Applied Material Analysis

7/27/2019 Applied Material Analysis

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STP 1429

Pr ed icti ve M ater ia l M odel i n g :Comb in in g F u n damen ta l Physi csU n der stand in g, Com pu tat iona lM ethods an d Empi r i cal l yObser ved B ehavior

M. T. Kirk and M. Erickson Natishan, editors

ASTM Stock Number: STP1429

/N l rm l t I ~NA/ .

ASTM International100 Barr Harbor DrivePO Box (2700West Conshohocken, PA 19428-2959Printed in the U.S.A.


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Library of C ongress Cataloging-ln-Publication Data

Predictive ma ted~ model ing; combining fundamental physics understanding, computat ional m ethodsand em pir ical ly observed b eh av ior /M .T. Kirk and M. Er ickson Nat ia l~an, eddors.p. cm. - (STP ; 1429)Includes bibliographical references."ASTM Stock Number: STP1429."ISBN 0-8031-3472-X1. Steel--Metallurgy-Congresse s. I, Kirk, Mark, 1961-11.Nat ishan, M. E f lc~:son. t i , AS TMspeciaJ Izchr~cal publication ; 1429.

2003062889TN701.5.P74 2003669'.142--dc22

Copynght 9 200 4 AS TM International, West Conshobocken, P/L Al l dghts reserved. This matedalmay not be reproduced or copied, in whole o r in par t , In an y pr inted, mechanic~d,electronic, t i lm, orOther dfstdbution and storage media, w ithout the written consen t of the pub lisher.

Photocopy RightsAu t h o r i z a t i o n t o p h o t o c o p y i t e m s f o r I n t e r n a l , p e r s o n a l , o r e d u c a t i o n a l c l a s s r o o m u s e ,o r t h e I n t e r n a l, p e r s o n a l , o r e d u c a t i o n a l c l a s s r o o m u s e o f s p e c i f i c c l l o n t e , i s g r a n t e d b yAST M I n t e r n a t i o n a l ( AS ' r M ) p ro v i d e d t h a t ~e a p p r o p r i a te f e e I s p a id t o t h e Co p y r i g h tC learanc e Cente r , 222 Rose wo od Dr ive , Dan vem , MA 01923; Te l : 978-750-8400 ; on l in e :h t t p : / / www. c o p y r i g h t . c o m L

Peer Rev iew Po l icyEach paper publ ished In th is volume w as evaluated by tw o peer reviewers and at least one edi tor .

The authors addressed al l o f the reviewers' comments to the sat isfact ion o f both the technicaledi tor(s) and the ASTM Intema~nal Commit tee on Publ icat tons.To make technical information available as quickly as possible, the peer-reviewed papers in thispubl icat ion were prepared "ce ntre -rea dy" as submit ted by the authors.The qu ar ~ of the papers in th is publ ication ref lects not only the obvious ef for ts of the authors andthe technical edi tor(s) , but a lso th e work of the peer reviewers. In keeping w ith long-standingpubl icat ion prac~cas, AS TM Internat ional mainta ins the anonymity of the pee r reviewers. The A STMInternat ional C ommit tee on Publ ications acknow ledges with appreciat ion their dedicat ion andcontribut ion of t ime and ef fort on behalf of A ST M International .

Prinled n May~etd,PAJanua,'y2004


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Foreword

T h e S y m p o s i u m o n P r e d i c t i v e M a t e r i a l M o d e l i n g : C o m b i n i n g F u n d a m e n t a l P h y s i c sUnders tanding, Computa t iona l Methods and Empir ica l ly Observed Behavior was he ld in Da l las ,Tex as on 7- 8 No vem ber 2001. AST M Internat iona l Com mit tee E8 on Fa t igue and Frac ture spo n-so re d t he sym pos i um. S ym pos i um c ha i rpe r sons a nd c o -e d i to r s o f t h i s pub l i c at ion w e re M a rk T . K i rk ,U. S . Nuc lear Regula tory C om miss ion, Rockvil le , M aryland and M arjor ieArm Erickson Nat ishan,Pho enix Engin eering As sociates , Incorporated, Sykesville , Maryland.

iii


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Contents

OVERVIEWFEm~rr~C STEZLS

T r a n s i t i o n T o u g h n e s s M o d e l i n g o f S t e e ls S i n c e R K R - - M . T . K IR K,M. E. NATISHAN,ANDM. WAGENHOFER

T r a n s f e r a b i l i t y P r o p e r t i e s o f L o c a l A p p r o a c h M o d e l i n g i n t h e D u c t i le t o B r i tt l eT r a n s i t i o n R e g l o n - - A . LAUKKANEN, K. WALLIN, P. NEVASMAA, AND S. T~HTINEN

C o n s t r a i n t C o r r e c ti o n o f F r a c t u r e T o u g h n e s s C T O D f o r F r a c t u r e P e r f o r m a n c eE v a l u a t i o n o f S t r u c t u r a l C o m p o n e n t s - - F . M ~ A M I AND K . APaM OCHI

A P h y s i c s - B a s e d P r e d i c t i v e M o d e l f o r F r a c t u r e T o u g h n e s s B e h a v i o r - - M . E . NATISHAN,M. WAGENHOFER,AND S. T. ROSINSKI

S e n s i t iv i t y i n C r e e p C r a c k G r o w t h P r e d i c t i o n s o f C o m p o n e n t s d u e to V a r i a b i li t yI n D e r i v i n g t h e F r a c t u r e M e c h a n i c s P a r a m e t e r C * - - K . M . NIKBIN

O n t h e I d e n ti fi c at io n o f C r i ti c al D a m a g e M e c h a n i s m s P a r a m e t e r s t o P r e d ic t t h eB e h a v i o r o f C h a r p y S p e c i m e n s o n t h e U p p e r S h e l f - -- c . POUSSARD,C. SAINTE CATHERINE, P. FORGET, AND B. MARINI

