s tp l a c e . T h e r e f o r e, t h eq u e s t i o n“ l i n ec r a c ko rac r a c kb a n d ? "i smootfromt h ev i e w p o i n to f a p p r o x i m a t i n gr e a l i t ybyc o m p u t a t i o n a lm o d e l i n g .Theo n l yp o i n tworthyo fd e b a t ei sc o m t h etwom o d e l sa p p e a r p u t a t i o n a le f f e c t i v e n e s sandc o n v e n i e n c e . Bute v e ni nt h a tr e g a r d, t obee q u a l . h o w e v e r, d e s e r v ea t t e n t i o n .L e i b e n g o o d ,Darwin V a r i o u sn u m e r i c a lm o d e l i n ga s p e c t s, 叫 t sf o rs t r e s s -d i s p la . c ementands t r e s s s t r a i nr e l a t i o n s andDodds( 1 9 8 6 )showedt h a tt h er e matchc l o s e ! yi ft h ec r a c k i n gd i r e c t i o n sa tt h ei n t e g r a t i o np o i n t sw i t h i nt h ef i n i t ee l e m e n t s a r ef o r c e dt obep a r a l l e lt o. e a c ho t h e r( a n dt ot h ea c t u a lc r a . c k ) .I ft h ec r a c k i n gd i r e c t i o n JS n o もk nownap r i o r i, 剖i nt h eg e n e r a lc a s e, i ti sd i 伍c u l tt oa c h i e v e出 i sp a ra . l l e l n e s s . But ,i fc r a c k sa r ea l l o w e dt oforma td i t f e r e n to r i e n t a t i o n sa te a . c hi n t e g r a t i o np o i n t, t h e t h e n r e s p o n s eo ft h ef i n i t ee l e m 回 tm odelw i t hs m e a r e dc r a ε k i n gi ssomewhats t i f f e rt h 叩 t h a t w i t hd i s c r e t ei n t e r e l e m e n もc r a c k i n g, e v e ni ft h ee l e m e n ts i d e sa r epa . r a l l e lt ot h et r u ec r 配 k . 3 . 3 . 2P a r a l l e lCracks:ThirdParameter Asa l r e a d ym e n t i o n e d, i ft h es h a p eo ft h et e n s i l es o f t e n i n gc u r v ei sf i x e d, t h e nt h el i n e c r a . c k( f i c t i t i o u sc r a ι k )modeli sd e f i n e dbytwopa 陪 m e t e r s, Gf and, 刀 whilet hec r a c k 貧n edbyt h r e ep a ra . m e t e r sG" andWc ・F o rt h e貧c t i t i o u sc r a c kmodel , bandmodeli sde t o o, a .l e n g t hp紅a. m e t e r( c a l l e dt h e“ c h a r a c t e r i s t i cl e n g t h " )hil.9b e e nd e f i n e d ;e EG , / 刀 2 : : : o ( w h i c hi ss i 凶 l a rt oE q .3 . 7a sw e l l回Ir w i n ' sE q .2 . 1 0 ) . However,出i sp a r a m e t e ri sa . d e r i v e dparam e t e r ,notanindependentone, w h i l eωci sani n d e p e n d e n tp訂 a m e t e r . Wh y 四回? t h i sd i f f e r F o ri s o l a t e dc r a c k s, i tt u r n so u tt h a tt h ee t f e c to f叫 ont h er ,回 u l も si sa l m o s tne g l i g i b l e , : f , p r o v i d e dt h a tt h es o f t e n i n gp a r to ft h e8 t r e s s s t r a i nd i a g r a mi sad j u s t ed8 0国 t oa l w a y sy i e l d t h es a . m ef r a c t u r ee n e r g yG f o r組 Y叫・ However , t h ep a r a m e t e rWcdo 回 m akead i t f e r e n c ei n s i n c ei tp r e v e n t sa d j a c e ntc r a c k sもobec l o s e rt h 姐 t h ec a s eo fd e n s e l ys p a c e dp a r a l l e lc r a c k s, t h ed i s t a n c e叫 ( o t h e r w i s et h ea d ja . c e n tc r a ι kb a n d swouldo v e r I ap, whichi si na d m i s s i b l e) . l i a c a n c eofU 1 ci sn o tr e a l l yul ew i d t hofthea c t u a lcmkingzo nea tth $0t h ep h y s i c a l勾 I f r a c t u r ef r o n t, b u tt h eminimuma d m i s s i b l es p a c i n go fp a r a l 1 e lc r 民 k s .I si tne 偲 s s a r yt h 抗 t h i ss p a c i n gbeam a t e r i a lf r a c t u r ep a r a . m e t e r ? w e r ,c o n s i d e rt h a te l a s t i cl o n g i t u d i n a l f i b e r sa r ebondedt oal ong,出 i nc o n c r e 旬 F o r組 組s b a rs u b j e c t e dt otension.Thec r o s ss e d i o r l o ftheabersi Bs olugeshbt h ec o m p o s i s e 四 e x h i b i t snos o f t e n i n gu n d e rt e n s i o n, a l t h o u g h出 . ec o n c r e t ep a r td o e s .Thent h e s p e c i m s t a t eo fu n i f o r ms t r a i ni sa l w a y ss t a b l e . A c c o r d i n gt ot h el i n ec r a . c kmodel,c r a c k sc 姐 f o r mω a r b i k a r i l yc l o s es p a c i n gi ns u c has y s t e m ,a ndd e p e n d i n gonもhes p a c i n g , もh el o a . d e l o n g a t i o nd i a . gramP(u )mayf o I l owa町 o f s h ed i a g r a m sshownas0129, 0 1 3 9, O149.orO159 i nF i g .3 . 7 . Thus ,t h e r ei sana m b i g u i t y , 町 0 r泊 i 加 n ぬb o j 片 e c 凶 t t i 吋 v i t 句 yo fr 田 e s 叩 ' p ∞ o n s 紙 Theproblemc a no f COusebee l i m i n a t e dbye n r i c h i n gH i l l e r b o r g ' sl i n ec r a c kmodelw i t hat h i r di n d e pen d e n t t h eminimumc r a c ks p a c i n g, s( i n d e p e n d e n to fe ) . Thi si s ,ofcourse, m a t e r i a lp a r a m e t e r, e q u i v a l e n tもoparameterωc・ “ 3ふ 3RelationtoMicromechanismsofFr acture Then o r m a lm i c r o 蜘 a hsamsst h ef r a c t u r ep r o c 倒 z o n emaybed i s t r i bu t e dr o u g h l y舗 showni nF i g .3. 8 a.Thel i n ec r a c kmodels i m p l i f i ω t h i ss t r a i nd i s t r i b u t i o nt a k i n gi ta sa ∞nstaMplustheDiracdeltafunction, Fig.3.8b.Thec r a c kbandmodels i u p l i f t e si t . t o o t a k i n gi ta sac o n s t a r 句 l u sa n o t h e rc o n 山 叫w i t h i nttm mckband(mta n g u l a zd i s t r i b i o F i g .3. 8 c .Then O I 伽 a lcontinuummodelg i v e sasmoothb e l l -s hapedd i s t 出 u S i o nacmsst h l 如 d , a sshowni nFig.3.8d(BahMandP i j a u d i e r C a b o t, 1 9 8 8 a ), andi nthebiteelemmt b . 3 .8 e . f o r ma sFほ M e a s u r e e n t so ft h el 午前I o n so f乱c o u s t i ce m i s s i o n sd u r i n gt h ef r a c t ur ep r o 閃 5 8( Labuz ,Do?d時 , 1985;MaliandShah,1988)indi c a t e ,despi t et h e i ri n e v i b a b l es c a t t ; r ,that S h a h t h ee m i s s i o ns o u r ω sa r el o c a t e do v e rar e l a t i v e l yw i d ebandi nt h ef r o n s a lr e g i o no ff r & c t u r e a si nt h ec r a c kbandm o d e l .Ont h eo t h e rh組 d, v a r i o u smeasurementso fs t r a i 回 ( F i g .3 . 8 f ), バ 目 立 ,Y ; 出官民; : ? ; ; z J 2 2 足立己主; J : ; : 4 2 2 ' z r ; r よ : 忠 良 z o n ea tt h ef r o n tf r a c t u r e, w h i c hm i g h tbeb e b t e rmodeledbyal i n ec r a c kmodel.I もma vbe n o t 仙 h a tt h ef r a . c t u r e町 a i n sm i g h tbemorel o c a ! i z e da tt h es 凶 c eo fas p e c i m e n m , duet ot h ew a l le t f e c tando t h e re f f e c t s . t h ei n t e r i o r Asf o rt h ev i s i b l ec o n t i n u o u sb a c h I r eb e h i n db h ef r a c t u r epr o c e s sz o n e , i tmustbeE m b e d ω i bi sfmumslyh i g h l yt04uoumm e a n d e r i n gS oeachddeo ft h esymmetryl i n ebya 姐 句o fupt oa b o u tt h emaxmuma g g r e g a t esim.Eveni fdlm i c r o c r a c k i n gw e r e∞n c e n d i s t h a t e dona . l i n e , i nv i e wo ft h et o r t u o s i t yo ft h i sl i n e, t hef r a c t u r ei sr e p r e s e n t e dnob e t t e rby ?sMgMImecmkth 叩 b yacmkbando fwidUldoneorMageqatesi z e si however p u t a t i o n a lr e s u l t st h i sw i d t h陀 a l l yd o e sn o tma 伽 i nmost仙 討i o n s, 前 4 e n t i o n e J I o r∞m 山 ふ 47 46 b o v e ) . r o f i l ea c r o s st h ef r a c t u r ep r o c e s sz o n e, m i c r o c r a c k i n gi sma . n i f e s t e dbyani n A l o n g乱 p 品 s ei nt h ea . v e r a g ema . g n i t u d eo ft h et r 姐 s v e r s es t r a i n s .T h i si n c r e a s ei sp r o b a b l yg r a . d u a l, c r . a sshowni nF i g .3 . 8 a, o ri naf i n i t ee l e m e n tformi nF i g .3 .8 e .O b v i o u s l yt h ew i d t ho f t h ef r a c t u r ep r o c e s sz o n eo b t a i n e dbym e a s u r e m e n t sd e p e n d sont h ec h o i c eo f出 . ec u t o f f . a l u eo ft h es t r a i n .T h i sm i g h te x p l a i nt h ed i 品 目n c e si ni n t e r p r e t a t i o no fv a r i o u st y p e so f v o b s e r v a t i o n s . t h e r es 田 m st obenoc o m p e l l i n gr e a s o n sf o rr e j e c t i n ge i t h e rt h ec r a c kband Tosumup, model, c h a r a c t e r i z e dbys o f t e n i n gs t r e s s s t r a i nr e l a t i o n s, o rt h el i n ec r a . c kmodel, c h a r a c a . t i o n s . Thec h o i c ei se s s e n t i a l l yam a t t e ro f t e r i z e dbys o f t e n i n gs t r e s s d i s p l a c e m e n tr e l c o n v e n i e n c eo fa n a l y s i s . 年一一一一平副ー P 3 品 + ー ム Mesh1ト→ベ四空当出企 2H It ート:→ィ宍 =0.05 ア / 3肝 隅 隅 密 画 / σt / ↑ f ' . 司 Et=-Ec1 8 / v 、 、 ε / フ → H = 4 内 . a-GOOF CrackBandModef , 2.3 Meshes1 C 〆 / / / _ グ く uoo a . 3. 4Non!inearTr i a x i a lConstitutiveModelsf o rS t r a i n S o f t e: n ing / 1川 t i山 r 4 之1 f Mesh3 │ M。¢αω 剖 e i は 。n d d /'¥Mesh2 commiteDGmqge ¥ / ___Mesh 1 ! odel ~~______,.,'-_," ,J M ¥μ-" . r 〆 「 ¥ / V σA ト¥ | displacement ム ( j n . ) ~ 8 0 . 0 1 F i g . 3 . 7F i n i t eE l e m e n tRe s u l t sf o rD i f f e r e n tC r a c kS p a c i n , gsi naT e n a r s i o n e dR β i n f o r c e db. Thes t r e s s s t r a i no rs t r e s s d i s p l a c e m e n tr e l a t i o n sf o rt h ef r a c t u r ep r o c e s sz o n ewhich h a v ebe 回 d e s c r i b e ds of a rt a k ei n 色oa c c o u n to n l yt e n s i l ec r a c k i n gi noned i r e c t i o n .However , t h ef r a c t u r ep r o c 問 sz o n emayu n d e r g om u l t i p l ec r a c k i n gi nmanyd i r e c t i o n sandmaybe 回 s i v es t r e s s e sp a r a l l e lt ot h ec r a c kp l a n e, a sw e l la ss h e a rs t r e s s e s . s u b j e c t e dt oh i g hcompr Toh a n d l es u c hs i t u a t i o n s, f i n i t ee l e m e n ta n a l y s i sn e e d st obeb a s e donag e n e r a ln o n l i n e a r nt h ef r a ε t u r ep r o c e s sz o n e ,thematerialundergoesdegradationo fi t s t r i a x i a lmode . l I m e c h a n i c a lp r o p e r t i e s, whichi sb a s i c a l l yo ftwok i n d s : a . d a t i o no fm a t e r i a ls t i f f n e s sdueもodam a . g es u c ha sc r a c k i n g, whichi sd e s c r i b e d ( 1 )d e g r byc o n t i n u u mdamagem e c h a n i c s( o rt h ef r a c t u r i n gma . t e r i a lt h e o r yo fD o u g i l l, 1 9 7 6 ), andi s c h a r a c t e r i z e dbyu n l o a d i n ga c c o r d i n gt ot h es e c a n tm o d u l u s jand ( 2 )d e g r a . d a t i o no ft h es t r e n g t ho ry i e l dl i m i t, w h i c hi sd e s c r i b e dbymoderna d a p t a t i o n s o fp l a s t i c i t y . e s c r i b e d Ther e a lb e h a v i o ri sac o m b i n a t i o no fb o t h.Suchcombinedb e h a v i o rh a sb田 nd c r e , 旬e . g ., byt h ee n d o c h r o n i ct h e o r y , p l a s t i c f r a c t u r i n g出切r yo rt h edamageth 田 r y f o r∞n o fO r t i z .F o rar e v i e w, 附 Ba . i a n t( 1 9 8 6 ) . ,t r i a x i a . lc o n s t i t u t i v em o d e l sc a n n o tbea . c commodatedi nt h es t r e s s G e n e r a ln o n l i n e a r 自 f o rs h a r pl i n ec r a c k s, whicha r eb a s i c a l l yc o u n t e r p a r 旬 o fu n i a x i a l d i s p l a c e m e n tr e l a t i o n s t r e s s s t r a i nr e l a t i o n s .Thec r a c kbandmodelc 佃 b ee x t e n d e dt oa . c commodates u c hm o d e l s, b u tf o rap r o p e rもr e a t m e n t, t h en o n l o c a la p p r o 叫 hs 田 m st obene 田 s s a r y . a nh a n d l eb o t ht h ep r o c e s sz o n eo ft e n s i l e Av e r yg e n e r a lc o n s t i t u t i v er n o d e l,whichc , a sw e l l出 ∞m p r e s s i o nands h e a rdamage , andd e s c r i b e sw e l lt h en o n l i n e a rt r i a x i a l f r a c t u r e b e h . a v i o ri ng e n e r a l, i st h em i c r o p l 却 e mode l .Thatt h i smodelg i v e s出 ec o r r e c tr 田p onsef o r f r a c t u r e si se v i d e n c e dbyt h ef . a c tt h a ti ty i e l d s. ac o r r e c tt r a n s i t i o n a ls i z ee f f e c t js e eF i g .3 . 9 . i a n t組 dO z b o l t( 1 9 8 9 ) . t a k e nf r o mBa ' zp a r a l l e lt ot h ec r a . c k, a tl e a s ts i m p l ea d j u s レ Whent h e r ei sal a r g e rc o m p r e s s i v es t r e s sσ mento f出 es o f t e n i n gs t r e s s d i s p l a c e m e n tc u r v eo f円 v s ・d rs t r e s s d i s p l a c e m e n tc u r v eo f co 円 v s .fl n 白 d st obemade.B i a x i a lf a i l u r ee n v e l o p eda . t a .i n d i c a t et h a tt h et e n s i l es t r e n g t h l i m i ts h o u l dbes c a . l e ddownf r o mf :approximatelytothevaluef igivena l r 回 . d yi nE q .3 . 9 . 49 48 Whethert h e円 ( o )c u r v eo r円(ら)c u r v es h o u l dbes c a l e ddownp r o p o r t i o n a l I y, o rs h i f t e d c ,o rs u b j e c t e dt osomemorec o m p l e xt r a n s f o r m a t i o nt or e s e c tt h ee f f e c to fむ i sno も down knowna tp r 田e n t .Whenσ zく o andF i g .3 . 9i so b s e r v e d, t h e n, o f∞u r s e , t h ea r e ou n d e r t h e円 ( ι )curvedoesnotr e p r e s 四 t t h ef r a . c t u r ee n e r g yGJ, b u tI s1 ω st h a nG, fi . e .E q .3 . 2 I si n v a l i d . . 。 3.5RandomParticleSimulationofMicrostructure emmlsslons F i g . 3 . 8F r a c t u r eP r o c e s sZo ne, ondS t r e s sP r o f i l e sA c r o s s1 t :( a )A c 叫 , ( b )ー( c )V a r i o u sR e p r e s e n t 乱t i o n t u 、 、 、 ー0 . 3 ¥ ー0 . 4 / " 画 、 ¥ 切 ー ¥ t 5 、 " 、 て 二i 、 、 -0.5 ¥ ¥ ¥ 。 -0.6 ¥ C J l ¥ ¥ 、 、 ¥ 。 -0.7 -0.8 0.4 0.6 ¥ Size Effecl Law FEM,Nonl.Microplane Model Test results 0 . 8 ¥ . 0 1 .2 1 .4 log(d/dg) Fig.3.9MaximumL o a d so fNotchedBeamS p e c i m e n sC a l c u l a t e dby , Comparedt oS i z eE f f e c tLawandt oT e s t N o n l o c a lM i c r o p l a n eModel . z a n t叩 dO z b o l t, 1 9 8 9 ) Datι(Ba , y e tc o m p u t a t i o n a l l ye x t r e m e l ydem 組 d i n g , modelf o rn o n l i n e a rf r a c Themostr e a l i s t i c t u r eb e h a v i o ro fc o n c r e t ei sp r o v i d e dbyt h erandomp a r t i c l es i m u l o t i o n .E x t e n s i v ei n v e s t i st y p eh a v eb e e nc a r r i e do u tbyWittmanne ta l ., ( 1 9 8 4 ) ;蹴 a 1 8 0R o e l f s t r ae t g a t i o n so fぬi , . 1(1985), andRo e l f 前回 ( 1 9 8 7 ) .C r a c kf o r m a t i o nandp r o p a g a t i o ni ncomputerg e n e r a t e d a c o m p o s i t es t r u c t u r e swas8 t u d i e dn u m e r i c a l l y .Thesei n v e s t i g a t o r su s e ds t a n d a r df i n i t ee l e mentt e c h n i q u e s, s u b d i v i d i n ge a . c ha g g r e g a t ep i e c eandt h em o r t a rr e g i o n si n t omanyf i n i t e anda l s oc o n s i d e r i n gw e a k e ra g g r e g a te -mortari n t e r f a c e s.Thec a l c u l a t i o n sp r o e l e m e n t s, v i d e dv a l ua . b l ei n s i g h t sb u ta r ee x t r e m e l ydemandingonc o m p u t e rt i m eands t o r a g ewhich madei ti m p o s s i b l et omodelv e r yl a r g es t r u c t u r e s . a l s oc a l l e dt h ei n t e r f a c ee l ement I nt h es i m p l e s tv e r s i o no frandomp a r t i c l es i m u l a t i o n, model( Z u b e l e w i c zandBa . z a n t, 1 9 8 7 ;Ba . z a n t, T . めb a r a , KazemiandP i j a u d i e r C a b o t, 1 9 8 9 ), t h ea g g r e g a t ep i e c 回 a r ec o n s i d e r e da sr ig i dd i s c so rs p h e r e s .Animport a n t副 p e c ti st h a tt h e i r os a t i s f y c o n f i g u r a t i o ni sg e n e r a t e drandomlybyt h ec o m p u t e r .Amethodt odot h i ss o回 t もe d t h ep r e s c r i b e dmixr a t i o so fa g g r e g ω e so fv a r i o u s' s i z e sh a sb e e ns u c c e s s f u l l yf o r m u l a a . z a n t, Tabbara, KazemiandP i j a u d i e r C a b o t, 1 9 8 9 ) . (B T h e s em o d e l sr e p r e s e n tana d a p t a t i o no fane a r l i e rmodelo fC u n d a l l( 1 9 7 1) , S e r r a n oand Rod r i g u e z -O r t i z( 1 9 7 3 ), C u n d a l landS t r a c k( 1 9 7 9 ), Kawai( 1 9 8 0) , andP l e s h aandA i f a n t i s s a n do rg r a v e l .l nc o n t r a s tt oC u n d a l l ' smodel , ( 1 9 8 3 )f o rg r a n u l a rm a t e r i a l ss u c ha sr o c k, もh es i m p l ef r i c t i o n a lt e n s i o n r e s i s t i n gc o n n e c t i o nb e t w e e np a r もi c l e smustber e p l a c e dbyan I n e l a s t i ci n t e r a c t i o nw i t hf r a c t u r e .T h i sh a sb e e ndon ebya s s u m i n ga 1 1t h ei n t e r pa r t i c l e 七 七h em i d d l eo ft h ei n 旬r p a r t i c l ec o n t a c tl a y e r d e f o r m a t i o n st obec o n c e n t r a t e da tap o i n ta o ft h em a t r i x(mo 山 r ) .I nt h e0 柑n a lv e r s i o no ft h i smodel , bo t ht h en o r m a lands h e a r i n t e r a . c t i o n s( b u tn o tmoments)w e r et a . k eni n t oa c c o u 爪 t h es h e a ri n t e r a c t i o nb e i n ge l 副 i c , 組 dt h en o r m a li n t e r a c t i o ne x h i b i t i n gas u d d e nl o s so fs t r e n g t ha f t e r乱c h i e v i n gt h ep r e s c r i b e d も hl i m i tf o r出 ei n t e r p a r t i c l ef o r c e . As i m p l e rv e r s i o n, i nw h i c ht h es h e a ri n t e r a c t i o n s t r e n g b e t w e e np a r t i c l e si sn e g l c c t e dandt h en o r m a l( a x i a l )i n t e r a c t i o ni sc h a r a c t e r i z e dbyaf o r c 仔 d i s p l a c e m e n tc u r v ew i t hp o s t p e a kg r a d u a l ω f t e n i n gi nt e n s i o n,h a sb e e nde v e l o pedby Ba Za n t, T,晶b b a r a ,Kazemi組 dPijaudier-Cabot(1989)andshownt obes u f f i c i e n tf o rmost p u r p o s e s, e x 伺 p tもh a ti tt e n d st og i v eaf r a c もu r ep r o c e s sz o n et h a ti sn a r r o w e rands h o r t e r 叩 t h a to b t a i n e dwheni n t e r p a r t i c l es h e a rs t i f f n e s si st a k e ni n t o乱c c o u n t( a n da l s ot h 組 t h t h a te x p e c t e dfroms i z ee f f e c td a t af r o m∞n c r e t ef r a c t u r et e s t s ) .T h i sv e r s i o no ft h emode l i~ e q u i v a l e n tt oarandomt r u s sw i t hs o f t e n i n gmembersandr e p r e s e n t sag e n e r a l i z a t i o no f o u g i l l( 1 9 7 7 ) .Theprogrammingi sq u i t es i 凶 a rt on o n l i n e a rf I n i t e t h emodelo fB u r t副 D e l e m e n tp r o g r a m s . 5 1 50 J : : ; t z ぷ ♂ ; ; ; 二 : ; : : r 222:c:2c:21;:22:7iJZ 2222IZ : : 1 2 2 p q i J F F X : 諮右 鵠話器 5 2 詰 芯 . p L ;::P;:;2:(;;;;)i1nipiJ1Ac ,fromwhich pe 8 o r st h 8 0 tt h ew i d t ho ft h ebando f r 叫 t u r et e n d st obea b o u tt h r 田 m aximuma g g r e g 8 0 t e m i c r o c r a c k sa h e 8 0 do ft h ec o n t i n u o u sf 主ねじず:ぷ ロ ロ エ ヱ : : 川 222:;;t p r e p o s t e r o 凶 ydem . a n d i n gf o r∞m p u t e rt im J73Z22LTL 。 ' 0 ' 50 . ミ 1 20 3A 、 s . 、 c l . 回 。 w ( 0 ) 吋拍 30 。 ∞ O. 。‘。 0.02 0 06 Displacement U ( 1 7 ¥ 1 7 ¥ ) 2C ' : l 了n r e e Doi n t Be ~ di 刊 (Notc門 eョ) D i r e c tT e n s i o n( U n n o t c h e d ) . a 句 ﹁ ♂ ぷ/イ " c .0 電 今 、 町 ¥AU 。 toV 。 判 ラ ヘ ' ミ ミ 1口 、 、 、 - . 4=0.378 C=2.143 s A=1 .176 C=2. 005 仰h u 4 - ( b ) ・ 10 ,~ d/d o 7 5 10 d/da F i g . 3 . 1 0Ra . n domP 8 0 r t i c l eS i m u l a t i o no fC o n c r e t e( 8 0 ), Re s u l t i n gL o a d D e f l e c t i o nC u r v e sandS i z eE f f e c t( b ), a n dC r a c k i n gPa t t e r n( c ) ( c o n t i n u e d ) 5 2 53 C h a o t e r4 .SPECIALNONLINEARFRACTURE 恥i O I J E L SBASEDONADAPTATIONSOFLEFM S p e c i a ln o n l i n e a rf r a . c t u r emode l sdon o tωtemptt omodel出 eb e h a v i o ro ft hef r a c t u r e p r o 伺 8 Sz o n et h r o u g hs t r e s s d i s p l a c e m e n to rs t r e s s s t r a i nr e l a t i o n s .I n s t e a d, t heyi n t r o d u c e r o x 凶 a t e l yr e f l e c t 80mea d a p t a t i o n so fl i n e a re l a s t i cf r a c t u r em e c h a n i c s(LEFM)which叩 p a . c t u r eb e h a v i o r . t h en o n l i n e a r i t yo ff r 4 . 1E f f e c t i v eCrackModels Thee f f e c to fal a r g e s i z ef r a c t u r ep r o c e s sz o n ea もt h et i po f出 en o t c ho rc o n t i n u o u s 山田 t h e5もr e s s e sn e a rt h ec r a . c kt i pandp u s ht h epeaks t r e s sf a r t h e ra h e a d . c r a c ki st or e Co n s e q u e n t l y , t h es p e c i m e nb e h a v e sr o u g h l ya sane l a s t i cs p e c i m e nw it hal o n g e r , e f f e c t i v e c r 叫 k1 叩 g t h, ae • T h i sl e n g t hc a nbed e f i n e di nv a r i o u sways ,givi n gsomewhatdi f f e r e n t r e s u l t s .1 nD u g d a l e B a r e n b l a もt ・t y p em o d e l s( o rd u c t i l em e t a l s, t h ee f f e c t i v ec r a c kl e n g t hi s c a l c u l a t e du s u a l l yfromt h ec o n d i t i o nt h a tt h esumo ft h es t r ぉ si n t e n s i t yf a . c t o r sa tt h et i p o (t h ee 晶c t i v ec r a c kduet ot h ef a rf i e l ds t r e s s e sandt o出 ey i e l ds t r e s s e si nt h ef r a . c t u r e 倒 s z o n ebez e r o( D u g d a l e , 1 9 6 0 jB a r e n b l a t t, 1 9 5 9 ) . p r o Butt h i sd e f i n i t i o ns e e r n sn e i t h e rq u i t ea p p r o p r i a t en o rt o oc o n v e n i e n tf o rb r i t t l emat 令 r i a l s .F o rt h e s em a t e r i a l s, ane l a s t i c a l l ye q u i v a l e n t(Gr i 伍 . t h t y p e )e f f e c t i v ec r a c kh a sb e e n d e f i n e da st h ec r a c kf o rw h i c hもhee l a s t i c a l l yc a l c u l a t e dc o m p l i 組 問 o ft h es p e c i m e ni st h e samea st h emeasuredu n l o a d i n gc o m p l i乱n c eCuf o ru n l o a d i n go ft h es p e c i m e nfromt h ep e a k i v ec r a 心km o d e l sf o rc o n c r e t eu s i ngt h i sd e f i n i t i o nh a v eb e e n l o a dp o i n t( c r i t i c a ls t a t e ) .E賀 町t 叩o s e dbyNa l I a t h a . mbi加 dK a r i h a . l o o( 1 9 8 6 a )andS w a r t zandR e f a i( 1 9 8 9 ) .Thee f f e c t i v e p r c r a c kl e n g t h ,a.,dependson出 emicrostructure,a . sw e l l制 t h eg 曲 m etryo ft h es p e c i m e n . WhenKlci sc a l c u l a t e du s i n gt h ee f f e c t i v ec r a c k, t h er e s u l t ss 肥 mt obea pp r o x i m a t e l ys i z e a ta .andJ(Ic m i g h tbeu s e da stwof r a . c t u r ep訂 a m e t e r s . i n d e p e n d e n t . Thusi ta p p e a r sもh Tot h i send , e m p i r i c a lr e l a t i o n sf o rr e l a t i n gt h ee f f e c t i v ec r a c kl e n g t ht ot h en o t c hl e n g t hi n af r a c t u r es p e c i m e nhaveb e e nd e v e l o p e dont h eba . s i so fe x p e r i m e n t a ld a t abyN a l l a 山a mbi 泳叫∞ ( 1 9 8 6 a ) . andKar Then e e df o re m p i r i c a le q u a t i o n sr e l a t i n gt h ee f f e c t i v ec r a c kl e n g t hもot h en o t c hl e n g t h r e d u c e st h es c o p eo fa p p l i c a b i l i t yo ft h i sm e t h o d .I na d d i t i o n, onen e e d se m p i r i c a le q u a t i o n s h e s el i m i ta t i o n sa r eo v e r ∞mei nt h et w o p 訂 a m e t e rmodel f o re a c hs p e c i m e nge~~etry. T p r o p o s e dbyJ e n qandS h a h .T h i smodeli sd e s c r i b e dn e x t . 4 . 2Two-ParameterModelofJenqandShah F i g .3 . 1 0c o n t i n u a t i o n T h i smodel( J e 叫 a ndShah, 1 9 8 5 a, 1 9 8 5 b jShah ,1988), w h i c ha p p e a r st og i v ear a th e r r e 引 i 抗i cp r e d i c t i o no fc o n c r e t ef r a c t u r eb e h a v i o r, i nv o l v e stwof r a . c t u r eparame t e r s :( 1 )t h e cr~ticaI s t r e s si n t e n s i t yf a c t o rKlca tt h et ipo fane f f e c t i v ec r a . c ko fl e n g もha tP u, andt h e ea ~~ítical v a l u eOCTODo ft h ec r a c kt i po p e n i n gd i s p l a c 泡m ent, wh i chi sc a l c u 1a t e da tt h et i po f e e x i s t i n gc~~ck o rn o t c h, wh~se iengthi sd~noted 出 ao, anda t凡.Thee f f e c t i v ec r a c k t h ep r 54 4 . 3Geometry-DependentR-CurveDeterminedfromS i z eE f f e c t Law Aq u a s i e l a s t i ca n 8 0 l y s i so fn o n l i n e a rf r a c t u r ec 姐 bea c c o m p l i s h e dbyc o n s i d e r i n gt h e e n e r g yr e q u i r e df o rc r a c kg r o w t h ,R,tobevariabler 8 0 出e rt h a nc o n s t a n t . Thec u r v eo fR v s .t h ec r a c ke x t e n s i o nci sc a U e dt h eR c u r v e( o rr e s i s t 組 問 c u r v e ) .Thec r a . c kp r o p a g a t i o n c o n d i t i o ni sG = R ( c )r a t h e rt h a nG= G" andi ft h ev a l u eo fR ( c )i sknown, t h er e s p o n s e i sc a l c u l 8 0 t e da c c o r d i n gt ol i n e a re l a s t i cf r a c t u r em e c h a n i c s, r e p l 8 0 c i n gG w i t hR ( c ) .A l t e r , b u te q u i v a l e n t l y , onec a nd e f i n et h eRc u r v ei nt e r m so ft h ec r i t i c a ls t r e s si n 句n s i t y n a t i v e l y f a c t o rKlt!w i も h! ( IR= =( E 'G )1/2, a . ndu s et h ec r a c kp r o p a g a t i o nc o n d i t i o nK1= =J(IRi n s t e a d o ff{1=J(lc・Thep h y s i c a lr e 邸 o nt h a tt h ee n e r g yr e l e a s er a t e,R, r e q u i r e df o rc r a c kg r o w t h 臼 e swi 山 ci sもh a ti ti sd e t e r m i n e dbyt h es i 回 o ft h ef r a c t u r ep r o c e s sz o n e , whichg r o w s . inαe Al a r g e rp r o c 回 sz o n ed i s s i p a t e smoreen e r g y .Howt h ez o n eg r o w s, o fc o u r s e , depend sa l s o oni n t e r a c t i o nw i t ht h eb o u n d a r i e s . fR c u r v e swass u g g e s t e dbyI r w i n( 1 9 5 8 )andf o r m u l a t e df o rm e t a l si nd e t a i l The∞n切 pもo 色r t a i nt y p e so f∞n c r e t es p e c i m e n s, R c u r v e sw e r em e a s u r e d byK r a f f te ta l .( 1 9 6 1 ) .F o rc byWecharatanaandShah( 1 9 8 0, 1 9 8 2 )andJ e n qandShah( 1 9 8 5 a ) .F o ral o n gt i m ei twas 回 be t h o u g h tt h a tt h eR c u r v emeasur e de x p e r i m e n t a l l yonas p e c i m e no foneg e o m e t r yc u s e du n i v e r s a l l y , 副 am a t e r i a lp r o p e r t ya p p l i c a b l ea p p r o x i m a t e l yt oo t h e rg e o m e t r i e s . But , ε f f ec ; t l v eC ; , . lti t hC " OC;K pr& ・ c n l l c a tcrackQr , 帥 1 1 t h C r i t i c a lP O l n l i nt K,,,~ CTODcC T O D c d “ ( a ) 。 唱 l e n g t h, a sm e n t i o n e db e f o r e, c a nbec a l c u l a t e dfromt h ec o m p l i 姐 c eCu r e c o r d e da tu n l o a d i n g 四 i sl o a d e dt ot h ec r i t i c a ls t 8 0 t e( F i g .4 . 