ELECTRONIC MATERIALSI n t e r f a c e S t r e n g t h E v a l u a t i o n o f L S I D e v i c e s U s i n g t h e W e i b u l l S t r e s s - - F . M INA MI,

W. TAKAHARA, AND T. NAKAMURA

COMPUTATONALTECHNIQUESC o m p u t a t i o n a l E s t i m a t i o n o f M n i t i a x i a l Y i el d S u r f a c e U s i n g M l c r o y i e ld P e r c o l a t io n

Analysls---A. B. GELTMACHER,R. K. EVERETI', P. MATIC, AND C. T. DYKAI m a g e . B a s e d C h a r a c t e r i z a t i o n a n d F i n i t e E l e m e n t A n a l y s i s o f P o r o u s

S M A B e ha v io r - - M . A . QIDWALV. G . DEGIORGI,AND R. K. EV ~RETI'

v ii

2 2

4 8

6 7

8 1

103

12 3

1 3 5

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Overv iew

A n A S T M I n te r n a t io n a l S y m p o si um c o n c e m i n g Predictive Ma terial M odeling: Com biningFundamen tal Physics Understanding, Co m putational M ethods , and Em pirically ObservedBehavior w a s h e l d o n 7 - 8 N o v e m b e r 2 0 0 1 i n D a l l a s , T e x as in c o n j u n c t i o n w i t h t h e s e m i -annua l m ee t ings o f A S TM In te rna t iona l Co m m i t t ee E8 on F rac tu re and Fa t igue . The sympo -s ium w as m o t iva te d by t he focus o f m any indus tr ies on ex ten d ing t he design l i f e o f s truc tu res.S a fe l i fe ex tens ion depends on t he ava i lab i l i t y o f robus t me thodo log ies t ha t accu ra te ly p red ic tbo th t he f undam en ta l m a te r ia l beh av io r and the s truc tu ra l response unde r a w id e range o f loadc o n d i t i o n s . H e r e t o f o re , p r e d i c t i v e m o d e l s o f m a t e r ia l b e h a v i o r h a v e b e e n b a s ed o n e m p i r i c a lde r i va t ions , o r on f un dam en ta l phys ics -based m ode ls t ha t desc r ibe ma te r ia l beh av io r a t t henano - o r m ic ro -sca le . B o th app roaches t o m ode l ing su f fe r f rom issues t ha t l im i t t he i r p rac t i ca la p p l i c a t i o n . E m p i r ic a l l y - d e r i v e d m o d e l s , w h i l e b a se d o n r e a d i l y d e t e r m i n e d p r o p e r t ie s , c a n -n o t b e r e l i a b l y u s e d b e y o n d t h e l im i t s o f t h e d a ta b a s e f r o m w h i c h t h e y w e r e d e r i v e d .Fundamen ta l , phys ica l l y -de r i ved mo de ls p rov ide a sound bas is f o r ex t rap o la t ion to o the r ma -te r ia ls and c ond i t ions , bu t re ly on pa rame te rs t ha t a re measu red on t he m ic rosca le and thusm a y b e d i f f i c u l t a n d c o s t ly t o o b t a i n . I t w a s t h e h o p e t h a t th i s c o n f e re n c e w o u l d p r o v i d e a no p p o r t u n i t y f o r c o m m u n i c a t i o n b e t w e e n r e s ea rc h er s p u r s u in g t he s e d i f fe r e n t m o d e l i n g a p -proaches.

The pape rs p resen ted a t th is S ympos ium inc lude d s ix co nce rn ing f e r r i ti c s teel ; these ad -d ress frac tu re in t he t rans i t ion reg im e , on t he upp e r she lf , and in t he c reep range . Th ree o f t hesep a pe rs u s ed a c o m b i n a t i o n o f t h e G u rs o n a n d W e i b u l l m o d e l s t o p r e d i c t fr ac tu re p e r f o rm a n c eand a cco un t f o r con s t ra in t loss . W h i le successful a t p red ic t ing con d i t ions s im i la r t o t hose rep -resented by the c a l ib ra t io n da tasets, a l l investiga tors fo un d the param eters o f the (p re do m i-nan tl y) em p i r i ca l W e ibu l l mo de l t o dep end s ign i f i can t l y on f ac to rs such as t empe ra tu re , s tra inra te , in i t ia l y ie ld s t rength , s tra in hard en ing expo ne nt , and so on . These strong dependenciesma ke mo de ls o f t h is t ype d i f f i cu l t t o ap p ly bey ond the i r ca l ib ra ted range . Na t ishan p ropos edthe use o f phys ica l l y de r i ved mode ls f o r t he t rans i t ion f rac tu re t oughness o f f e r r i t i c s tee ls .W h i le t h is app roa ch shows be t t e r s im i la r i t y o f pa rame te rs ac ross a w id e range ma te r ia l , load -ing , an d temp era tu re con d i t ions t han does t he W e ibu l l app roa ch , i t has no t ye t been used toassess cons t ra in t loss ef fec ts as t he W e ibu l l mo de ls have .

Th ree pape rs a t t he S ympos ium add ressed top ics un - re la ted t o s tee ls . O ne pape r app l iedthe W e ib u l l m ode ls used ex tens ive ly f o r s tee l f rac tu re t o assess he in tedac ia l frac tu re o f e lec -t ron ic com pone n ts . A s is t he case fo r s tee l f rac tu re , t he W e ibu l l mo de ls p red ic t w e l l c ond i t ionss im i la r to t he c a l ib ra t ion da tase t. I n t he rem a in ing two pape rs resea rchers a f f i li a t ed w i t h t heNava l Resea rch Labo ra to ry used advanc ed com pu ta t iona l and expe r ime n ta l t echn iques t o de -v e l o p c o n s t i t u ti v e m o d e l s f o r c o m p o s i t e a n d s h a p e m e m o r y m a t e r i a ls .

v i i


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v i i i O V E R V I E WW e w o u l d l i k e t o c l o s e th is o v e r v i e w b y e x t e n d i n g o u r th a nk s n o t o n l y t o t h e a u th o r s o ft he pape rs you f ind in t h is vo lume , bu t a lso t o t he man y pee r rev iewers , and to t he mem berso f t he A S TM In te rna t iona l s taf f w ho m ade pub l i ca t ion o f th i s vo lum e poss ib le.

Mark T KirkNuclear Regulatory Com missionRoekville, MarylandSymp osium chairperson and editor

MarjorieAnn Erickson NatishanPhoenix Engineering Associates, Inc.Sykesville, MarylandSymp osium chairperson and editor


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Ferritic Steels


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M a rk T Kirk, 1 . . . . 2arjorteAnn Erzckson Natzshan, and M at thew W ag en ho fe/T r a n s i ti o n T o u g h n e s s M o d e l i n g o f S t ee ls S i n c e R K RReference: Kirk , M. T. , Nat ishan , M . E., and Wa genh ofer , M . , " T r a n s it io n T o u g h n e s sM o d e l in g S i n c e R K R , " Predict ive M aterial Modeling: Combining Fund am ental Phy sicsUnderstanding, Computational Me thods an d Empirically Observed Behavior, A S T M ST P1429, M . T. Kirk and M. Er ickson Nat ishan, Eds . , A ST M International, W es tConshohocken , PA , 2003 .Abstract : In th is pap er w e t r ace the deve lopm en t o f tr ansi tion f rac tu re toughnes sm ode l s f rom the l andmark p aper o f R i t ch ie , Kno t t, and R ice in 1973 up th rough thecur ren t day. W hi le such mode l s have beco m e cons ide rab ly m ore soph i s ti ca ted s ince1973, none have ac hieved the go al of b l ind ly predic t ing f rac ture tough ness data . Inth is paper w e sugges t one po s s ib le wa y to ob ta in such a p red ic tive mode l .K e y w o r d s : Ritc hie-K no tt-Ric e, cleav age fracture, transit ion fracture, m ode ling,ferritic steels.B a c k g r o u n d a n d O b j e ct iv e

A longdm e goa l o f the f r ac tu re mechan ics com m uni ty has been to under s tand thef racture process in the t rans i tion region o f fer ri tic s tee ls so that i t m ay be quant i f ied wi thsuff ic ient accurac y to e nable i t s conf iden t use in sa fe ty assessm ents and l i fe ex tens ioncalcula tions . W atana be e t a l. ident i f ied two d i f ferent approac hes tow ard th is goal : themechan ics ap proach and the m ate r ia l s approach [ 1 ]. The c las sica l mec han ics , o r f rac tureme chanics , approac h is a semi-em pir ica l one in w hich so lu t ions for the s t ress fie lds nearthe crack tip are use d to draw co rre la tions betw een the near - t ip condi t ions in labo ratoryspecim ens and f rac ture condi t ions a t the t ip of a crack in a st ructure . Con versely , themate r ia l s approach a t t empts to p red ic t f r ac tu re th rough the u se o f mo de l s desc r ib ing thephys ica l mec han i sms invo lved in the c rea t ion o f new su r face a reas . W atana be ' s"m ater ia ls appro ach" is ident ica l to what K not t and Boc cacc in i [2] refer to as a "m icro-scale approach." K not t and Bo ccac cin i a lso ident i fy another approach to trans i t ionfracture characterization, the nano-sc ale approach, wh ich at tem pts to des crib e thecompet i t ion be tw een c rack p ropaga t ion and c rack b lun t ing th rough the use o f d i s loca t ionmechan ics . In m any ways , the mic ro - sca le (o r ma te r i a ls ) approach p rov ides a b r idgebe tw een the c las sica l f rac tu re m echan ics and nano- sca le approaches .

1 S en i o rMaterials Engineer, U nited States Nuclear Re gulatory Com mission, 115 45 Rockv ille Pike, Rock'ville, MD , 20852 , U SA([email protected]). (The views expressed herein represent those of the author an d not a n of f icia l posi t ion o f the USNRC.)2 Pres lden t, hoenix Engineenng Assomates, Inc., 979 Day Road, Sykesville, MD , 21784, USA ([email protected]).3 G rad ua te tudent, Department of Mechanical Engineering, University of Maryland, College Park, MD , 20742, USA.

C o p y r i g h t * 2 0 0 4 b y A S T M I n t er n a ti o n a l3

w w w . a s t m . o r g


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4 PREDICTIVE MATERIAL MODE LING

Ri tch ie , Kno t t and R ic e ' s [3] l andmark 1973 paper (RK R) i s a c l a ss ic exam ple o f themic ro - sca le approach . The RK R m ode l has ga ined w idesp read accep tance as anappropr ia te descr ip t ion o f the condi t ions ne cess ary for c leavage f rac ture ( i.e.,ach ievem en t o f a c r it ica l va lue o f st re s s no rmal to the c rack p lane ove r a cha racte ri st icd i s t ance ahead o f the c rack t ip ) a t t emp era tu res w e l l be lo w the t r ans it ion t empera tu re .Ev en thoug h R K R them se lves we re unsucces s fu l in app ly ing the i r mo de l a t h ighertem peratures ( i.e. tem peratures approaching the f rac ture mod e t rans i t ion tem perature) , thes t reaml ined e legance o f the ir mode l has p rom pted m any r esea rcher s to expa nd on R K R inat tem pts to desc r ibe f rac ture up to the t rans i tion temperature . These mo dif ied / enhance dR K R a p p r o a c h e s h a v e p r o d u c e d v a r y i ng d e g r e e s o f s u cc e s s, y e t th e y h a v e n e v e r a c h ie v e dthe u l t ima te go a l o f be ing fu l ly p red ic t ive because , be ing base d on an under ly ing m ode lthat do es not de scr ibe fu l ly the precursors to c leava ge f rac ture , the parameters o f them odi f i ed /enhanced R K R m ode l s invar iab ly m us t b e em pi r ica l ly calibrated .In th i s paper we t race the deve lopm en t o f RK R- ty pe m ode l s f rom 1973 th rough thep resen t day , and p rov ide ou r pe r spec t ive on the s t eps neede d to ach ieve a fu l ly p red ic t ivet rans it ion f rac ture m ode l for fer ri tic s tee ls, a goal w ho se achievem ent can now be c lear lyenvisaged.R K R : T h e 1 97 3 M o d e l