1 ) .J e n qandShahe x p e r i m e n t a l l y whent h es p e c i m o b s ぽv edt h a twhena .wωcalculatedfromt h ec o m p l i a n c em伺 s u r e m e n t sf o rn o t c h e dheam t h ec a l c u l a t e dv a l u e so fdCTODw e r emoreo rl e s si n d e p e n d e n to ft h es i z e部 w e l la s s p e c i m e n s, g e o m e t r yo ft h es p e c i m e n s .T h i si n d i c . a t e dt h a tdCTODandKlcmaybef u n d a m e n t a lm a t e r i a l J e n q組 dShaha l s op r e d i c t e dt h eu n i a x i a lt e n s i l e p a r a m e t e r s .U s i n gt h e s etwop a r a m e t e r s, s t r e n g t h, t h es p l i tc y l i n d e rs t r e n g t h , t h es i z ee f f e c ton∞n v e n t i o n a lK1e, andt h es i z ee f f e c t ont h emoduluso fr u p t u r e . Thee f f e c l i v ec r a c kl e n g t h , α e, c a l c u l a t e dont h eb a s i so ft h e s etwof r a c t u r ep a r a m e t e r si s ont h ec o m p r e s s i v es t r e n g t ho fc o n c r e t e, and f o u n dt odependont h es i z eo ft h es p e c i m e n, ont h es t r a i nr a . t e . Thev a l u eo fa .d e α e a s e sw i 出 i n c r e a s i n gs t r e n g t handw i t hi n c r 回 s i n g 田 ,t 出 e o fane l a . s t i cp e r f l yb r i t t l em a t e r i a l, dCTODa p p r o 品c h e sz e r oand s t r a i nr a t e .F o rt h ec a .a p p r o a c h e sa o ・ Thev a l u eo fKlcfromt h i smodelh a sb開 nshownt obee s s e n t i a l l yi n d e p e n d e n to ft h e wedges p } i t t i n g g e o m e t r yo ft h es p e c i m e n s .R β s u l t so b t a i n e du s i n gcompactt e n s i o nt e s t s, c u b et e s t sandl a r g e ,tapered,doublecantileverbeamsconductedundertheauspi田 so f RILEMCommitt 伺 FMT8 9showedt h a tt h etwof r a c t u r ep a r a m e t e r s(Klc anddCTOD) m i g h tbec o n s i d e r e d剖 g 回 m e t r y i n d e p e n d e n tm a t e r i a lp a r a m e t e r s( J e n qandSh a . h, 1 9 8 8 8 0 ), a l t h o u g hal a t e rs t u d y( J e n qandSha . h, 叩 1 9 8 部8 吋 b )indicated8 0s 討 i g 凶 n 1 1 f 包 i c a . 組 n 叫 1 ti n f f 丑 l l 田 u l e n c 白e o fg ∞metげ 凶 1 I 恥t s h o u l dben o t e d出 t ha t出 t h i ぬ smodel臼 nbe叩 a p凶 p l i 吋 e d t ω os p e c i m e n sand“ s 古 位 加 r u c t 町 u 1 汀r 田 e sw 叩i t 出 h o ω rw i 比 t h o 叫 u b n o ω t c h e ω s .Themodelh a sb e e ne x t e n d e dt omixed-model oa . d i n gandt oi m p a c tl o a . d i n g( J o h n 制 dS hah, 1 9 8 5,1 9 8 6, 1 9 8 9 ) . い¥ 55 CMOD T y pi c ;o IP l o t0 1L o c :V~ . C r oC :kM outl ' lO o e n t n gD ' s : l Ic : c e m e n t T w o . p a r a m e t e rf r a c t u r : cr n o d e Jf o rc o n c r : c t e ( b ) F i g . 4. 1TwoP a r a m e t e rF r a c t u r eModelo fJ e n qa n dSh ぬ( 1 9 8 5 ) t L へ R(c) , P 1' 1 コ コ ← 3向 コ コ 』 エ ーE crackI ength C c F i g . 4 . 2RC u r v e sf o rD i f f e r e n tSpecimenG e o m e t r i e sa n dT l w I rP o s t PeakA d j u s t m e n t s C 56 57 a s e .Tod 8 o Yweknow( e . g .Wecharatana. a ndSh 油, 1 9 8 3 8 0 ; t h i sh a sn o tt u r n e do u tt obeもhec Z 姐 も , KimandP f e i f f e r, 1 9 8 6 )出 品 f o rc o n c r e t e , R c u r v e sdepends t r o n g l yont h es h 叩 eo f Ba もr u c t u r eo rs p e c i m e n .T h e r e f o r e , t oen a . b l et h eu s eo fR c u r v e swhichc e r ta . i n l yr e p r e s e n t t h es o n en e e d ssomemethodt oo b t a . i nt h eR c u r v ef o rag i v e ns p e c i m e n 8 0v e r ys i m p l ea p p r o a c h, g e o m e t r yfromsomeb. a s i cf r 拭 t u r ec h a r 8 o d e r i s t i c s . Asp r o p o s e dbyBa Zant. a ndC e d o l i n( 1 9 8 4 ), andB a z a n t, KimandP f e i f f e r( 1 9 8 6 ), t h eR . nyg i v e ng e o m e t r yma . ybed e t e r m i n e dont h eba . si so ft h es i z ee f f e c tl a . w( S e c .5 . 1 ) . c u r v ef o ra . 4 )app 回1" Sa d e q u a t ef o rt h. a tp u r p o s es i n c e( u n l i k et h eR c u r v e )i ti sa . p p l i ・ T h i sl a w( E q .1 c a b l e ,i nt h es 8 0 m ef o r m, t omostg e o m e t r i 回 w i t h i nas i 民 r 組 g eupt oa . b o u t1 : 2 0, which 印 鑑c e sf o rmosta . p p l ic a . t i o n s .TheR c u r v ei st h e nobtainedωthee n v e l o p eo ft h ef r a c t u r e 四 f o rg e o m e t r i c a l l ys i m i l a rs p e c i m e n so fva . r i o u ss i z e s . Ane 回 yc a l c u l a t i o n e q u i l i b r i u mc u r v o ft h eR c u r v ei sp o s s i b l eon 臼 t h ep a r a m e t e r so ft h es i z ee f f e c tlawh a v eb e e nd e t e r m i n e d . 8 o r , t h eR c u r v e, g i v i n gt h ef r 8 0 c t u r ee n e r g yr e q u i r e dp e ru n i tc r 8 0 c kgrowth , c 組 be I np a r t i c u l o b t a i n e d( 低cordingt oBa . z a n tandKa . z e mi,1 9 8 8 )出 f o l l o w s( f o rd e r i v a . t i o n,蹴 Appendix 1 ) : l ヱ (α=2 , αo=22, c=a-ao) 、 d' d R ( c )=Gfz: 色 J g (α。 ) cJ --V ' ( α )(g ( α ) g (α ¥ g ' ( α ) 。 ) -, - , w h i c h ー =一一一 l一一一一 α+α。 I Cf . ・ U ) ( 4 . 1 ) ( 4 . 2 ) whereGJ= f r a c t u r ee n e r g y( o b t a i n e df r o mt h es i z ee f f e c t! a w ), c=a- ao=c r a c ke x t e n s i o n f r o mt h en o t c ho ri n i t i a lc r a c kt i p, g ( α ); ;n o n d i m e n s i o n a le n e r g yr e l e a s er a . t ed e f i n e dbyt h e 2 t o t a le n e r g yr e l e a . ser a . t eG i nt h eformG =g (α) P 2 / { E b d )( u s i n gG =K }/E', g ( α)c 組 be f o u n df r o mt h eKl-v a l u e sw h i c ha . r ea v a i l a b l ei nha . n d b o o k ss u c h酎 Tadae ta 1 . ,1 9 8 5, and Murakami, 1 9 8 7, f o rt h eba . s i cs p e c i m e ng e o m e t r i e s, andc a na . lw a . y sbeo b t a . i n e db y e l a s t i c 祖 a l y s i s ) ; α =a o / d ,andCf = d o g ( α) / g ' ( α w h e r ed sac o n s t a n tfromt h es i z ee f f e c t oi l a w ;CJi st h ee 恥c t i v el e n g t ho ft h ef r a c t u r ep r o c e s sz o n ei n叩 i n f i n i t e l yl a r g es p e c i m e n( s e e E q .5 . 5 ), d e f i n e da st h ed i s t a n c efromt h en o t c ht i pt ot h et i po ft h ee q u i v a l e n te l a 凶i cc r a c k . 側 z o n el e n g t h( d e f i n e da st h ed i s t 姐 c ef r o mt h en o t c ht i pt ot h ep o i n to f Thea c t u a lp r o c ma.x泊四もe n s i ! es t r e s s )i sa . bou も2 c j , a . si n d i c a . t e dby∞m p a r i s o n sw i t hv a r i o u sm o d e l so f t h ep r o c e s sz o n e(eιHorii, 1 9 8 8 ;H o r i i, e ta l ., 1 9 8 7, 1 9 8 8 ;P l 姐剖 andElices, 1 9 8 8 a ;Ba Z8 0 n t andKazemi, 1 9 8 8 ) . . 2d e f i n et h eRc u r v ep a r a m e t r ic a . l l y .KnowingGfandd , onemayc h o o s e E q s . 4 . 1叩 d4 o . as e r i e sofα-v a I u e s, a . ndf o re a c ho fthemc a l c u l a . t e員 凶 cf r o mE q .4 . 2組 dt h e nR (c )from E q .4 .1 . TheR c u r v e swhich. a r eo b t a i n e df r o mEq s .4 . 14 . 2f o rv e r yd i f f e r e n ts p e c i m e n 回 m e t r i e sa r ev e r yd i f f e r e n tf r o me a c ho t h e r, 部 i l l u s t r a t e di nF i g .4 . 2a . . g Re c e n t l yi 比 tw 晶制 sf o u n d但 ( Ba 量 z . a n 帆 G e t t 凶 u組 dK 拙 e 叫 m 1 l o a d ι d e H e c “ t も i ぬ ∞ onc u r v ei お st ω obec o r r e c t l yp r e d i c t e d, t h e泊 i n c α r 位 e a s 犯eo fRmustbea r r e s t e da tt h e 戸e a k l o a dp o i n t( p o i n tp i nF i g .4 . 2 ), 回 i n d i c a t e dbyt h eh o r i z o n t a Il i n e1 3i nF i g .4 . 2 . p Ther e a s o nt h ec r i t i c a le n e r g yr e l e a . s er a t eRc a n n o te x c 田 dt h a . tf o rt h epeakI o a ds t 8 0 t ei s t h a t :( 1 )t h ei n c r e a s eo fR i sduet oani n c r e a s eo ft h ef r a c t u r ep r o c e s sz o n es i z e( al a r g e z o n ed i s s i p a t e smoree n e r g y ), 叩d( 2 )i nt h ep o s t p e a kr e s p o n s e, t h ef r a c t u r ep r o c e s sz o n e 。 。 ) 。 ・ a . v e l sf o r w 8 o r dwi 山o u tg r o w i n g d e t a c h e si t s e l ff r o mt h et i po ft h en o t c ho ri n i t i a lc r a c k8 0 n dt r 戸 ・ T h i sbeh a . v i o rh剖 p r e v i o u s l yb e e ne x p e r i m e n t a l l yi d e n t i f i e d, e . g ., f r o mm伺 s u r e m e n t s i ns i a Z 組t,LeeandP f e i f f e r ,1987)飢 dwasv印 刷 especia . l l ybyt 倒 so fr o c k onalumiDum(B (B 話 相t , G e t t uandKazemi, 1 9 8 9 )andh i g h -s t r e n g t h∞n c r e t e(Ge も t u, B話 回t姐dK. a r r, 1 9 8 9 ) . Thes t u d yo ft h eR c u r v e si n d i c a t e st h a tt h eωn 田p to fe l a . s t i c a l l ye q u i v a . l e n te f f e c t i v e c r a c klmgthc a n n o tg i v ev e r ygoodr e s u l t si f a c o n s t a n tf r a d u E ee n e r g yv a l u ei s回 目o c i a t e d w i t h出 ee f f e c t i v eロa c k .C a l c u l a t i o na c c o r d i n gt oE q s .4 . 1 4 . 2i m p l i e st h a tn o to n l ymust t h ec r a c kl e n g t h, a=ao+c ,d i t r e rf r o ma o ,butalsotheenergyreleaseraterequiredf o r . s ew i t ht h ec r a c kl e n g t hi fa .s o l u t i o nt h a . ti s α ' a c kg r o w t hmustbec o n s i d e r e dt oi n α ' ea . a p p r o x i m . a t e l ye q u i v a l e n tt on o n l i n e a rf r a c t u r em e c h a n i c ss h o u l dbeo b t a i n e d .T h i si sn o t 白 川l yp r o p o s e dm o d e l s( e . g .K a r i h a l o o ' smodel ,S e c .4 . 1and6 . 6 )・ d o n ei nsomer e 59 58 。 = C h a p t e r5_.__SI~E EFFECTANDBPJTTLENESSOF STRUCTURES s e q u e n c eo ff ra . c t u r em舵 h叩 i c st h e o r y .T h e r e f o r e , Thes i z ee f f e di st h emosti m p o r t a n t∞n i ti s1 0 g i c a . It od e t e r m i n ef r a . c t u r ep r o p e r も i e sf r o mi t .Knowledgeo ft h es i 田 e f f e c ti sn e e d e d ゆo r a t o r y s i z et o f o rt h edesimo fs t r u c M r e hwhichg e n e r a l l yi n v o l v e se x t r a p o l a t i o nfroml r 叫 “ 4 白 g ( αJ )~ g ( α 0 )+g ' (αo ) ( c J / d ) 5 . 1SizeEffect LawforMaximumNominalStress 凡 一M en N σ - a . I l ys i m i l a rs t r u c t u r e so rs p e c i m e n sw i t hg e o m e t r i α l l ys i m i l a r Wec o n s i d e rg e o m e t r ic n o t c h e so Ec r a c k s .Wew i l lnowi l l u s t r a t et h ea n a l y s i sf o rs i m i l a r i b yi ntwod i m e n s i o n s, hthoughthemultsm t h esmeformihi も yinもh r e e -d i m e n s i o n s(Ba Z a n t, 1 9 8 7 a )・ The no 凶n a . l山 e s sa tf a i l u r e(maximuro1 0 0 0 )ma . ybed e f i n e da s ( 5: 1 ) ら し い ) 九 ( 句 ) a ) T h i s四 p1'e s e n t st h es i z ee f f e c tl a . we x p r ' 田 s e di nt e 1 ' mso fm a t e r i a lp a r a .m e t e r s(B 話a.n tand Kazemi, 1 9 8 8 ) .E q .5 . 5i se q u i v a l e n tt ot h es i z ee f f e c tl a wσ' N= B刀( 1+s)-1/2, β=d( ゐ .4 )i fonemakest h en o t a t i o n s: ( E q .1 。τり 一 q α i j 、 u fr 一 , s ' 、 司 ・ c w d n3 n ( 5 . 3 ) 内切 。 +(c/d) 1 a B ~ ~(蒜7y/2 ( 5 . 6 ) 1 tm晶yben o t e dt h a tamoreg e n e r a ls i z ee f f e c tl a . wh a sb e e nd e r i v e d ; t σN =B I : ( 1+ s r 12 r / , b u trd e p e n d songeomet 1 ' y, andf o ra l lg 句 m e t r i e scombinedt h ev a l u er~ 1wasf o u n dt o bea p p r o x i m a t e l yoptimum(Ba Z a n tandP f e i f f e r, 1 9 8 7 ) . I ti sf u r t h e ru s e f u lt or e w r i t et h es i z ee f f e c tl a wi nt h eform / 叩= ( 部y ムィ i nw h i c hg ( α )i saknownf u n c t i o no ft h e凶 t i v ec 1 ' a c kl e n g t h α =a / d . Be c a . u s eG = K]/ E, f u n c t i o ng ( α)maybeo b t a i n e df r o mt h eexpmSionsf o rths t r e s sintemtyf a c t o r s, h i c ha r ea l w a y so ft h eformKI= Pk(α) /. , f dwherek(α)=non-dimensionalfunction.For t y p i c a ls p e c i m e ng e o m e t r i e s, f u n c t i o nk (α )c 叩 bed e t e r m i n e dfromt h eKIva . l u e sa v a i l a b l e i nhandbooksandb e x t b o o k s, andf o ro b h e rg E o m d r i e si tcmbeo b t a i n e dbyl i n e a re l a s U c a n a l y s i s N o n l i n e a rf r a c t u r ew i 七h a .s o f t e n i n gc o h e s i v e( f r a c t u r ep r o c e s s )z o n ei sa . pproxim a . t e l y e q u i v a . I e n tt ol i n e a . re l a s t i cf ra . c t u r eo f回 i n c r e a s e dc r a c kl e n g t h ( 5 . 5 ) σN= 一 一 i nw h i c hP . . u l t i m a t el o a d, b=t h i c k n e s so ft h es t r u c t u r e , d=c e r t a i nc h o s e nc h a r a c t e r i s t i c fもhes t r u c t u r e , andC o e 伍c i e n ti n t r o d u c e df o r∞n v e n i e n c e .( F o rexample , i n n=c 蹴 o fb e n d i n go f乱 s i m p l y叫 p o r t e dbe a . mo fs p a nla . n dd e p t h, d, w i t h∞ncentmtedlO 吋 t h ec 2 抗 出d span, wemayi n t 1 ' oduceσN品目 t h emaximumb e n d i n gs t r e , 回i . e . σN =3 凡l / 2 b d= ιpjbdwhereら = 1 .5 l / d= c o n s t a n t ;b u twec a ne q u a l l yw e l lt a k et h ec h a r a . c t e 1 'i s t i c 34hsidnω d=l=自pan ,anddenotingh=beamdepbhwethenhaveσN = c . . Pu/b I . wherecn=322/2M=consunt, p r o v i d e d b h ebeamsa r es i m i l a r ) . 斗 σゐ /2Er e p r 飴 e n t st h enomina . Is t ra . i ne n e r g yd e n s i t y , もh e町 a i ne n e r g yi 5 N o t i n gt h u= V(σゐ/2E)ψ(α),whe1'eV = 匂 bd 2= volumeo ft h es t r u c t u r e( e o= c o n s t a n t )姐 d =GId-Thee n e r g yr e l e a s er a t ei sg i v e nbybG=-OU/θa=一δ (U/ θa ) / d .Nowwitht h e n o t a t i ( α ) C ! e o /2=g ( α), t h ee n e r g yr e l e 蹴 凶eo ft h es t r u c t u r ei sf o u n dt oh a v et h e form P 2 g (α)σμ ( 5 . 2 ) G=一 一 一 =--g(α) 2d c!E Eb α= 向 +C o r α=α ( 5. 4 ) w h e r eαJ= c J / d組 dg ' ( α。 )= d g ( α。 ) / d α .1 ti sr e a s o n a b l et oassumet h 抗 出eG v a l u ea t . lt ot h eg ( α ) -va . I uea tt h ep e a k1 0 0 0, i . e . t h ep e a k l o a di sn o tc o n s t a n tb u ti sp r o p o r t i o na G= G J g ( α) / g ( αJ ) .S u b s t i t u t i n gt h i sandEq.5. 4i n t o5 . 2, s e t t i n g凡 =σ (N b d / c 2, and n) s o l v i n gf o rσN, o n eo b t a i n s r e v i e wi t sf , お orm叫 u 1a . 叫 b i ∞ o nandc o n s e q u e n 偲 5 i nf u r t h e rd e t a i l . 。 w h e r eα t h ea c t ua . l l e n g t ho ft h ec o n t i n u o u sc r a . c ko rn o t c h, a . n dc= d i s t a . n c eo ft h et i p o ft h ee q u i v a l e n te l a . s t i cc r a . c kfromt h et i po ft h ei n i t i a lc r a . c ko rn o t c h( F i g.4 . 2 ) . . t9 ' ( α )>0,w h i c ha p p l i e st ot h ema . j o r i t yo fs p e c i m 佃 gωmetri , 田 the Assumingt ha f r a d u r ep r o c e s sz o n ef i r s tg r o w si ns i z ew h i l er e m a i n i n ga t t a c h e dt ot h et i po fn o t c ho r i n i t i a . 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It oc Jo n l yf o rani n f i n i t e l yl a r g es p e c i m e n( d→∞), buti ss m a l l e rt h a nc Ji nf i n i t e . n s i o n , weh a v e s i z es p e c i m e n s .BasedonT a y l o rs e r i e se x pa 2 nw h i c h ( 5. 7 ) 】 九 ロ7 てPu n_ g ( α。 ) =. j g ' ( 的)話 , D =訳可d ( 5. 8 ) ~ere.7'N a . n dD maybei n t e r p r e t e da st h es h a p e i n d e p e n d e n tn o m i n a ls t r e s sa tf a i l u r eand a . pe i n d e p e n d e 凶 c h a r a c t e r i s t i cdimen 討o no ft h e山 u c t u r e(Ba.孟姐も姐dK a z e r r u, 1 9 8 8 ; t h es h Z a . n t , G e t t u姐 dKazemi, 1 9 8 9 )・ Ba 60 61 1 nt h ec 回 eo ft h r e & d i m e n s i o n 晶l 自i 凶l a r i t y , weh a v eU = V(σ ゐ/2E)ψ(α),V =匂♂ and C l d G= o U / δa=一 θ (U / δa ) / di nw h i c he o組 dC la r esomec o n s t a n t s .D e n o t i n g 0 . 0 - ・ 4 巳 . 宅 -u E.M z b a ・ 1 thq •. 点M dnu a‘, a { , , ・ , , .t a u - a - - 白 u' •• .• - M • 内 f t ' ( α ) e o /2C tc !=g(α)weob凶 inagainEq.5.2,andalltheremainingderivationi st h es a m e . t h er e s u l も8a r ea g a i nt h esameaBE q .5 . 5 5 . 8 . T h e r e f o r e, . f e c tl a w( E q .1 ム5ム 5.7)国 na180bederivedverygenerallybymeansofd i Thes i z ee m e n s i o n a la n a l y s i sands i 凶 l i t u d ea r g u m e n t s( s e eBa Z a n t, 1 9 8 4 a , 1 9 8 5 b, f o rもwod i m e n s i o n s ; andB話 a n t, 1 9 8 7 a , f o rt h r 田 .d i m e n s i o n s )i ft h ef o l l o w i n gh y p o t h e s i si sadop もe d : ぽ g yr e l e a s ef r o mt h es t r u c t u r eo fany日i z edi n t ot h ef r a c t u r ep r o c e s sz o n e Thet o t a len d e p e n d sonb o t h :( 1 )t h el e n g t h ,ao,oftheωntinuouscraιkornotch,and( 2 )al e n g t h ∞n s t a n to ft h em a t e r i a l. 1 ぽ e i omayrepresentthematerialconstantcombination 0 ・H l o= E'G J !f ?(Irwin'scharacteristics i z eo fn o n l i n e a rn e a r t i pz o n e ), o rt h ee . f e c t i v e1 田 g t h C Io ft h ef r a c t u r ep r o c e s sz o n ei n姐 i n f i n i t e l yl a r g es p e c i m e n ,o rt h ee f f e d i v ew i d t hωco f fo n l yp a r t( 1 )o ft h i sh y p o t h e s i si sa d o p t e d , oneo b t a i n st h es i z ee . f e c to fl i n e a r t h i sz o n e .I e l aB t i cf r a c t u r em e c h a n i c s(σNα d-1! , ) l ' 組 di fo n l yp a r t( 2 )i sa d o p t e d, oneo b t a . i n st h a もo f p l a s t i cl i m i ta n a l y s i s( i . e .nos i z ee . f e c t ) . 舵 k Fromt h i sh y p o t h e s i si ta p p e a r st h a tt h es i z ee f f e c tl a wi sr e s t r i c t e dn e i t h e rt ot h ec r bandmodeln o rt ot h el i n ec r a c kmodelb u ti sv a l i df o ranyn o n l i n e a . rf r a c t u r emodelw i t h al a r g ep r o c e s sz o n e .S i n c et h el e n g t ho rw i d t ho ft h ef r a c t u r ep r o c e s sz o n ei sr e l a t e dt o t h ec h a r a c t e r i s t i cl e n g t hio fa .n o n l o c a lc o n t i n u u m , t h es i z ee f f e c tlawa l s oa p p l i e sf o rt h e n o n l o c a lcontinuuma p p r o a c ht of r a c t u r e .T h i sh a sb e e nd e m o n s t r a t e dbyf i n i t ee l e m e n t s f o ras i m p l en o n l o c a lsmearedc r a c k i n gmodeli nBa Z a n t祖 dL i n( 1 9 8 8 b )( s e eF i g .3 . 6 )姐 d f o rt h en o n l o c a . Jm i c r o p l a . n emodeJ( w h i c ha l s od e s c r i b e sw e l ln o n l i n e a rt r i a x i a lb e h a v i o ri n c o m p r e s s i v es t 前田)b yB a z a n tand0量b o l t( 1 9 8 9 )( F i g . 3. 9 ) .P a r t i c l es i m u l a t i o no fr a D Tabbara , KazemiandP i j a u d i e r C a b o t, 1 9 8 9 )山 os h o w sg o o d domm i c r o s t r u c t u r e( B a z a n t, a g r e e m e n tw i t h出 es i z ee f f e c tl a w . cc r i t e r i o nf o ra c c e p t a b i J i t yo fn o n l i n e a rf i n i t ee l e m e n もc o d e sf o r∞n c r e t es t r u c t u r e s AbaBi i st h a tt h e ymustd e s c r i b et h et r a n s i t i o n a ls i z ee . f e c t( E q .1 . 4 ) . f e c t] a ww i t hf r a c t u r et e s td a t af o rd i . f e r e n ts p e c ・ Ane x t e n s i v ec o m p a r i s o no ft h es i z ee imeng e o m e t r i e sf o rc o n c r e t eωwell剖 m o r t a ri sf o u n di nB a z a n t叩 dp f e i 恥 r( 1 9 8 7 ) ;s e e o / d , , ) .Theset 田 t sh a v eshownt h a tv e r yd i . f e r e n tt y p e so fs p e c i m e n s F i g .5 . 1( w h e r eA o=d , c 回 t r i ct e n s i o n , e c c e n t r i cc o m p r e s s i o n )y i e l da b o u tt h esamef r a c t u r ee n e r g y( e v e n ( b e n d i n g t l 町 p a r a m e t e rv a l u e si nF i g .5 . 1d i . f e rc o n s i d e r a b l y ) . t h o u g h出 eo 1 . 1 . 5 •• ( , , ) 0 . 0 0 . . . . . . . . . __J ι . 。 J e c tLaw F i g . 5. 1C o m p a r i s o no fS i z eE1 . ・L r . 伺tR . e a u l t son E q .4 . 1 )w I t h' ミ・O.IO~IS叫 制 B・08 8 8 . . Y'.Y ‘ . : I e t r i e s, S p e c i m e n so fV a r i o u sG回 m f o rh o t hC o n c r e t ea n dM o r t a r( a . f t e r ~伺・ 18.80 8・O.15i . o ・ '0 . 8 ・ 加. 1 _T)}})vJn:J 20 . . . . 0 . I . . . . . ・4凶ー~ ・0.25・ 0 . . 0 . 8 Ba.Z姐t姐 . dPf e i 1 J e r1 9 8 7 ) I 0-COnet.le ! : I -¥ (orb.p 0 . 8 . . ー ・ 1 . 0 1 . 2 l o C( d / da) ・ . J 1 . 8 ," ・ t1 d l v l c l . . . 1T . . u 0.0 、 ・ , c '・ o . z 2 、 、 0 . ' : Z : b v ・ ~ -0. ・ -0. 1 . 0 0. ' ・ ・u10&(d;a ) ・ 0 O . U 1 . a 1 .8 ・ 1 5 . 2BrittlenessNumber ( b ) E q .5 . 6e x p r e s s e s七hes i z ee . f e c tl a wi nt e r m so fm a t e r i a lf r a c t u r ep a r a m e t e r s , GIandcJ ( c o n v e r s e l y , t h e i rd e f i n i t i o nandmωurementmaybeb a s e dont h es i z ee . f e c tl a w , E q .1 . 4 ) . T h i sf a c twase x p l o i t e dbyBa Z a n t( 1 9 8 7 乱 )t od e f i n et h 陀e 砂 c a l l e 吋d凶 b r r i 川 も “ 川 t も l e 問 n e s 凶SD l 1 即 a p p r o x i m a t e l yd e s c r i b e st h eb r i t ω t l e n e s s0 ぱf t 仕 出 hes t r u c t u r a lr e s p o n s er e g a r d l e s so ft h es t r 邸 u 1 κ c t u r 印e s h a p e .BasedonE q '5 . 7, t h eb r i t t l e n e s snumberi se x p r e s s e daB(Ba Z a n tandKazemi, 1 9 8 8 ) : “ 1 0 mall da' ' ' ' ' . 90 ' N ~i__sma t1da F i g . 5 . 2C u r v e so fS t r u c t u r a l J d 一勺 α ‘, ' F 、ta、 a -JaE 何一 一 一 , 、 ‘ , g コ 切 、 g D 一勺 ρ μ 一 一 官 S i z eE f f e c tf o rD i f f e r e n ( 5 . 9 ) A邸 r e g a . t eS i z e s logd 63 6 2 w h i l eont h eb a s i so ft h es i z ee f f e c tl a . we x p r e s s i o ni nE q .1 . 4i ti se q u i v a l e n t l ye x p r e s s e d酪 (Ba . Z 叩t , 1 9 8 7 a ;BazantandP f e i f f e r , 1 9 8 7 ) : , r t 2 s=え=内州合;=AB2fd ( 5 . 1 0 ) )'i vD , . ,,.‘、 a sく 0 . 1 p l a s t i cl i m i ta n a l y s i s 0 . 15 .s5 .10 nonlinearf r a . c t u r em e c h a n i c s s>10 l i n e a re l a s t i cf r a c t u r em e c h a n i c s F o rs<0 . 1, t h ee r r o ro fもhep l a s t i cl i m i ta n a l y s i si sl e s st h a . n4.7%o fσN・ andf o r , comparedt ot h e β>loth ee r r o ro fl i n e a re l a s t i cf r a c t u r em e c h a n i c si sa l s ol e s sthan4.7% n o n l i n e a rf r a c t u r em e c h a n i c ss o l u t i o n .I fane r r o ru n d e r2%i sd e s i r e d,t h e nt h en o n l i n e a r ngemustbeexp a . n dedt o1 / 2 5壬9三2 5 . O t h e rd e a n i t i o n so ft h eb r i t t l e E l e s snumbero fa s b r u c t u r ehaveb e e np r o p o s e db e f o r e . r e s e a r c h e r si nr n d a l Eknewt h a ts t r u c t u r a lb r i t s l e n e s si sb a s i B e g i n n i n gw i t hI r w i n ( 1 9 5 8 ), c a l l yc h a r a c t e r i z e dbyt h er a t i oo ft h es t r u c t u r es i z et oI r w i r i ' ss i z eo ft h en o n l i n e a rz o n e , E q .2 . 1 0 .T h i sd e f i n i t i o nw回 c o o p t e df o rc o n c r e t es t r u c t u r e sbyH i l l e r b o r g( 1 9 8 5 b )( c f . a l s oE1fgml, 1989)whop r o p o s e dt oc h a r a c t e r i z et h es t r u c t u r a lb r i t t l e m s sbyt h er a t i od / ゐ 九一 何 日 C Meme1hanicB Eq59makesitpossiblehoedcuMepsolelyonthebuisofalinearelasticf 、f o rwhicht h es h a p eandt h el e n g t ho ft h eο c凶 n u o u sc r a . c ka tu l t i m a t el o a . d( o rt h e s o l u t i o n n o t c hl c n g t hi naf r a c t u r es p e c i m e n )mustbeknown. 伺 ? witht h et r a l s i t i o no f t h es i Z E Eq.510yieldst h eb I i t t l e n e s BE 1 1 I n b e r w h i c h e x a c t l ya g E e f T e c hl a wt ot h ep l a s t i ca n a l y s i ss o l u t i o nwhent h es t r u c t u r e日i z ei sv e r ys m a l l .Coe 伍c i e n もB n P u / b d f :whereP u .= u l t i m a . t el o a d蜘 e s sa . c c o r d i n gt oap l 越も i c c a nbed e t e r m i n e da sB=c l i m i ta n a l y s i sform 山 f o rt h es t r u c t u r a . lf a i l u r e ,s 叫 a semploy 出 nt h ec u r r e n tωdesや r d i a g o n a ls h e a zf a i l u r e , p u n c h i n gs h e a rf a i l u r e , t o r s i o n a lf a i l u r e , etc.Eq.510ismoEea c c u r a 巴 f o r自m a l ls , w h i l eE q .5 . 9i smorea ι c u r a t ef o rl a . r g es . Asat h i r dmethodf O i :d e t e r m i n i n gb, i f出 emaximuml o a d sf o rag i v e ns t r u c t u r a lgeome t r y紅 eknownf o rf : li g n i 五c a n t l yd i f f e r e n ts i z e s, e . g ., ont h eb a s i so fl a . b o r a t o r yt e s t so ront h e b a s i so fS I l i t ee l e m e n ta I l a l y s i sbasdonsomemethodo fn o n l i n e a rf r a c t u r emιh 姐 i c s, t h e n r e g r e s s i o no fもhet e s tr e s u l t sbym位 n so ft h es i 民 e f f e c tl a wy i e l d st h ep a r a m e t e rd, from whichb=d / ゐ AItemauveiyi も品I s os u 伍C 国 t oo b t a i nt h ef a i l u r el o a dP uf o rav e r ys m a l ls i z e( s→ 0 )by plMicl i m i ta n a i y s i s, whichy i e l d sB=du/ 叫, a . n dt h e nd e t 釘 m ineσN fromt h eLEFM whichy i e l d sd o / d=σ (N/B f D2. o l u t i o n, Thev a l u eF=1(ord=do)correspondsinU1esimefectp l o ho fl o g σN v s .l o gdt ot h e r i z o n t a l掛 Y 町 t o t ef o rt h es t 回 附hc 山 r i o nandt h ei n d i n e d出 ymFo 七 e p o i n twheret1叫o f o rもhel i n ea . re l a s t i cf r a c t u r emech 姐 i c si n t e r s e c t ;F i g .1 .4 .F o rs<1 , t h eb e h a v i o ri sc l o t op l a s t i cl i m i t叩 乱l y s i s ,andf o rs> 1i ti sc l o s e rt ol i n ea . re l a s t i cf r a c t u r emech 岨 l C S・ I f s< t : :1o rs> >1, n o n l i n e a rf r a c t u r em e c h a n i c si sn o tn e 印 s s a r y .Themethodo fa n a l y s i smay . Z a n t, 1 9 8 7 ;Ba . Z a n tandP f e i 狂e r, 1 9 8 7 ) : bec h o s e na sf o l l o w s(Ba w h e r ee f ?(hecalleae h e“ c h a r a c t e r i抗 i c l 叩 g t h " ;b u tt h i sc o n s i c t sw i t hp児吋・ o= EGJI ot o u st e r m i n o l o g yi nn o n l o c a lc o n t i n u u mt h e o η ) .T h e r e f o r ewew i l lc a l l l ot h ec h a r a c t e r i s t i c . r p i n t e r i( 1 9 8 2 )p r o p o s e dt h eb r i t t l e n e s snumbers= Gj/f : d .Simi1a . r p r ω e s sz o n es i z e . Ca d e f i n i t i o n swぽ e回 r l i e rp r o p o s e df o rc e r a m i c sbyG o g o t s ie ta l .( 1 9 7 8 )岨 dHomenye ta l . ( 1 9 8 0 ).Theb r i t t 1 e n e s snumberso fm 削 凹l l 的 e E 出b 叫 o r 喝 g, Cω むa 叫 r 叩 巾 戸 p 叩 阿 i 泊 n t 凶 釘 凶i ,Gゆ o g 酔o t 旬 凶 s 討i凶 a n 吋d恥 H Io附m l 紅 en o ti n d e p e n d e n to ft 出 hes t r u c t u r eg 印 e ome 抗t r 可 y却 dt 山 h凶u sc a n n o tbeu s e da su n i v e r s a l , a . b so l u t ec h a r a . c t e r i s t i c s( e . g ., s= 3couldthenm品 naveryb r i t t l eb e h a v i o rf o rones t r u c t u r e gωmetry組 dav e r yd u c t i l eb e h a v i o rf o ra n o t h e rg e o m e t r y ) . A l t h o u g hE q s .5 . 9and5 . 1 0d e f i n et h eb r i t t l e n e s snumberi nr e l a t i o nt ot h es i z ee f f e c t もi m a t el o a d, もh eb r i t t l e n e s snumber. a l s od e t e r m i n e st h en a t u r eo ft h ep o s t p e a . k ont h eu l r e s p o n s e , p a r t i c u l a r l yt h es t e e p n 悶 o ft h ep o s t p e a . ks o f t e n i n g( 1 0 0 0d e c r e a s e叫 i n c r e a s i n g d i s p l a c e m e n t ) .Thatwasa l r e a d yi l l u s t r a t e dbyt h es e r i e smodeli nF i g .1 .1 3 . . Z 叩t , i tappe 町 st h a t I nv i e wo ft h eu n i v e r s a l i t yo ft h eb r i t t l e n e s snumherp r o p o s e dbyBa as i m p l ea d j u s t m e n to ft h ec u r r e n tl i m i t a n a l y s i s -b a s e dc o d ef o r m u l a s, t a k i n gi n t oa c c o u n t a . c t u r em配 h a n i c s , c a nb emOOebyr e p l a c i n gt h en o m i n a ls t r e s sa tu l t i m a t el o a a, n o n l i n e a rf r , 九a sg i v e nw i t h o u tc o n s i d e r a t i o no fs i z ce f f e c t, byt h ee x p r 四 S l o n (三 v~n) ( 5 . 1 2 ) However ,themethodtocalculated o, onw h i c hsd e p e n d s, s t i l ln e e d st ober e s e a . r c h e d . E q .5 . 1 2i n d i c a . t 回 t h a tt h e r em i g h tbeal o w e rl i m i tv: : a mon出 enominalstrength, duet oa 心l u r em配 h a n i s m .Anexampleo fs u c ha p o s s i b l et r a n s i t i o nt osomen o n b r i t t l ef r i c t i o n a lf l i m i ti ss 田 ni nBr 招 i l i a ns p l i t -c y l i n d e rt e s t s( s e ee . g .B話 相t , 1 9 8 7 a ) . 5 . 3OtherSizeE: f f ectsandLimitations 5 . 3 . 1EffectofResidualDuctileorFr ictionaiLimit I nB r a z i l i a nt e s t so fs p l i もc y l i n d e rs t r e n g t h,Hωegawae ta l . ( 1 9 8 5 )ob 悶 v edt h a tt h e , 乱 山r e( i . e ., t h es p l i tc y l i n d e r抗r e n g t h )d e c r e a s e dw i t hi n c r e a s i n gd i a m e t e r n o m i n a ls t r e s sa tf o ft h ec y l i n d e r, h o w e v e r , beyondac e r t a i n1 a r g es i z eo fc y l i n d e r, nof u r t h e rd e c r e a s eo ft h e b s e r v e d( t h i sb e h a v i o rh a snowb e e nc o n f i r m e dbyt e s t so fBazant, Kazemi, s t r e n g t hw白 o e ta l( 1 9 9 0 ) ) .T h i ss u g g 回 t 5t h a tf o rac e r t a i ns u f f i c i e n t l yl a r g es i z e, t h e r ei sat r a n s i t i o n も 0叩 men o n b r i t t l ef a i l u r em氏 h a n i s ma tt h emaximuml o a d .Ther e 晶 o nmightbet h a tf o r l a r g es i z c st h emaximuml o a di sd e c i d e dbyf r i c t i o nonas m a l lwedg e -s h a p e dr e g i o nu n d e r t h el o a da p p l i c a t i o np o i n t s .F o rs m a l ls p e c i m e nd i a m e t e r s, t h el o a dt oc a u s e出 es p l i も t i n g h e1 0 0 0t oc a u s ef r i c t i o n a ls l i po ft h ew e d g e s h a p e dr e g i o n , 組d c r a c ki smuchh i g h e r出 ant ft h esi~ i s~ery l a ; g ea b o u t t h e r e f o r et h eb r i t t l emechanismo fc r a c k i n gd e c i d e s .How~ver, i 1 0 0d ", t h en o m i n a l自t r 回 S 机 w h i c ht h es p l i t t i n gf r a c t u r eo c c u r sb e c o m e sv e r ys m a l l, s m a l l e r 柏 t h en o m i n a l自t r e s s σ hw h i c hc a u s e st h ef r i c t i o n a ls l i po ft h ew e d g e ;s e eF i g .5 . 2 .Thus, t h ~t 1eas~_ f o rt h i st y p eo ff a i l u r e, i もs e e m sa p p r o p r i乱t et op u tal o w e rl i m i tont h e5 i z ee : f e c t ! a . w, σ ; . " beyondwhichσN C 姐 n o tbed e c r e a s e d .T h i smigh もa l s obet r u ef o rsome0 仙e r 65 64 f ♂l u r emechanisms , e s p e c i a l l yt h o s ei n v o l v i n gs p l i t t i n gc r a c k si nc o m p r e s s i o n . n . 1 . 5 . 3 . 2S t a t i s t i c a lS i z eE f f e c tDuetoRandomStrength S t a t i s t i c a lh e t e r o g e n e i t yo ft h em a t e r i a lnod o u b tp l a y sani m p o r t a n tr o l ei nt h em i c r o m e c h a n i s m sd e t e r m i n i n gt h em a t e r i a ls t r e n g t h .I td e s c r i b e st h ee f f e c to ft h es i z ed i s t r i f o rr e a s o n s b u t i o no ft h ef l a w si nt h em i c r o s t r u c t u r eont h em a t e r i a ls t r e n g t h . However, ,山er andomn 自由 o fs t r e n g t hduet ot h eh e t e r o g e n e i t yo ft h e a l r e a d ye x p l a i n e di nChapter1 m a t e r i a l, 回 describedbyWeibull-typeprobabilisticmodels, d o e sn o tseemt oh a v eamajor 田 o nt h es i z ee f f e c to h s e r v e di nb r il ef a i l u r e so fmostc o n c r e t es t r u c t u r e s, ex 田 p 七 i n f l u e n t h o s ew h e r et h emaximums 色E伺 si su n i f o r mo v e ra .l a r g epa : 吋 o ft h es t r u c もu r e( e . g ., al o n g u n i f o r m l y目t r e s s e ds p e c i m e ni nt 民 e n 四 凶 s s 討 山 i o n 小o 町rw heret h es t r u c t u r ef a i l sa t五r s tm a c r o c r a c k i n i t i a t i o n . “ 2 T E S T SB Y8 A L A N T& K A Z E t v 1l , ‘9 89 5 . 3 . 3Diffusion-TypeS i z eE f f e c t s 、 、 I tmustbekepti nmindt h a ts i g n i f i 伺 n ts t r u c t u r a ls i z ee f f e c 凶 C姐 b eo b t a i n e dduet o t h ed i f f u s i o np r o c e s so fd r y i n go fc o n c r e t ei ns t r u c t u r e s, t h ec o n d u c t i o no fhe a . tp r o d u c e d andt h en o n u n i f o r m i t i e so fc r 田 pp r o d u c e dbyd i f f e r e n c e si nt e m p e r a t u r eand byh y d r a t i o n, m o i s t u r ec o n t e n tt h r o u g h o u tt h es t r u c t u r e . .~ 句 . . ! e 。 で 。 ". . 0 ま8 1 5 . 3 . 4W a I lE f f e c ta saSourceofS i z eE f f e c t a J S t i l la n o t h e rt y p eo fs i z ee f f e c ti sc a u s e dbyt h ef a c tt h a taboundaryl a y e rn e a rt h e s u r f a c eo fc o n c r e t ei n e v i t a b l yh a sad i f f e r e n t∞m p o s i t i o nands t r e n g t ht h a nt h ei n t e r i o r o fc o n c r e t es t r u c t u r e .T h i sl a y e r , whoset h i c k n e s si sa b o u tonea g g r e g a t es i z e ,containsa , 邸r e g a t e sandah i g h e rp e r 田 n t a g eo fmorta : r .T h i sphenomenon l o w e rperαntageo fl a r g ea i sknowna s“ w a l le f f e c t " .I nas m a l ls t r u c t u r e, t h ee f f e c to ft h i sl a y e ri sl a : r g e rもhani na l a r g es t r u c t u r eb e c a u s et h eboundaryl a y e rt h i c k n e s si si n d e p e n d e n to fs t r u c t u r es i z e .F o r t h i se f f e c tbecomesn e g l i g i b l e . v e r yt h i c kc r o s ss e c t i o n s, R J ミ = b 2 4 6 丞 勺争 4 1 a• 6' d ( i γi ) ・ c 2 4 F i gふ 3D e v i a t i o nfromt h eS i z eE f f e c tLaw( E q .4 . 1 )Ob 町 v edi nB r a z i l 岨 C o m p r e s s i o nSp 1 i t t i n gT e s t s i ~.~.5 S i z eE f f e c tDuet o3DS i n g u l a r i t ya ttheEndsoftheCrackEdge A c c o r d i n gt ot h r e ed i m e n s i o n a ll i n e a : re l a s t i cs o l u t i o n s, t h es t r e s si n t e n s i t yf a c t o ro fa 出o g o n a lf r o n te d g ei sn o tc o n s t a n ta l o n gt h ee d g e( i . e . c r a c ki nap l a t ew i t has t r a i g h to r a c r o s st h et h i c k n e s so ft h ep l a t e )butd r o p st oz e r oa tt h ei n t e r s e c t i o n sw i t ht h es u r f a c e so f a.Z組も叩dE s t e n s s o r o, 1 9 7 9 )( e x c e p ti ft h eP o i s s o n ' sr a t i owerez e r o ) .Thec o n t h ep l a t e(B 符 q _ u e n c ei st h a tt h ef r o n te d g eo fap r o p a g a t i n gc r a c kmustbec u r v e d, s u c ht h a . tt h es u r f a c e 吋 pointsofthecrackedgemustbetrailingbehindtheinteriorofapropagatingcrack. f f e c to fp l a t et h i c k n e s sont h ea v e r a g ev a l u eo ft h es t r e s si n もe n s i t yf a c t o r T h i se n g e n d e r s組 e 記 e f f e c tmightn o tbel a r g eb u tc a nbec o m p l e t e l y o v e rt h ee n t i r ep l a t et h i c k n e s s . Thi ss i 6 6 6 7 e l i m i n 七 . ae di ft h et h i c k n e s s e so fs p e c i m e n so fa l ls i z e sa r et h es a m e . t h a . nt h a tw i t hau n i f o r ms . a n d . 5 . 3 . 6EffectofAggi"egateSize 5 . 3 . 7Effect ofMatrixStrengthandHighSt: r engthConcretes “ The合 fa 拭 蹴 訓£ f o ro 田 n ea 組.n dt h es 乱m ec ∞ ο n c α r e t , e 鳥w h i c hi m p l i e sも 由 he拍s ame喝 a . g g r e 句: , g a t e討 s i 配 .I ft h ea g g r e g a . t es i z ei s 姐 g 叫 t l 吋 凶u r ep a r a m e t e r sa . ndもh es i z ee f f e c tl a . wpa . ram 山 sc h 回 g e.ABproposdv c h Bdmt , t h esimefeetl a wn e e d st obedjusted掴 f o l l o w s( 8 白 色1 8 0Baz 姐 t 阻 dK im , 1 9 8 4 ) : 品 ,: f= . f t 1 ( + 伺 , s=え σN= ( 5. 1 3 ) 廿 i c h伊 f ?i 拘 st 由 hed i r e 氏c 七t e n s i l es t r e n g t hf o rac h o s e nr e f e r e n c e∞n 即c r 陀e も 旬 e,刀 i st 出 h巴 d i r 伐 e c t i nw h i i 匂 e 悶 n s 討i l e蜘 附h f 知 i o rma x 対 泊 i 加 m π m u ぽ 包 s姐 a . l o 伊 g 。u 凶sも 旬 ot h eP e t c hf o r m u l a .f o rt h ee 宜 f f e c to fg r 釘 a i ns 討 i 問 ∞ ont h ey i e l ds t r e n g t ho fp o l y i r y s t a l l i n em e t a l s , whichi sd e r i v e dbyt h edishcauonthmEy.Eq.5.13hasbeenshownt o 田 四 回o n ぬl yw e l lw i t ht h ef r a c t u r et e s td a t ao fCh. a n a .( 1 9 8 1 ), T a y l o r( 1 9 7 2 )姐 d1 9u r o a g r f f i c i e n t向 h a sb e e nc a l i b r a . t e d . e ta l .( 1 9 8 5 ), 組d∞e F i g .5 . 2( B a i a . n t, 1 9 8 6 )∞mparest h es i z eebdp l o t sa c c o r d i n gt oEq . 5 . 1 3 f o rB p e c i m ? n s 叩 t maximuma g g r e g a . t es i z e sd . . .The∞m p a r i s o ni smadeu n d e rt h ea s s u m p t i o n w i t hd i f f e r thudo=dawheren=constant, whichi so n l yapproxmate(andmightbei n v a l i di fv e r y p e s, e . g . roundandv e r ye l o n g a t e d. a gg 時耐久 o rc r u s h e d組 d山 e r d i f e r e n tagFegateもy t h es i z ee t r e c tc u r v e si nt h ep l o to fl o gσN v sl o g d a g g r e g a t e sMeu s e d . A c c o r d i n gt oE q . 5 . 1 3, . ve出 esames h a p eforanyd, i . e ., o n etransfOEmsMmottleEbybranSIMba-Ani n c r e a s e ha o fd "n o to n l ys h i f t st h es i z ee 宜e c tc u r v ev e r t i c a l l ydownward(dueωtheterm匂 j d . .)b u t d . .i sc o n t a i n e di nt h eb r i t t l e n 田 s numbers )・ a l s ot ot h er i g h t( b e c a u s ed o=n C o n s e q u e n t l y , t h es i z ee f f e c tc u r v e sf o rd i π e r e n td . .mayi n t e r s e c t, a sshowni nF i g.5 . 2 ( t h i smusthappeEu n l e s st h em a t e r i a lc o n s t a n tc u i slugeeno 略 的 ・ Thus, f o rasu 直c i e n t l y s m a l ls p e c i m e ns i z e ,品 h i g h e rn o m i n a ls 七r e n g t hi so b ta . i n e dw i 出 s m a l l e ra g g r e g a t es i z e sd w h i l ef o ra .s u 伍d e n t l yl a r g es p e c i m e ns i z e , ah i g h e rn o m i n a l s t r e n g t hi so b t a i n e dw i U 1 l a r g e r a . t es i z e s .F o ri n t e r m e d i a t es t r u c t u r es i z e s, t h ea g g r e g a t es i z em山 田 l i t t l ed i f f e r 叩 c e . a g g r e g 何回 i nF i g.5 . 2i sn o tknownv e r ya c c u r a . ωly , S i n c et h ei n t e r s e c t i o np o i n to ft h etwoc u t h e r emaybe乱 l a r g er a n g eo fs t r u c t u r es i z e si nw h i c ht h ee f f e c to ft h emaximuma g g r e g a t e s i z ei su n c e r t a i n .F o rs m a l lbeamsands l a b s,出i sf r. a c t u r e岨 a l y s i swoulds u g g e s tu s i n ga sm a . l la . g g r e g a . t e .C o n c r e t eda . msa . r es ol a . r g et h a tt h el a . r g e rt h ea g g r e g a t et l 叫 e t t e r , ωfar 剖 t h edams t r e n g t hi sc o n c e m e d .However , r e c e n tf r a c t u r et e s t so fdamc o n c r e t 田 o fSaouma a t e. as u r p r i s i n g l y ' s m a l le f f e c to f. a g g r e g a t es i z ea . scompa . r e dt ot h ee f f e c t e ta l .( 1 9 8 9 )i n d i c. o fa g g r e g a . t es h a p e . a . t es i z ea . n dg r. a d a t i o ne f f e c t sh. a v e叫s ob e e ns t u d i e df o rpolymer∞nCretes(ViP? A g g r e g l 組and 岨 a ndDharm . a r a j. a n, 1 9 8 7, 1 9 8 8, 1 9 8 9 a 丸山 Dharmaraja . n 組dV i p u l a . n a . nd a . n, 1 9 8 8 ) a . m p l e , t h e yf o u n dt h a ta .p o l y m e r∞n c r e t ew i t hwe l lg r a d e ds andh a s20%h i g h e rKlc F o rex 刷 , Ther e 目 n tspec 同 四l a ra d v a n c e si nt h es t r e n g t ho f∞n c r e t ehaveb e e na c h i e v e dm a i n l y byi n c r e a s i n gt h es t r e n g t ho ft h em a t r i xa . ndt h ea . g g r e g a te -matrixbond. 1nhighs t r e n g 出 c o n c r e t e s, t h ed i f f e r e n c e sb e t w e e nt h es t r e n g t hande l a s t i cmoduluso ft h ea g g r e g a . t ea . n dt h e m. a t 山 ( m o r t 紅)a r emuchs m a l l e rt h a . nt h e y訂 ef o rn o r m a l抗 r e n g 出 ∞n c r e t e .C o n s e q u e n t l y , h i g hs t r e n g t hc o n c r e も eb e h a v e sa samorehomogeneousm a t e r i a l ,a . n dt h er e s u l ti st h a tt h e f r a c t u r ep r 侃 , 伺 sz o n ebecomess m a l l e r( J o h nandSh a . h, 1 9 8 9 b ;G e t t u, BaZa n tandK a r r, 1 9 8 9 ).l nv i e wo ft h ed i s c u s s i o no ft h es i z ee f f e c t( 伺p e c i a l l yt h ef a c tt h a . tc o e f f i . c i e n td o i nt h es i z ee f f e c tl a wi sr e l a t e dt o出 es i z eo ft h ef r a c t u r ep r o c 悶 z o n e ), i ti sc l e a rt h a t, f o rt h es a . mes t r u c もu r a ls i z e, t h eb e h a v i o ro fh i g hs t r e n g t hc o n c r e t ei sc l o s e rt ol i n e a re l. a s t i c f r a c t u r em e c h a n i c s, i . e ., moreb r i t t l e , t h a . nt h ebeh a . v i o ro ft h es a . mes t r u c t u r emadeo fn o r m a l ∞n c r e t e .T h e r e f o r e ,f r a c t u r em e c h a n i c s姐 a l y s i sands i z ee f f e c t sa r emuchmorei m p o r t a n t f o rh i g hs t r e n g t hc o n c r e t e st h a nf o rnorm a . ls t r e n g t hc o n c r e t e s( e . g ., Shah, 1 9 8 8 ) . 5ふ 8SuppressionofSizeEffectbyYieldofReinforcement . r to ft h el o a d, A s i d efromp r o v i d i n gana d d i t i o n a ldu c t i l emechanismwhichc a r r i e spa t h ee f f e c もo fr e i n f o r c e m e n ti st os p r e a dt h ef r a . c t u r ep r o c e s sz o n e .T h e r e f o r e ,reinforced a ndt h ea . c companyings i z ee f f e c t s c o n c r e t ei sg en e r a l l yl e s ss u s c e p t i b l et of r a c t u r ee f f e c t s. t h a nu n r e i n f o r c e dc o n c r e t e .F u r t h e rs t u d i e sa r en e e d e di nt h i sr e g a . rd . I 七m ustbek e p ti nmindt h o u g h, t h a td e s p i t er e i n f o r c e m e n t, m姐 yf a i l u r e so f∞n c r e t e u r r e n tp h i l o s o p h y , t h eu l t i m a . t el o a di sasumo f s t r u c t u r e sa r eb r i t t l e .A c c o r d i n gt o出 ec ・ ,they i e l d i n go fs t i r r u p si nd i a g o n a ls h e a r )a . n dt h a tdue t h a tduet oy i e l dmechanisms( e . g t oc o n c r e t ea l o n e ,withoutreinforcingba . r s .T h i smigh もo f t e nbeac o n s e r v a t i v ea p p r o a c h, w h i c hi nf a c ti m p l i e st h a tt h ec o n c r e t ec o n t r i b u t i o nt ot h eu l t i m a t el o a . ds h o u l dbea . n a . l y z e d a c c o r d i n gt of r a c t u r eme c h a n i c s .l ti sl i k e l y ,however,tha . tt h ep r e s e n 回 o fs t e e landi t s i n c r e a s i n gt h ec o n t r i b u t i o nt of a i l u r el o a d y i e l d i n ga l t e r st h ef r a c t u r eb e h a v i o ro fc o n c r e t e, d u et oc o n c r 山 a l o n e(Ba 雲仙も a ndSun, 1 9 8 7 ).Th 悶 p r o b l e m s{leedt obes t u d i e df u r t h e r . 6 8 69 (α) b Chapter6 .EXPERIMENTALO RA NALYTICALDETERMINATIONOFMATERIALFRACTUREPARAMETERS Af t e ras l o wね 叫 madea se a r l y部 1 9 6 1( K a p l a n, 1 9 6 1 ), f r a c t u r et e s t i n go fc o n c r e t eh a s d e v e l o p e dt r ' 四 l e n d o u s l yd u r i n gt h e1 9 8 0 ' s . A numbero fe x p e r i m e n t 3 1t e s t i n gt e c h n i q u 偲 姐 ds p e c i m 四 t y p 伺 h aveb e e nt r i e d, andt h ed e v e l o p m e n t sc r y s t a l i z e di n t osomee f f e c t i v e method 目. d F 6 . 1NotchedBeamTests .1NotchedBeamT estsUsingLEFM 6 .1 Be . a mt e s t shav ebe 四 m ostp o p u l a . r ;p r o b a b l yb e c a u s eo fs i m i l a r i t i e sw i t ht h es t a n d a r d 回 t組 db e c a u s et h ef i r s tLEFMt e s ts t a n d a r df o rmeta . l su s e dt h i s moduluso fr u p t u r et (ASTM, 1 9 8 3 ) .I na d d i t i o n ,testingproceduresむ esimplerth姐 f o ro t h e rg 回 m e t r i 田 a nd p i c a lbe 釘 ng 四 m etryandt h r 骨 p o i n tl o a d i n g紅 ep r e a l i g n m e n te r r o r sa r em i n i m a l . A もy s e n t e di fF i g .6 . 1 a . Thes p a nt od e p t hr a t i of r e q u e n t l yI s4 : 1whichc o n f o r m sw i t ht h e r e q u i r e m e n t so fもheASTMs t a n d a r dE399( 1 9 8 3 ) .T h i si sb e c a u s ef o r m u l 回 f o ro p e n i n g mode耐 e s s i n t e n s i t yf a c t o r sf o rbeamsa r eg i v e nf o rSjd=4o r8-Tadae t3 1 ., ( 1 9 8 4 )and 倒 a l t h o u g hf o r m u l a sc a . na 1 s obeo b t a . inedf o ro t h e rSjd Murakami( 1 9 8 7 )a r et y p i c a ls o u r Go, S w a r t zandHu, 1 9 8 6 ) . r a t i o s( e . g ., Thet e s tu s u a l l yc o n s i s t so ft h ef o l l o w i n gs t e p s . 1 .N otcht h ebeamt od e p t h向 a tm i d s p a n .T h i sc o u l dbedonebyas a w c u to rby姐 i n s e r tc a s ti n t ot h ebeamandl a t e rr e m o v e d . ( b l P d Lood p 9~" 0 'I n l t l o l s l o p . 2 .U s i n gac o n s t a n tr a . t e ,about1t o1 0m i n .t of a . il u r e ,increa5etheload,desectiono r c r a c ko p e n i n gu n t i lf a i l u r e . Re ∞rd七he1 0 吋 -d e f l e c t i o n(OLPD)o rI o a d c r a c kmouth 8 田p o n s ec o n t i n u o u s l y . o p e n i n gd i s p l a c e m e n t( CMOD)r 3 .Thep e a kl o a d凡 o rt h el o a dPQ a . tt h ei n t e r s 民 t i o nw i t ha .s e c 姐も o fs l o p e95%o f i n i t i a ls l o p e ,s 開 F i g .6 . 1 h, i st h e nu s e dt oc a l c u l . a t et h ef r a c t u r e七o u g h n e s sfromt h e a . t i o n s h i p(ASTM, 1 9 8 3 ;Ta . dae ta , . 1 1985;Go,SwartzandHu,1986;Murakami, r e l 1 9 8 7 ) K1c=F(S , d , b , αQ, PuorP Q) ( 6 . 1 ) 0d e t e r m i n e di spresumedt obet h ec r i t i c a l, o p e n i n gm o c l es t r e s si n t e n s i t y Thev 3 1 u eo fKlc5 o c i a t e dw i t huns t a b 1 ec r a . c kg r o w t h組 di sa .m a t e r i a . lp r o p e r t y . f a c t o r剖 s I fKlci s, i nf a c t, ama . t e r i a . lp r o p e r t yo b t a i n a . b l ef r o mt h eu n m o d i f i e du s eo fLEFM, t h e f o l l o w i n gs h o u l dbet r u e :( 1 )K1cs h o u l db ei n v a r i a n tw i t hr e s p e c tt obeams i z ea . ndn o t c h 出. ( 2 )Anyz o n eo f“p l 回 t i c i 句"o r“凶α0 ・c r a c k i n g, "i . e .,t h ef r a c t u r e d e p t ho rc r a c kd e p 間 z o n e, shou l dbev e r ys m 3 1 lcomparedt ot h en o t c hd e p t h .ASTME399( 1 9 8 3 )g i v e s p r o c Displacement 凶 叫a 山 :2r(a)No 8 or 8 CMOD LPO T h r e e P o i n tBend b )L叫 叩 nd( D i s p l 舵 印 刷 7 0 7 1 af o r m u l at od e t e r m i n ei ft h i sr e q u i r e m e n ti ss a t i s f i e d .T h i sh a sb e e na d a p t e dt oc o n c r e t e ( S e c .6 . 8 ) .( 3 )Thec r i t i c a le n e r g yr e l e a s er a t ei sr e l a t e dt oK1cby T . r2 す c= Gl ( 6 . 2 ) 。 I nwhicht h e( s l i g h t )i n f l u e n c eo fP o i s s o n ' sr a t i oi 自n e g l e c t e dandEi st h emoduluso fe l a s t i c i t y . I n姐 a t t e m p tt oe s t i m a t et h ec r a c kl e n g t hi np l a i nc o n c r e t ebeamss u b j e c t e dt ot h r e e n d i r e c tmethodb a s e dont h ec o m p l i a n c eo fn o t c h e dbeamswωproposed p o i n tb e n d i n g, 姐 i HuandJ o n e s( 1 9 7 8 ) .T h i st e s tp r o c e d u r ei sa sf o l l o w s : byS w a r t z, 1 .Notchabeama tm i d s p a n . 2 . Mountad i s p l a ι e m e n もg a g et omeasuret h ec r a c k m o u t h o p e n i n gd i s p l a c e m e n tdCMOD ( s e eF i g .6 . 1 a )andc y 必 t h el o a dont h ebeamt h r e et i m e sw h i l ep l o t t i n gl o a dP抑 制 dCMOD. T hema . ximuml o a di ss e l e c t e dt obel e s st h a n1 / 3t h ee x p e c t e df a i l u r e10M f o rt h a tn o t c hd e p t h . 、 ,i0・8 o-MA.一 a¥.c-υ.@υceこa Eυ 。 6 . 1 . 2CcmplianceCalibrationMethod 60 @ ・ C! o rGlc, w制 C ο n s t a n tw i t hr e s p e c tt on e i t h e rbeam Thee a r l yr e s e a r c h e r sf o u n dt h a tKI 1 9 7 2, 1 9 7 6 )n o rn o t c hd e p t h( S w a r t z, Hu, F a r t 卸 h姐 dH uang, 1 9 8 2 )f o rc o n c r e t e s i z e( W a l s h, 抗i o nf o rt h i swasp r o v i d e dbyShah回 dMcGarry( 1 9 7 1 )who b e a m s . Onep o s s i b l ee x p l a n n o t e dt h a tc r a c k sa r ea r r e s t e dbyt h ea g g r e g a t ep a r t i c l e s .A d d i t i o n a le n e r g yi sr e q u i r e dt o h e s ep a r t i c l e s . Anotherp o s s i b l e阿 部onf o r p r o p a g a t et h ec r a c k st h r o u g ha n d / o r訂 oundt a . t i o ni nKlcme a . s u r e dbyLEFMw凶 s u 邸e s t e dbyS w a r t zandc か w o r k e r s( S w a r t z, t h ev a r i Hu, F a r t a s handHuang, 1 9 8 2 ;S w a r t z ,HuandJones,1978;SwartzandGo,1984). They tt h ed a t ae v a l u a t i o n si nt h ee a r l yt 倒 s( N a u s姐 d1o t t, 1 9 6 9 ;Walsh ,1972, 1 9 7 6 ) n o t e d出 a α Q .T h i sv i o l a t e so n eo ft h er e q u i r e m e n t so ft h e werer e f e r e n c e dt ot h eo r i g i n a ln o t c hd e p t h, namelyt h a tat r u ec r a c k-n o tan o t c h-i st obeu s e d .Ofc o u r s e , ASTMmethodf o rm e t a l s, me a . s u r i n gt h ea c t u a lc r a c kl e n g t hi nc o n c r e t ei sq u i t ed i f f i c u l t' and, i nfad, ap r e c i s er e s u l t , s i n c et h ec r a c kf r o n ti ss t r o n g l yn o n u n i f o r mt h r o u g ht h es p e c i m e nt h i c k n e s s . i si m p o s s i b l e A i n . / l b .= 0.57Ix xTheoretlcal 40 20 e r s u sdCMODp l o t .T h i si st h ec o m p l i a n c eCf o rt h e 3 . Me a . s u r et h ei n v e r s es l o p eo ft h ePv ∞tchdepth.(Thisi s, o fc o u r s e , o n l yan勾 p a r e n ti n d i r e c tmeasureo fc o m p l i 乱n c e .The t r u ec o m p l i a n c ewouldbeo b t a i n e dbym e a s u r i n gt h ev e r t i c a l l o a d p o i n td i s p l a c e m e n t dLPD, o rbyl o a d i n gt h es p e c i m e na tt h ec r a c km o u t h . ) 4 .Re p e a ts t e p sa cf o rd i f f e r e n tn o t c hd e p t h su s i n gt h esamebeam. ,theexperimenもa l l yd e t e r m i n e d∞m p l i a n c ep l o ti su s e dt oe s t i m a t et h e S u b s e q u e n t l y HuandJ o n e s, 1 9 7 8 )・ i n i t i a lc r a c kl e n g t ho fp r e c r a c k e dbeamsl o a d e dt of a i l u r e( S w a r t z, Theu s eo fp r e c r a c k e dbe a . mswasf e l tt obe町 田s s a r yi no r d e rもoo b t a i nv a l i df r a c t u r e 。 0.2 0.4 o d 。 5 .P l o tC v e r s u s α I d .At y p i c a lp l o ti sshowni nF i g .6 . 2 . 0.6 . _ , . . . 喧 _ , J " . . " 皿 塗 ・ ・ 0.8 F i g . 6 . 2ComplianceC a l i b r a t i o nC u r v ef o rNotchedB白 血 cm./N 72 73 da . t ab e c a u s et h ephenomenono fc r a c kc l o s i n gs t r ω s e smak 回 i ti m p o s s i b l et omodela r e a lc r a c kw i t han o t c h . Thep r o c e d u r eもop r e c r a c ka( d i f f e r e n t )beamf o ras u b 叫 l e n tl o a dt e s tw回 a sf o l l o w s : 6 .C凶 as m a l l , s t a r t e rn o t c ha tm i d s p a n .T h i sw硝 t y p i c a l l y α口/ d三0. 1 0 . 。岨由 oadt h ebeamt oal o a dbeyond九 t ogrowac r a c k企omt h en o t c hu n t i lt h eP-OCMOD 7 .L u n l o a d i n go rr e l o a d i n gs l o p em a t c h e st h a ta s s o c i a t e dw i t ht h ed e s i r e dc r a c kl e n g t h . A f t e rt h ebeami 自p r e c r a . c k e d, t h e n 守 値 。 ︻ .εE0.0ω-.三 o-.向 調 。 .auoυ 8 .L o adt h ebeamt of a i l u r e姐 dme 回 u r e九 o rPq ・C a l c u l a t eK1cfrom ( 6 . 3 ) 。。ミ 同日 0・ 。 ・ 悶 • HE胃3 。 U句宮崎 前 N 惜 。 , “ 明 司 。制同 ﹃ i n u r o n u • 582u崎 -o-x ・ 。柑両 , 。 Z ω マ.守"・ a ﹄ 。 6 .1 .3M odisedComplianceC a l i b r a t i o nMethod Them o d i f i c a t i o nwast ou s ep r e c r a c k e dbeamsw h i c ha r ei m p r e g n a t e dbya .d yet or e v e a l andt h e nu s et h ee x t e n d e dc r a c kl e n g t ha .t od e t e r m i n e](1. , t h es h a p eo ft h ec r a c kf r o n t, Themethodwasf i r s tp r o p o s e dbyGo( S w a r t zandGo, 1 9 8 4 jGoandSw a . r t z, 1 9 8 6 )岨 d S w a r t zandR e f a i, 1 9 8 9 ) .P r e c r a c k e dheamsa r eu s e dbo も h s u b s e q u e n t l yr e f i n e dbyR品 i( f o rt h ec r 品もi o no fac o m p l i a n c ec a l i b r a t i o nc u r v eandf o rf a i l u r et e s t s .Themethodh a st h e d i s a d v a n t a g et h a . ti ti sq u i t et i m e c o n s u m i n gandanumbero fbeamsa r er e q u i r e dt oo b ta . i o . sf o l l o w s : ac a l i b r a t i o nc u r v ei n s t e a do fj u s to n eb e a m .Thep r o c e d u r ef o re a c hbeami sa c Ee 守.ωω". e-nwhωOON.寸 一 宇 。 -咽 N o t i c et h a tt h eo n l yd i f f e r e n c eb e t w e e nt h i sandt h ee a r l i e rLEFMp r o c e d u r ei st h eu s e o ・ o f α ii n s t e a do fa U s i n gt h er e l a t i o no fc o m p l i a n c et oc r a c kl e n g t hc a l c u l a t e dfromLEFM , onec a nde 同minet h ei n i t i a lc r a c kl e 時出向 f r o mt h es l o p eo ft h eP -dCMOD c u r v e( F i g .6 . 3 )a f t e rt h e i n i もi a . lc r a c kc l o s u r es t r e s s e sa r eo v e r c o m e . 田 t i n gr e q u i r e st h eu s eo fac l o s e d I o o pe l e c t r o -hydrodynamics y s t e mi n T h i smethodo ft w h i c ht h ec o n t r o l l e d, f e e d b a c kv a . r i a b l ei st h eo u t p u to ft h et r a n s d u c e ru s e dt om伺.su r et h e dCMOD・T h i sh a st r a d i t i o n a l l yb e e nc a l l e d“ s t r a i nc o n t r o l . "T h i stermmaya . pproxim a . t e l y a p p l yi ft h ec o n t r o l l e dv a r i a b l ei st h edCMODo rsomeo t h e rbeamd i s p l a c e m e n t, b u tn o tt h e s t r o k ed i s p l a c e m e n to ft h et e s t i n gs y s t e m . 白も W 臼 m adet oshowad i r e c tc o m p a r i s o nb e t w e e nn o t c h e dbeamsandp r e c r a c k e d At beamsf o l l o w i n gt h ep r o c e d u r e sd e s c r i b e da . b o v eandu s i n gE q .6 . 1f o rt h en o t c h e dbeamsa n d twasf o u n dt h a tt h ecomputedKlcf o rt h ep r e c r a c k e d E q .6 . 3f o rt h ep r e c r a c k e db e a m s .I beamsw踊 a l w a y sh i g h e rt h 組 f o rt h en o t c h e dbe 釦 1 5( Swa 巾, Hu ,Fa 巾.s h組 dHua 時, 1 9 8 2 ).However , t l 由 a p p r o a c hw ω c r i t i c i z e db e c a u s en o t c h e d-n o tp r e c r a c k e d-beams 悶s t e p s日).Af u r t h e rω i t i c i s m w e r eu s e dt oc o n s t r u c tt h ec o m p l i a n c ec a l i b r a t i o nc u r v e s( wast h a t, duet os l o wc r a c kg r o w t h, t h ei n i t i a lc r a c kl e n g t ha im i g h tn o tbea p p r o p r i a t ef o r h e r e f o r et h i smethodw出 modi 自e d . d e t e r m i n i n gK1c'T 守 。 Klc= F(S , d, b , αh九 , orP Q) 7 4 75 1 . Cuta .s m a l ls t a r t e rn o t c ha tm i d s p a n . 2 .U s i n garampingf u n c t i o nands t r a i nc o n t r o l,l o a dt h ebeambeyond九, growac r a c k , p l o tP v e r s u sdCMODu s i n gamaximuml o a d andt h e nremovet h el o a d .S u b s e q u e n t l y 壬1 / 3t h emaximuml o . a da s s o c i. a t e dw i t ht h ec r a c k . 3.I n t r o d u c et h edyeandc y c l et h el o a dt oworkt h edyei n t o山 ec r a c k e ds u r f a 偲・ T he bea . mmustbel o a d e dw i t h出 ec r a . ckp r o c e e d i n gf r o mt h et o ps u r f a c edownw a . r d .The ∞mecrackc 1 0 s u r es t r e s s e sandmusth e 1 0 0 0mustbeg r e a t e rt h a nt h a tn e e d e dt oo v e r . b o u t1 / 3o fP u . l e s st h a na 4 . Dryt h edyeds u r f a 田 a ndl o a dt h ebe a . mt of a . i l u r e .P l o tP . a ndsCMOD. Thei n i t i a l . . s l o p ea f t e rc r a c kc l o s u r ei sov e r c o m eg i v e st h ei n i t i a lc o m p l i a n c eC 5 .A f t e rf a i l u r e, m伺 s u r et h edyeds u r f a c ea r e a .Thei n i t i a lc r a c kl e n g t hi s 向 =( a r e ao fdyeds u r f a 田 ) / b ( 6. 4 ) T y p i c a ldyeds u r f a 田 sa . r eshowni nF i g .6. 