Ritchie , Kn ot t , and Ric e (RK R) [3] we re the f ir s t to l ink explanat ions for the causefo r c l eavage f r ac tu re based on d i s loca t ion m echan ics w i th the concep t s of LEFM. B y1973 both me chan is t ic [4] and d is locat ion-ba sed [5-6] m ode ls sugges ted that c leavagefracture required achiev em ent of a cri tica l s t ress level. The R K R mo del com bined th iscr i teria wi th th e ( then) re cent ly pub l ished so lu t ions for s t resses ahead o f a crack in anelas t ic-p las t ic so l id [7-9] to predic t su ccess fu l ly the var ia t ion o f the cr i t ica l s t ressin tens i ty fac tor wi th temp erature in the low er trans i tion reg im e o f a mi ld s tee l (see F ig .1). The se researchers a lso in t roduced the co nce pt that achievem ent o f th is cri tica l s t ressa t a s ingle poin t ahead o f the crack tip was no t a suf f ic ient cr i ter ion for f rac ture. T heypos tu la ted , and subse que nt ly dem ons tra ted , tha t the cr it ica l s tress value had to beexc eed ed ove r a micro-s t ructura l ly re levant s ize scale (e .g ., m ul t ip les o f gra in s izes ,m ul t ip les o f carb ide spacing) for fa ilure to occur .

The R K R m ode l p rov ides a d esc r ip t ion o f c l eavage f rac tu re tha t, a t l east in the low ert rans it ion regim e, i s both c ons is tent wi th the phy s ics o f the c leav age f rac ture proce ss andsucces s fu l ly p red ic t s the resu l ts o f f rac tu re toughnes s exper imen ts . Ho we ver , the m ode lhas l imi ted eng inee r ing u t il i ty because the p red ic t ions dep end s t rong ly on two pa ramete r s( the cr i t ical s tress fo r cleav age fracture, or cr j; and the c r i t ical distance, ~, ov er w hic h ~ i sachieved) that are both d i f f icu l t to m easure an d can on ly be determ ined inferential ly . Inthe fo l lowing s ec t ions we d i s cus s va r ious r e f inemen ts to RK R- typ e m ode l s tha t havebeen pub l ished s ince 1973 . W e define a "RKR-type" m ode l a s one tha t a t t empts tocharacter ize and /or predic t the c leavage f rac ture ch aracter is t ics o f ferr it ic s tee ls andadopts the ach ievem ent o f a cr i tica l s t ress ove r a cri t ica l d is tance ahead o f the crack t ip asthe fa i lu re c ri te rion . W e beg in by d i scus s ing ea r ly a t tempts to app ly the RK R m ode l to


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K I R K E T A L . O N M O D E L I N G O F S T E E L S S I N C E R K R 5

t empera tu res h igher in the tr ans it ion r eg im e than a tt empted by R K R themse lves . W e thenrev iew e f fo r ts under taken in the 1990s and the rea fte r to ex tend the t empera tu re r eg imeove r wh ich R K R app l ies th rough the use o f m ore accurate ana lys i s o f the s t re s ses aheado f the de fo rming c rack tip . W e con c lude the paper w i th a d is cus s ion o f the advan tagesand l imi ta tions o f these cu r ren t mo de l ing approaches, and p rov ide a pe r spec t ive on ho wthese l imi ta t ions can be overco me .

Z=E

5 0

4 0

3 0

2 0

1 0

i ' " t I i t i - I [ ' IO 9 Com puted from Ostergen stress distribution

~ F r o m Rice & Rosen grenlHutchinso n stress d istribution(Open sym bols re fer to a cha racteristic dislance o f onegrain diameter, 60~. Closed symbols refer to acharacterist ic distance of two grain diameters, 120p.)

/ J~" Ko values, Measured experimentallyK~ values, Fr om H.S.W. analysis / ,

O.

00

L . J t I I f . , I , k- 1 4 0 - 1 2 0 - 1 0 0 - 8 0 - 6 0

T e m p e r a t u r e [ ~F I G . 1-Comparison o f RK R mo del pred ict ion (symbols) with experimental Kic data (SolidCurve) sho win g good agreement fo r a characteristic distance o f two-grain diameters.No te the low stress intensity fa ct or values, indicating that these fra ctu re toughness dataare in the low er transition.


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6 PREDICTIVEMATERIAL MODELING

E a r l y A p p l i c a ti o n o f th e R K R M o d e l t o U p p e r T r a n s i ti o nA p aper by Te te lman , W i l shaw and Rau (TW R) [10 ] he lps to p rov ide a pe r spec t ive

on w hy the RK R mo de l appear s to be ine f f ec t ive a t t empera tu res approach ing thef racture m od e trans i tion temperature . In thei r paper , TW R con clud e that the m icroscop icf r ac tu re s t re s s mus t be e xcee ded over a g ra in d iamete r and a ha l f fo r f rac tu re to occur. Ina r riv ing a t th i s co nc lus ion they iden t i fy th ree even t s tha t mus t occ ur p r io r to the onse t o fc leav age f rac ture in s tee l:

1 . M icrocrac k nucleat ion ,2 . P ropaga t ion o f the mic roc rack th rough the g ra in in wh ich the c rack was nuc lea ted

( i.e. the crack remains sharp an d doe s not b lun 0, an d3 . M ic rocrack p ropaga t ion th rough the boundar ies tha t su r round the nuc lea tinggrain.