4 . α i 6 . Re p e a ts t e p s15f o rd i f f e r e n tc r a . c kd e p t h sande s t a b l i s hc a . l i b r a t i o nc u r v e sr e l a . t i n gC ; anda i,凡回dα4・Thel a t t e rr e l a t i o n s h i pa l l o w sonet oo b t a i nane s t i m a t eo ft h ec r a . c k l e n g t ha . ss o c i a . t e dw i t hanyl o a . dont h es o f t e n i n gp a r to ft h el o a d d i s p l a c e m e n もp l o t . ・ Iu c=F(S , d, b , a ., 0 . 9 5 P ) u i す・ 0.511 SeriesB 。 i 0.580 す・ 0.761 SeriesC ( 6 . 5 ) Thev a l i d i t yo ft h ep r o c e d u r et oe s t i m a t et h ee x t e n d e dc r a c kl e n g t ha .maybea . r guedb y r e f e r r i n gt oa .l o a d u n l o a d r e l o a dd i a g ram( F i g .6 . 5 ) .Theo b j e c t i v ei st od e t e r m i n et h ec r a . c k l e n g t ha tsomep o i n tont h es o f t e n i n gb r a n c h-s a . yp o i n tC( w h i c hmaybeanyp o i n t ) .I f もh e a . c t u a . lu n l o . ad i n gt r a . c ei sa . va . i l . a b l e, t h eu n l o . ad i n gc o m p l i a n c eCu c a nr e a d i l ybeme a . s u r e d a . ndu s e dw i t hac o m p l i 組 c ec a l i b r a t i o nc u r v et od e t e r m i n et h ee x t e n d e dc r a c kl e n g t ha . . A l t e r n a t i v e l y , t h e九一 a;r e l a t i o n s h i pma . ybeu s e dwheret h el o a d以 p o i n tCi su s e df o rP u . I nc o n s t r u c t i n g出 e凡 -a;c u r v e , i ti sn o t e df r o mF i g .6 . 5t h a tana p p r o x i m a t i o ne x i s t si n uand implyt h eu s eo ft h es l o p eo fl i n eOBi n s t e a do ft h ea c t u a l l i n eOA.Thee r r o r t h a tP r a c kl e n g t hfromt h i sa p p r o x i m a t i o nwasd e t e r m i n e dt obel e s st h回 6% i nd e t e r m i n i n g山 ec w i t hac o e f f i c i e n to fv a r i a t i o no f8.5%( 6 2s a m p l e s )( S w a r t zandRe f a i, 1 9 8 9 ) . Ther e s u l t so b t a i n e dbyu s i n gt h i smethodon8I n .and1 2i n .d e e pbe a . m sw i t ha . / d至 t 0 . 6 5showJ(Ict obei n v a r i a n tw i t hr e s p e c tt ot h ec r a c kl e n g t handbeams i z e , w i t hac o e f f i c i e n . a 。・ τ 。 : 7 .Att h ep o i n tont h eu n l o a . d i n gp l o tc o r r e s p o n d i n gt oもheo n s e to fu n s t a b l ec r a c kg r o w t h a t0 . 9 5九 d e t e r m i n et h ee x t e n d e dc r a c k1 叩 g t ha .f romt h e九一向 p l o t -t a k e nt obe. 伺 t e rt h 叩 a . / d= 0.65. Using0.95凡 ( s t e p6 ) . Thee x t e n d e dl e n g t hmustn o tbeg r 組 da .computeK1cfrom F i g . 6. 4DyedC r a c k e < lS u r f a c e s p m A B . 95凡 P 8CMOO F i g . 6 . 5Lo adv e r s u s6CMODw i t hRe l o a da t0 . 9 5凡 77 76 v a r i a t i o no f5.5%(N= 1 9, 8i n .b e a m s )and3 . 5 %(N= 1 9, 1 2i n.beams)( R e f a iand ~wartz, 1 9 8 8 ;S w a r t z姐 dR e f a i, 1 9 8 9 )・ 6 .1 .4E f f e c tofFr i c t i o n . d v a n t a g eo fn o t c h e dbeamt e s t si st h a tf r i c t i o ne f f e c ti ss m a l l . Toe x p l a i nt h i s Ana d, l e t ηdbet h ed i s t a n c ef r o mt h ebottomf a c e( F i g .6 . 1 )t ot h ep o i n tont h ec r a c kp l a n e a . t姐 a x i a lf o r c ep制 i n g出 r o u g ht h i sp o i n twouldc a u s enode f l . e 仰 no ft h ebeamj u c ht h t l b v i o u s l y向 / dく ηく 1(typicallyη=0. 7 5 ) .I ft h eh o r i z o n t a . lf r i c t i o nf o r c 回 配t i n ga . tt h e 同e a . ms u p p o r t sa r ed e n o t e d剖 F, t h eb e n d i n gmomenta tmidspann~姐ed t oc a u s ec r a c k a . t i o ni sM =( P / 2 ) ( S / 2 )-FηdwhereS=sp a . n, P= a p p l i e dl o a . d, F = kP/2組 d : J l o p a g ki s -t h ec o e f f i c i e n to ff r i c t i o n .D e n o t i n g九 =4M/S, w h i c hr e p r e s e n t st h ef o r c en e e d e dt o t : a u s et h ec r a c kt op r o pa . ga . t ei ft h e r ew e r enof r i c t i o n( k=0 )oneg e t s九 =P-d . Pj w here P =m e a s u r e da p p l i e df o r c e組 d( a c ∞r d i n gt oB話相も): u恥 , 圏 、 ロE 吋~y 、 ー ' -EZ身S 。 " 司 、 、 . . 〆 " ‘ 、 孟且 , 個 、 旬 、 ー ' , 圏 、 、 ・ ・ ' (d) 、 - ,E白 皆 宮 OA古 () H O E Sg 0 2 4 u ︿S Uぷ S3ht〆2 d骨同︻的安否ES司令ム)6azmg百万色白宮市三一身・品志津 句。・凶日 Q , 目今 喧園 , 司 、 @ ・ 、 4 2ηd thRf=3-kP T h i sr e p r e s e n t st h ep o r t i o no ft h ea p p l i e df o r c en e e d e dt oo v e r ∞methef r i c t i o n . The l 釘 g e rt h eS/ dr a . t i o ,thesma . l l e ri sd . 乃.Forη =0. 7 5a . ndk= 0 . 0 0 5f o rr o l l e rbe a . r i n g s r sg i v eanupperv a . l u eo fk= 0 . 0 1 )a . ndf o rS/d= 2 . 5( u s e dbyBazant姐 d ! m a n u i a c t町 e 1 9 8 7 ), d .Pj = 0. 0 0 3 P . Thu8, we5 田 t h a tt h en o t c h e dbeamt 旬e s 錦h 旬sa . 閃 r e l a . t i v e l y P f e i f f e r, i 凶 n 悶s 民 附 悶 e 釘叩 則 n 凶 l h 旬yP 凹l 却 a n 剖 a ndE 凹l i c e 剖5 , 1 9 8 8 b ) 6 . 2Wedge-SplittingTest A n o t h e ru 5 e f u lt e s tf o rf r a c t u r eo fc o n c r e t ei st h ew e d g e s p l i t t i n gt e s t( F i g .6 . 6 ) .I ti s a . c tt e n s i o nt e s tu s e df o rm e t a l s . Wedges p l i t t i n gt e s t sweres t u d i e d i m i l a rt ot h ecomp o rc o n c r e t ebyH i l l e m i e r組 dH i l s d o r i( 1 9 7 7 )andt h ep r e s e n ts h a p eo ft h et e s ts p e c i m e n, I l a r a . d e r i z e dbyaBもa r t e rn o t c handag u i 必ngg r o o v ewh i chc a nbee i t h e rmouldedo rs a . wn, a sp r o p o s e dbyL i n s b a u e randTschegg( 1 9 8 6 ) . Thet e s tw出 s u b s e q u e n t l yr e f i n e dby andB r u h w i l e randWittmann( 1 9 8 9 )whoc o n d u c t e d( a tt h eS w i s sF e d e r a l H r u h w i l e r( 1 9 8 8 ), l '3 0 0s u c ht e s t sonn o r m a lc o n c r e t e, damc o n c r e t eand0七h e r I n s t i t u t eo iT e c h n o l o g y )O V e . m e n t i t i o u sm a t e r i a . l s . Veryl a r g ewedges p l i t t i 碍 s p e c i m e n s, o fs i z e supt o1 . 5m ( 5f t . ), 附 r e c e n t l yb e e nt e s t e dbyS ω四 叫 B r o z, B r u h w i l e randBo 鰐 ( 1 9 8 9 )a tt h eU n i v e r s i t y C A l o r a d o , t os t u d yt h es i z ee f f e c ti ndamc o n c r e t e . F i g・6 . 6( a d )s h o w sv a r i o u sp o s s i b l ew e d g e s p l i t t i n gs p e c i m e ns h a p e s .S p e c i m e n( F i g . もh e ra .d e e pn o t c ho ral o n g i t u d i n a . lg r o o v eonb o t hs i d e s, i no r d e rt op r e v e n t ' . c lr e q u i r e se i 訂 f a i l u r eo fo n eo ft h ec a n t i l e v e r s .F i g .6 . 6( e , f )i l 1u s 位以e st h emethodo ft e s t i n g.The 悶 twol o w f r i c t i o nr o l l e ro rn e e d l ebωings( o n ! n b l yo ftwow e d g e si sp r e s s e dbetw 副 e )w h i c hd e v e l o pap a i ro ff o r c 白 Nt h a tt e n dt os pl i tt h es p e c i me n( F i g .6 . 6 g ).The l I ,~" a s s e m b l yi sl o a d e di nas t a t i c a l l yde t e r m i n a t em姐 n e rs ot h a te a c hwe dger e 田 i v 郡 山e 78 79 samel o a d . Thed i m e n s i o n so ft h en o t c handt h eg r o o v emustbec h o s e n8 0t h a tt h ec r a c k a . ga . t e ss y m m e t r i c a l l y . prop During出 et e s t, t h es p l i t t i n gf o r c eN ( F i g .6 . 6 g )mustbem回 s u r e dw i t hs u f f i c i e n ta c 出 y . Thec r a . c kmoutho p e n i n gd i s p l a c e m e n もOCMOD i smeasuredbyat r a n s d u c e ro ra c u r もa c h e da tt h el e v e lo ft h es p l i t t i n gf o r c 田, i nw h i c h dipgage( F i g .6. 6 f )whichs h o u l dbea t 掛 eo 一CMODrepresentstheloOO-pointdispl民 町lentOLPD舗 回Cla胞 dwiththehorizontal C e s ti sc o n もr o l l e dbyOCMODi nac l o s e d I o o ps e r v o componento ft h es p l i t t i n gf o r c eN.Thet a8 t a b l et e s tc a na l s obep e r f o r m e dundera c t u a t o r h y d r a u l i ct e s t i n gm a ε h i n e . However, s t r o k eC O I l s m lo runderc r o s s h e a dd i s p l a c m I e n tc p n t r o lu s i n gc o n v e n t i o n a 1 4 e s t i n gm a c h i n e s . I nt h a tc a s e , t h ea p p r o p r i a t en o t c hl e n g t hne 偲 s s a r yt oe n s u r es t a b i l i t ymustbei d e n t i f i e dby s p e c i m e ns t i f f n e s sandm a t e r i . a l c o n s i d e r i n gt h ei n t e r a c t i o nbetweent e s t i n gmachines t i f f n e s s, p r o p e r t i e s( B r u h w i l e r, 1 9 8 8 ;B r u h w i l e r副 Wittmann, 1 9 8 9 ) . 組 . t a g e so f出 . ewedges p l i t t i n gt e s ta r ea sf o l l o w s : TheOOv i g h t, s i n c et h er a t i oo ff r a c t u r ea r e aもot h es p e c i m e n 1 )Thes p e c i m e n sa r e∞mpactandl 山 (eι5 ユtim回 largerthanthatforthethree-poirル bend volumei sl a r g e rthanf o ro t h e rt e 1985)Thisi se s p e c i a l l yu s e f u lf o rt h es t u d yo fs i z ee f e c b, s i n c e t e s ta c c o r d i n gt oRILEM, 出 C 担 beo b t a i n e dw i t hs m a l l e rs p e c i m e nw e i g h t . Duet ol e s s e rw e i g h t, l a r g e rf r a c t u r ea r e l a r g e rs p e c i m e n sa r ee a s i e rt oh a n d l e, 乱ndt h e r ei sal e s s e rr i s ko fb r e a k i n gthemd u r i n g h a n d l i n g . 2 )Thec u b i c a lo rc y l i n d r i c a ls p e c i m e n s( F i g .6 . 6a ・c )c a nbee a s i l yc a s ta tt h ec o n s t r u c t i o n h esamemoldsasf o rs t r e 碍t ht e s t s, andt h ec y l i n d r i c a ls h a p e s( F i g .6 . 6b d )c 組 s i t e戸時 t a l s obeo b t a i n e dfromd r i l l e dc o r 回 f rome x i s t i n gs t r u c t u r 田. 3 )Theu s eo fwedgesf o ri n d u c i n gt h el o a di n c r e a s e st h es t i f f n e s so ft h et e s ts e t 叩 日d makingi tp o s s i b l et oc o n d u c tt h et e s teveni namachine t h u se n h a n c e ss t a b i l i t yo ft h et e s t, t h a ti sn o tv e r ys t i f f . nc ο n t r a s tt on o t c h e dbeamt e s t s( w h e r et h e 4 )t h ee f f e c to fs e l f w e i g h ti sn e g l i g i b l ei 回 t duet oownw e i g h tc 拍 beo v e r50%o ft h et o t a lb e n d i o gmoment) b e n d i n gmom tmustben o t e d七h a tt h ewedge1 0 晶d i n gh組 乱1 5 0ad i s a d v a n t a g e Ont h eo t h e rhand,i tP = a p p l i e dv e r t i c a ll o a d, N= =specimenr e a c t i o n s a si ti n t e n s i f i e sf r i c t i o n a le f f e c t s . Le 償対e dt op r o p a g a t et h ec r a c k swhicha r enormalt ot h ewedges u r f a c ei n c l i n e dbya n g l ea n =f r i c t i o nc o e f f i c i 叩 t o ft h eb e a r i n g s .Then, t h ee q u i l i b r i u mc o n d i t i o no f ( F i g .6 . 6 g ) andk= 田 a c t i n g00t h ewedgey i e l d sP =2(Ns i o α +kN∞sα)=九 (1+kcotα ), w h e r e v e r t i c a lf o r c 凡 =2Nsinαist h ;f o r c eneed;dt op r o pa . g a t et h ec r a c ki ft h e r ewerenof r i c t i o n( k= 0 )・ S i n 田 k ∞t α< t : :1 , wehave九ど P /(l+ kcotα)=P(l-kcotα)o rP+ Q= =P-I : : .P where ( a c c o r d i n gもoBa Z a n t ) : ( 6 . 6 ) I : : . P = P/(c o tα t h 姐 f o rt h es h o r tn o t c h e dbeamso fBa Z a n tandP f e i f f e r( 1 9 8 7 ), andabout20 ・t i m e sl a r g e r t h a nf o rt h el o n g e rn o t c h e dbeamsr e ∞mmendedbyRILEM,for出 esamevalueofk(see . 4 . ) .T h i sd i sa . d v a n ta . g eo ft h ewedges p l i t t i n gt e s ti ssurmounta . b l e, 岨df r i c t i o n a l S e c .6 .1 姐 ber e d u c e dby( 1 )a t t a c h i n gh a r d e n e ds t 田 1i n s e r t sa l o n gt h ei n c l i n e dwedge e f f e c t sc , 品 目 ,( 2 )u叫 n e e d l eb e a r i n g s,and( 3 )c a r e f u i l yp o l i s h i n gt h ewedges u : r f a c ea sshown s u r f i l l e m e i ぽ a ndH i l s d o r f( 1 9 7 7 )whoe x p e r i m e n t a l l yd e t e r m i n e da/ ( v a l u e= 0 . 0 0 0 3 1f o r byH t h e i rwedgel o a d i n gs e t u pw i t hn e e d l eb e a r i n g s . I ft h ev a l u eo fki sn e a r l yc o n s t a n tandw e l lr e p r o d u c i b l e, onemayi n t r o d u c et h ec o r r e c t i o o l l Pi nt h e組 a l y s i s .However, s i n c et h ev a l u eo ff r i c t i o nc o e 伍c i e n ti so f t e nq u i t eu n c e r t a i n, i ti sb e t t ぽ t om' 品 s u r et h es p l i t t i n gf o r c eN d i r e c t l ybyi n s t r u m e n t i n gt h ewedgesandt h e 四. s h a f t st h a tc a r r yt h eb e a r i n g sw i t hs t r a i ngag Thef o r e g o i n ga n a l y s i sshowst h a tav e r ys m a l lwedgeangleαisu n f a v o r a b l efromt h e t h es m a l l e rt h ea n g l e, t h es t i f f e ri st h es p e c i m e n v i e w p o i n to ff r i c t i o n .Ont h eo t h e rh祖 d, sa .r e a s o n a b l e∞m p r o m i s e . m a c h i n ea s s e m b l y .Thea n g J eα =1 50 i 叫 al a r g ewedgea n g l e(α>3 00) i su n d e s i r a b l eb e c a u s ei tl e a d sもoas i g n i f i c a n tnormal A l 田 o fs u c hs t r e s s e s s t r e s sp a r a l l e lt ot h ec r a c kp l a n ei nt h ef r a c t u r ep r o c e s sz o n e .Thep r e s e n r i b e dbyE q .3 . 9 . The maya f f e c tt h es o f t e n i n gc u r v ef o rt h ef r a c t u r ep r o c e s sz o n e, 制 d田 c , G jn o ri st h e a r e au n d e rt h es o f t e n i n gc u r v ei st h e nn o tl o n g e re q u a lt ot h ef r a c t u r ee n e r g y a r 目 u n d e rt h el o a d d i s p l a c e m e n tc u r v e . ! ( l ci so b t a i n e dbyt h esamemethoda sd e s c r i b e di n Thea p p a r e n tf r a c t u r et o u g h n e s s, . c kl e n g t h, whicha c c o u n t sf o rt h ef r a c t u r e S e c t i o n6 . 1f o rn o t c h e dbeamt e s t s .Thee f f e c t i v ec ra 恥 凶sz o n e , i sd e t e r m i n e dbyt h ec o m p l i a n c emethod, b a s e don五n i t ee l e m e n tc a l i b r a t i o n . p r , u n l o a d r e l o a dc y c l e sa r ep e r f o r m e dd u r i n gt h et e s t .Othermethodss u c h F o rt h a tp u r p o s e ee v a l u a t i o no ff r a c t u r ee n e r g yfromt h ea r e a .undert h el o a . d d i s p l a c e m e n tdiagramand a sもh a sd e s c r i b e di nt h es e q u e l . t h es i z ee f f e c tmethoda r ea p p l i c a b l e, 唱 , , Pi st 恥 heme 也a 邸 s 叩町 r e d1 0 0 0叩副 dI : : .丹 Pf 0, 恒 討c 叫t i 叩 o l l .I fα<45 I : : .P i slarg~r t h a nkP, whichmeanst h a tf r i c t i o n a le f f e c t sa r ee n h a n c e d f byt h ewedgel o . a d i n g . s I : : .PJ= 3 .7 3 k P .Themanuf . a c t u r e r so fr o l l e rb e a r i n g F o rt h et y p i c a lwedgea n g l eα =1 50, 組も田 0 . 0 1a st h el i m i t )・ Assuming g i v ek v a l u e srωgingfrom0 . 0 0 1t o0 . 0 0 5( a n dg u a r k= 0 . 0 0 5, I : : .P = O . 0 1 9 P .T h i sf r i c t i o n a le f f e c ti ss i g n i f i c a n tandi sabout6 t i m e sl a . r g e r , , , , 6 . 3 .Work-of-Fr actureMethod(悶LEM, H i l l e r b o r g ) T h i smethod, whichw剖 o r i g i n a l l yd e v e l o p e df o r田 r a . 凶ω(Nak a . y 制 a , 1 9 6 5 ;Ta . t t e r s a l l dTappin, 1 9 6 6 ), i st h ef i r s tmethodo ft 田t i n gf o rf r a c t u r ep r o p e r t i 回 o fconc 同 et obe 凶 a sa .s t a n d a r d(RILEM, 1 9 8 5 ) . Theb a s i sf o ra p p l y i n gt h i smethodt oc o n c r e t e p r o p o 1 9 8 5 b ) .T h e i rmethodu s e s w a sd e v e l o p e dbyH i l l e r b o r gandh i sc o w o r k e r s( H i l l e r b o r g, f i c t i t i o u sc r a c k "c o n c e p t( H i l l e r b o r ge ta , . l 1976;H i l l e r b o r g ,1980jPetersson,1981) t h e“ sno もa nLEFMmethod. ( F i g .6 . 7 )i m p l i c i t l yandもhusi I no r d e rもoc o n t r a s tt h i sw i t hLEFM00t h eba . si so fe n e r g yp a r a m e t e r s, r e c 乱It h a tt h e 町g yr e l e a s er a t eGIci st h ee 且e r g yr e q u i r e dp e ru n i tc r a < ユke x t e n s i o ni nam a t e r i a l c r i t i c a le n ~n ~hich t h e r ei snop r o c e s szone ,thati s, a l lt h ee n e r g yi ss u r f a 田 e n e r g y組 dnoe n e r g y h ec r a c k' t i p .I nf a c t, aproces~- z o n ed o e se x i s t~d t h e r e f o r eth~ i sd i s s i p a t e dawayfro~ t t ? t a le n e r g yo ff r a c t u r ei n c l u d e sa 1 1t h ee n e r g yd i s s i p a t e dp e ru n i tp r o p乱. g a t i o nd i s t叩 c eo f 側 zone剖 aw h o l e .T h i si sc a l l e dt h ef r a c t u r ee n e r g yGJ( F i g .6 . 7 ) . t h ef r a c t u r ep r o c ,themethodcanbeappliedtoavarietyoft 田 ts p e c i m e ng ∞metriesbut C o n c e p t u a l l y s e sabeamsp~cimen l o a d e di nt h r e ep o i n tbending ~withacent r a l t h ep r o p o s e dst~ndard u 佃 8 1 e d l ! : en o t c h( F i g .6 . 1 a ) . Co m p l e t ed e t a i l so ft h ep r o p o s e ds t a n d a r da r eg i v e ni nt h eRILEM 副 da r en o tr e p e a t e dh e r e .B r i e f l y , t h et e s tp r o c e d u r ec o 凶s t so f t h ef o l l o w i n gs t e p s . b∞ mmendation( 1 9 8 5 ) 1 .Theb e a r np r o p o r t i o n sa r es e l e c t e di nr e l a t i o nt omaximuma g g r e g a . t es i z e .Them i n i mumd e p t hdi sa p p r o x i m a t e l ys i xt i m e st h es i z eo ft h ea g g r e g a t e . Ther a t i o so fS/d .Thebeams p e c i m e ni sn o t c h e dt oadep 出 向/ d=0ふ v a r yf r o m8t o4 .S e eT a b l e6 .1 A l 2 .Thev e r t i c a l l o a . d p o i n td e f l e c t i o no ft h ebeam( c a l l e ddLPDi nF i g .6 . 1 a )i st obem e a s u r e da . n dp l o t t e dc o n t i n u o u s l ya l o n gw i t ht h ea p p l i e dJ o a dP. Ther e s u l t i n gt r a c ei s showni nF i g .6 . 8 . st obec o n d u c t e di nam組 問rもop r o d u c es t a b l ec r a c kg r o w t h .I fc l o s e d l o o p 3 .The色白色 i 加 t i n gi su s e dt h e ns t r a i nc o n t r o ls h o u l dbes e l e c t e d.I fac I o s e d l ∞psystemi sn o t a v a i l a b l e, t h e nas t i f ft e s t i n gmachinei sr e q u i r e d( s t i f f n e s sr e ∞mmendationsa r eg i v e n 1 9 8 5 . ) i nRILEM, 4 .Thef r a c t u r ee n e r g yi sc a l c u l a t e da s Al R _Wo+mgdo 一 一 日 J! 一 円 = c ) E σ σ ,''' a 戸 ‘ ,s ﹃ 4 E ( 6 . 8 ) 一 回 ' ~I・g ' z . , , U f t / E c = i nw h i c hWo a r e au n d e rP -dLPDc u r v eupt odo;d o d i s p l a c e m e n twhenP r e t u r n st o m l+2m2)g組 dm19=b e a r nw e i g h tb e t w e e ns u p p o r t s, m29=w e i g h to ff I x t u r e s 0 ;mg=( r i g i n a l, u n c r a c k e dl i g a m e n ta r e a= b (d-a o ) . : : : :o w h i c hi sc a r r i e dbyt h ebeam;andA1ig : T h i sf o r m u l ai sv a l i di ft h emovemento fl o a da . n dhPDa r edownward. I ft h ebeami s 陥 t e d“ oni t ss i d e "白 山a tt h ea p p l i e dl o a . dP i sn o r m a lt o山 eb e a m ' sseHw e i g h tv e c t o r , t h e nt h et e r mmgdoi sn e g l e c t e d .A l s o, i ft h ed e a dw e i g h ti so t h e r w i s ec o m p e n s a t e d, t h i s t e r mi 自n e g l e c t e d . F u r t h e r , i ft h emovementso fl oadanddLPDa r eupward-t h u so p p o s i n gt h es e l fw e i g h t h a t( S w a r t zandYap, 1 9 8 8 ) v e c t o r-t h e ni tお shownt G I J= 尚一 ~mgðo !- - - A - f { f { d J w ( 6 . 7 ) w F i g . 6 . 7F i c t i t i o u sC r a c kM o c l e lD e s c r i p t i o no fT e n s i l eF ra c t u r e nw h i c hd i s p l a c e m e n ta t出ep o i n ton出eu n l o a dp o r t i o no ft h ep l o t whenP = = o= d 屯t /2+m2)g. Eq~. 6 : 6a . n d6 . 7w e r ed e r i v e dbyS w a r t zandYap( 1 9 8 8 ).Thes e 旺w e i g h もt ermmaybe 1 I 1 i t es i g n i f i c a n t, e s p e c i a l l yi fy o u r i g∞n c r e t ei sb e i n gt e s t e do rt h es p e c i m e ni sl a r g e . 4laboratori~ i n ∞r p o r a t i n ga b o u t7 0 0be 畠m sw e r e E x t e n s i v er o u n d r o b i nt e s t sfro~ 1 l 戸rtedbyHillerborg(1985c)Withregardtovariationo fr e s u l t swidmag i v e nbmu 叶匂,~t~e c o e f f i c i e n to fv a r i a t i o nr a n g e df ; o ma b o u t2.5%t o25%w i t hmos 七r e s u l t sa . round h e悶 凶i v i t yo ft h es t r e n g t ho fas t n 川u r ew i t hr e g a r d ω 115%.Itwωnotedt h a t仁 .t 略; e si nG 7i sn o r m a l l yl e s st h a n1 / 3o ft h es e n s i t i v i t yw i t hr e g a r dt och a . n g e si n~ormal 83 82 P(lood) TABLE6 . 1 . SpecimenS i z e sRecommendedf o r H i l l e r b o r g ' sMe 出 od( RILEM. 1 9 8 5 ) 明' i d t h b mm Le ngth L mm 1 ・1 6 100土5 100ま5 840: t10 800土5 1 6 . 1・32 200土5 100土5 1190土 10 1130士5 3 2 . 1・48 300土5 150土5 1450士官 O 1385土5 4 8 . 1・64 t5 400: 200: t5 1640土 1 0 自 1. 0 , m TAB 日 6 . 2 . Re g r e s s i o nC o e f f i c 1 e 臨 ん .C l .司 .Dj f o rKar1ha l o oandN a l l a 血 ambl ・ sMe 血od. ( i= O .... . 4 :j= O ..• .. 5 ) 1 / j 。 L 凶 1 3I l d 2 . 5d . ー 4 s=d/d o 6 " F i g . 6 . 9Th 即 時t i c a 1S i z eDependenceo f悶 LEMFr . a c t u r eEnergyG~ AI C I 同 D j 3.6460 1 .5640 0. 4 607 1 .9560 l -6.7890 -8.3200 0.0484 0.3982 2 39.2400 52.9500 0. 0063 -0.0553 3 -76.8200 -124.9000 -0.0003 0.0027 4 74.3300 122.9000 -0.0059 0.0202 0.0003 -0.0055 5 2 Span S mm r nu a﹄ OAW Depth d mm 一 一 F i g . 6 . 8Lo adV e r s u sdLPD Max1mum Agg r .S i z e mm . 84 85 “噸 s t 児 r e n l 紅 ea b o u t3t i m e s国 h i g ha si nmosts t r e n g t ht e 山 ( H i 1 l e r b o r g, 1 9 8 5 c )" ・I twa . sc o n c l u d e d ont h i sba . s i sandbye x a m i n i n gt h ec h a n g e si nG 7叫 ues割 問atedwith仰 g田 inbeam deptht h a tt h es p u r i o u si n f l u e n c eo fs p e c i m e ns i 田 o nme a . s u r e dv a l u e 目o f ,G 7i sa c c e p t a . b l e . (However , t h es i z ee f f e c tmethodd i s c u s s e dl a t e re l i m i n a 同 t h i si n f l u e n 団・) Thus, eventhought h e r ewωanu n d e s i r a b l es i z edepend e ncyi nt h et e s tr e s u l t sf o rG 7 i twωconsideredt obeo fnog r e a t e ri m p o r t a n c eonbhc a l c u l a t e ds t r e n g t hd4hes t r u c t u r e t h a nt h es i m i l a rs i z edependencyi no r d i n a r ys t r e n g t ht e s t s . R怠 叫t 8from姐 o t h e rs e r i e so ft e s t s( S w a r t z組 d R 出, 1 9 8 9 ;R e f a iandSw 叫 1 9 腿) showas i m i l a rs i z ee f f e c ta 刈 a l s oav a r i a t i o no f w i t hn o t c hd e p t h . l nsummary , t h eRILEMw o r k o f f r a . c t u r emethodi sa .p r a . c t i c a la . ppro a . c h, whichi ss u i ι a b l ef o ru s ei nl a b o r a t o r i e swhichdon o thavev e r ye l a b o r a t eequipmentandi n c o r p o r a t e s a .numbero fωmpromisesi no r d e rt os i m p l i f yt e s t i n gw h i l es t i l lo b t a i n i n gu s e f u lr e s u l t s (H i l 1 e r b o r g, 1 9 8 5 b ) .Themethodi sb a s e donat h e o r ywhichh a sh品dw i d e 'a . c c e p t a n c e , t h e d c t i t i o u sc r a ι kmodel( H i l l e r b o r ge ta , . 11976).Thef r a . c t u r ee n e r g y∞n c e p ta l s oh a sav e r y s i m p l emeaning, w i t h o u tu s i n gLEFM. There釘 e, however , c e r t a i na . s p e c t si nwhicht h eR1 LEMw o r k o f f r a . c t u r emethodn e e d s t obef u r t h e ri m p r o v e d .T h i smethodd o e sn o tg i v er ω u l t si n d e p e n d e n to ft h es i z eo ft h e fp r e c r a c k e db白 血sa r eu s e d, andt h u st h ev a l u e c r a c k e da r e af o rag i v e nbeamd e p t heveni }mustbec o n s i d e r e dt obeo n l yana p p r o x i m a t i o no ft h et r u ee n e r g yp a r a m e t e rw h i c h o f01 char~cte出es t h es u r f a 田 e n e r g yandt h ee n e r g yo fp r o 倒 s z o n ef o r m a t i o n.1 ti ss i m p l i 五edb u t no もu n r e a . s o n a b l et oc o n s i d e r出 i ssuma l s ot ob ec o n s t a . n tw i t hr e s p e c tt ot h ec r a c kl e n g t h . T h i si m p l i e st h a tt h eshapeands i z eo ft h ep r o c e s sz o n edon o tc h a n g e .F u r t h e r m o r e , r e f i n e m e n もo ft h eR1LEMmethodi sn e e d e d回 ar e s u l to ft h es i z ee f f e c t,whichw i l lb e d i s c u s s e dn e x t . 0 7 6. 4SizeEffectinWork-of-Fr actureMethod 0 7 o b ta . i neda . c c o 凶時t ot h ee 泊s t i n gRILEM Thes i z edepe n d e n c eo ff r a c t u r ee n e r g y r e c o n u n e n d a t i o nont h eworko ff r 乱c t u r emethodh a sb e e ni n v e s t i g a もe dbyP l a n a s姐 dE l i c e s ( 1 9 8 8 b )ont h eb a s i so f日o l u t i o no fani n t e g r a lequa . t i o nandbyBa . Z a . nもan dKazemi( 1 9 8 9 b ) ont h eb a s i so ft h es i z ee f f e c tl a w . Thep r e m i s eo ft h el a t t e rana . l y s i swωtha ももh el a . w i sa p p l i c a b l et os i 田 r a n g 回 u pt oa b o u t1 : 2 0andh a st h esameform( f o rt h i ss i z er a . n g e ) fordls p e c i m e ng e o m e t r i e s ( w i h ho n l yn e g l i g i b l ee r r o r s ) . T h ea n a l y s i su t i l i z e st h emethod f o rc a l c u l a t i n gt h eRc u r v efroms i z ee f f e c tandl o a d d e f l e c t i o nc u r v efromt h eR c u r v ea s a l r e O O ye x p l a i n e d( S e c .5 . 1 ) .TheR叩 r v eo b t a i n e di nt h i sm姐 n e ri ss t r o n g l yd e p e n d e n to n t h egeometryo ft h es p e c i m e n .Thebωicr e l a t i o n( F i g .6 . 9 b, c ) o~= 旦p , m=bfR(帆 Wz = bR(cm)(1-c " , ) ( 6 . 9 ) whereb= s p e c i m e nt h i c k n e s s, l: : ;d-ao = l i g a . mentl e n g t h( d= be a . md e p t h ), 組dc",= c r a . c kl e n g t hca tpea kl o a d .同 組dWzr e p r e s e n t出 eworkso ff r a ε t u r eb e f o r eanda I'岡山E 戸a k1 0 0 0 . NotefromF i g6 . 9 b, ct h 以 Wzd ep e ndss t r o n g l yons p e c i m e ns i z e ,becausethe v a l u eo fc", d e p e n d sont h es i z e . 山 o ft h i si n v e s t i g a . t i o nshowt h a tG 7i sn o ts i z e i n d e p e n d e n t, 出 onemight Ther e s b u td e p e n d ss もr o n g l yont h es p e c i m e ns i z ea sshowni nF i g6 . 9 a .T h i sdependen 田 i s d e s i r e, S 田 nt obeev e ns t r o n g e rt h a . nt h a to f山 eR-c u r v e .Whent h es p e c i m e ns i z ei se x t r a p o l a t e d , G?c o i n c i d e 自e x a c t l yw i t hBa . Z 制' sd e f i n i t i o no ff r a c t u r ee 肘 r g yG/o b t a i n e dby t oi n f i n i t y t h es i z ee f f e c tmethod. T h i si n v e s t i g a t i o na l s oshowedt h a . tt h ep r e p e a kc o n t r i b u t i o no ft h eworko ft h el o a dt o t h ef r a c t u r een ぽ g yGIi sr e l a t i v e l ys m a l l( g e n e r a l l yunder6 % ) .T h i sc o n c l u s i o nba . s i c a . l l y a g r 間 w i t h出 a . to b t a i n e dbyP l a n a s叩 dE l i c e s( 1 9 8 8 b )i nad i f f e r e n tm油 田r . 6.5Two-ParameterF r actureModelofJenqandShah U n l i k et h ef i c t i t I o u sc r a c kmodel ,thetwoparame ほ m odelo fJenqandShah( 1 9 8 5 , 品 1 9 邸b ), a l r e a d ye x p l a i n e di nS e c .4 . 2, d o e sn o tr e q u i r eap o s t p e a . k( s t r a . i ns o f t e n i n g )c o n y e ti もc a nd e s c r i b et h en o n l i n e a rs l o wc r a c kgrowthp r i o rt opeakl o a d . The s t i t u t i v elaw, twopa . r a . m e t e r sa r et h ec r i t i c a ls t r e s si 山 悶t yf a c t o rKc !andthecritica . lc r a c kt i po p e n i n g d i s p l a c ementdCMODr e l a t i o n s h i pshowni nF i g .6 . 1 0.T h i sr e l a t i o n s h i pi se s s e n t i a l l yl i n e a r s c e n d i n gp o r t i o no ft h ec u r v efromP= 0upt oa b o u tt h el o a dc o r r e s p o n d i n gt oh a l f onぬea t h emaximuml o a . dP u・Att h i ss ta . g e, t h ec r a c kもi po p e n i n gd i s p l a c e m e n ti sn e g l i g i b l e姐 dKl i sl e s s山 岨 0.5K c !(Fig.6.lOa).Astheloa . dP e x c e e dt h ev a l u eo f0 . 5凡 ,i n e l a s t i cd i s p l a c e m阻 も 佃ds l o wc r a c kgrowtho c c u rd u r i n gt h en o n l i n e a rr a n g e( F i g .6 . 1 0 b ) . Att h ec r i t i c a l p ぬも(同.6 . 1 0 c ), t h ec r a c kt i po p e n i n gd i s p l a . cementr e 叫 e sa伺 t i c a 1v 山 e叫 ん =K f c . F o rs t a n d a r dp l a i nc o n c r e t ebeamst e s t e di nt h r e φ p o i n tb e n d i n g, t h ec r i t i c a . lp o i n tc 組 be a p p r o x i m a t e db e t w e e nt h ep o i n to fPua . ndt h ep o i n to f0 . 9 5 凡 ont h ed e s c e n d i n gb r a n c ho f t h eP-dCMODp l o t .Thec o n c e p ti sa 1 s oshownf o rabeami nb e n d i n gi nF i g .6. 1 1 . Dependingupont h eg 回 me t r yo ft h es p e c i m e n ,therateofloading, andt h emethodo f l o a d i n g , f u r t h e rc r a c kg r o w t hmayo c c u r. a tas t e O O y s t a t ev a l u eo fK f c .Denoting出 ec r a c k t h el o a dP w i l lh a v ea .v a . l u ee q u a lt o九 whenK1 r e a c h e sI < ! ci fd I < I / d α> O . l e n g t ha, I fd I < J /d aa . tt h i sl o a d i n gs t a g ei sn e g a t i v e, t h e nbeyondt h ec r i t i c a 1p o i n tt h ec r a c kw i l l a . t ea tac o n s t a n tv a l u eo f[ ( ! candtheappliedloadwillincrea . s eu n t i ldK I / d a=O . p r o p a g 1 nt h em a j o r i t yo f p r a c t i c a lc a s e s, t h ev a l u eo fdKl/dai sp o s i t i v e, i . e. , I < li sa m o n o t o n i c a l l y . Thr e e -andf o u r p o i n tb e n t, ands i n g l e -edgenotchedandd o u b l e l n . c r e a s i n gf u n c t i o no fa e d g en o t c h e ds p e c i m e n ss u b j e c t e dt ot e n s i l el o a d i n ga r ef o u n dt osati~fy t h i sc o n d i t i o n( J e n q 拍 dS h a . h, 1 9 8 5 b ) .F o rt h i sc a 民 t h ev a l u e so fK f c拍 ddCTOD C組 beo b t乱i n e dfrom出 e a . kl o a dandfromt h ek n o w l e d g eo fana s s o c i a t e de f f e c t i v ec r a . c kl e n g t ha .・ m e a s u r e dpe Themethodh a sb e e np r o p o s e dt oR1 LEM( K a r i h a . l o oandN a l l a t h a m b i, 1987~) 出a. s t a n d a r dt e s tmethodf o rd e t e r m i n a t i o no f]{先制46cTODi np l a j nc o n c r e t eu s i n gbeams p 民 1 m e n s . Thef o l l o w i n gd i m e n s i o n sa r es u g g e s t e di fa .maximuma g g r e g a t es i z en o te x c e e d i n g on~ i n c h( 2 5. 4mm)i su s e d js e eF i g .6 . 1, i nwh i c hbxdxL= 3 i n .X6 i n .x2 8 i n .( 7 6 . 20 l I I 1 耳 1~2.4 m mx711. 2mm);S= 2 4in.e609.6mm)jao/d= 1/3;S/d=4. ‘ r o rla . r g e rffiaXimuma . 邸r e g a t es i z e st h ed i m e n s i o n ss h o u l di n c r e 剖 ep r o p o r t i o n a . t e l y . Ac l o s e d l o o pt e s t i n gs y 蜘 mw i t hs t r. a i nc o n t r o lu s i n gt h edCMOD gag eo re l s e乱 s t i f f m a c h i n e( amachinet h a もi sa . tl e a s t1 0 t i m e ss t i f f e rt h a nt h e凶 s p e c i m e n )i sr e ∞mmended 乱di sr e a c h e di n t oa c h i e v es t a b l ef a i l u r e . T h er a t eo fl o a d i n gs h o u l dbesuchshMpeakl o a b o u t5m i n . p P P しV -E .E ,・・ EE E ' a duoo 一 , ・ ・ e-e- kd u n C ; 105KLdEぷ 1CTOD<CTODc E da-- Go LVo , to T・ 0c I CMOD αコ σ 3 〆 ノ ,噌 P m ノ J r . u o o一 , ,,nuoo n _ n _ ii ∞Tt P M 一 一 D 11 0 1 戸 し M oj, ω。 O Q I I~O!tl 9ヨ ( a ) / kzd;C I CTOD=CTODc CMOD CMOD ( b t - ( c ) ,明 o 臼 F i g . 6 . 1 0F r a c t u r eResi s t a n Stag fP l a i nCo n c r et e _ , , . _ CMODt CMOD3 CMOD2 ド 4 ~ド‘- CTOO.O ( a J t b J I c ' α 3 P h t . St c foloblωe!"Gc~ g r a t l t h . " ・ , 弓 ・ 2. p , oC&O~ Z ne p~, tt ・lI y dev.lopod. ' 1 5 ・ f d o y o l o p e c l . ・ ! I t!!IP I・ ・ 川 ' " a .8tortO~ unotob!earackgr wtll• 戸円 pr I J d 胸1) ' T O O F i g. 6. 1 1dCMOD anddCTODBehavior& tD i f f e r e n tLo a d i n gS t A g e s 司 89 88 Tod e t e r m i n et h etwof r a c t u r ep a r a m e t e r sK f canddCTODf o rp l 司n∞n c r e も e も ,hem a x i mum凡 andt h ec o r r e s p o n d i n ge l a s t i c∞mponen もo ft h edCMOD, d e n o t e ddbMODa r em品 s u r , 巴d .T h e s ev a l u e s釘 e七henu s e dt oe s t i m a t et h ee f f e c t i v ee l a s t i cc 即 kl e n g t haes u c hもh a t t h ec a l c u l a t e ddb ' M O D( bωedonLEFMe q u a t i o n s )i se q u a lt ot h emeasuredv a l u e .F o rag i v e n i n i t i a ln o t c hl e n g もha oanddependingupont h et y p eo ft e s t, t h etwof r a c t u r ep a r a m e t e r sa r e 田 F i g .6 . 2組 d6 . 1 1 ) : d e t e r m i n e da sf o l l o w s( s Themoduluso fe l a s t i c i t yE i sd e t e r m i n e du s i n gi n i t i a lcompJ i a n c eC jfrom E-6SaoM(α0) C j b a l ( 6 . 1 0 ) 0 . 6 6 Y i .(α0)=0.76-2.28αo+3.870'~ -2・040'~ +万て0'0 ) 2 ( 6 . 1 1 ) 内 u-e α α 一 同 一 vn n M G q α 一 一 一ae andα0 +( a o= h o ) / ( d+h ) , h o=c l i pg a g eh o l d e rt h i c k n 倒・ o Thee f f e c t i v ec r a c kl e n g t hi sd e t e r m i n e dfromE q .6 . 8byr e p l a c i n ga owithαhαow i t hα e 組 dC w i t h C " w h e r e 的 = ( a . + h o ) / ( W + h o ) 岨 dC i s t h e u n l o a d i n g c o m p l i a n c e a t 0 . 9 5 u i P U 'Thus, ( 6 . 1 2 ) w h i c hmaybe8 0 1 v e df a i r l ye a s i l ybyi t e r a t i o n . Then山. ec r i t i c a ls t r e s si n 何 回i t yf a c t o ri s =E主 主 主Yi.( α ) [ ( 1 , 8) 2+( ー1 . 149α+1 .0 8 1 ) ( , 82 ) ] 1/ 2 2E b d 。 ( 6 . 1 5 ) a .・ whereβ=α / I nu s i n gt h i smethodf o rp r e c r a c k e d-i n s t e a do fn o t c h e d-beams, r e p l a c ea obya j. I nJ e 問 a ndShah( 1 9 8 5 b )i ti sshownt h a tt h i smethodg i v e s] ( [ c ( =] ( ! c )resultswhic~ a r ei n d e p e n d e n to fs p e c i m e ns i z ewhenn o t c h e dbeamsa r eu s e d . Asshowni nS w a r t z姐 d R . e f a i( 1 9 8 9 )出 esamei n v a r i a n c ei so b t a . i n e dwhenp r e c r a . c k e dbeamsa r eu s e danda l s o Kr .i si n d e p e n d e n to fc r a c kl e n g t hf o rα. / dく 0. 6 5 .I ti sa l s oshownt h a tt h em o d i f i e d -hliM-method( s e c t i o n6 . 1 . 3 ) g i v e sv i r t u a l l ythsamemultf o rkICf o EpmMked e 制 p l e,f o rd= 1 2i n .,anda . / d三0 . 6 5,t h e乱. v e r a g beamsa st h eJ e n q Shahm e t h o d .f o re x Kc != 1206kNm-3/2 (C.V.= 9.2%,N = 11)andthecompliancemethodaV釘 ageKlo: ; 3/2 ( 1 l51kNmC . V .=3.5%, N= 1 9 ) . I ti sf u r t h e rshownbyJ e n q叩 dSh a . h( 1 9 8 5 b )t h a . tt h er e s u l t sf o rdCTOD a r er e l a t i v e l y c o n s t a n tw i t hr e s p e c tt obeams i z e .T h i sr e s u l twa . sa l s oo b t a i n e df o rbeamst e s t e dbyRe f a i 岨 dS w a r t z( 1 9 8 7 )b u tav a r i a t i o nw i t ha iw国 f o u n d . -3 w i t h α =α . /d .Thec r i t i c a lc r a c k t i po p e n i n gd i s p l a 四 m enti s 、A ( 6 . 1 4 ) 何百一守 -. , f i + 一 ト 1 1.99α(1-α)(2.15-3.93α+2. 7 α 2) ( 1+2 α ) ( 1-α) 3 / 2 F( α )ー ー M s一 a T and ( 6 . 1 3 ) , R一 A 一 r h i smethod( N a l l a t h a m b iandK a r i h a l o o ,1986a,1986b) hasa 1 8 0b e e np r o p o s e dt o 1 9 8 7 ) .l nt h i sa p p r o a c has i m p l et e c h RILEMa sas t a n d a r d( K a r i h a l o oandN a l l a t h a m b i, , l Iq u ei su s e dwhereもhec r i t i c a ls t r e s si n t e n s i t yf a c t o r] ( [ c ( =K 1 0 )andc r i t i c a le n e r g yr e l e a s e c ( =G 1 c )a r ef o u n df o rb白 血 s p e c i m e n sbyLEFMmethodsa f t e rt h ec 蹴 kh 副 g rown r d t eGl f r o ma .n o t c ha ot oa . ne f f e c t i v el e n g t hae . Thef o r m u l a swhichf o l l o wl a t e rw e r ed e v e l o p e df o rc e r t a i nr a n g e so fs p e c i m e nd i m e n s i o n s . r h e s er a n g e s, w h i c ha r et obeu s e df o rma . x imuma . g g r eg a t es i z e sda r a n g i n gfrom5t o2 5 mm.w i t hnoc r i t i c a ld i m e n s i o nb e i n gl e s st h a nf i v etim 回 d a r e :b=40・8 0mm( 三3d jd a, a) =5 0-3 0 0mm( と3 d 00-1 8 0 0m m( 三3 d iwi仕14壬S/d壬8 .A l s oa o / d=0.2-0.6 ;S=2 a) a) 0 . 3o r0. 4i sp r e f e r r e d ) . e a . l i z e di nF i g .6. 1i sshowni nF i g .6 . 1 2wheret h edLPD i s Thel o a d i n gs e t u pwhichi si d m e a s u r e dbyanLVDT.I na d d i t i o n, t h edCMOD c o u l dbemeasuredbya d d i n gac l i pg a g e a . c r o s st h eno もc h .P r e f e r a b l yad o s e d l o o pt e s t i n gs y s t e ms h o u l dbeu s e dw i t hs t r a i nc o n t r o l f P . e db a c kf r o me i t h e rt h edLPD o fdCMODga . g e .O t h e r w i s e ,as t i f ft e 日t i n gmachines h o u l dbe v e n t, ac o n t i n u o u sr e ∞rdofPa . ndh P Di smadeupもopeakl o a dw h i c hs h o u l d n s e d .I n組 ye her e a c h e di n1 ・1 0m i n u t e s .t h es o f t e n i n gr 田 p o n s ei sno もr e q u i r e dw i も ht h i smethod . I fi ti sn o tp o s s i b l et oo b t a i nac o n t i n u o u sl o a d d i s p l a 田 m entr e ∞rd, t h emethodc a ns t i l l t ) p u s e db u tt h emoduluso fe l a s t i c i t y , E, mustbed e t e r m i n e dfromu n i a x i a lt e s t sono t h e r p e c l m e n s . t i n u o u sr e ∞rdo fP v s .dLPDi sa . va . il a b l e , d e t e r m i n eE f r o m I fa∞n E ' d ; 3九 S. K l c=五 J i i 7 i ' e F( α ) 6.6E f f e c t i v eCrackModelo fK a r i h a l o oandNallathambi ( 6 . 1 ) l h e r eP i• d ia r eme a . s u r e da . も 姐yi n t e r m e d i a t el o c a . t i o n00t h ei n i t i a la s c e n d i n gp o r t i o no f : n ec u r v eandwi st h es e 1 f w e i g h to f出eb e a m .Thetermλisac o r r e c t i o nf a c t o rr e l a t i n gde 市 t i o n so f品 n o t c h e dt oanu n n o t c h e dbeama . ndi so b t a i n e dfromf i n i t ee l e m e n tc a l c u l a t i o n s 入=ηlexp{ ゎ( a o / d ) 2+T J 3 ( S / d ) 2+仇 ( a o / d ) ( S / d )+η5 ( S / d ) 3 } ( 6 . 2 ) 3,1 ' / 3=ー0 . 0 3 6, り' 4= 0 . 2 0 1,and1 / 5 =0 . 0 0 4( K a r i h a l o o 1 9 8 7 ) . Thent h ee f f e c t i v ec r a c kl e n g t hαei so b t a . i n e dfrom九 組dt h e N a l l a t h a m b i, .r r e s p o n d i n gd i s p l a c e m e n td "舗 A, d' 1 1; : :1 . 3 7 9, 市 : : : : : : -1必 a c / d= 1- r 凡S 3 ( 1+説 4 bd 36 c uE , 2 ' ¥ 1 ' " ( 6 . 3 ) u .sdi s“ s m a l l, "t h ee f f e c to fs h er .d e f o r m a t i o nc a nbe o n s i d e r e dbyu s i n g 乱 t ; E =~i r~3(1 +義)↓ (1+ν)~L. j l 6 4 b ♂( 1ー守 ) 3 ・ 2x :b d ( 1ー守 , . J ) ( 6. 4 ) 9 0 9 1 P ﹁ード﹂ b e r e民 =s h e a rωnstant= 1 0 ( 1+ν)/(12+ 1 1 1 1 )f o rar e c t a n g u l a rs e c t i o nand1i sP o i s ' I ' Sr a t i o .T h i sv a l u eo fE i st h e ns u b s t i t u t e diI印刷.6.16ωobtainae a r i h a l o o回 d ・K J l a t h a m b i( 1 9 8 7 )c o n t a i n sr e g r e s s i o nf o r m u l a st od e t e r m i n eaei n 尚 a do fu s i n gE q .6 . 1 6 . a rhec r i t i c a ls t r e s si n t e n s i t yf a d o ri st h e ng i v e na s I < I c= σ r t Y ζF (α) ¥ t e r eF ( α )i sg i v e nbyE q .6 . 1 6, α d u d i n gs e l f w e i g h te f f e c t si s =a./d 姐 ( 6 . 5 ) dUn = 6M/b ♂. Them idspanmomentM, M=( 凡+ ω/ 2 )S / 4 ( 6 . 6 ) h ec r i t i c a le n e r g yr e l e a s er a t ei s Gt= ( K i c ) 2 lC-E 一 ( 6 . 7 ) I m p r o v e d( b u tmorec o m p l i c a t e de x p r e s s i o n sf o rK I c { =J ( j c )andG1 c (=G i c )areobtained i 凶ぬ e l e m e n tmethodsw h i c hc o n s i d e rt h et e n s i l es t r e s snormalもot h ec r a c kf a c e s, vu s eo ff ・ h et e n s i l es t r e s si nt h ep l a n eo ft h ec r a c kandt h es h e 紅 i n gs t r e s s .Thef o r m u l a so fJ ( ,cand ' ; 1 rare J ( I C= σn { α . ) 1 / 2Y j( α )日( α, s) ( 6 . 8 ) s p a n .S ー 舎 (ι )問 I ? i c F i g . 6 . 1 2L o a d i n gA p p a r a t u sa . n dF i x i n gArr angemento fLVTD [ nt h e s eequa もi o n sa g a i n ( 6 . 9 ) = α=a . / d, s S / d ; ぺ 14 I N F I N I T ES I Z E ﹄ 4 NORMALS I Z E γ, ( α )-Ao+Alα+A2 α2+A3α3+A4α〈九 α (, s )=Bo+Bls+B2s2 +B3 β3+B4α β +Bsαβ2, Z l (α )=Co+Cα C C 2 + 3 + 1 +C α α α 2 3 4 Z 2 ( α, β)=Do+Dlβ+D2β+D3β3+D4αs+Dsαs 2 . 1 )ELASTICFIELD ,, ・ ・ ・ ,4 ;j=0, 1, . •. , 5 )a r eg i v e ni nT a b l e6 . 2 . Av e r ye x t e n s i v et e s t i n gprogramwasc a r r i e do u tbyK a r i h a l o oandN a l l a t h a m b i( N a l ; ; th a m b ia n dK a r i h a l o o, 1 9 8 6 a ;K a r i h 乱 . 1 00組 dN a l l a t h a m b i, 1 9 8 7 )u s i n gt h i smethod戸o 叫u a t ed a t af r o mmoret h 叩 9 5 0beamsi nw h i c hmaximuma g g r e g a t es i z e,beams i 回 , 姐d s tr e 自u l t sw e r eg e n e r a l l yv e r yc o n s i s t e n tw i t h ' d a t i v en o t c hc r a c kd e p t hw e r ev a r i e d .Theもe pc a l c u l a t e dv a l u e s ; t h ecoemdento fv a r i a b i o ng e n e r a l l yr a n g e dbebween6%and10%.T 田 t w i t hg e n e r a l l ye x c e l l e M a g r e e m e n tbdweent h e a L af r o mo t h e ri n v e s t i g a t o r sw e r ea l s ou s e d, ,thedatafrom J t l i t so ft h i smethodandJenqmdS h a h ' smeq10df o rKIe-Forexample 釘 t z( 1 9 8 7 )f o rd= 3 0 4m mg a v eI ? l c: :1074kNm-3 / 2 •, K~" : :1206kNm-3/2 l e f a ia n dSw m t la n dShah)andI f I c=1151kNm3 / 2 ( c o m p l i i n c em e t h o d ) . '{崎悶s i o nc o e 日c i e n t sA;, Bj , Ci, D j ( i= 01 ・ めASYMPTOTIC ト 玉 E L A S T I CF I E L D 功NONL l NEAR HARDENINGZONE 4 )FRACTURE PROCESSZONE ニ (NONL l NEAR SOFTENINGZONE1 H . iD eterminationofMaterialParameters byS i z eE f f e c tMethod . 7 . 1AsymptoticD e f i n i t i o nf o rI n f i n i t eS i z e 、 mcet h es i z ee f f e c t F i g . 6 . 1 3F r a c t u r eP r o c e s sZo n田 i nD i f f e 陀 n tSp 民 I J D e n sandE x t r a p n f i n i t eS i z e o l a t i o nt oI i s出 emosti m p o r t a n tp r a c t i c a lc o n s e q u e n 伺 o ff r a c t u r em e c h a n i c s, i t 92 93 b . ω Z 2 d d s 4 ω ω 切gω‘ Jz sz a sp r o p o s e dby i sn o ti l l o g i c a lt oe x p l o i ti tf o rd e t e r m i n i n gt h em a t e r i a lf r a c t u r ep r o p e r t i e s, Ba . Za n t .I np r i n c i p l e, t h es i z ee f f e c tp r o v i d e st h eo n l ym品 nsf o ranun a . mbiguou5d e f i n i t i o n . t e r i a lf r a c t u r ep r o p e r t i 伺. T hef r a c t u r ee n e r g ya sw e l la . so t h e rn o n l i n e a rf r a . c t u r e o fma i nw h i c hwehavee x p r e s s e dP "a c c o r d i n gt oE q .1 .2 .T a k i n gt h el i m i td→∞, f o rw h i c h o / d, weo b t a . i nt h ee x p r e s s i o n α→ αo= a g (α。 ) GJ= 一 一 ! : 2 d o g ( α 0 )=一 ηE ' " A一 E B2 . < / 2J _f_ ¥ - V J¥-'V' ( 6 . 1 1 ) p r o p o s e dbyB話 相t( 1 9 8 5 b )i nw h i c hAi st h es l o p eo fal i n e a rr e g r 田s i o np l o to ft e s tda t & ( F i g .6 . 1 4 )o b t a i n e dbyalgebr ぬc a l l yr e a . r r 姐 g i n gt h es i 民 e f f e c t}aw( E q .1 . 4 )i nt h ef o r m Y=AX+C, inwhich X =d, Y=(en/σ~) , , do=C/A B= C-1/2 "。何ιIM~R 、 、 ‘ 、 2 ¥ じ、 ・・'.---~ FRACTUAE 舗ECH 且陶I C S F i g ふ1 4TheS i z eE f f e c tLaw( E q .1 . 4 ;B ai 組 t , 1 9 8 4 ) Y-AX+C 6l _ -1_ _ , -2 A-7.2~3 玄10. : -i n .-:'psi 6 ~-=.-2 c=22.34x. 10-'"psi ωY 1 r "0.1090 0 司 i o . s τ 一 ( 6. 10 ) , 。 P J (α ) B2! t r J .1 ! _ d GJ=J i mR=J i m 一 一 一 =一. J _ ! _ lim一一 一 → ∞ Ef i Jd c~E . . _ -1+ d / ゐ limg(α) ~・ LOG(SIZE) ﹁︼︻圃的向田} I nE q .5 . 2weg a v et h ee x p r e s s i o nf o rt h een ぽg yr e l e a s er a t ei nt e r m so ft h ea p p l i e dl o a d 紅 白s .S e tもingP=, p川 wehaveG = R = specificenergyrequiredforcrack o rn o m i n a ls , a . n dbye x t r a p o l a t i o nt oi n f i n i t es i z eweha . v e growth ﹄ ﹁ 6 . 7 . 2Bas! cRelations /出足 、 、 / ; ,, . f, I M O S TE X l s r l 附 T ESTS {C} , d T I N - - 、N(Zb¥吋)日ト . r a . m e t e r smaybeu n i q u e l yd e f i n e dbyt h e i rv a . l u e se x t r a . p o l a . t e dもoa .s p e c i m 田 o fi n f i n i t e pa 回 s o ni st h a . t , i nani n f i n i t e l yl a . r ges p e c i m e n , t h ef r a c t u r eprocωsz o n eo c c u p i e s s i z e.Ther a . l lf r a c t i o no ft h es p e c i m e nvolume( F i g .6 . 1 3 ) . 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I np a r t i c u l a r ,i tmus 七 d i s s i p a . t et h esamee n e r g yandha . v et h es a . mel e n g も handw i d t h .T h e r e f o r e ,剖 p r o p o s e db y Ba . z 姐 t ( 1 9 8 7 a ), 組 unambiguousd e f i n i t i o ni sa . sf o l l o w s : . ndt h ee f f e c t i v ef r 拭 t u r ep r o c e s sz o n el e n g 出 CJa r e出 ee n e r g y Thef r a c t u r ee n e r g yGJa a . s er a t er e q u i r e df o rc r a c kgrowtha . ndt h ed i s t a n c efromt h en o t c ht i pt ot h et i po ft h e r e l e a . l e n tLEFMc r a . c ki na . ni n f i n i t e l yl a . r ges p e c i m e no fa . n ys h a . p e . e q u i v . s y m p t o t i cd e f i n i t i o n, d e t e r m i n a ι i o no ft h ef r a c t u r ep r o p e r t i e si sr e d u c e dt o Witht h i sa a . w .I fweknewthes i z ee f f e c tl a we x a c t l y ,wewouldg e t t h ec a l i b r a t i o no ft h es i z ee f f e c tl . t e l y ,theexacts i z ee f f e c tlaw ,a p p l ic a . b l eupt oi n f i n i t es i z e ,i sn o t e x a c tr e s u l t s .U n f o r t u na t h i smethod, l i k eo t h e r s, y i e l d si np r a c t i c eo n l ya p p r o x i m a t er e s u l t s . known. 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AST'C'TY 。 。 。 O-TestData x=d( i 孔) 10 ( 6. 1 2 ) g r e s s i o nP l o tf o rS i z eE f f e c tMethod F i g . 6 . 1 5Re Thes e c o n dn o n l i n e a rf r a . c t u r ep a r a m e t e r, namelyt h ee f f e c t i v el e n g t hcfo ft h ef r a c t u r e 四 s sz o n e , d e f i n e df o r姐 i n f i o i t e l yl a r g es p e c i m e n, i so b t a i n e dfromE q .5 . 6a sf o l l o w s : p r o L h ep r o p o s a lh a sb e e nworkedo u ti nde 同i lf o r山i st y p eo fs p e c i m e n.( 8 伺 F i g .6 . 1, which A l s oe x p l a . in st h en o t a t i o n s . ) AUt h ep r e s c r i b e dd e t a i l sf o rs p e c i m e np r e p a r M i o n, t e s t i n ganddataevdmuona r eg i v i nh r i h a l o oandNdlathmbi(1987)andWHIn o tber e p e a t e dhere.BNeay ,thet白 ting p r o g r a mu s e ss p e c i m e n so fa t1 助 成 仏r e e-b u tp r e f e r a b l ymore・ 自i z e sc h a r a c t e r i z e dbybeam も 1 1 sd 3, d 2, etc.ands p a n sS E, S 2, ....Thes m a l l e s td e p t hdImustn o tbel a r g e rthan5 t f e p da n dt h el a r g e s td e p t hdn mustn o tb es m a l l e rt h 佃 1 5d~ ・ The r a t i odn/d m u s tbea t 1 s h o t 耐 b e k e p μ t ∞ c ∞ on 路 s 旬加n t酎 s h o u l dt h 児. e叩 spa 叩n/μdept出 hh rriふt“ , 一ioS めi/dj • A l e a s t4 .Thebeamw i d t hb油 i I I inimumo ft h r e es a m p l e 哩p e rg i v e ns i z es h o u l dbet e s t e d . Oneo ft h eadv 叩 t ag-ofthemethodi s t l 国 a no r d i n a r yu n i a x i a lt e s t i n gmachinew i t h o u t ・ c o n t r o lo rh i g hs t i f f n e s sm品, ybeu s e d . Theo n l ymeasuredr e s p o n s e sn田 deda r et h e arvo p e a kl o a d s乃 f o ra l ls p e c i m e n s, j= 1 , 2, . . .n.Theloadingrateshouldbe日electedsothat t h emaximuml o a dwouldber e a c h e di no n et ot e nminute~ 4 g ( α 0 )C E _ 。 ‘ 内 g ( α 0 )J 訳可7 ゐ=只石}五 、 , CI - ) 95 n o ・ 唱 , , 94 f= A g a i n, wes e et h a tcf c 姐 bed e t e r m i n e dfromt h el i n e a rr e g r e s s i o np a r a m e t e r sA andC ( B a z 姐 t andKazemi, 1 9 8 8 ) .Ani m p o r t a n tp o i n ti st h a tt h et e s td a t amustωverasu 伍c i e n t l y b r o a dr a n g eo fs i z e s,自ot h a tt h es t a t i s t i c a ls c a t t e rando t h e ri n f l u e n c e swouldno もo v e r s h a d o w t h es i z ee f f e c t. E q s .6 . 2 4and6 . 2 6f o r mt h eb a s i sf o rt h ed e t e r m i n a t i o no ft h ef r a c t u r ee n e r g y組 de f f e c t i v e 倒 s z o n el e n g t hf r o mt h es i z ee f f e 比 Ba Z a n t組 dP f e i f f e r( 1 9 8 7 )v e r i f i e dt h a tv町 d i f f e r e n t p r o t y p e so fs p 回 i m e n s( F i g .5 . 1 ), i n c l u d i n gt h r e e p o i n tbends p e c i m e n s, e d g e n o t c h e dt e n s i o n s p e c i m e n s回 de c c e n t r i cc o m p r e s s i o ns p e c i m e n s, y i e l da p p r o x i m a t e l yt h esamev a l u e so ft h e 出a.I ooandN a l l a t h a m b i( 1 9 8 7 ), S w a r t z f r a c t u r ee n e r g yf o rt h esamec o n c r e t e .S t u d i e so fKa andRe f a i( 1 9 8 7 )andP l a n 回 初dE l i c e s( 1 9 8 8 b )f u r t h e re s t a b l i s h e dt h a tt h i smethody i e l d s a p p r o x i m a t e l yt h esamer e s u l t sf o rt h ef r a c t u r ee n e r g y舗 JenqandS h a h ' stwo ・ p a r a m e t e r m e t h o d . Ont h eo t h e rhand, t h ev a l u eo ff r a c t u r ee n e r g yo b t a i n e dbyt h e s emethodsi s 乱 AMwo 位d 巾 k 心o i f 企 r 拭 a 叫 . t cu r emethod( i . e . g e n e r a l l yq u i t ed i f f e r e n tfromt h a to b t a i n e dfromt h eR l l H i l l e r b o r g ' sm e t h o d ) .P l a . n ωandE l i c e s( 1 9 8 8 b )andP l a n a se t札 ( 1 9 8 9 )a n a l y z e dぬe s e d i f f e r e n c e se x h a u s t i v e l yandc o n c l u d e dt h a tt h er e a s o nwhyHi 1 I e r b o r g' . smethody i e l d sd i ι 乱l u e si sma . i n l yt h eassumeds h a p eo it h es o f t e n i n gs t r e s s d i s p l a 田 m entc u r v e . By i e r e n tv 前e e p e ri n i t i a ld e c l i n e , e x t e n d e dt a i l ), ab e t t e ra g r e e m e n t m o d i i y i n gt h es h a p eo ft h i sc u r v e( i nt h ev a l u e so iGJc o u l da p p a r e n t l yb eo b t a i n e d . I tmustben o t e dt h a tE q .6. 2 6f a i l si ig ' ( α。 )a p p r o a c h e sO.T h i sha . ppens, eιforac e r t a i n 回 ea c c o r d i n gt o c r a c kl e n g t hi nac e n t e r c r a c k e ds p e c i m e nl o a d e dont h ec r a εk .1 nt h a tc E q .6 . 2 6, Cwouldhavet ov a n i s hi fcJi sc o n s t a n t, andt h e na l s od o→ O .Duet os t a t i s t i c a l , t h er a t i oC / g J ( α。 )i nE q .6 . 2 6becomesm e a n i o g l e s 8when9 '( α。 )i st ∞ small.For s c a t t e r o n eo b t a i n sg '( a o )<0, andi nt h a tc a s ea g a i n s t i l ls h o r t e rc r a c k si nt h i sk i n do is p e c i m e n, E q .6 . 2 6c a n n o tbeu s e d . However, f o rt y p i c a lf r a c t u r es p e c i m e n ss u c hs i t u a t i o n sdon o t an 8 e . q .1 . 4,t h es i z ee f f e c tl a wi nt h eformo iE q .5 . 6i n v o l v e so n l ym抗 e r i a l I nc o n t r a s tt oE p a r a m e t e r s,GJandc J .T h e r e i o r e, もh e s ep a r a m e t e r 8c 姐 b eo b t a i n e dd i r e c t l ybyoptimum f i t t i n go fE q .5 . 6t ot h emeasuredv a l u e so f1'N i o rv a r i o u sv a l u e so fD. Suchf i t t i n gc a nb e a c c o m p l i s h e de a s i l ybyn o n l i n e a rr e g r e s s i o no ranyo p t i m i z a . t i o ns u b r o u t i n e . Th es p e c i m e ns h a 附 doo o tn e e dt obeg e o m e t r i c a l l ys i m i l a ri iE q .5 . 6( r a t h e r伽 B Eq .1 . 4 )i su s e d . However, t h ep a r a m e t e r山 川 t a . k e si n t oa c c o u n tt h es p e c i m 四 s h a p e , namelyt h er a t i og ' (αo ) / g ( α 0 ), i so n l ya p p r o x i m a t eandinvolvesωmee r r o r . Toa . v o i dt h i s , i ti sp r e f e r a b l et ou s es p e c i m e n sw h i c ha r eg ' ω m e t r i c a l l ys i m i l a r . e r r o r Thes i z ee f f e c tmethodh a sa 1 8 0b e e np r o p o s e dbyRILEMCommitteeTC89i o raRILEM ∞mmendation. Re i np r i n c i p l eanys u i t a b l ef r a c t u r es p e c i m e nc a nbeu s e d .I n Asf o rt h es p e c i m e ns h a p e, v i e wo it h ep o p u l a r i t yandc e r t a i np r a c t i c a la d v a n t a g e so it h et h r e e -p o i n tbends p e c i m e n s , G 6 . 7 . 3C a l c u l a t i o nProcedureo fS i z eE f f e c tMethod: I ft h el e n g t hL jofspecimenji sa 1 mostt h esamea s8ji t ssp a . n, t h e ncompute 可=尽 +jmjg ( 6. 1 4 ) mw h i c hmjgi st h ew e i g h to ft h es p e c i m e nandt h es p e c i m e ni sl o a d e ddownward.HLJis muchJ a r g e rt h 加 S jt h e n 。 2-J S . i- L P ; U = =P ;+ー 3m ・ 0 2S j . & . . 1 J '''J3 • ( 6 . 1 5 ) [ ft h es p e c i m e n s訂 eg ' 田 m e t r i c a l l ys i m i l a r制 P ; *=1 1 ・I fn o t . . . d _ S : P : *=P.・~ J Jd 問 . S j ( 6 . 1 6 ) w h e r emr e f e r st ot h em i d s i z es p e c i m e n .Next, p l o t} j( o r d i旧民)v e r s u sXj ( a b s C I s s a )where r 号 ( 巧= Xj=dj ( 6. 1 7 ) a n dddermineb h es l o p eo ft h el i n eAa sshowni nFig.6.i15lH l b 也 heY i 凶 n t 町 e 'f 偲 C e p μむC.The f 任 r a 総 叫 ぬ c ぺ 叫 t h h 1 u ? ; ; 斗 ζ な : よ : t r A r m o m恥 6 . 却 2 6 . The叫 i eo fE f o rh m n e e d sb obed e t e r m i n e df mC04 i rs t l E I l a s e dfro-zu s i n ge s t a b l i s h e de m p i r i c a lf o r m u l a s .TheexpmSionf o y g ( α 0 ) 24;;: r e n c e sf o rLEFMformulωThef o 加 i 叩.r ef r o 足 立 な沼野にで山長 F ( α)=よ 1 . 9 9-α(1ー α ) ( 2均 一 3 . 9 3 α +2 . 7 (2) 、 f i ( 1+2 α ) ( 1ー α) 3 / 2 ( 6 . 1 8 ) 9 6 も " . ‘ AWMV ' ・ & ) = f h , αo [ 1 . 5F( αoW ) g ( 句 ρ oa 、 F ( α )= 1 . 10 6-1 .552α+7 . 7 1 α 2-13.53α3+1 4 . 2 3 α 4 0 .F i n a l l y , L i n e a ri n t e r p o l a t i o ni sa c c e p t a b l ep r o v i d e d3~ S/d壬1 , , F o rS/d=8 : 97 • ゐ ,; :5. 4 d and( 3 )f o rc e n t r i ct e l 凶o ns p e c i m e n sd p . 8 d " .C o n s e q u e n t l y ,theminimum m, o= 1 悶 s p e c i m e n sw h i c ha r en e e d e df o ra p p l i 叫 i l i t yo fLEFM( w i 出 " r o S ss e c t i o nd e p t h so ft h . h eもh e o r e t i c a . le r r o ro f2%)a . r e. a sf o l l o w s : ( 6 . 2 0 ) I nBa . Z a n tandP f e i f f e r( 1 9 8 7 )andK a r i h a l o o組 dN a l l a . thambi( 1 9 8 7 )t h e r ea r ed i s c u s s i o n s o fc a l c u l a t i o n sn e e d e dt ov e r i f yt h er e s u l tf o rGJ岨 dt oo b t. a i nt h es t a n d a r dd e v i a t i o n .I t . ts c a t t e r e dr e s u l t sc 組 a r i s ef r o mo t h e rs i z ee f f e c t 自 ( S e c .5 . 3 ) .T h i s i sa l s op o i n t e do u tもha c e r nf o rv e r yl a r g eandt h i c ks p e c i m e n s, s i n c eh y d r a t i o nh e a tw i l l maybeo fp a r t i c u l a r∞n p r o d u c ed i f f e r e n tt e m p e r a t u r e si nt h i nandt h i c ks p e c i m e n sandt h u sl e a dt o叩 a d d i t i o n a l 8 1 田 e f f e c tt h a ti ss u p e r i m p o s e dont h a td e s c r i h e dbyt h es i z ee f f e c tl a w .I nt h a tc 描 et h i s method∞u l df a i. lS i m i l a r l y , i tc o u l df a i lduet od i f f e r e n c ei nm o i s t u r ec o n t e n t, s i n c et h i n c ko n e s.Thesee f f e c t s, 剖 wella st h ew a l le f f e c t, a r em i n i m i z e d s p e c i m e n sd r yf a s t e r出 組 出i byu s i n gt h esames p e c i m e nt h i c k n e s sf o ra l lt h es i z e s . 