T W R s ta te tha t the f i r st two even t s occur m ore eas i ly wh en g ra in bou ndary ca rb ides a r epresent . The d eterminat ion o f a gra in d iameter and a ha l f as a "cri tica l d is tance ' co m esfrom assum ing that i f the s t ress perpendicular to the p lane o f the c rack is less than them icrosco pic f rac ture s t ress a t the cri t ica l gra in bou nd ary of the 3rd eve nt, then unstab lec rack g row th w i l l no t occur .

R K R 's wo rk s eem s to bu i ld on these ideas f rom TW R. B y se t ting thei r charac te ri st icd i s tance a t two g ra in d iamete rs , they p lace the focus o f the i r m ode l on the thi rd TW Reven t . The R K R mo de l thus as sum es impl ic it ly that the fi rs t and s econ d TW R even t soccur w i th su f fi c ien t ease and f r equency to m ake the tbArd T W R even t alone control theoccurrence , or non-occurrence , o f c leavag e frac ture . At the low tem peratures ( relat ive tothe f r ac tu re mo de t r ans it ion t empera tu re ) tha t RK R w ere concerned w i th , theseassum pt ions are appropr ia te . Ho we ver , a t temp eraat res h igher in t rans it ion crackb lun t ing bec om es a m ore impor tan t i s sue to conside r. Bec ause c racks b lun t due toem iss ion o f d is locat ions f rom the t ip of the crack , b lunt ing is c ontro l led in large par t bythe f r ic t ion s t ress o f the mater ia l . Con sequent ly , b lunt ing is eas ier a t h igher temp eratures(whe re the f r ic tion s tress is low er) . At these h igher temp eratures i t cannot be assum edthat TWR's second e vent can occ ur e ither eas i ly or f requen t ly so the potent ia l for crackb lun t ing needs to be add res sed quan t it a tive ly . Thus , the as sumpt ions ma de b y R K Rregard ing c rack tip b lun t ing a re s een to have g rea t ly impa i r ed bo th the mo de l ' s accu racyand i t s phys ical appropr ia teness a t temp eratures app roaching th e f rac ture m od e trans it iontempera tu re . A t tempts to " f ix" the R K R mo de l to w ork a t h igher t empera tu res byadjus t ing o nly the parame ters o f the R K R m ode l (e~ and cry)and no t i ts fundamen tal naturehave the re fo re never en joyed succes s beyon d the spec i f i c ma te r i a ls on w h ich they werecalibrated.R K R - T y p e M o d e l s F e a t u r i n g I m p r o v e d S t re ss A n a l y s is

B y the la te 1980s and ear ly 1990s, m uch o f the indust rial inf ras tructure fabr ica ted


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KIRK ET AL. ON MODELING OF STEELS SINCE RKR 7

f rom fer r it ic steels fac ed im pen ding limi tat ions - e i ther des ign, econom ic, or reg ul at or y -on i t s cont inued useful l ife . Exa m ples include s t ructures such as oi l s torage tanks [11]and pet roc hem ical t ransmiss ion pipel ines [ 12]; i .e . s t ructures fabr icated long ago and/orus ing old techniqu es that som et im es exper ience d spectacu lar fai lures , and that invar iablyhad toughness proper t ies tha t we re e i ther no t w el l quan t i fi ed and/or f eared to be low.Other examp les inc lude nuc lear r eac tors , which w hi le hav ing wel l d ocum ented toughnessproper t ies faced regulatory l imi ts on operabi l i ty based on con cerns about serv ice relatedprop er ty degradat ion (i .e ., ne utron em bri t t lem ent) [13]. Also in this t im efram e s ignif icantadvances in com puta t iona l pow er ava i l ab le to engineer ing researchers led to a r en ewe din te res t in the appl ica t ion o f RK R- type model s . M any researcher s be l i eved the Ach i l les 'hee l o f the RK R m odel to be i ts use o f an asym pto t ic so lu tion for the c rack- tip s tr ess f i e ld( i.e . H utchinson Rice R oseng ren (HRR ) solutions , or i t s c lose equivalents ) , and sov iewe d the advent o f desk top f in i te e lemen t capabi l i ty as a w ay to ex tend the t empera turereg ime over which the mo del applies. In th is S ec t ion w e rev iew the r esu lt s o f RKR- typem ode ls that seek im prove m ents in predict ive capabi l i ties and/o r range o f appl icabi l i tythrough the use o f bet ter near- t ip s t ress solut ions than w ere avai lable to R KR in the ear ly1970s.Two-Parameter Characterization o f Cleavage Fracture Toughness

In it ia l e f for ts o f th i s type bor rowed f rom R KR the idea tha t the c r it e rion for c leavagefracture i s the a chieve m ent o f a cr i tical s tress ahead o f the crack-t ip . These ef for tsfocuse d on q uant i fying the leading non-s ingular term s in the n ear- tip s t ress f ie ld solut ionas a m eans to expand grea t ly ( r ela tive to the H RR so lu t ion used b y RK R) the s ize o f theregion aroun d the crack- tip over wh ich the ma them at ical solut ion is accurate . Thisapproach accura te ly descr ibed the deformat ion condi t ions as soc ia ted wi th m uch h ighertoughness va lues thereby enabling appl ica t ion o f the m odel s to h igher t empera tures in thet rans i tion r eg ime. Num erous approaches of thi s type we re proposed , inc lud ing theelastic-plastic, FE-based, J-Q appro ach [ 14], the elast ic J -T app roach [15], the elast ic-plas t ic asym ptot ic solut ion for J-Ae [16], and the "engin eer ing " J-yg techn ique [17] tona m e jus t a few. The se ideas di f fered in detail , but we re s imi lar in conc ept in that thesecond param eter was used to quant i fy the degree of cons t r a in t los s, which was invar iab lydef ine d as a depar ture o f the n ear- tip s t resses f rom sma l l scale yielding (SSY) condi tions .A l l o f these t echniques succeede d a t be tt e r parameter iz ing the condi t ions under w hichc leavage fa ilu re occurs , bu t none p rovided any im provem ent in p red ic t ive capa bi l i t i esbecause o f the r equi rement to per form ex tens ive t est ing o f spec imens hav ing d i f f e ren tcons t raint condi t ions to ch aracter ize wh at ca m e to be cal led the "fai lure locus" [18].Prediction o f Relative Effects on Fracture T oughness