6.7. 4ComparisonwithOtherMethods K a r i h a l o o加 dNa . l I a t h a r n b i( 1 9 8 7 )l i s tmosto ft h eknownf r a c t u r et e s tr e s u l t s, i nw h i c h Ba . ia n t ' ss i z ee f f e c tmethodi sa l s ocomparedw i t hK a r i h a . l oo姐 dN a l l. a t h. a m h i ' sm e t h o d . s eo ft h eLEFMr e l a . t i o n s b i p sp u r p o s e山 ev a l u 田 o fGJ w e r ec o n v e r t e dt oK1 F o r出 i cbyu / ) 1 / 2 .F o rt h ebeamr e s u l t sr e p o 凶e d, t h eK1 . l u e sw e r e0.847-0 . 8 9 2 k N ; ; . 3 2 K l c: ; ;(G cva JE e= a . n dt h ev a l u e sfromt h eK a r i h a l o oandNa . la . t h a r n b i ' smethodw e r eKic= 0 . 8 6 7andKI c n.Similara 1 .0 0 5 k N ; ; . 3 g r . 四 m en もw asf o u n df r o mt h eda . t . ao fBa . Z a n t姐 dP f e i f f e r( 1 9 8 7 ) . 吋 b yB r u h w i l l e r( 1 9 8 8 )副 知umae ta 1 .( 1 9 8 9 )t o Thes i z ee f f e c tmethodwasu s measuret h ef r a c t u r ee n e r g yo fdamc o n c r e t e . 6 . 8S i z eRequiredf o rA p p l i c a b i l i t yofLEFM Ane a r l ya . t t e m p tt os p e c i f yt h eminimums i z eo fも回ts p e c i m e ns u c ht h a tLEFMma . yb e h. a to fWalsh( 1 9 7 2, 1 9 7 6 ) . Hea r g u e dt h a . t , byc o m p a r i n gt h en o r n i n a ls t r e s sa t u s e dw回 t t h en o t c ht i pf o rabeams p e c i r 悶 ( c a l α l 1 . a t e dbyu s i n ganLEFMform 由 f o r] ( [ , )w i t ht h e at r a n s i t i o nd i m e n s i o nma . ybef o u n d .Hes t a t e dt h. a tt h e moduluso fr u p t u r eo ft h em a t e r i a l, c h a r a c t e r i s t i cbeamd i m e n s i o ns h o u l dbea tl e a . s td o u b l et h i si no r d e rf o rLEFMt obev a l i d . Ba . s edonh i st e s tda . t . ah es u g g e s t e dt h en o t c h e dbeamt obet e s t e ds h o u l dha . v e. ad e p t bn o t l e s st h a . n2 3 0m m( 9i n . ) . I nt h el i g h to fr e c e n te x p e r i m e n t a ls t u d i e sont h es i z ee f f ,民t i nf r a . c t u r e , t h er e c o m r n e n d a t i o nf o rt h es p e c i m e ns i z ean. a l y z. a b l ebyLEFMnowa p p e a r st obeg r o s s l yu n d e r e s t i m . a t e d . A c c o r d i n gt ot h es i z ee f f e c tl a wo fB話 a n t, t h eb r i t t l e n e s snumbersmustbea tl e a s も2 5i f 抗i o nfromt h es t r a i g h tl i n ea s y m p t o t ef o rLEFMs h o u l dbel e s st h a n2%. Ba . i ωb t h ed e v i 却 dP f e i f f e r ' s( 1 9 8 7 )七e s t so fv a r i o u st y p e so fs p e c i m e n sshowni nF i g .5 . 1i n d i c a t et h a t:( 1 ) f o re c c e n t r i cc o m p r e s s i o nf r a c t u r es p e c i m e n sdo=1 .8 5d ( 2 )f o rt h r e e p o i n tbends p e c i r r 削 S m, F o re c c e n t r i cc o m p r e s s i o ns p e c i m e n s : dと4 6 d " F o rt h r e e -p o i n tbends p e c i m e n s : d三 1 3 5 d " F o rc o n c e n t r i ct e n s i o ns p e c i m e n s : dさ 4 2 0 d " ( 6 . 2 1) a . t o r y七e s t s,a . n dt h e i ru s ewouldb e T b e s ed i m e n s i o n sa r ei m p r a c t i c a b l yl a r g ef o rl a b o r f i n a n c i a . l l yw a s t e f u ls i n c ek n o w l e d g eo (t h es i z ee f f e c tc a np r o v i d ea d e q u a t ei n f o r m a t i o non a t e r i a lf r a c t u r ep a r a . m e t e r su s i n gmuchs m a l l e rs p e c i m e n s . m. 6 . 9I d e n t i f i c a t i o nofNonlocalC h a r a c t e r i s t i cLength F o rb h em n l o c a lcon?inuummodelw i t hl o c a I叫 r a i n( o rn o n l o c a ldamagec o n t i n u u m ) . . h ef r a c t u r een 句 yGJi sp r o p o r t i o n a lt ot h ea r e aW.u n d e rt h e∞m p l e t et e n s i l e町 制ー ポr a i nc u r v e ( w i s h B o f t e n i n g ) t i m e sb h ec h a r a c t e r i s t i cl e n g t h , t , andc fi sp r o p o r t i o n a lt ot h e c o n c e i v e dbyBdant姐 d c h u a c t e r i s t i cl e n g U 1 . T h i sf a c ti se x p l o i t e dbyas i m p l emdhod, P i j a u 必ぽ-C. a b o t( 1 9 8 8 b, 1 9 8 9 ) .I tu s e st h er e l a . t i o n : i =皇位竺! 3] W .[ J / m ( 6 . 2 2 ) , t k r e i s a m u l t i p l i c a u v e e m p i r i c a l c o e 缶c i e n tont h er i g h もh ands i d e, which Z M o r e p r e c i s e l y 1 8h o w e v e r, 刊r yc l o s et ol ( i bs l i g h t l yd e p e n d sonb h et y p eo ft h en O B l o c a lmodeladop もe d , , . u df o r0 田 p a r t i c u l a rmodelu s e di twas1 .0 2 ) .Sot h eo n l yp r o b l e mi showt om倒 的 出 a 日e r g yd i s s i p a t e dp e ru n i tvolume , W.・ Tot h i send , onen e e d st os t a b i l i z eS h es p e c i m e nbybondedr e i n f o r c i n grodsmthatc o n 噌 E eund e f o r mh o m o g e n e o u s l yandr e m a i ni nam a c r o s c o p i c a l l yhomogeEmuss b a b ed u r i n g , w i s h o u tanyl o c a l i z a t i o no fc r a c k i n g . T h i smaybea c c o m p l i s h e du s m s o f t e n i n gb e h a v i o r ~t h es p e c i m 佃 s howni nF i g . 6 . 1 6 , w h i c ha l s os h o w sacompanionf r a c t u r es p e c i m e nu s e d n f . d e t e r m i n a b i 0 1 1 o fGI-Thel o n g e rs i d e so ft h er e c t a n g u l a rc r o s ss e c t i o na r er e s t r a i n e dbv . u n gt othem, w i t hepoxy , as y 蜘 no fpa 叫 e lden 均 s p a . c e d仙 s t e e lrods-Thegaps ・ J ・ a ? な ど な; 3 2 J 3 2 ぷ : : : : : : ニ::22ztL222ぷ: : t r z 一 n?feetlnconcrete-Byc}loomgt11ecrossmtionsofumodstobemchmdlertharl f 1 m m喝 r e 仲間,t h er 仙 c a n n o ta f f e c tt h en o n l o c a . lbeha . v i o ro ft h em a t e r i a li n a n s v e r s ed i r e c t i o n T h er e c t a n g u l a rc r o s ss e c t i o ni se l o n g a t e d, s oaE4om i n i m i z et h e : 官会1 1 f 器Z 2 1 5 2 5 S 2 5 1 5 主主主忠誠i ; k n 州 99 98 Thec o m b i n e dc r o s ss e c t i o no f3 0 1 1t hes t e e lr o d si ss e l e c t e ds oa . st o出 s u r et h et a n g e n t i a l , ・u n e s so ft h ec o m p o s i t es p e c i m e nωbea l w a y sp o s i t i v e .T h i sg u a r 阻むe e st h es p e c i m 四 h. ~ s もa b l e makingl o c a l i z a t i o ni n s t a b i l i t yi m p o s s i b l e . Thef o r c ei ns o f t e n i n g∞ c o 町 n c r 問e t 怜e , , ' げ 1 : 1. . 川 ‘ 1 . t e 舵e l r o 也 d sc a l c u l a t e dfromt h emeasureds t r a i ni ss u b t r a . c t e dfromt h em他 s 叫u r ,e 陀d ω も 。 叫 4 同a lf o r c e . e . g ., bym e t a l l i cp l a t e sg l u e dbyepoxyt ot h es u r f a . 目 f h es p e c i m e ni sg r i p p e da tt h ee n d s, , r ¥~ hesteelrods.Thespecimeni ss u b j e c t e dt ot e n s i o ni nac l o s e d l o o pt e s t i n gma . c h i n e , a . nd L トサ l o to ff o r c eVB. r e l a t i v ed i s p l a c e m e n to v e rg a u g e1 田 g t hL i nt h em i d d l ep a r to ft h e , ;ri I meni sr e ∞rded;seeFig.6.16c. Theindinedstraightlinei n出 i sp l o tr e p r e s e n t st h e 慢n e s so ft h er o d sa l o n e , w i t h o u t回 yc o n c r e t e .F o rv e r yl a r g ed i s p l a c e m e n t s, t h er 田 p o n s e 仰r 叫 c l 悶 t h i sl i n ea s y m p t o t i c a l l y . Work W~ , t h ed e s i r e dr e s u l 色 , i sr e p r e s e n t e dbyt h e 吋 川・b a t c h e da r 品 b etweent h er e s p o n s ec u r v ea . ndt h er i s i n gs t r a i g h tl i n ef o rt h es t e e lr o d s 山.川.‘ f ・ b 1 62 . . .圃 , TI--T-14 (a) ⋮ u l F o rt h epa . r t i c u l a . rc o n c r e t eu s e dbyB話 回tandP i j a u d i e r C a b o t( am i c r o c o n c r e t ew i t h , i n t oE q .6 . 3 4 h i si sr a t h e rc l o s et owhath a sb e e n ) ' t l. l n e dt h en o n l o c a lc h a . ra c t e r i s t i cl e n g t hf= 2.7da • T け J ! ν nedf r o mf i n i t ee l e m e n tf i t t i n go ft h es i z ee l f e c td a t afromf r a c t u r et e s t s . a . n so ft h ef i c t i t i o u s [ tm i g h tben o t e dt h a te v a l u a t i o no ft h ep r e s e n tt y p eo ft e s tbyme fonet a . k e sc a r et op r o d u c ea .n e a r l yp e r f e c tr e s t r a i n t, r M ; km o d e l. a p p e a r sp r o b l e m a t i c .I fllI, ~tiple p a . r a l I e lf i c t i もi o u sc r a c k sw i t ha r b i t ra . r i l yc l o s es p a c i n gc o u l dbeo b t a i n e di ncompu~い 川s, ma . k i n gt h er e s u l t sa . m b i g u o u s .Asa . l r e a d yp o i n t e do u t, t h i sproblemc a nbea v o i d e d I 川 崎r i c h i n gt h ef i c t i t i o u sc r a c kmodelw i t hat h i r di n d e p e n d e n tm a t e r i a lp a r a m e t e r ,the l 1 ' " I l JI I I U ms p. a c i n go ft h el i n ec r a c k s( s u c hs p. a c i n go f∞u r s ei se q u i v . a l e n tt ot h en∞ o 叫 n 1 1 o ca . l 凶 Eh a 瓜: 川.: ¥ ' l In 叫 l 比 t e ぽr e 飽s “ 叫 t i 暗 n gpoω 叫 m もt ω o n ω o t 旬e l お st 由 h抗 a tt h ec h a 釘x 拭 a c t ω 釘 e n 叩s “ 叫 も “i 比 cl e n g t h, , . fcannotbei d e n t i f i e dfrom 倒 So fu n r e s t r a i n e ds p e c i m e n so fd i f f e r e n t1 田 g t h s( e . g .t e s t so fv a nMier, 1 9 8 4, l ' o ll ' I a . lt I P" b 'u rd i f f e r e n tg a g el e n g t h s( S h a . h. a ndS a n k a r ,1987). Suchd. a t a .c 姐 bef i t t e de q u a l l y n yv a l u eo f , . fi n c l u d i n gf .=0(Ba . i a n t,1 9 8 9a . ) . I !,I.~ing a 1 1 . '' ( i m u ma g g r e g a t es i z e9. 5mm) s u b s t i t u t i o no ft h emeasuredva . l ueo fW~ 1 1s t 僧 1 r 噛 P (c . ) i 堅 ' 1nIdentificationofTensilePost-PeakSofteningStress.Strain ⋮臨 ーーム 1l il l - U﹀翠 叫﹁ →い霊 rU 引コい引μlhui Cpol}' .て.I'~te. t A ¥ . Ou r v e s 1 ・・ i r l e n t i f yas t r a i n s o f t e n i n gt r i a x i a lc o n s t i t u t i v er e l a t i o nfromt e s td a ta . ,onemusti n ,. i n v e r s eboundaryv a . l ueproblemt a k i n gi n t oa c c o u n ts t r a i nl o c a l i z a t i o n , ' :! 9 8 8 ) .T h i si sat a . s ko ff o ; m i d a b l ec o m p l e x i t 子 H owever, i d e n t i f i c a t i o no fau n ia . x i a l ( O r l ' , u a i l ' '.Jf t e n i n gs t r e s s s t r a i nr e l a . t i o ni sr e l a t i ; e l ysi~ple, p r o v i d e dt h a tt h ec h a r a c t e r i s t i c k-tl~! 1, I、 d e t e r m i n e di na d v a n c e 卸 , alreadyexplained.AsshownbyBa.i組 内 (1989a)analysis I . ni a !t 凶 d a t a .f o rs p e c i m e n sofVa . r i O U 8l e n g も h s, l o c a . l i z a . t i o ni nu n i a x i a lt e s ts p e c i m e n s 0 1u l1 ~: I , r .' : '!~u.a.t~ly d e s c r i b e dbyt h e 's e r i e sc o u p l i n gm~del, i nwhichas t r a i n s o f t e n i n gl o a d i n g I C 4・ fI . ' n g t hli sc o u p l e di n附 i e st oanu n l o a d i n gz o n e .I ft h ev a l u eo f. fi sknown, t h e間 i a l es t r a i n s o f t e n i n gz o n ec o r r e s p o n d i n gt ou n i a x i a ls t r 田 ssmayb eapproxim a . t e l y h車問・" ~ h 伊 1 1r 1 l K > l v e姐 Figs.16Measuremento fEBergD i s s i p a t i o nDuet o C E a C I t i n gPerUnis 阻 A c c o r d i n gt oBaiantandP i ja . u d i e x . Volumei naS t a b i l i z e dSpecim Ca . b o t( 1 9 8 8 ) 1 0 1 1 0 0 _L .E' d、 r ‘ 、 σ ,z E α u 、 ・ 、‘‘ , , , , f'h¥ L一 t EE σ , , , 、 、 一 一 σ ‘ 、 , ‘ ( ﹄ , F ‘ . , L一 t c a l c u l a t e da s( B話 回t1 9 8 9 a ) : t uppersupporti ngbeam ( 6 . 2 3 ) 寸lid← 一 whereL ~ 1, 可 f 市 刷 ( 司 σ 伊 例 )=me 叩&胡 n& 凶x 対i 叫 a 1柑 s t 仕r 凶a 附 l e r 凶 o 肝v e 白r t h eg a g el e n g ぬL , 姐 df u ( σ )= a x i a ls t r a i ni nt h eu n l o a d i 時 p a r tc o r r e s p o n d i n g m . 組 c he m a n a t i n gf r o m出 ep e a ks t r e s sp o i n t. toσforanu n l o a d i n gb r LD 6 . 1 1MaterialParametersf o rModeI IandPlanarMixedMode Fr acture F o rtwod i m e n s i o n M s t r e s s B b a t e s , i ng e n e r a li tmaybepresumedt h a tc r a c kpropagdon c t i o no fMode1( o p e n i n g )andModeI I( s l i d i n g ) u l dt a k ep l a c eu n d e rもhecombineda d e f o r m a t i o n . I nc o n c r e t e ,however, t h ee f r e e b o fa g g r e g a t ei n t e r l o c k , wbichl 品 d st ov o l u m e d i l a t i o n .t e n d st or e s i s tc r a c kp r o p a g a t i o ni nanYmode0 出e rt h 組 o p e n i n g( B a z a n t祖 d j o v a -1 98 0} .Ne ve r t h e l e s s , t h e r eha sbe e nane x 対t e 叩n 隠 s 抑 e晶m 釦o G釧 ba d 白o 且邸 e仰o v 釘 e r古 仏 ド ey 戸e 一 …s, 組dcontinuingtod品y,i nan叫 emptt oc a u s ec 即 k i n g姐 df 間 印 刷n c r e t et ooc c u ri nmode1 1o ra tl e a s ti nmixedmode. TheI o s i p e s c us p e c i m e ngeomdry , showni narnodigdformi nF i g . 6 . 1 7, h a sb明 nu 詑d byv a r i o u siwestigaws(Barre ta l ., 1 9 8 7 iBahMandP f e i f e r ,1985h1985b,1986;Swartz ndT a h a .1 9 8 7 )ands ohωas i m i l a rgωmetryw i t ho n l yo n en o t c h( A r r e a叩 dI n g r a : f e a , 1981-SW4 巾, TMlEMidR e f a i,1 9 8 8 )・ Beamsi nt h r e e p o i n tb e n d i n gw i t ho f f c e n t e r otcheshave--usedbyJ e 叫 a ndShah( 1 9 8 7 a ,1987b)andSwartz, LuandTang( 1 9 8 8 ) ( s e eF i g . 6 . l a LB i a x i a lt e s t i n gs y s t e m shaveb e e nd e v e l o p e di nwhichc o m b i n a t i o n so fd i 1989)ord i r e c tt e n s i o nands h e a r r e c tt e n s i o nandcompmSiona r eu s e d(binhardte44, 叩 l z a d e he ta l ., 1 9 8 7 ) ( H a s N u m e r i c a lm o d e l i n go ft h eI o 均e sc us p e c i m e nwωdonebyA r r e a乱ndI n g r a f f e a( 1 9 8 1 ) , I n g r a f f e a叩 dP a n t h a k i( 1 9 8 5 ),I n g r a f f e a . 組dG e r s t l e( 1 9 8 5 ), BazantandP f e i 鉦 ' e r( 1 9 8 5 b ), Ro t sanddeB o r s t( 1 9 8 7 ), 組dSwen 則 ( 1 9 8 6 ) ・Ra t he ri n t e r 偲 t i n gwast h ef i n d i n gb y . Z 姐tMldP f e i f r e rt h a ti ft h es h e a rz o n ei nt h ebeami sv e r yn a r r o w , t h ec r a c kp r o p a g a 4 6 Ba .thec r o s s s e d i o np l a n er a t h e r S h a ninani n c l i n e ddiredion-Thisp r o v i d e dac h a l l e n g e a . lm o d e l i n g .A c c e p t a b l ef i t so ft e 叫 r e s u l t sw e r eo b t a . i n ed( B話相も andP f e i : f e r , i c 1 9 8 5 b )笛 sumingaModeI Is 附 a tt h ec r a c kt 恥 mod e l 山 aband山 伽dMod 拭c k s 目. O t h e r so b ta . i n e da 叫 . c c 白e p μ t 功 a b l e自 臥t ω sa s s u m i n gも出】 hec r a c kp 戸r 叩 o p喝 a g “ a 叫 t i ぬ ont ooc c a n l yMode1d e f o r m a t i o natt h ec 叫 t i pa lt houghModeI I t y p ed e f o r m a t i o n( ∞m w i t hModeI le x i s t si m m e d i a t e l yb e h i n dt h ec r a c kfront.Thea p p l i c a t i o no fS w e E l g o ns -d e lt obe-st e s t e dbySwutze talinthreepoiMbending(Fig.618)mdfoumiM 紅 同 Lu , Ta . ngandR . e f a i, 1 9 8 8 )(si出l~r t oF i g .6.U, ' b t r lw i t h ' o o l yon en o t c h ) b e o d i n g(Sw , a l 刷 l a . t i v e s h o w sc l e a r l yt h a tp r o p a g a t i o ni nt h e i rt y p e o f t e s to c c u r si nM04elHowever e x p l a n a s i o n s o ft h ef r a c t u r ep a t t e r no b s e r v e dbyBdMltandPfahr(1985b, 1 9 8 6 ), w h i c h h a v eb e e no f e r e dbyInudea姐 dP a n t h a k i( 1 9 8 5 ), a n d r e q u i r eo n l yMod eItypecra ck i n g, ・( S w a r t z回 dTah , . a1987), w h i l eR o も 5andd eB o r s t( 1 9 8 7 )c o n s i d e r e dt h ec r a c k S w a r t zeta l 仔吋 ILPD Test Specimen h 'owersupporti ngbeom L u r e a c ti o n A,d 孟1 ・ - 凶 ザ F i g . 6 . 1 7 1 ω i p e s c u ・ TypeS h e a rT,白tG 回 m etry 。 ' 行 x~ k -x S F i g _ 6. 1 8B e a r nw i t hO f f -C e n t e rN o t c h e s 103 102 nMode1ands u b s e q u e n t l yt or e s p o n dp o s s i b l yi nModeI Ia l s o .I n g r a . f f e a o p e n i n gt oi n i t i a t ei dPan もh a k ih a v ea r g u e dt h a tt h eo b s e r v e df r a c t u r eo r i g i n a もe dfromt e n s i l es p l i t t i n g8 t r e s s e s 対s t e n c eo ft h e s e錦 町s s e sw槌 叫 句 i nt h er e g i o nb e t w e e nt h en o t c ht i P 8( F i g .6 . 1 7 ) . Thee showna n a l y t i c a l l ybyR o t s組 ddeB o r 8 t .S i m i l a re x p e r i m e n t sbyS w a r t ze ta l .( S w a r t za n d ,1 9 8 7 )showedt h ep r i n c i p a lf r a c t u r eo r i g i n a . t i n gi nt h i sr e g i o na . ndn o ta tt h en o t c h Taha f e i f f e r( 1 9 8 5 b )c o n f i r m e dt h ee x i s t e n c eo f t i p s . Thef i n i t ee l e m e n ta n a l y s i so fBaza:叫 andP s i g n i f i c a n tt e n s i l es t r e s s e sb e t w e e nt h ec r a c kt i p sb u tshowedt h ei n c l i n e dp r i n c i p a lt e n s i l e s t r e s 御成 t h ec r a c kt i p st obee v e nh i g h e r .R民 e n tt e s t sa tN o r t h w e s t e r nU n i v e r s i t y(K 回 e 凶 組 d Bai 姐 t ,1 9 8 8 )i n d i c a t e dt h . a ti ft h es 戸c i m e ni nF i g6 . 1 7i sl o a . deds y m m e t r i c a l l ybyP / 2 a te a c hs i d eo fe a c hc r a c k m o u t hr . a t h e r出 叩 a n t i s y m m e t r i c a l l y ,t h e ni ts p l i t sa ta .1 0 0 0n o t muchh i g h e rt h a nt h emaximuml o a . df o rt h ea n t i s y m m e t r i cl o a d i n g自h o w n .T h i ss u g g e s t s t h a tt h es p l i t t i n gt e n s i l es もr e s sm i g h tbei m p o r t a . n t. Froms i z ee f f e c tt 由民 B a i a n tandP f e i f f e r( 1 9 8 6 )f o u n dt h eModeI I台a c t u r ee n e r g yめ bem 加 yt i m e sh i g h e rt h a nf o rMode1( B a i a n t組 dP f e i f f e r, 1 9 8 5 a, 1 9 8 5 b ) . C l e a r l y ,aggregatei n t e r 1 0 c kands h e a rf r i c t i o np l a yamajorr o l ei np r o v i d i n ge n e r g y h et i pi sduet oMode1 t or e s i s tc r a c kp r o p a g a t i o ne v e nt h o u g ht h ed r i v i n ge n e r g y8.tt 伺 db emodeledp r o p e r l ybya p p r o p r i a t eu s eo fac r a c k d e f o r m a t i o n . Whethert h i sc a ni n d bandw i t hi n c l i n e dm i c r o c r a c k so rf r i c t i o ne l e m e n t si sn o tc l ea . ra tp r e s e n t .Att h esamet i m e , t h e r e晶r enogoodm i c r o s c o p i co b s e r v a t i o n sa v a i l a b l et oi n d i c a t et h ep r e s e n c e, s h a p eands i z e 田 o ft h i si sc l e a rfromModeIb e n d i n gandt e n s i o nt e s t s . o fap r o c e s sz o n ea l t h o u g he v i d e n a . r dt oModeno rmixedmodec r a ε kp r o p a g a t i o n, t h e r ei st h u sa .g e n e r a l l a c ko f Withr e g i n f o r m a t i o nandu n d e r s t a n d i n g .T h e r ei snog e n e r a la g r 田m ent副 知 t h es u i t a b i l i 句o ft e s t a t ae v a l u . a t i o no rf a i l u r et h 切 r yt op r e d i c tmixed-modec r a c kp r o p a g . a t i o na n d methods,d f r a c t u r ei nc o n c r e t e . 組 12 1 10暗 唱 同 ヘ s ¥ ドミ 、 ? 一 帽 6 4 2 6.12MaterialParametersforModeI I IFracture I ntheI o s i p e s c ut y p emixed-modes p e c i m e n sa l r e a . dyd i s c u s s e d, ap e r { . 舵 ta n t i s y n u n e t r y i sn o ta c h i e v e d .Thes p e c imenf a i l sw i t hap l a n a rc r a ε ki nt h ec r o s ss e c t i o np l a n e , a l t h o u g h a . r dt ot h ec r o s ss e c t i o np l姐 e .H o w e v e r , t h ep r i n c i p a ln o r m a ls t r e s s e sw e r ei o c l i n e dw i t hr e g ap e r f e c ta . n t i s y m m e t r y ,r e q u i r e df o rModeI I If r a c t u r e,c a nbea c h i e v e di nac y l i n d r i c a l .c i r c u m f e r e n t i a ln o t c hi nt h em i d d l e , s u b j e c t e dt ot o r s i o n.Suchas p e c i m e n s p e c i m e ow i t ha 組dBaiant, P r a tandTabbara( 1 9 8 9 ) ;F i g .6 . 19 . wasi n t r o d u c e dbyB a i a n tandP r a t( 1 9 8 8 a ), . rs p e c i m e n so fd i f f e r e n ts i z e s, 出 t h 句 e yf o u n dap r o n o u n c e ds 釘 i z ee f . “ ωt T e s t i n gg e o m e t r i c a l l ys i m i la 姐 du 国s e 吋di 比 t句 t od e t e 釘r 紅 n I 凶 f お o r c 印e ,ぬe yf o u n di tt obea b o u tt h r e e t i m e sl a r g e rt h a nt h a tf o rMode1 .However,t h e y o b s e r v edt h a td e s p i t ep e r f e c t l ys a t i s f y i n gt h ea n t i s y m m e t r yo fg 回 m e t r i cc o n d i t i o n s姐 d I Is i t u a t i o n, aModeI I If i e l dm i g h tn o th a v eb e e n l o a d i n ga r r a n g e m e n tr e q u i r e df o r乱 ModeI a c h i e v e dI o c a l l yduet ovolumee x p a n s i o no ft h ef r a c t u r ep r o c e s sz o n e . Theya l s oo b s e r v e d t h a tt h eModeI I If r a c t u r ee n e r g yi sa p p a r e o t l yv e r ys e n s i t i v et ot h en o r m a ls t r e s sa c r o ss t h ei r a c t u r ep r o c 邸 sz o n e .I ta p p e a r st h a tModeI I If r 乱c t u r ee n e r g yc a n n o tbeama も ' e r i a l c o n s t a n t jb u ti tc o u l dbeam a t e r i a lf u n c t i o n, d e p e n d i n g00t h et r a n s v e r s en o r m a ls t r e s sa o d p o s s i b l yo t h e rva . r i a . b l e s . y , s i m i l a . rt e s t so fXuandR e i n h a r d t( 1 9 8 9 )i n d i c a t e do n l yan e g l i g i b l es I z ee f f e a R怠 田 川l I Is p e c i m 佃 s .The. r モa s o nf o rt h i sd i f f e r e n c ei sn o tc 1 e a r . i nt o r s i o n a lModeI A, . 0.312 C. . 4.167 • 。 5 10 15 d/da 、 r で¥ー0.3 ¥ S t r e n g t hG r i t e r i o n 。 ¥ ¥ "¥・ぞ 、; ; / \・~ トミー0.4 ー 、 0 ' ) ¥ 。 ¥ 、 ~ -0 . 5 日 = 0.490 λa= 13.35 -0.6 0.4 0 . 0 1 . 0 1 .2 1 . 4 log(d/dQ. ) tR . e g r e s 討0 0P l o tf o rM錨 u r e m e n to fModeI I I( a f t e r F i g . 6 . 1 9S i z eE長 c B a i a n t, P r a t組 dTa . bb 訂 a . ,1 9 8 9 ) 104 105 s l r e n a l h c r i l e r i o n -・恥白¥zb C h a p t e r7 .FACTORSINFLUENCINGFRACTURE PARAMETERS 7 . 1E f f e c to fLoadingRa t eandCreep I nnom a t e r i a lc a nf r a c t u r ehappeni n s t a n t l y .C r a c k st a k eaf i n i t et i m et of o r mandt o .f i n i t ed i s t 佃 c e .Ther e a s o ni st h a tt h ep r o 印 S 8o fbondr u p t u r e sp r o c 田 d s p r o p a g a t eo v e ra a ta .c 位 t a i nf i n i t er a . t e .Theba . s i cp h y s i c a lthωryd e s c r i b i n gt h ep r o c e s si st h er a t ep r o c e s s 切 r y , i nwhicht h er a t eo fbondr u p t u r 白 i sd e t e r m i n e dbyt h ea c t i v a t i o nen 町' g y , U .The t h p r o ba . b i l i t y( o rf r e q u e n c y )o fbondr u p t u r ed e p e n d sont h em i c r o s t r e s sl e v e la . tt h ec r a . c kt i p , whichi nt u r ni sd e t e r m i n e dbye i t h e rt h es t r e s si n t e n s i t yf a c t o r, K], o rbyt h een ぽ g yr e l e ω e r a t e ,G. Basedonthesephysicalobservations, t h ef o l 1 owingapproxim a . t er e l a t i o ne n s u e s 叫 1 9 7 4 ) : ( E v a= I I r / K l c ) " e -U/R T ( 7 (K . 1 ) c ! !fost,peok i n1 . 4s + +~suòl: peokIn"SOOs ~~ ~Iow. p eak i n1 4 0 0 0 s * キV . r y$ I o w .p e o kI n 253000$ 0 . 1 0 . 1 B r i t f l e n e s s number,P=d/do 。, Fig.7.1hmueofB d t t l e n e s sNmber(Shia句 t h e町g h tont h eS i E f f e c tC u r v e )Duet oI n c r e a . s i n gTimet 唱 。 , (F oFa . i l u r r a c t u r eS p e c i m e n s I . Z 3 = G :的 .,,(I/T- へ G~f 36.48 N/m . d r y= Gf f.O 70 J ・ 20 c = ・ ﹁、、 恥 -h 4 E u-U'Vw ¥HV -ge a u a v , o . 旬 - 、 、 、 ( b ) 0 . 0 , . . ﹄﹃ i nwhicha= v e l o c i t yo fc r a c kt i pp r o p a g a t i o n, T= a b s o l u t etempera~ure, R = u n i v e r s a lg a s c o n s t a n t, andI I , n= e m p i r i c a lc o n s t a n t s .Rοughly , n~ 3 0, w h i l e110 c a nh a v ev e r yd i f f e r e n t c t h er e l a t i o n s h i pb e t w e e nl o ga姐 dl o gK]s h o u l dbel i n e a r . v a l u e s .A c c o r d i n gt oE q .7 . 1, Butt h i si st r u eo n l ya p p r o x i m a t e l y .E x p e r i m e n t si n d i c a t eam i l d l yn o n l i n e a rr e l a t i o n s h i p b e t w e e nlogKIandt h el o a d i n gr a t e, b u tnoc l e a ri n f o r m a t i o ne x i s t sw i t hregardωι T h e r ee x i s to t h e ru s e f u lr e l a t i o n 8r e l a t i n g七her e m o t es t r e s st ot h er e m o t es t r e s sr a . t eo r r e m o t es t r a i nr a t e,o ru n l o a d i n gr a t e( e . g .Re i n h a r d , 七1 9 8 4 ) .Theya r ehoweverv a l i do n l yf o r a . b l e .( “ R e mote"a 削 h el o c a . t i o D S c e r t a i np a r t i c u l a rs i t u a t i o n sanda r en o tg e n e r a l l yi n t e r p r e t a . c k8 0t h a tt h es も r e s si sn o ts i g n i f i c a n t l y a f f e c t e dbyt h ec r a c k ) . 5 0f a rfromt h ec r JohnandShah( 1 9 8 5, 1 9 8 6, 1 9 8 7, 1 9 8 9 a ja l s oJohn, Sh a . h組 dJenq, 1 9 8 7 )a p p l i e dt h et w φ r a c t u r et e s t sc o n d u c t e da . td i f f e r e n t p a r a m e t e rf r a c t u r emodel七oMode1andMixedModef 5 t r a i nr a t e s . Theyc ο n c l u d e d七h a tK l oi sr a t e -i n d e p e n d e n tb u tt h a t8 踊 w e l lωthe CTOD, a . c kl e n g t ha tp e a kl o a d, d e c r e a s e田 t h el o a d i n gr a t ei n c r e a s e s . e f f e c t i v ec r I ts h o u l da l s obeo h s e r v e dt h a tt h ee f f e c t so fc r a c kg r o w t hr a t eandl o a d i n gr a t ea r e, i n t h e i rp h y s i c a lmech 組 i s m, r e l a t e dt ot h es t r e s sc o r r o s i o ne f f e c ti nf r a c t u r e . I na n a l y z i n gt h ef r a c t u r eb e h a v i o ro fs p e c i m 田 s o rs t r u c t u r e su n d e rv a r i o u sl o a d i n gr a t e s, t h ephenomenao ff r a . c t u r e姐 dc r 田 pa r ei n s e p a r a h l e .Ther e s u l t sa r ei n f l u e n c e dbyt h es h o r t terml i n e a rv i s c o e l a s t i c i t yo f∞n c r e t e, whichi sq u i t ep r o n o u n c e d, andi nt h eh i g hs t r e s s r e g i o n sn e a rt h ef r a c t u r ef r o n t, byt h ea d d i t i o n a ln o n l i n e a . rc r 田 po f∞n c r e t e .F u r t h e r m o r e , t h es t r a i n -s o f t e n i n gbeha . v i o ri nt h ef r a c t u r ep r o c e s sz o n ei sl i k e l yt obee v e nmoret i m e d e p e n d e n tt h 組 t h en o n l i n e a rc r 田 pi nt h eh a r d e n i n gh i g hs t r e s sr a n g e .R e c e n ts t u d i e so f . tv a r i o u sr a t e so fl o a d i n g( B a i a n t飢 dGettu,1 9 8 9 )i n d i c a t eav e r ys t r o n g t h es i z ee f f e c ta s ,田po rs t r e s sr e l 日 以i o ni nt h es t r a i n s o f t e n i n gr a n g ewhichp r e v a i l si nt h ef r a . c t u r ep r o c e s c r z o n e .T h i sphenomenons e e m st or e d u c et h es i z eo ft h ef r配 t u r ep r o c e s sz o n e出 t h el o . a d i n g e a . s e s ), w i t ht h e∞n s e q u 回目 t h a . tl o n g -七i r n r a t ed i m i n i s h e s( i . e .t h et i m et of a i l u r ei n c r e a . st i cf r a c t u r em e c h a n i c st h a ns h o r t t i m er e s p o n s e .T h i s r e s p o n s emightbec l o s e rt ol i n e a re l s 田 n tda . t ao fBaza 瓜 叩 dG e t t u( 1 9 8 9 )i nF i g .・ 7 . 1i nwhicht h er e s u lt i si l l u s t r a t e dw i t hr e 2 . 5 d QO fZO tlf T e r n p eγa t u T e( ・ 0 ZDO :U f ~jg. ~.2 F r a . c t u r eEnergyDependenceonTemp 釘 a t u r eandM o i s t u r e Co n d i t i o n s( a f t e rBa . Z a n t姐 dPr a . t , 1 9 8 8 ) 1 0 6 1 0 7 f o rd i f f e r e n tt i m e st of a i l u r e , p l o t t e da s σN v s . もheb r i t t l e n e s snumbero fB a z a n t( S e c .5 . 