Dodds , Anderson , and co-worker s p roposed improvem ents to these 2-parameterapp roach es [ 19]. Th eir f ini te elem en t com putation s reso lve d the elastic-plast ic s tressstate at the crac k tip in detail , an d used the se results to eva luate the cond it ions for


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cleav age f rac ture on the bas is o f the R K R fa i lure cri ter ia ( i.e ., achievem ent o f a cr i t ica ls t ress ove r a cr i t ica l d is tance) . B y comp ar ing the ca lcula ted near-t ip s t ress f ie lds ford if ferent f in i te geom etr ies to a referenc e so lu t ion for a crack t ip loaded unde r SS Ycond i t ions these inves t iga to r s quan t i f ied the e f f ec t o f depar tu re f rom SSY cond i t ions onthe app l ied-J value ne ede d to generate a par ticu lar dr iv ing force for c leavage f rac ture (asdef ined by a RK R- ty pe f a ilu re c ri te rion ). Th i s approach enab led p red ic tion o f theapp l ied - J va lue nee ded to cause c leavage f r ac tu re in one spec im en geom et ry base d ontoughness da ta ob ta ined f rom ano the r spec im en geomet ry .In the cour se o f the i r re sea rch , D odd s and A nder son de te rmined tha t the s t r e ss f i e lds infmi te geom et r ie s r emain s e l f- s imi la r to the S SY re fe rence so lu t ion to qu i t e h ighdefo rmat ion l eve ls . Bec ause o f th is , the par t icu la r va lues o f the R K R paramete r s ( i.e ., thecri t ical s tress and cri t ical dis tance, o-fand e~, respect ively) se lec ted exer ted no inf luence onthe d i f f e rences in f r ac tu re toughness p red ic ted be tw een tw o d i f f e ren t c rack geomet r ie s .Th is d i s c overy tha t the d i f f e rence in toughness be twe en two d i f fe ren t geomet r ie s d id no tdepend on the ac tual va lues o f the c r i ti ca l ma te r ia l pa ramete r s in the RK R m ode l pave dthe w ay fo r the use o f f in it e e lemen t ana lys is to accoun t fo r geom et ry and los s o fcons t ra in t e f f ec ts . In th is man ner the D odds /An der son t echn ique pe rmi t t ed toughnessva lues to be s ca led be tw een geom et rie s , the reby el imina t ing the ex tens ive t e s ting bu rdenassoc ia ted wi th the tw o-param eter techniques d escr ibed earl ier.

In sp i te o f these advan tages , the p rocedure p ropo sed by Do dds and Ander son a l so hadthe fo l lowing d rawbacks :

9 A s the deform at ion level increased , the se l f -s imilar i ty o f the s tress f ie lds in fin i tegeom et r ie s to the S SY re fe rence so lu t ion even tua l ly b roke down , m ak ing theresul ts again depe nden t on th e spec if ic values o f cr i tica l s t ress / c r i t ica l d is tanceselec ted fo r analysis .

9 The Dodds / And er son mode l a ssumes tha t an RK R- ty pe f a ilu re cr it e rion iscorrect, i .e . tha t c leav age f rac ture is contro l led so le ly b y the ac hievem ent o f acr i t ica l s tress a t som e f in i te d is tance ahead o f the crack t ip . In thei r papers , Do ddsand And erson adm it ted that th is micro-mec hanical fa i lure cr i terion wa s adoptedfor i t s convenience , and i ts s im pl ic i ty re la t ive to o ther proposals . Nev er theless , asd iscussed ear lier , the R K R fa i lure cr iter ion is in fac t a specia l case o f a m oregeneral cr iter ion for c leava ge frac ture prop osed by TW R. Thus, theDod ds /Ander son w ork d id no th ing to improve , r ela t ive to RK R, on the r ange o ftempera tu res ove r wh ich the mo de l cou ld be phys ica l ly expec ted to genera teaccurate predic t ions o f frac ture toughness .

9 Exper imen ta l s tud ies dem ons t r a ted tha t the Do dds / Ande r son t echn iquesucc ess fu l ly quan t i f ied the ef fect of cons tra in t loss on f rac ture toug hness fo r testsperform ed a t a s ingle tem perature and s train ra te [20]. H ow eve r , such resul tscould not b e used to predic t f rac ture toughn ess a t o ther temperatures / s t ra in ra tesdue to the lack o f an un der ly ing phys ical re la t ionship that included these ef fec ts in


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t h e D o d d s / A n d e r s o n m o d e l .Prediction o f Relative Effects on Toug hness: Accou nting fo r the Effects o f Both F initeCrack-Front Length and Lo ss o f C onstraint

Bec ause i t was de f ined on ly in t e rms o f s tr e s ses ac t ing to open the c rack p lane , theDo dds / Ander son m ode l canno t, by de f in it ion , cha racte ri ze the we l l r ecogn ized "wea kes tl ink" ef fect in c leav age f rac ture , w he reb y specim ens having longe r crack f ront lengthsexh ib i t sys temat ica l ly lower toughness va lues than those de te rm ined f rom tes t ing th innerspecim ens [21] . Character iza t ion of th is inherent ly three-d imens ional ef fe ct requiresadop t ion o f f a i lu re c r it e ri a tha t accoun t fo r bo th v o lum e e f fec t s and the va r iab i l ity o fcrack f ront s t resses depe nding up on proxim ity to a free sur face . Therefore in 1997Do dds , e t al . adop ted the "W eibu l l S t r es s " deve loped by the Berem in r esea rch g roup inFrance as a local f rac ture param eter [22]. This m od el beg ins wi th the assum ption that arandom dis t r ibut ion o f m icro-scale f laws that ac t as c leavage in i tia t ion s i tes ex is tsthroughou t the mater ia l , and that the s ize and de ns i ty of these f law s cons t i tu te prope r t ieso f the material . Th ese f law s are fur ther assum ed to have a d is t ribut ion of s izes descr ibedby an inver se power - law , a s fo llows :

wh ere to i s the ca rb ide d iamete r and a and f l a r e the pa ramete r s o f the dens i ty func tion g .The pro bab i l i ty o f finding a cr it ica l m icro-crack ( i .e . on e that leads to f rac ture) in som es m a ll v o l u m e Vo is then s im ply the integral o f eq. (1), as i l lustrated g raph ically in Fig.2 (a ) and desc r ibed m athemat ica l ly be low:

( 2 )

whereLo c is the cri t ical ca rbide diameter .


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10 P R E D I C T I V E M A T E R I A L M O D E L IN G

g ( I o )\ t , . , / ! " 1 0 Probabi l i ty of~ / - o J f in d in g a c rit ic a l/ m icro-crack ( i .e. , a" ~ / m icro -crack o f s ize~ o ~ _ _ ~ la rg er th an I o ) in Vo.

w - m 0t o

O"

( a ) ( b )F I G . 2 - ( a ) D i s t ri b u ti o n o f m i c r o - sc a l e f l a w s a s s u m e d b y t h e B E R E M I N m o d e l [2 2 ] , a n d(b) i ll u s t r a ti o n o f G r i f fi th f r a c t u r e c r i te r i o n a p p l i ed t o c r a c k i n g o f a ca r b i d e .

T h e B E R E M I N m o d e l f u r t h e r a s s u m e s t h a t t h e G r i ff i th f r a c tu r e c r it e ri o n [ 23 ] a p p li e sto t he c rack ing o f a ca rb ide ( s ee F ig . 2 (b )) , i . e ., t o one o f t he p re -ex i s t ing f l aw s who ses i ze dens i t y i s cha rac t e ri zed by eq . (1 ) . S t res s and f l aw s ize a re conseq uen t ly r e l a t ed asf o l l o w s :

KeY K~Y z~rc = - - , or equ iva l en t l y l~ =- - - - -5 - - (3)

w he re ere is the cri t ical s t res s , K c i s the cr i t i ca l s t ress in tens i ty fac tor , and Y is thege om et ry f ac to r fo r a c racked ca rb ide . S ubs t i t u t i ng eq . (3 ) in to eq . (2 ) a l l ow s the f a i l u rep rob ab i l i t y to be exp res sed on a s t re s s bas is , a s fo l l ows :

p : ( , : r > _ a c ) = V ( 4 a )

w here m and o -, a re the pa rame te r s o f a W eibu l l d i s t r ibu t i on , de f ined as fo l low s :m = 2 f l - 2 (4b),~ . = ( # - I ) " ' a ~'"K,Y ( 4 c )

F o r t he s im p le case o f un i ax i a l l oad ing con d i t i ons wh en t he t o t a l s t r e s sed vo lum e , V , i sv e r y m u c h l a rg e r t h a n t h e s m a l l v o l u m e Vo eq . (4a) becom es

[ ( 5 )


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KIRK ET AL. ON MODELING OF STEELS SINCE RKR 1 1

Ho wev er , to so lve f r actu re p rob le m s one n eeds to eva lua te eq . (5 ) fo r the h igh ly non-uniform s t ress s ta te ahead o f a deform ing crack-t ip . In th is s ituat ion the volu m es ( V o ) a r ema de sm al l enough tha t the as sum pt ion o f a un i fo rm s tr es s s t a te ove r the vo lum e i sreasonable , and the s tresses are evalu ated us ing the fin i te e lem ent technique. Fo r theso lu t ion o f c rack p rob lems , eq . (5 ) takes on the fo l lowing fo rm, w i th eq . (6d ) be ing theanalogue to the uniaxia l vers io n g iven in eq . (5) :

PI (~ ) = 1 - ~xp vo o-,E = E(% ,,,,cr,,,...) (6 b)

F 1 T I -a ~ = | - - I E m d D ! (6e)LVo jP I ( a . ) = 1 - e x p - a " (6d)

He re f~ rep resen ts the "p roces s z one , " wh ich i s typ ica l ly de f ined as e i the r the p las ti c zoneor a s the r eg ion w i th in w h ich the m ax im um pr inc ipa l s tr e s s exceeds som e integ ra lm ul t ip le (usual ly 2 or 3) of the y ie ld s tress. Add i t ional ly , in the so lu t ion of c leavagefracture prob lem s c7 is typica l ly taken as to the m axim um pr incipal s t ress (o-1 . In thismo de l , the pa ramete r s m and a , a r e taken to be charac te r is ti c s o f the mate r ia l r e la ted ,r espec t ive ly , to the shape o f the p robab i l i ty dens i ty func t ion desc r ib ing m ic ro -c rack s i ze(see eq . (4b)), and to the character is tics o f the probab i l i ty den s i ty funct ion descr ib ingmicro-crack s ize a s w e l l a s to the m agni tude o f the s t ress in tens i ty fac tor required tof racture a carb ide (see eq . (4c)) . H ow eve r , in pract ice m and o-, a re not def ined f rommeasurem en ts o f these mic ro - sca le pa rameter s , bu t r athe r a r e back-ca lcu la ted f rom theresul ts of mu l t ip le f rac ture tough ness ex per imen ts [24].