1 and5 . 2 ), c a nbes e e nt os r u f tt ot h er i g h ta st h et i m ei n c r e a s e s . a . rf i n d i n g s, i n d i c a t i n gt h a tt h ep r o c e s sz o n es i z e姐 df r a c t u r ee n e r g ydeαeaseおもh e S i m i l t i m et of a i l u r ei n c r e a s e s, ha . veb田 nr e c e n t l yr e p 側 e dbyWittm . a nne ta l .( 1 9 8 7 ), a l t h o u g h e -i n d e p e n d e n tf r a c t u r ec h a . r a c t e r i s t i c s . t h i ss t u d yd i dn o td e a lw i t hs iz L e sti tbep e r c e i v e dt h. a tt h ea f o r e m e n も i o n e df i n d i n g so fBa . z a n ta . n dG e t t uc o n t r a d i c t も h o s eo fJohna . n dShah , ones h o u l dr e a l i z et h a tJohnandShah' smethodd e t e r m i r 悶 b a . a c ho ft h ef r a c t u r ep r o c e s sz o n ewheret h e r ei ss i g n i f i c 姐 td i s p l a c e m e n td u e s i c a l l yt h er e t om i c r o c r a c k i n g .Buta . tt h el o c a t i o n sw h e r et h em i c r o c r a c k sa r eopenedmostw i d e l y , t h e . vea . l r e . a dyr e l a x e dt oz e r o .T h i sp o r t i o no ft h em i c r o c r a c k i n gz o n ewouldn o t s t r e s smayha bema . n i f e s t e di nt h ep r o c e s sz o n es i z eo b t a i n e dbyt h es i z ee f f e c tmethod, whiche v e n t u a l l y 回 s m i t もe d .I ti s g i v e st h el e n g t ho ft h ez o n eo v e rw h i c hs i g n i f i c a n tb r i d g i n gs t r e s s e sa r et r duet os t r e s sr e l a x a t i o n, t h i sl e n g t hr e p r e s e n t so n l yasm a . l l p紅 to ft h e q u i t ec o n c e i v a b l et h a t, l e n g 出 o ft h em i c r o c r a c k i n gz o n e, 回 o b t a i n e dbyJohnandS h a h ' smethod. Morer i g o r oU8 s t u d i e sa r en伺 ded, h o w e v e r . . r e Thee x p e r i m e n t a lt e c h n i q u e sf o rm e a s u r i n gf r a c t u r ep r o p e r t i e sa th i g hl o a d i n gr a t e sa d i s c u s s e dbyJohnandShah( 1 9 8 5 ),Re i n h a r d t( 1 9 8 4 ),M i n d e s s( 1 9 8 4 ),ando t h e r s . 7 . 2E f f e c tofTemperatureandHumidityonFr actureEnergy , と W1 R U一 n 2一 吋, 一 一 T h i se f f e c ti sl i k e l yt oh a v ea tl e a s もp a r t l yt h esamep h y s i c a lmechanisma st h ee f f e c to f c r a c kv e l o c i t yonl o a d i n gr a t e .Ther a t eo fbondr u p t u r e sd e p e n d sn o to n l yont h es もr e s s1 肝 e l b u ta . l s oont e m p e r a t u r e ,a c c o r d i n gt oもhea c t i v a t i o ne n e r g yt h e o r y .U s i n gE q .7 . 1a st h e p o i n to fd e p a r t u r e, B a z a n ta . n dPr a . t( 1 9 8 8 b )ha . veshownt h a t, a c c o r d i n gt ot h ea c t i v a . t i o n e n e r g yt h e o r y , t h ef r a c t u r ee n e r g yo fc o n c r e t es h o u l ddependont h ea b s o l u t etempera もu r e a sf o l l o w s : ( 7 . 2 ) Gf= ~exp(ヱーユ) , ~~nT To " i nwhich九 = r e f e r e nc et e m p e r a t u r e, G~ = f r a c t u r ee n e r g ya . tr e f e r e n c etemper a . t u r e, a n d γ =c o n s t. a n t .E q .7 . 2a g r 田 sw e l lw i t ht 田も r e s u l t so ft h es i z ee f f e c tmethodo b t a i n e df o r v a r i o u stemper a . t u r e sf r o mr ∞m upt o2 0 00C js 田 F i g .7 . 2 . Somer e s u l t sha . vea l s ob e 畑 o b t a i n e df o rt e m p e r a t u r e si nt h ef r 田 z i n gr a n g e, i n d i c a t i n gt h et r e n di nF i g .7 . 2t oc o n t i n u e o downt oa b o u t-20 C,叩dt h e nr e v e r s i n g( M a t u r a . n a .e ta . l . ,1 9 8 8 ) . As i m i l a . rd e p e n d e n c eo fG onTwase x p e r i m e n ta . l l yd e t e r m i n e dbyBrameshuber( 1 9 8 8, 1 9 8 9 )who, h o w e v e r, u s e dd i f f e r e n tf o r m u l a . s もod e s c r i b et h eo b s e r v e dt r e n d . 鉦 ' e c tt e s t sw e r ec o n d u c t e da . tv a r i o u st e m p e r a t u r e sons . a t u ・ Thea f o r e m e n t i o n e ds i z ee r a t e d( w e t )∞n c r e t e鎚 w e l la son∞n c r e t ew h i c hwasd r i e db e f o r et h et e st .Thed i f f e r e n c e d i t i o n sappe a . r e di n s i g n i f i c a n ta troomtemper a . t u r e, however , a th i g h旬 mi nh u m i d i t y∞n p e r a t u r e st h ew e tc o n c r e t eh. a damuchl o w e rf r a c t u r ee n e r g yt h a nもhedryc o n c r e t e( F i g .7 . 2 Z 必 ね a r 叫 a ndPr 悶a 抗 , ム A1 ω 9 邸 8 8 吋 b ) .Assumingl i n e a rd e p e n d e n c eOst h es p e c i f i cw a t e rc o n t e n t , a f t e rBa p a r a m e t e r" 1wa . se x p r e s s e da s ( 7 . 3 ) " I 1ー γ 0 ) γ=γ0+( i nw h i c hω =s p e c i f i cw a t e rc o n t e n t,Wl i t sv a l u ea ts a t u r a t i o n ,andγ 0,γ 1= v a l u eo fγ 0 i ndryandwetc o n d i t i o n s( f o rもhe∞n c r e t e同 t e d, i tw硝 f o u n dt h a t" 1 0~ 6 0 0 [ (, 組d 1 1~ 1 9 0 0 0[ ( ) . i nwruchv i s c o e l a s t i c Thee f f e c to ft e m p e r a t u r ei sq u i t ep r o n o u n c e di np o l y m e rc o n c r e t e s, i t yo ft h epolymerb i n d e ri snod o u b ta . ni m p o r ta . n tf a c t o r( V i p u l 姐a.n danandDh a . r m a r a j a n, 1 9 8 7, 1 9 8 8, 1 9 8 9 a 丸c ) . 7 . 3E f f e c tofCyclicLoading R β p e a t e dl o a d i n gt e n d sもog r a . dua . l l yi n c r e a s et h ec r a c kl e n g t he v e ni ft h emaximum s t r e s si n t e n s i t yf a c t o ro ft h ec y c l e si sw e l lb e l o wi t sc r i t i c a lva . l uef o rs t a t i cl o a d i n g . The r a . t eo fi n α e a s eo ft h ec r a c kl e n g t h ,a, w i t ht h enumbero fcyd , 四 N, h a sb e e nshownf o r . p p r o x i m a t e l yd e s c r i b e dbyt h eP a r i slaw( P a r i sa . ndErdogan, 1 9 6 3 ) : manym a t e r i a . l st obea da/dN= C(! : J .Kd [ ( I c ) r ヘinwhichC andm areempirica . lc o n s t a n t s . AlthoughE q .7 . 4i s .f u n d a m e n t a lp h y s i c a . lb a s i ss u c ha sE q .7 . 1, i ti s e s s e n t i a l l ye m p i r i c a landd o e sn o thavea n e v e r もh e l e s s組 e q u a t i o no fr a t h e rg e n e r a la p p l i c a . b i l i t y .F o rc o n c r e t e , h o w e v e r, i tw剖 f o u n d ( Ba.Z叩t組 dXu, 1 9 8 9 )t h a td e v i a t i o n sf r o mP a r i sl a wa r i s eduet ot h es i z ee f f e c t .Theyc 姐 b ed 脱 出e dbyt h ef o l l o w i n gg e n e r a l i 叫 i o n(B a.Z姐t 叩 dXu , 1 9 8 9 ) : 表= c ( 2 5 ) m, 出 i t h[ ( 1 m= ι ( r / 2 ( 7 . 4 ) w h e r e( 1 0i sa .c o n s t a n th a v i n gas i m i l a rmeaningt od nt h es i z ee f f e c tl a . w( E q .1 .4 ), [ ( 1 mi st h e oi a . r e n t( s i z e d e p e n d e n t )f r a c t u r et o u g h n e s sc a . l c u l a t e dfromt h epeakl o a d~f t h es p e c i m 佃 a p p 組 dL EFMr e l 叫i o n s,叩ddi s山 es i z eo ft h es p e c i m e n .T e s tshowedt h a td 0 d hen ow o~ 1 t h e1 0 0 0i sc y c l e dbetwe 回 O阻 d8 0%o f出 emaximuml o a df o rm o n o t o n i cl o a d i n g .E q .7 . 4 , s h o w ni nF i g .7 . 3byt h et h r e es もr a 抱h tl i n e s , i ss e e nt oa g r e eq u i t ew e l lw i t ht h eda . t . af o r トp o i n t b e n dn o t c h e dbeamso fs i z e s1 : 2 : 4 . P. a r i sl a w, byc o n t r a s t, y i e l d si nF i g .7 . 3a . t h r e e s i n g l es t r a i g h tl i n e. a ndc a n n o td e s c r i b et h eo b s e r v e ds i z 氾 e f f e c t . a b o u tP a r i sl a wi s出 a tt h er a t eo fc r a ι ke x t e n s i o nd o e sno もd epend Ani n t e r e s t i n gp o i n t. onぬ. eu p p e ra . n dl o w e rl i m i t i n gv a l u e so ft h es t r e s si n t e n s i t yf a c t o r, b u ta . pproxim a . t e l yo n l y 四 回, ! : J . J ( 1・T h i smigh もb ee x p l a . i n e dbya .8 0 r to fa .shakedowni nt h ef r a c t u r e O Dt h e i rd i f f e r , w i t hd e v e l o p m e n to fr e s i d u a ls t r e s s e sa f t e ranumbero fc y c l e s .V a r i o u so t h e r p r o c e s sz o n e i n t e r e s t i n gr 回 u l t sonc y c l i cf r a c t u r e' o fc o n c r e t e , w r u c hp e r t a i nt op a r t i c u l a rt 回 t c o n d i t i o n s, h a . v eb e e no b t a i n e d ,f o rex a . mple ,bySwartz,Hu相 dJ O l 悶 ( 1 9 7 8 ),S w a r t z ,Hu組 ga . n dHu ( 19 8 2 ), S w a r t zandGo(1984)~ P e r c U k a r i s組 dCalomino( 1 9 8 7 ), ando t h e r s . Thei n t e r p r e t a t i o no ft h e企 Krvaluewhich8 h o u l dbes u b s t i t u t e di nE q .7. 4i ss u b j e c tt o ぬ~e q u e s t i o nw i t hr e g a r dt ot h es i z ee f f e c tont h ev a l u eo fK l c .Thev a l u eo f! : J .Kls h o u l dbe c a l c u l a t e df o rt h ee l a s t i c a . ll ye q u i v a l e n te f f e c t i v ec r a c kl e n g t ha= ao+c, whichmeans! : J .[(l s h o u l dbem o d i f i e dw i t hr e s p e c もt ot h el e n g t ho fc r a c ke x t e n s i o nf r o Int h en o t c h( F i g .4 . 2 )・ 108 1 0 9 ロ d .1 . ! 5i n . . 句 C_ 1 6 . 7 ・ 1 . 0{・ d. . 3i n . , 同c . ・ 1 8 . 2 +“叫川 n-10. 6 - Chapter8 .EFFECTOFREINFORCE 乱 1 :ENT 2 . 0 2 司 ち. 1E f f e c tofReinforcingS t e e lBars e 乏-3 . 0 。 明 向: 4 . 0 5 . 0 t . 1 1 .3 1 . 7 1 .5 109(dKI / KI f ) F i g . 7 . 3F r a c t u r eo fT h r e e トP o i n tBendSpecim 租 s unde rC y c l i cL o a d i n g ( 0 ) ( b ) .0 II I f I I ( c ) : 1 L .一 =一6一 1-P a-U1-p+YEP ・ 6 Whenc o o c r e t ei sr e i n f o r 伺 db yar e g u l a rg r i do fr e i n f o r c i n gb a r s, i ti so f t e na p p r o p r ia . t e . rgridωoneh e t e r o g e n o u sc o m p o s i t ema . t e r i a . lwhichi s ιc o n s i d e rc o n c r e t ew i t ht h eba 辻Cf c r h o m o g 回 目USo nal a r g e rs c a l e .O b v i o u s l y , i ns u c hac o m p o s i t e , t h eb a r si n h i b i tt h e r o p a g a t i o no fc r a c k s .I ft h e r ew e r enobonds l i p, t h e nt h eo p p o s i t ef a c e so f r m a t i o n回 dp , : r a c kw o u l dn o tbea b l eもoo p e na p a r t, ands ot h e r ewouldbenof r a c t u r eu n t i lt h eb a r f P . a k s .T h i ss i m p l ec o n s i d e r a t i o ni m m e d i a t e l yc l a r i f i e st ha . tbonds l i pmustt a k ep l a c eand 凶 t b et a k e ni n t oa c c o u n t . Thep r o p e r t yw h i c hm抗 t e r si st h es t i f f n e s so ft h ec o n n e c t i o n 関 1 ba . r sb e t w e e nt h eo p p o s i t ef a c e so ft h ec r a c ko rc r a . c kb a n d . u v i d e dbyt h es t A s s u m i n gt h eu l t i m a t ebonds t r e s s e sUb t obea p p r o x i m a t e l yc o n s t a n t, t h eb a rs t r e s s, . ' 1 i c hi se q u a lt o山 叫 t h ec r a c kc r o s s i n g, d e c r e ωe sl i n e a r l yo v e rt h ed i s t a n c eo fbonds l i p , , _o ne a c hs i d eo ft h ec r a . c k . Fromt h i ss t r e s sd i s t r i b u t i o n, o n ec a ne s t i m a t et h er e l a t i v e . ra g a i n s tc o n c r e t eo v e rl e n g t h2 L . .F u r t h e r m o r e , i na .ma . c r o s ∞p i c a l l y p l a c e m e n to ft h eba ' I l f o r mo rq u a s i u n i f o r mf i e l d, t h ef o r c ec a r r i e dbyt h eba . ra tt h ec r a c kc r o s s i n gmust o r c ep e rb a rc a . r r i e dj o i n t l ybyc o n c r e t e組 ds t e e la tl o c a . t i o n sbeyondt h e u i l i b r a t e七hef , n ds l i p1 田 g t h .F r omt h e s ec o n d i t i o n si tf o l l o w st h a t( B a z 姐 t andC e d o l i n, 1 9 8 0 ;BaZa n t ; : l i c ) : l' I F i g .8 . 1E f f e c to fBondS l i po fRe i n f o r c i o gB a r sC r o s s i n gC r a c kBands o rC r a c k s S ( 8 . 1 ) " " h i c hp=s t e e lr a t i o , n士 s t e e l t o c o n c r e t er a t i oo ft h ee l a s t i cm o d u l i, andA6=b a rc r o s s もee l e m e n ta n a l y s i s, onemayf u r t h e ri m a g i n eane q u i v a l e n t ' o i o n .F o rt h ep u r p o s eo ff i n i w h i c hi s叩 c h o r e dt oc o n c r e t ea tt h en o d e so ft h ef i n i t ee l e m e n tmeshandh a snobond "'",~es b u tas h o r t e re q u i v a l e n tf r e e -s l i pl e n g t h .Anothers i m p l ef o r m u l af o rt h i sl e n g t hh剖 .四 d e v i s e dbyBaZa n tandC e d o l i n( 1 9 8 0 ) . ¥d e t a . i l e dd e s c r i p t i o no fc r a c k i n gi nr e i n f o r c e dc o n c r e t er e q u i r 田 am ores o p h i s t i c a t e d , : . 'y ;S白 e . g .P i j a u d i 位ー C abotandMa . z a r s( 1 9 8 9 ), andB r e y s s eandMazars( 1 9 8 8 ) .These 回 t i cmodelf o rs t e e lb a r s . l i e su s e ddamaget h e o r yf o rc o n c r e t ecombinedw i t hane l a s t o p l r e s u l t sw e r ei n t e r e s t i n g, g i v i n gi n f o r m a t i o nont h ee v o l u t i o no fdamagez o n 回a.r ound C ' , 8 .d e v e l o p m e n to fc r a c k sandt h eg l o b a lr e s p o n s eo ft h es t r u c t u r e . Bonds t r e s s e s姐 d 民dt obea l s oi n c l u d e di nad e t a i l e da n a l y s i s ;t h i sh a sb e e ndonebyP i j a u d i e r Cabot i l 9 8 9 )i nt e r m so fan o n l o c a ldamagemodelw h i c ha l s oy i e l d st h es i z ee f f e c t . ~ F r a c t u r ei nFiber-ReinforcedConcrete ~ ~,nd ~lip o !t h ef i b e r si saphenomenons i m i l a rt ot h es l i po fr e i n f o r c i n gb a r s, anda g a . i n m a j o re f f e c tonf r a c t u r e, a l o n gw i t ht h ephenomenono ff i b e rpull-~ut duet os h o r t , . n t i a l l yu s e f u li m p r o v e m e n t si nt h em e c h a n i c a lb e h a v i o ro fもe n s i o n -weakc o n c r e t e "c a nbee f f e c t e dbyt h ei n c o r p o r a t i o no ff i b e r s .S i m i l a rt o七heb e h a v i o ro fp l a i n 1 1 1 110 c o n c r e t e, ∞mpositef , 品 . il u r eu n d e rmostt y p e so fl o a d i n gi si n i t i a t e dbyt h et e n s i l ec r a c k i n go f . t r i xa J ongp l a . n e swheret h enorm a Jt e n s i l es t r a . i n se x c e e dt h ec o r r e s p o n d i n gp e r m i s s i b l e t h ema l I owedbym u l t i p l ec r a c k i n go ft h em抗 r i xp r i o rt oc o m p o s i t ef r a . c t u r e . v a l u e s .T h i smaybef o i ft h evolumef r a c t i o no ff i b e r sa 同 副 館c i e n t l yh i g h( o ri ft h ef i b e r sa r e∞n t i n u o u s )(A v e s t O D e ta J . , 1 9 7 1 ) . However,whens h o r tf i b e r sa r eu s e d( s t 田1 ,p o l y p r o p y l e n e,g l 剖 5 ,e t c . ),o n 句 t h emat r Ixh a . sc r a c k e d , o n eo ft h ef o l l o w i n gt y p e so ff a . il u r ew i l lo c c u r : 。 。 。 " “ 岡 田 1 1 0 2 ' 単 白 l 伽 同E e , ( a )Thec o m p o s i t ef r a c t u r e si m m e d i a t e l ya f t e rm a t r i xc r a . c k i n g .T h i sr e s u l t sfromi n 叫 q u a t ef i b e rc o n t e n ta tt h ec r i t ic a . ls e c t i o no ri n s u f f i c i e n tf i b e rl e n g t h st ot r 姐 s f i 釘 8 t 陶鞠 a c r o s s仙 em a t r i xc r a c k . ( b )A l t h o u g ht h emaximum1 0 0 0onもhe∞mposi 匂 i sn o ts i g n i f i c a n t l yd i f f e r e n tfromt h a to f t h em a t r i xa J one , t h ec o m p o s i t e∞n t i n u e s t oc 紅 r yd e c r 田 8 i n gl o a d sa f t e rt h ep e a k .T h e p o s t c r a c k i n gr e s i s t a n c ei sp r i m a r i l ya t t 伽 t e dt og r a . du a J( r a t h e r伽.nB u d d佃)f i b e r p u l l o u ta . ss o c i a . t e dw i t hc r a ε kb r i d g i n g . Whilenos i g n i f i c a n ti n c r e a . sei nc o m p ω i t e ,thecompositefractureenergyandtoughnessa . r ec o n s i d e r a b l y s t r e n g t hi so b s e r v e d e n h a n c e d( F i g .8 . 2 )(Gop a J a r a t n. a m andSh. a h, 1 9 8 7 a ) . ( c ) Evena . ! t e rm a t r i xc r a c k i n g ,t h ec o m p o s i t ec o n t i n u e sもoc a . r r yi n c r e a s i n g1 0 0 0 5 .T h e a . r r y i n gc a . p a c i t yo ft h ec o m p o s i t eandt h ec o r r e s p o n d i n gd e f o r m a . t i o na r e p e a kl o a d -c s i g n i f i c a n t l yg r e a t e rt h a nt h a to ft h eu n r e i n f o r c e dm a t r i x .D u r i n gt h epr e -p e a ki o p r o g r 飽 s i v ed e b o n d i n g6 ft h ei n t e r f a c e組 d e l a s t i cr e g i m eo ft h ec o m p o s i t er e s p o n s e, d i s t r i b u t e dm i c r o c r a c k si nt h em a t r i xm乱yber e s p o n s i b l ef o rt h ee n e r g yd i s s i p a t i o D 側 (Mobasher, C a s t r o -MonteroandShah, 1 9 8 9 ) .I ti sc l e a rt h a t出 i smodeo f p r o c ∞mpositef a i l u r er e s u l t si nt h ee f f i c i e n tu s eo fb o t ht h ec o n s t i t u e n t s(Mob a . s h e r叩 d Shah,1 9 8 9 ;S t a n g ,M obasherandShah,1 9 8 9 ;Mobasher ,S t a n gandShah,1 9 8 9 ) . 1 f 6 . u sl'l倒閣 。 ' 崎司自 圃圃圃目.占申司曙司 l C I O O 規制 1 0 1 > > 。 10伺 DOPUCI 闇 調T , 6 (悼制 F i g .8 . 2T y p i c a . lR怠s u l t ao fS t r e s s D i s p l a c e m e n tC u r v e sO b t a i o e dFrom D i r e c tT e n s i o oT e s t s00P l a i nMort a . rM a t r i xa . n dS同 1F i b e rRe i o f o r c e dC o n c r e t e 1 3 2 L_ ' t t , . . 丁lsll 一 , AL J血 0 w向断固 ム TI--ー a J y t i c a lm o d e l su s e df o rt h ep r e d i c t i o no f出 em e c h a n i c a lb e h a v i o ro ff i b r o u scompωAn i t e sma . ybes u i t a b l ef o ro n eo rmoreo ft h ea b o v et y p e so ff a . il u r em e c h a n i s m s . Bas e < ii o pa . r tont h efundament a Ja p p r o a c hi nt h e i rf o r m u l a t i o n, a n a l y t i c a . lm o d e l sc 姐 be日 匂 伊 出 e d( G o p a l a r a t n 制 副 S hah,1 9 8 7 a )剖: m o d e l sb a s e dont h et h 悶 yo fm u l t i p l efra.c~ t u r e,c o m p o s i t em o d e l s,s t r a i n r e l i e fm o d e l s,f r a ι t u r em e c h a n i c Bm o d e l s,i n t e r f . a c emech 回 i c s a . n dm i c r o m e c h a n i c sm o d e l s .Fa . ir l y e x h a u s もi v er e v i e w so f t h e s em o d e l sa r ea v a . i l . a b l e m o d e l s, e l s e w h e r e( G o p a l a . r atnamandShah, 1 9 8 7a . ;M i n d e s s, 1 9 8 3 ) .B r i e fr e v i e w so ft h ef r a . c t u r e m配 h組 問 ba . s edm o d e l sandt h ei n t e r f a c em e c h a n i c sm o d e l sa r eg i v e nh e r e出 thωea . r e t y p i c a l l yt h emωts u i t a b l ef o rm o d e l i n gもhei n e la . s t i cp r o c e s 5 田 i ns h o r t f i b e r∞m p o s i t e s . Twob r o a dc a t e g o r i e so fm o d e l sc a nbei d e n t i f i e df r o mamongt h ef r a c t u r emech a . n i c s ba . s edm o d e l s . Themorefundamenta . lc l a s so fm o d e l su s et h ec o n 田p t so fLEFMt os o l v e t h eproblemo fc r a c ki n i t i. a t i o n, g r o w t h , a r r e s もa nds ta . b i l i t yi nt h ep r e s e n 伺 o ff i b e r st h r o u g b a J d iandB a t s o n, 1 9 6 3 ;Ro m叫 d i a p p r o p r i a t ec h a n g e si nt h es t r e s si n t e n s i t yf a c t o r(Romu andMandel, 1 9 6 4 ) .T y p i c a l l yt h 偲 em o d e l sa . s sumep e r f e c tbondbetweent h ef i b e r組 dt h e m a t r i x, anda r eon e p a r a m e t e rf r a c t u r em o d e l s .U n l i k et h ec l a s s i c a . lLEFMm o d e l s 自 ,omeo! t h el a t e rm o d e l si m p l i c i t l ya c c o u n tf o rt h ei n e l a s t i ci n t e r f a c er e s p o n s ed u r i n gc r a c kg r o w t b c e s 唱 ha .n o n l i n e a r伽 e s s -d i s p l a c e m e n tr e l a t i o n s h i pf o rt h ef i b e r b r i d g i n gz o n e( p r o t h r o z o n e ) .T h i sa p p r o a c h , w h i c hh a s∞met obeknown舗も hef i c t i t i o u sc r a c kmodel( H i l l e r b 叫 1 9 8 0 ), i s∞n c e p t u a l l ys i m i l a rt oもh a td e s c r i b e di nS e c .3 . 1f o rt h ef r a c t u r eo fu n r e i n f o r c e l l c o n C l ' 叫e .Themajord i f f e r e n c 届 i nt h ef i c t i t i o u sc r a c km o d e l s( H i l l e r b o r g, 1 9 8 0 ;P e t e r s 似 ・ ' 1 0 一一一 トー- 0 - " - - - - 0 -4 脚 . . . . . . . . . . . I 脚山山~-l hF W ~ig. 9 . 1A r r e s to fP a r a l l e 1CrackDuet oI n s t a b i l i t yandChangeo f Sp a . c i n gf o rL e a d i n gCr a . c k s 1 1 2 1 1 3 1 9 8 0 ;WecharatanaandShah, 1 9 8 3 b jV i s a l v a n i c handNaaman, 1 9 8 2 )a r et h es i 碍u l a r i T . h ec 即 k t i p, t h ec r i t e r i au s e df o rc r a c ki n i t i a t i o nandgrow もh , 組dth~ a s s u m p t i o n s前 t s t a b i l i t yo ft h ec r a c kg r o w t h .Morer e c e n t l y , JenqandShah( 1 9 8 6 )havep r o p o s e daf r a c t u l , f m e c h a n i c smodelt op r e d i c tt h ec r a c kp r o p a g a t i o nr e s i s t a n c eo ff i b e rr e i n f o r c e dc o n c r e t et h a r p p r o 配 h e s .F r a c t u r er e s i s t a n c ei nf i b r o 思 i ssomewhatd i f f e r e n tfrome i t h e ro ft h e s eもwoa o o m p o s i t e sa c c o r d i n g七ot h i smodeli ss e p a r a t e di n t of o u rr e g i m e sw h i c hi n c l u d e :l i n e a l e g i n n i n g e l a s t i cb e h a v i o ro ft h ec o m p o s i t e js u b c r i t i c a lc r a c kg r o w t hI nt h em a t r i x組 dもheb e rb r i d g i n ge f f e c t jp o s t c r i t i c a lc r a c kg r o w t hi nt h em a t r i xs u c ht h a tt h en e t8 t ! 馳 o ft h e五b i n t e n s i t yf a c t o rduet ot h e乱p p l i e dl o a d叩 dt h ef i b e rb r i d g i n gs t r e s s e sr e m a i nc o n s t a n l 酔 w h e r et h er e s i s t a n c et oc r a ι ks e pa . r a t i o ni s ( s t e a d ys t a t ec r a c kg r o w t h ) ;andt h ef i n a ls t a p r o v i d e de x d u s i v e l ybyt h ef i b e r s . Themode1u s e stwop a r a m e t e r st od e s c r i b et h em a t r i x f r a c t u r ep r o p e r t i e s( K I c 'm o d i f i e dc r i t i c a 1s t r e s si n t e n s i t yf a c t o rb a s e donLEFMa n dt h e anddCTOD, t h ec r i t i c a lc r a c kt i po p e n i n gd i s p 1 a c e m e n t, 掛 described e f f e c t i v ec r a c kl e n g t h, andaf i b e rp u l l o u ts t r e 砕 c r a c k w i d 七hr e 1 a t i o n s 1 中 ωthe e a r l i e rf o ru n r e i n f o r c e dc o n c r e t e ), 品i ci n p u ti n f o r m a t i o n b a Ouyang叩 dShah( 1 9 8 9 )r e c e n t l yd e v e l o p e danR c u r v ea p p r o a c hf o rf r a c t u r e Mobasher, -curvecanbeuniquel yd e t e r . o fc e m e n t b a s e df i b e rc o m p o s i t e s . Thep a r a m e t e r so ft h eR s u c ha sK I candOCTOD d e f i n e dbyJenqandS h a h mineda c c o r d i n gt om a t e r i a lp r o p e r t i e s, e r si si n ∞rpor叫 edintotheR-curvebyf i b e r ( 1 9 8 6 ) .T o u g h e n i n go ft h em a t r i xbyt h e伍b n gp r e s s u r ei nt h ee q u i l i b r i u mc o n d i t i o n sd u r i n g b r i d g i n gp r e s s u r e _Byi n c o r p o r a t i n gt h ec l o日i , もh eo n s e to fi n s もa b i l i t yo fm a t r i xi sshownt obed e p e n d e n to n t h es t a b l ec r a c kp r o p a g a t i o n t h ef i b e r s . A l lo ft h ef i c t i t i o u sc r a c km o d e l sr e l yont h es t r e s s c r a c k w i d t hr e l a t i o n so b t a i n e de x p e r . i m e n t a l l y .Thereh a v eb e e nsomea t t e m p t sa tp r e d i c t i n gt h em a c r o s c o p i cs t r e s s c r a c k w i d t h e r m a t r i xi n t e r f a c e .T h e y r e l a t i o n so ft h ec o m p o s i t efromas t u d yo ft h em e c h a n i c so ft h e五b C叩 b eg r o u p e d剖 m o d e l sb a s e dont h es h ea . r 1 a g山 田ryo rm o d i f i c a t i o n st 1 悶 ωf 仏 ( La 刑w r e 凹E 町r α 悶 s 止 ;Lawse ta , . l1972jGopa1aratnamandShah, 1 9 8 7 b ;G o p a l a r a t n 如 1a ndCheng , 1 9 8 8 ) , 1 9 7 2 f r a c t u r em e c h a n i c sb a s e di n t e r f a c em o d e l s( S t a n gandShah,1 9 8 6 ;M o r r i s o ne ta l 1 9 8 8 ) 1 9 8 8 ;S a h u d i n, 1 9 8 7 ) . Manyo ft h e s em o d e l sh a v e andn u m e r i c a lmodels( M o r r i s o ne ta l ., be ens u c c e s s f u lt ov a r y i n gd e g r e e si np r e d i c t i n gt h ep e a kp u l l o u tl o a d sandt h el o a d s i l p 叩 もr e m e n d o u s 1 y r e s p o n s eo fi d e a 1 i z e da l i g n e ds i n g 1 ef i b e rp u l l o u t .T h e s em o d e l sh a v ebe も もh ei n t e r f a c eands h o w i n g u s e f u li nu n d e r s t a n d i n gt h eb a s i cm e c h a n i c so fs t r e s st r a n s f e ra もa n tr o 1 ei nt h ef r a c t u r eo fs u c hc o m p o s i t e s .S i g n i f . t h a ti n t e r f a c i a ld e b o n d i n gp l a y sanimpor how 巴v e r, ben e e d e dt omodifyt h e s emodelst op r e d i c t出 ep u l l o u t i c a n tr e s e a r c he f f o r t sw i l l, e ' l s c h a r a c t e r i s t i c so fi n c l i n e df i b e r st h a ta r erandomlyo r i e n t e da tam a t r i xc r a c k(random n 町 吋 也 d 袖 lme i nb o t ht h ea n g u l a ro r i e n t a t i o na sw e l la st h e巴mbe ね a 釘 加時 I 乱姐 ndShah( 1 9 8 9 )p r o p o s e dadamagemodelf o rt h es t u d yo fd i 批 i b u t e dm i c r o c S t p o s i t es u b j e c もe dもou n i a x i a lt e n s i o nw出 i n i t i a l l yd e r i v e d r a c k i n g .Thec o m p 1 i a n c eo ft h e∞m a c c o r d i n gt ot h es h e a rl a gt h e o r y . A damagev a r i a b 1 e, whichi saf u n c t i o no fc r a c ks p a c r a c ks e c t i o n,wasi n t r o d u c e d .A f t e rs p e c i f y i n gt h e i n gandf i b e rd e b o n d i n gl e n g t ha t山 ec t h es t r e s sands t r a i nr e s p o n s eo ff i b e rr e i n f o r c e d damagee v o l u t i o nl a wbyadamages u r f a c e, c o m p o s i t e ss u b j e c t e dt ou n i a x i a lt e n s i o nc a nb ec a l c u 1 a t e d . 吋 a sa l s ob e e ns t u d i e df o rg l a s s f i b e rpolymer( p o l y e s t e r )c o n c r e t e .Suchm a t e r i ιl s F r a c t U I a . k1 o a d, 組dもheR c u r v e s showc o n s i d e r a b 1 en o n 1 i n e a r i t yands t a b l ec r a c kgrowthp r i o rt ope , 1 9 8 7, 1 9 8 8 . t h a thaveb e e nm e a s u r e dr e f l e c tt h i sb e h a v i o r( V i p u l a n a n d a n組 dDharmarajan ムc ) . 1 9 8 9 a 吋 恥1 S ! h a p t e r9 . CRACKSYSTE l nc o n t r a s もも om e t a l l i cs t r u c t u r e s , ∞ncretestructure so o n t a i nnumerousl a r g 己cracks l o s eenought oi n もe r a c t .T h e r e f o r e, u n d e r s t a n d i n go fc r a c ki n t e r a c t i o ni smore h i c h釘 ec : 1 ; l p O 巾 n tt h a ni ti sf o rm e t a l l i cs t r u c t u r e s . { ' . lResponseofStructureswithI n t e r a c t i n gGrowingCracks C o n s i d e ra抗 r u c t u r ew i 出 c r a c k so fl e n g t h sα j ( i= 1, 2 . . .n )l o a d e dbyas ystemo ff o r c e s b o u n d 紅 yd i s p l a c e m e n t s, s h r i n k a g eo rもherma ld i l a t i o np r o p o r t i o n a lt opa . r ame t e rA.The 目e r g y( u n d e rI s o t h e r m a lc o n d i t i o n s, morep~ecÎs~ly _ t h _ eHel~oltz. fr~e e~ergy) o ft h e山 u c a . ls y s t e mi st h esumo f出 es t r a i ne n e r g yU, whichd e p e n d sont h el o a dp a r a m e t e r舗 w e l l I l r 、t h ec r a c kl e n g t h s, andt h ee n e r g yn e e d e dt op r o d u c et h ec r a c k s , i . e ., ! " F= U( a l, 川)+午 1 4 iR(αDぬ ; ( 9 . 1) p e c i 五ce n e r g y( p e ru n i tc r a c kl e n gt h )r e q u i r e df o rc r a c kg r o w t handα~ a r et h e L I .w h i c hR=s nt h ec 掛 e o fl i n e a re l a s t i cf r a c t u r em e c h a n i c s , R = G/ . r a c kl e n g t hv a l u e sbetween0andαt・I •C h ee q u i l i b r i u mc o n d i t i o no ft h es y s t e mi soF= E( U, i + R ( a i ) ) o a i=0f o r組 yv a r i a t i o no a j, h i c hi m p l i e s出 a t-U 刊 =R ( a i ) 'T h i si st h ew e l l k n o w nc o n d i t i o no fc r a c kp r o p a g a t i o n 七t h eb e g i n n i n g .Thes t a b i 1 i t yo ft h es t r u c t u r ewit ht h ec r a c k sr e q u i r e st h a t I r e a d ys t a t e da 2 a . r i a t i o no F bep o s i t i v ef o ranya d m i s s i bl eo a j(Ba 量姐ta ndOhtsubo, 1 9 7 7 , h es e c o n dv l f7 8 ;lJoUIlUl tandWahab, 1 9 7 9 ) .Fr omE q .9 . 1, wehave o R ( a i ) 2 2 ; _ U , i j O a i O α j+~一一一 o F= E rE; 2 ~ "J, J • ' 1 o a i (6d = H- AJMαj r ( 9 . 2 ) , 1: .w h i c hAi q u a r em a t r i x jformas Ai j+2R , i d i j H ( o a i ) j= U, ( 9 . 3 ) I , j" f et h es u b s c r i p t sf o l l o w i n ga∞mmar e p r e s e n tp a r t i a ld e r i v a t i v e sw i t hr e s p e c もt oa i, =lifi=j叩 d0i fi= Fj ;andH ( o a i )= 1i fo a jと 0, and0o t h e r w i s e . No t et h a t , r, 'j= Kl/E'whereKi= s t r e s si n t e n s i 勿f a c t o ro fc r a c ka i( a n ymode ) , 組d -E'UiJ= . 日 /(i, j= 2KjKj , i ' . ¥ sもh e1 0 a d i n gpa . r a m e t e rAi n c r e 回邸, t h ec r a c k sg r o w . Ata民 r t a i nc r a c kl e n gt hi tc a n i・ 岬 e nt h a tもh e r ea r emoret h a no n es o l u t i o nf o rt h ec r a c ki n c r ements, i . e .,t h ep a t ho f l o t t e di nt h es p a 田 w i t hc o o r d i n a t e sa ib i f u r c a t e s .I tcanbeshownt ha tt h e ' : . ~ystem p Ll r c a t i o ncond i t i o ni s detA ( 9. 4 ) i j= 0 1 ¥ i" rb i f u r c a t i o n , t h epa 出 w hichi sf o l l o w e di st h a tf o rw h i c h(Ba . Z a叫 2F =m F o rd i s p l a c e m e n to o n t r o 1 :o in 2 F o rl o a dc o n t r o l : oF = max ( 9 . 5) 114 (o) 5 6 2 115 o . . 吋 , . . . . . - T h i sc o n d i t i o ni sd e r i v e df r o mt h ef a c tt h a tt h ep a t hwhicho c c u r smustbet h a tw i t h出e ma . x imumi n t e r n a le n t r o p yi n c r e m e n t . 田n t l y ,Bazant(1989b)組 dBaz叩 tandTabbara .( 1 9 8 9 )h a v eshownもh a tt h e Morer e t a n g e n t i a l叫 i f f n e s sr n a t r i xc ∞t a i n i n gN i n t e r a c t i n gc r a . c k so fl e n g t h sa j ( i= =1, . . .N)i s 以 . 1 w h e r e S ~14 1 . 匂 司H α, 自国 = s= σ冴'X~ q = E' /5x_ 、 1 1 1o , , !r-‘唱『 於 :11 ・ ' " 3 5 ・ ーーーー O . S . . ・ ・ ・ . . - . . I I ; " ' t ) - ' . o u Q = B q 4 1, 0 F i g . 9 . 3D i s c r e t eC r a c k i n gi nF r a c t u r eP r o c e s sZ o n e岨 dT h ω r e t i c a l S t r e s s D i s p l a c e m e n tDiagramf o rC r a c kC伺 J e s 伺 : n c e Thep r e c e d i n gf o r m u l a t i o nh a s姐 i l l t e r e s t i n ga p p l i c a t i o nt os y s t e m so fp a r a l l e lc r a c k s c a u s e d, e . g ., byd r y i n gs h r i n k a g eo rc o o l i n gs t r 回s e s, o rbyb e n d i n g .I nt h ei d e a l i z e dc 出 e o f amω8ivew a l lt r e a t e da sah a l f s p a c e, t h es h r i n k a g ec r a c k s8 t a r ta tsomec l o s es p a c i n gsand ' , h e n , a st h ed r y i n gf r o n ta d v a n c e si n t ot h ew a l l , t h e yc o n t i n u et ogrowi n w a r d .C a l c u l a t i o n s も embe ω , m esu n s t a b l ewhent h ec r a c kl e n g t h o nt h eb a s i so fE q .9 . 2h a v eshownt h a tt h es y s r e a c h e st h ev a l u ea p p r o x i m a t e l y向 =1 .7 8, a twhichp o i n tt h ed r y i n gf r o n ti sa p p r o x i m a t e l y a tt h ed e p t h2 . 6 8( Ba . Z a n tandOhtsubo, 1 9 7 7 ;B話 回t , Ohtsubo組 dAoh ,1979;BaZant姐 d ~ahab, 1 9 7 9 ) .AUhisp o i n tt h e r ei sab i f u rt i o no ft h ee q u i l i b r i u mp a t hi nt h es p a c eo fa j . r o w i n gandc l o s e s , k e e p i ngt h ei~ngth a 2, ~hile t h ei n t e r m e d i a t e E v e r yo t h e r~rack stop~ g : . r &c k so fl e n g t hα jc o nt in u et og r o w, t h e i rs p a c i n gd o u b l e dt o2 s .Thea r r e s to ft h eg r o w t h si sa g a i nr e p e a t e dl a t e r ,andωon.This I md.c l o s i n go fe v e r yo t h e ro ft h el e a d i n gc r&ck &c ks y s t e m sshowni nF i g .9 .1 . ' ) r o d u c e st h ec r . Thes p a c i n go ft h ec r a c k s, 8 , i simpo~tant f o rd e t e r m i n i n gt h e i rw i d t h .Asi sw e l lknown, ti sd e s i r a b l et ok田 pt h ec r a c kw i d t hl e s st h a na b o u tO.lmmt oO . 4 m r n ,mainlybecause n a r r o wc r a c k sa r en o tr e a l l yc o n t i n u o u sanddo110も s e r v ea sgoodc o n d u i t sf o rm o i s t u r eo r c a : 、、 ( , . , r 7 . U " . ' 0.0 0 . 0 n = 9 . 2I n t e r a c t i n gP a r a l l e lCracks ・' . ' 1 T . Ir ;I 1 . 0 = 雷 同 =i n v e r s eo ft h em a t r i x φk m, i nwhich m=伊 F/ θa k 偽情 + d k md凡n /dαmiand φk 九 ‘( a g i v e nR c u r v eo ft h em-thc r a c k( i nLEFM, 凡 G" d R " . / d a m 0 ) . m) a . 6 q . 伊 FJ d a k d q "; : : 0f o ra 1 1m 壬 v and E q .9 . 6i sv a l i do n l yi f6 m = -EkWkmE E u 8 2 U J d a m d q k; : :0f o ra l lm >ν. I fn o t, t h es e to fp r o p a g a t i n gc r a c k si sd i f f e r e n tt h 姐 a s s u m e d .O b v i o u s l y ,variouss e t so fp r o p a g a t i n gc r a c k smustbet r i e du n t i lt h ef o r e g o i n g 2 ム2 6 F =E r E . K ; a 6 q r 6 q nwhichn e e d st obe c o n d i t i o n sa r emetf o ra l lm. BasedonE q .9 もe r m i n et h es t a b l ep a t ho fe v o l u t i o no ft h ec r 拭 ks y s t e m .F o rf u r t h e rd e t a i l s, s e e u s e dt ode t h et e x t b o o kbyB a z a n tandC e d o l i n( 1 9 9 0 ) . 1 . 2 . 0 旬 ( 9 . 6 ) 1 1= numbero fc r a c k s七h a tp r o p a g a t ed u r i n gl o a d i n g( c r &c k sa川 , . . . αN r e m a i n i n gs ta t i o n a . r y , ν壬N ) ; F i g .9 . 2EnergyR e l e 回 eZ o n e s00 時 四i n gC r a c kSp&ci n gi nT e n s i o n e d R e i n f o r c e db a r 3 . 0 θ2 F 否a ,,8 q . J ( : .=伊 FJδqrdq.=secantstiffnessma . t r i xo fc r a c k e ds t r u c t u r e j 三ζ+ s d2 F ゃ =d i s p l 配 e m e n t s{ r 1 , ・ .. Nd, Nd= =numbero fd i s p l 抑 m ents) j j S _ = k f a 25 Y呪 瓦 ~ ~ . ‘ ' ー ー ・ . J 116 117 o o r r o s i v ea g e n t s .I f もh ea v e r a g es h r i n k 品, ges t r a i no r∞o l i n gs t r a i ni sfO. 姐 dt h ebodyi s t h e nt h eo p e n i n go f出 ec r a . c k si s r e s t r a i n e d, i nw h i c hd .U =t o t a lr e l e a s eo fs t r a i ne n e r g y , I : 1Wf=GfAc=e n e r g yn e e d e dt op r o d u c e 品c t u r e( A c=a r e 乱 o fa l lt h ec r a c k s ), andI : 1Wb : n e r g yd i s s i p a t e dbybonds l i pw h i c h ; : ;e f r 出 c r a . c k i n g . o c c u r ss i m u l t a n e o u s l yw i C o n s i d e rnowc o n c r e t et h a ti sr e s t r a i n e dbys t e e lb a r swhicha r es u 伍c i e n t l ys t r o n g旬 D r e v e n tt h ep r e v i o u 日l yd i s c u s s e di n s t a b i l i t i e so fc r a c ks y s もem 自 i nwhichsomec r a c k sc l o s e 血dぬespacingo ft h eopenc r a . c k sm u l t i p l i e s .F o rt h es a k eo fs i m p l ei l l u s t r a t i o n, ∞nsi d e r ar o u n dc o n c r e t er o di nt e n s i o n, w i t has i n g l e批 e lb a ri nt h em i d d l e( F i g .9 . 2 ) .T h i sa l s o 伺 t h eb e h a v i o ro fもhe∞n c r e t ez o n es u r r o u n d i n goneo ft h eb a r si na a p p r o x i m a t e l ys i m u l a t . , i nw h i c ht h ec r o s ss e c t i o n乱r e ai ss u b d i v i d e di n t on o n -o v e r l a p p i n gz o n e so fc o n c r e t e , beam 蹴 h i 凶 n も 旬 加 e 創r a a n dOh, 1 9 8 3 b, 1 9 8 4 ) . ,thes t r a 品 a tw h i c h Case1 a . StrengthLimitandNoBondS l i p . I nt h i sc a s e c r a c k i n gb e g i n si s : ( 9 . 1 0 ) f.' 2 :f UE ι 竺 S fO ( 9 . 7 ) Thed r y i n gc r a . c kst h a th a v es t o p p e dg r o w i n gw i l lg r a d u a l l yc l o s e( F i g .9 . 1 ) .Th 民 o n e w h i c he v o l v e sa ss=s o, 280 , musts u b s t i t u t ei nE q .9 . 6t h es p a c i n go ft h el e a d i n gc r a c k s, 4s , 8s , . . . .Thespacingofthel e a . d i n gc r a . c k sg e n e r a l l yr e m a i n sb e t w e e ns= aJ / 1 .7姐 d o o 2ad1 .7, i . e ., ont h ea v e r a g es~ 0 . 9 a 1・ Thus, i nv i e wo fE q .9 . 6, t h eo p e n i n gw i d t ho ft h e s h r i n k a . g ec r a . c k si s, ont h ea v e r a g e , o d . 9 a1f ( 9 . 8 ) c~ O Theb e h a v i o ri sr a 出. e rs i m i l a rf o rp a r a U e lc r a . c k sc a u s e dbyc o o l i n g , a sw e l l回 p a r a l l e l ,the c r a c k si nbeamsc a u s e dbyb e n d i n go ra x i a lf o r c e .I nr e i n f o r c e dc o n c r e t es t r u c t u r e s . c k sw i t ht h ep r o g r e s so fd r y i n go rt h eb e n d i n gc r a c k s g r a d u a lo p e n i n go ft h es h r i n k a g ec ra r o g r e s so fl o a d i n g , i sp r e v e n t e dbyr e i n f o r c e m e n t .I nf a c t, i ti sp o s s i b l etoωIcul a . t e w i t h出 ep ( p r o v i d e dt h ebonds l i pl e n g t hi sknown)t h eminimumr e i n f o r c e m e n tw h i c hi sr e q u i r e dt o a . c k sp r o p a g a t i n gw i t he q u a ll e n g t h s抗 t h e i rs m a l li n i t i a ls p a . c i n g . Sucha .r e k e e pt h ec r i n f o r c e m e n tp r e v e n t st h eo p e n i n gw i d t ho ft h ec r a c k sf r o me x c e e d i n gac e r t a i nv a l u e .The ne 臼 s s a r yr e i n f o r c e m e n th a sb e e nc a . l c u l a t e df r o mt h es t a b 出t yc o n r u t i o nb a s e donE q .9 . 2 1 9 8 0 ), andi tt u r n e do u tt h a t, f o rat y p i c a ls i t u a t i o n, t h eminimum i nB話 回t姐 dWahab( r e i n f o r c e m e n twasa b o u t0.2%o ft h ec r o s s s e c t i o n a la r e a .o fc o n c r e t e . Thuswes e et h a も もh e a . lr u l e sf o rminimumr e i n f o r c e m e n ta r ei nf a c te x p l i c a b l ebymeanso ff r a . ct u r eme e m p i r i c c h a n i c s .Ani n t e r e s t i n gp o i n t, t h o u g h, i st h a . tt h ep r e c i s ev a l u eo ft h eminimumr e i n f o r c e m e n t n e e d e dt op r e v e n tp r o g r e s s i v e l yw i d e ro p e n i n go ft h ec r a c k si sn o tao o n s t a n tb u td e p e n d s もo u t l a y , t h es t r u c t u r a lg e o m e t r y , t h ed i s t r i b u t i o nandh i s t o r yo fp o r e ont h er e i n f o r c e m e n h u m i d i t yo rt e m p e r a t u r e , e t c . Thephenomenonj u s td e s c r i b e dmaya l s ober e g a r d e da s晶 k i n do fl o c a l i z a t i o no ft h e c r a c k i n gs t r a i ni n t osomep r e f e r r e dc r a c k s . 9 . 3CrackSpacingand可Vidthi nBeams A l t h o u g ht h ec u r r e n tc o d ep r o v i s i o n sr e g u l a . t i n gt h ew i d t ho fc r a c k sar ee m p i r i c a l, 品 d配 peru n d e r s t a n d i n g伺 nbeg a i n e dt h r o u g hf r a c t u r em e c h a n i c s .Onemus もd i s t i n g u i s hbe tw 悶 :( 1 )c r a c ki n i t i a t i o n, w h i c hc o r r e s p o n d st ot h ep e a kp o i n to f山 t h 児e s 蜘t r e 悶s 悶s s t 回凶 r a i n必 d1 噌 a . g r 副 組刷 d叩 山 t 恥 he伽 蜘r t 川o fd 伽e 肝刊 V叫 e l ω l o 阿 p m則 e n 叫tぱ 0fm 削i c r o c α r 畠拭 c 叫 k , s 只 姐叫d( ρ 2 ) μ c 日 c k∞m p l e t i o n, whichc o r 自p o n d st oar e d u c t i o no ft h et e n s i l es t r e s st oz e r oandc o a l e s c e n c eo fm i c r o c r a c k si n t o r e も i n u o u sc r a c k s . c o n Thec r a c ki n i t i a t i o ni sg o v e r n e dbyt h es t r e n g t hc r i t e r i o n ,whilecrackcompletioni s r a c t u r em e c h a n i c se n e r g yc r i t e r i o n .F o ra p p r o x i m a t ean a . l y s i s, o n en e e d g o v e r n e dby出 ef n o tw r i t et h ee n e r g yb a l a n c ec o n d i t i o nf o ri n f i n i t e l ys m a l lc r a c kl e n g t hi n c r e m e n t s, b u to n e 伺.nw r i t ei tf o rt h ee n t i r ec h a n g efromt h eu n c r a c k e dt o. af u l l yc r a c k e ds t a t e : I : 1U =I : 1Wf+I : 1Wb ( 9 . 9 ) Casel b . StrengthLimitandBondS l i p .-Byf o r c ee q u i v a l e n c ec o n d i t i o n, t h ef o r c e a α o s st h ec r a c kp l a n e,Ac , 刀 mustbeequalt ot h ebondf o r c e叫ん accumula t edo v e rt h e 四ゅん(巧=u l t i m a t ebond抑制).S i n α s' 2 :L 6. a ndAc: ; : ; 付2 / 4( b= ruame t e r b o n ds l i p1 ぱt h e∞n c r e t ec r o s ss e c t i o n, andt h ec r o s ss e c t i o no ft h eb a ri sn e g l e c t e d ), t h ec r a c k sc a n b e g i nt of o r mi nt h i smιnneri f 2 ( 9 . 1 1) s三π (刀/4U { , )b Case2 a .EnergyLimitandNoBondS l i p .-Asanapproxima t i ono n ec 叩 i m a . g i n e ef u l lf o r m a t i o no fac r a c kr e l i e v e st h es t r e s sfromt h et r i a n g u l a ra r e 制 c r o s s 占a t c h e d t h a tもh i nF i g .9 . 2 a .Thevolumeo ft h er e g i o no b t a . i n e dbyr o ta . t i n gt h i sa r e aa . bou t凶 eb a ra x i si s 町=宵l r / 1 2 k .T h e r e f o r e, d .U =v i .σ?/2E( σ 1=E f . ) .A I s o叫ん =A c f : .N e g l e c t i n gd .Wし web a v e企 U' 2 :sW Sowef i n dt h ec o n d i t i o nf o rc o m p J e t ec r a c kf o r m a t i o n : , , 2 らさ ( 6kGI E ) 1 / 2 b -1 / ( 9 . 1 2 ) Case2 b .EnergyLimitandBondS l i p .-I nt h i sc a s e, i n s t e a do f町 =E E .weha . v e, 2/ romI I I でc o n r u t i o no fe q u i l i b r i u mw i t ht h ebond山 e s s e so v e rl e n g t hs, 町 宵b 4=【J{ . s, a l l a . t i o n sb e i n gt h esamea si nc a s e2 a .I tf o l l o w st h a t~mplete cr~cks c 回 f orm t h eo t h e re q u w h i l et h e r ei sbonds l i pi f 32 / s' 2 :( 3 π2 kEG /8U t , )1 / 2 b ( 9 . 1 3 ) , . t i o n s(B 話a n tandOh ,1983b,1984)a r e凶 吋 ont h eassump t i on Theforegoingω!ωa t h a tt h es もr e s sr e l e a s ez o n e so fa d j a c e n tc r a c k sdon o to v e r l a p, ωshowni nF i g .9 . 2 a .I ft h e y d o .as l i g h t l yd i f f e r e n tc a l c u l a t i o ni sr e q u i r e d . もi o n(B aZa n tandOh, 1 9 8 3 b, 1 9 8 4 )y i e l d sc ∞ ー Asomewhatmores o p h i s t i c a t e dc a l c u l a 山内sr e l a . t i o n sb e t w e e ns p a c i n gsands t r a i nfc叩 宙e n t i n gat r a n s i t i o nfrommi c r o c r配 k ~?ítíatjon t oc o m p l e t ec r a c kfo~ation. T h e s ecalc~lations h~ve b e e nshownt oa g r 偲 w i t h 句 tr~s.ults o fC l a r k( 1 9 5 6 ),C h iandK i r s t e i n( 1 9 5 8 ), MatheyandW a t s t e i n( 1 9 6 0 ), t h et H o g n e s t a d( 1 9 6 2 ), 姐dkaarand , Mat t ock(1963).' 118 119 Asa n o t h 釘 i l l u s t r a t i o no ft h ef r a c t u r ea p p r o a c ht ot h ef o r m a t i o no fc o m p l e t ec r a . c k s ‘ c o n s i d e rt h ec r 品c k i n go ft h econcreteωvera s s o c i a t e dwi 出 s t r a i nf. i nt h ec o n c r e 句 b 紅 s h r i n k a 伊 a ndt e m p e r a t u r ee f f e c t sd i s r e g a r d e d ;F i g .9 . 2 b .F o rt h es a k eo fs i r 叩l i c i t y , c o n s i d e r o -d i m e n s i o n a la c t i o no n l y .F o r m a t i o no f晶 c o m p l e 句 c r a c kr e l i e v e st h ei n i t i a ls もr a i neneuv tw d e n s i t yEf~/2 f r o mvolume同:=kh2w h e r ey i .i st h ec r o s sh a t c h e da r e ashowni nF i g .9 . 2 b , 姐 dk=回 mec o n s t a n t .N e g l e c t i n gdWb, wemusth a v ed U主 d Wo rG, bh 三 (Ef~/2) Yt. H e r ewemayB u b s t i t u t eE.=d .F ur t h e r m o r e , a s s u m i n gt h eb a rt obes u 缶c i e n t l ys t r o n g , c/s weknowfromt h ep r e v i o u sa . na l y s i so fs t a b i l i t yo fap a r a l l e lc r a c ks y s t e mt h a tt h es p a . c i n g h e r ek ssomec o n 日t a n t .Thusweo b t a i n : o ft h eopenc r a c k sw i l lbes:= k 1hw 1i r e l a t i o n sf o rt h ef r a c t u r ep r o c e 8 Sz o n ei nn o n l i n e 紅 白n i t ee l e m e n ta n a l y s i ss h o u l de x h i b i ta a . i n )a twhicht h es t r e s ss u d d e n l yd r o p st oz 釘 o .However , i もI S maximumd i s p l a c e m e n t( o rs t r 山e rt h i sc o n d u s i o ni sa l s ov a l i di nt h ep r ,伺e n 田 o ff r i c t i o na . n do t h e ri n e l a s t i c n o t ω r t a i nwhe m e n a p h e n o . a . t h e rp o w e r f u lapproxim a . t et e c h n i q u ef o r∞n s i d e r i n gi n t e r 民 t i o n sb e t w e e nrandomly Ar l o c a t e dc r a c k si nal a r g ebodyh却 肥 田n t l yb e e nd e v e l o p e dbyKachanov( 1 9 8 5, 1 9 8 7 ) ;s 回 a l s oB笹 川 口i s t e, Dvorak , Z a r z o u randWung( 1 9 8 8 ) . ・ ι , 1 2 さん (2Gb / kE)1/2h / , ( 9 . 1 4 ) E q s .9 . 9 9 . 1 3a r ei n t e r e s t i n gi nt h a t出 eyr 抗i o n a l i z e出 ei n f l u e n c eo ft h et h i c k n e s sb o fc o n c r e t ea r o u n dt h eb a r( o rb a rs p a c i n g )andt h ei n f l u e n c eo ft h et h i c k n 邸 o fc o n c r e t e c o v e r .Withr e g a . r dt oE q .9 . 1 3, o n emayr e c a l lt h ee m p i r i c a lf o r m u l ae s t a b l i s h e db y e x t e n s i v e s t a t i s t i c a la . n a l y 由 o f t e s td a t abyG e r g e l yandL u t z( 1 9 6 8, s e ea l s oM e i e r岨 dG e r g e l y ,1 9 8 1 ), a c c o r d i n gt ow h i c ht h ec r a c kw i d t hOci sr o u g h l yp r o p o r t i o n a lt oA~/3. S i n c eAci sr o u g h l y t h i si so n l ys l i g h t l yd i f f e r 四 t fromE q .9 . 1 3 .UnderBomemores o p h i s t i c a t e d p r o p o r t i o n a lt oh, h ep r o p o r t i o n a l i t yo fd oA~/3 C組 b eo b t a i n e dbyf r a c t u r e田 町gy姐 a l y s i s a s s u m p t i o n s,t ct ( B a -z a n tandOh ,1983b, 1 9 8 4 ) . 田 i tc l e a . rt h a tv a r i o u se x i s t i n gp r o v i s i o n sr e g a r d i n gm i n i Thef o r e g o i n gd i s c u s s i o nmak 田 m entandc r a c kw i d t hc o u l dbea tl e a s tp a r t i a l l yj u s t i f i e d叩 dp r o b a b l ya l s o mumr e i n f o r 乱c t u r em e c h a n i c s . i m p r o v e dbyt h eu s eo ff r 9 . 4I n t e r a c t i n gMicrocracks M o d e l i n go ft h em i c r o c r a c ks y s t e mi nt h ef r a c t u r ep r o c e s sz o n eh e l p si nd e v e l o p i n g陪 t i o n a ls t r e s s d i s p l a . cemen もo rs t r e s s s t r a i nr e l a t i o n sf o rt h ef r a c t u r ep r o c e s sz o n e .I ns u c h m o d e l s, i ti si m p o r l a n tもot a . k ei n t oa c c o u n tt h ei n t e r a c t i o no fi n d i v i d u a lm i c r o c r a c k s .T h i s i n t e r a c t i o nmayl e a dt oi n s t a b i l i t i e sandl o c a l i z a t i o no fc r a c k i n g . n 山eもe r m i n a lp h a s eo fm i c r o c C o n s i d e rf o rexamplet h ei d e a l i z e ds i t u a t i o ni nF i g .9 . 3 .I r a c k i n g, twoa d j a c e n tc r a c kt i p sc o a l 目 白 出 t h el i g a m e n tb e t w e e nthemi sg e t t i n gt o r n .T h i s ∞ales田 n臼 mayberoughlymodeledastheterminalpropagationofacircumferentialcrack i nar o u n db a r( F i g .9. 3 )w h i l et h er a d i u so ft h ec i r c u l a rl i g a m e n ta p p r o a c h e sz e r o.B a s e d , onec 姐 c a l c u l a t et h e ont h eknowns o l u t i o n so ft h es t r e s si n t e n s i t yf a c t o rf o rt h i sp r o b l e m diagramo ft h ea p p l i e dr e m o t es t r e s sv s .t h ed i s p l 乱ε ementduet oc r a . c k i n g ;F i g .9 . 3 b . An 偲 s s e samaximumd i s p l a c e m e n ta f t e rw h i c h i n t e r e s t i n gp r o p e r t yo ft h i sd i a g r a mi st h a ti tp o s t h e r ei sas n a p b a c ki n s t a b i l i t y . もC組 b ep r o v e nmoreg e n e r a l l yt h a . tamaximumd i s p l a ε e m e n tw i t hs n a . pbacki n s t a b i l i t y I もo cωrf o re v e r yt y p eo fc r a . c kl i g a . mentt e a r i n g, p r o v i d e dt h el i g a m e n tt r 組 s m i t saf o r c e . mus i fo n l yamomentb u tnof o r c ei st r a n s m i t t e da c r o s st h el i g a m 叩 t , t h e nn o Ont h eo t h e rh日 d, a . Z 副 ,1 9 8 7 b ) .S i n 回 t h elatterωei su n l i k e l yt op r e v a i li n伽 f r a c t u r e s n a p b a c ko c c u r s(B 倒 z o n e,i tf o l l o w sf r o mt h i sa n a l y s i st h a tt h e町 側 ー d i s p l 紅 白n en t( o rs t r e s s s t 叩n ) p r o c CONCLUDINGRE 乱1ARKS Thep r e s e n te x p o s i t i o no ft h es t a . t e トo f t h e -a r ti nc o n c r e t ef r a c t u r ec o n c e p t sandd e t e r m i 目e a r c ha c t i v i も yi nr e 偲 n tye 釘 S n a t i o no fm a t e r i a lp r o p e r t i e sr e f l e c t sat r e m e n d o u ss u r g ei nr e a n ddocument 8al a r g ed e g r 田 o fp r o g r e s sw h i c hh a sb e e na c h i e v e do v e rar e l a t i v e l ys h o r t s p a no ft i m e . Whileo n l ya b o u tt e ny e a r sagot h ea p p l i 日 b i l i t yo ff r乱c t u r emech 組 i c st o m組 ye x p e r t snowa g r 偲 t h a tf r a c t u r em e c h a n i c sc a nh晶v e c o n c r e t es t r u c t u r e sWasd o u b t e d, ac o n s i d e r a b l eimp . a c t, i m p r o v i n gt h es a f e t y叩 de ∞nomyof∞ncretestructuresandma . k i n g newd e s i g n sp o s s i b l e . r 回 f o l l o w u pr e p o r t s, ACICommitt 田 4 4 6onF r a c t u r e~ech組ics w i l la t t e m p tもo I nもh r e v i e wt h e8 ta . t e -O ιth e -a r ti na p p l i c a t i o n so ff r a c t u r em e c h a n i c st os t r u c t u r a lb e h a v i o r, w i t h . av i e wt o w a r dp o t e n t i a lc o d ei m p r o v e m e n t s, t h es t a t e o f . ・ t h e a r ti nf i n i t ee l e m e n t企a c t u r e c r e t es t r u c t u r e s, andt h es t a t e o f t h e -a . r ti nm o d e l i n go ft h er a t ee f f e c t sand a n a l y s i so f∞n d y n a m i cf r a c t u r eo fc o n c r e t e . Acknowledgment. Thanksf o rh e l p w i t hc o m p i l a t i o no ft h er e f e r e n c 剖 a ndv a r i o u s m 泊i o n sa r ed u et oJ . -K .Kim, Y.Xi, M.R.Tabba 叫 a nd~.T. K回 e 凶 ( a l lgradua もer e s e a . r c h 制 i s t 組 t sa tN o r t h w e s t e r nU n i v e r s i t y ) . 1 2 0 REFERENCESANDBIBLIOGRAPHY 1 .ACI226R・8 0 :“ C o n t r o lo fC r a c k i n gi nC o n c r e t eS t r u c t u r e s , "reportedbyACICommittee 2 2 6( c h a . ir e dbyD .D a r w i n ) . 2 .ACI224.1R ・8 4 :“ C a u s e s, E v a l u a t i o nandR . e pa . i ro fC r a c k si nC o n c r e t eS t r u c t u r e s, "R勾 or凶 byACICommittee2 2 4( c ha . i r e dbyD .D a r w i n ) . 3 .A r r e a, M., andI n g r a 飾 品 , A.R., 1 9 8 1, ‘・Mixed-ModeCrackPropagationi nMortara n dC o n . "Report31・13, Corn e J lU n i v e r s i t y , I t h晶 c a . c r e t e, 4 . ASTM, 1 9 8 3, “ Sも岨d a r dT e s tMethodf o rP l a n e . 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K ,p p .3 9 い5 0 U . 任 Y u z u g u l l u, 0 ., andS c h n o b r i c h, W.C ., 1 9 7 3 “ ,ANumericalProceduref o rt h eDe t e r r n i n . a t i on "ACIJ ., v.7o , No.7, pp.47 4・4 7 9 . o ft h eBehwioro faS h e a rWanETameS y s t em, 二 ヱ ヰ よ 。 ii23JZ:::::;::;;;:;2JF;::?::::: Z1 2 2 J4LZ: ;tz::: a t e r ia J s , Kyoto, J . a pan, 7 0 5p . 1 9 7 3Symp.onM e c h a n i c a lB e h a v i o ronM. ,, , , “ ,I n t e r f a c eE l e m e n tM o d e l i n go fF r a c t u r ei nA g g r e 山 s , "J .o fEng 略 M e c h .D i v ., ASCE , V .日3, No.1 1, p p .1 6 時 1 6 3 0 . g a t eCompL 司2 ・Z u b e l e w i c zA . andB a i a n tZ .P .1 9 8 7 140 Appendix1 . DerivationsofSomeFormulas 圃 D e r i v a t i o no fE q .1 .4 : Thes i z ee f f e c tl a wi nE q .4 . 1c a nbed e r i v e dmostg e n e r a l l yb ¥ l y s i sands i m . i l i t u d ea r g u m e n t s( B a . Z a n t, 1 9 8 4 a )ont h eb a s i so ft h efollowin~ d i m e n s i o n a l姐 a h y p o t h e s e s :( I . )Thee n e r g yr e l e a s eo ft h e山 u c t u r ei saf u n c t i o no f b o t h :( a )t h el e n g t h a, 叩 d( b )t h ec h a r a c t e r i s t i cf r a c t u r ep r o 慣 sz o n e, Cj・ ( 1 1 . )Thel e n g t ha , a t o ff r a c t u r e, sp r o p o r t i o n a lt os t r u c t u r es i 配 d , wh i 1 ecJi sas t r u c t u r a J maximuml o a di sn o tn e g l i g i b l e姐 di p r o p e r t yi n d e p e n d e n to fd. Thet o t a lamounto fe n e r g yr e l e a s e dfromt h es t r u c t u r ei n t ot h ef r a c t u r emustb ee x . p r e s s i b l ei n出 eform 2 W =. . . . ! σ τ t b d F (" 川),。 8 2 2 E ' UN " . . . . . 1 . '¥ 1 ."2)> V1=E =d ' " 2=2 =d ( A. 1 ) J t ,8 r ei n d e p e n d e n tn o n d i m e n s i o n a lp a r a m e t e r sも (h e i rnumberf o U o Ws whereσN = P/bdj( 2a fromBuckingham 込 山 田remo fd i m e n s i o n a l組 a l y s i s ), andF i sac e r t a i nf u n c t i o nw h i c hma . y r omt h ec r a c kp r o p a g a t i o nc o n d i t i o n8W/ δ α =Gf b , o n eg e t s bee x p e c t e dt obes m o o t h .F σL= 竿 問ヲ~rl ( A . 2 ) WL--T ・ 1 1/ コa 、 、'f A 一 一 F q& c 、 . /'tin- σ 一 一N E - Wenowc h o o s et h es t a t e( J2 =0( w h i c hc o r r e s p o n d st od→∞)ωther e f e r e n 四 s t a t e , 組d exp 組 d8 F/ δ8 i n t o T a y l o r s e r i e s a b o u t t h i s s t a t e . i e . , aF / δ 0 = 凡 + F , ) ' 8 , ) ' + 九時+九時+ 1 1 …whereFl, 凡 一a r ec o n s t a n t si fg e o m e t r i c a 1 1 ys i m . il a rs h a p e s( s 叩 l e( 1 )a r ec o n s i d e r e d . q .( A . 2 )姐 dt r u n c a t i o no ft h es e r i e sa f t e rt h el i n e a rtermy i e l d s : S u b s t i t u t i o ni n t oE ( A . 3 ) , / j . T h i sy i 帥 E q .4 . 1i fo n eden o t e sB= ( 2 E ' G andn o t e 告側 o=凡C F1, J)/(九叫)佃dd B anddoa r ec o n s t a n t s . 田 e f f e c tl a w( E q .1 .4 ), weh a v ep2=σ (N b d /c . , .) 2= ( B ! u b d /c . , .) 2 d J ( d+ d . ) Fromt h es i o o S u b s t i t u t i n gf o rp2i nE q .5 . 2andn o t i n gE q .6 . 2 4f o rG weg e t , ( α ) G(α)=G/ 一 一 一 Jg (一一 α )d + d 。 o ( A. 4 ) Thec r i t i c a ls t a t eo c c u r si ff r a c t u r ep r o p a g a t i o ni sp o s s i b l ea l s of o rt h en e x ta d j a c e n t t a t e , i . e . ifθ(G-R ) J 8 d= O .B e c a u s eθR ( c ) / 8 d= 0, wemusth a v eθ G / θd= O .S i n c e α=a J d=( a o+c ) / d , wef i n d8 α / δd=c J ♂ =(αoー α) / d , 組 d80s u b s t i t u t i o no fE q .A . 4 i n t oθGJ θd=[ o G ( α) J θ α ] θ α / δd+θGJ δd=0p r o v i d e s 日 d+d o=dog(α)J !9'(α)(α 一 αo~ (~司 S u b s t i t u t i 碍 t h i sf o r( d+d )i nE q .Aム wef u r t h e ro b t a i nG(α)=G, 旬' ( α) J g ( α' o ) ] c f d o ・ o F i n a U y ,notingthatdo= cg (α) / g ' ( α 0 )( f r o mE q .5 . 6 ),weo b t a i nE q .4 . 1 (Bai 組 主 回d 凶, 1 9 8 8 ) .F u r t h e r m o r e ,s u b s t i t u t i n gE q .4 . 1i n t oE q .A.4i nw h i c h(R=G)回 d Kaze e l i m . in a t i n gdw i t ht h eh e l po fE q .A.5, E q .4 . 2e n s u e s . ,。 PartI I ConferencePapers