Th i s u se o f the max im um pr inc ipa l s t r es s to de f ine the BE RE M IN W eibu l l st re ss ineq . (6c ) iden t if i e s th is approach as be ing a RKR - type m od e l App l ica t ions o f th isapproach by Do dds and co -worke r s has succes s fu l ly p red ic ted toughnes s da ta fo r pa rt-through sem i-e l lip t ic sur face cracks from toug hness data obta ined us ing conven t ionals t ra ight- f ronted f rac ture tough ness tes t specimen s [25] ( see Fig. 3) . This success wo uIdn o t h a v e b e e n p o s s i b l e u s in g a n y o f t h e m o d e l s d e s c ri b e d p re v i o u s l y b e c a u s e t h e y h a d n om echa nism to deal wi th the cons ide rable var ia t ions in s t ress s ta te around the crack f rontthat characterize sem i-e l lip t ic sur face cracks. Ho we ver , Bass , e t a l. w ere unab le to useth is approach to predic t su ccess fu l ly the toughn ess o f cracks sub jected to b iaxia l loadingfrom conv ent ional toug hness resul ts , needing ins tead to use o f the m ean s t ress (i .e ., ~m,the average o f the three pr incipal s t resses ) ra ther than the ma xim um pr incipal s t ress in eq .(6b) to achieve a goo d predic t ion o f exper im enta l resul ts ( see F ig . 4) [26].


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1 2 P R E D I C T IV E M A T E R I A L M O D E L I N G

25.4 .~" I 152

51 102P i n - L o a d e d S C ~ I~)

l u ' [ z 5 2o o

2 5 ] 152

~0 c ~ .~

5 1 I g 2B o l t - L o a d e d 8C(T)

~ )

1.00 ,80.60,40,2'

i

o ~

er1. 01 , ~ . J ' ~ , ~ ' i 9 # . ~ , , 9 ,B a R - L o a d e d , " 6o.e ~% ~r,..,,,,,0 . 6 / :o . 4 . - I i o , - ' 5 ; ~0.2 . . . . . . . . . # '- , '/ _ '-~ ~ u ~ t ~ -

OJ'9 , _ . ~ , ~ - " . . . . . . . . . .o~ 20 40 6o 8o mo~2o~4o~ 8o~soz;,oJ C~lm~)

~ r

" - , , * > o l / / - ....

, / o / / s c ~j # / " : " _ _ w e a i . ~ r

2o 40 6o 8o loo ~2o~4o~ o ~ o ~ o oJ ( ~ / ~ )

F IG . 3-Successful prediction o f critical J values reported by G ao , et al. fo r surfacecracked specimens [25].A d v a n t a g e s a n d L im i t a t i o n s o f t h e B E R E M I N M o d e l i n g A p p r o a c h , a n d a P r o p o s e dW a y F o r w a r d

As de ta i l ed in the p reced ing S ec t ion , Do dds , ' e t al. imp lemen ta t ion o f the BER EM INapproach in the p red ic t ion o f c leavage f rac tu re succes s fu l ly p red ic ts the e f f ec t s o fcom plex chan ges in crack geom etry on f rac ture toughn ess ; predic t ions that had here toforebeen im poss ib le . Ho we ver , in i ts cu r ren t fo rm the m ode l does no t cap tu re the e f fec t s o fre la t ively s imp le changes in appl ied loading . Specif ica l ly , unde r condi t ions o f rem oteuniaxia l loading the m axim um pr incipal s t ress (cr1) m us t be taken as E in eq . (6) to obta inpredic t ions that agree wi th e xper imen ta l resul ts, where as under con di t ions of rem oteb iax ia l load ing E mu s t be s e t equa l to the mean s t res s (~% ) . Th is incons i s tency in thes t ress that appears to be contro l l ing f rac ture sugges ts that, l ike i t s predece ssors , theBE RE M IN approach does no t fu l ly desc r ibe the phys ica l p roces ses r espons ib le fo rc leava ge f rac ture . Ad di t ional ly , the Berem in approach conta ins no informat ion regardingho w the m ode l paramete r s (m and a , ) change w i th t empera tu re and st ra in ra te . Theseparam eters ca n be bac k calcula ted f rom the resul ts of repl ica te f rac ture toughne ss tes tsperfo rm ed under the temp erature and s train ra te condi t ions of interes t. H ow eve r , the needto pe r fo rm such de ta il ed exper imen ta t ion f rus tr a tes use o f the Berem in m ode l in ap red ic t ive capac i ty and , m ore p rac ti cal ly , m akes app l ica t ion o f the mode l bo th cos t ly andt ime-consuming .


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To progress toward fully predictive models that are not as costly to implement, itseems important to enhance the Beremin model in two ways. First, it needs toincorporate a constitutive model that accounts for the physical phenomena responsible fortemperature and strain rate dependency so that these trends don't have to be backcalculated from toughness data. Secondly, a physically based criterion for cleavagefracture is needed so that the model can be applied with confidence to any condition ofinterest within the transition fracture regime. These refinements are discussed in thefollowing Sections.

FIG. 4-Fo ur-inch thick biaxial load frac ture specimen (left) an d frac ture toughness data(right) fo r biaxial loading reported b y Williams, et aL [26] demon strating that betterprediction o f the trends in toughness data is achieved by using the mean stress as ~ in eq.(6) rather than the max imum principa l stress. The material tested is A S T M A533 B thatwas given a special hea t treatment to elevate its yiel d strength; it has a A ST M E1921 Tovalue of- 3 7 ~A P hysically Ba sed Constitutive M ode l fo r F erritic Steels

Dislocation mechanics-based constitutive models describe how various aspects ofthe microstructure of a material control dislocation motion, and how these vary withtemperature and strain rate. The microstructural characteristics of interest include shortand long-range barriers to dislocation motion. The lattice structure itself provides theshort-range barriers, which affect the atom-to-atom movement required for a dislocationto change position within the lattice. An inter-barrier spacing several orders of magnitudegreater than the lattice spacing defines long-range barriers. Long-range barriers includepoint defects (solute and vacancies) precipitates (semi-coherent to non-coherent),boundaries (twin, grain, etc), and other dislocations in BCC materials.

As a stress is applied to a metal dislocations begin to move, which results in plasticdeformation. For a dislocation to move it must shift from one equilibrium position in the


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14 PREDICTIVE MATERIAL MODELING

l a tt ice to ano the r , overcom ing an energy ba r ri e r to do so . The w ave leng th o f thesebarr iers i s equal to the per iodic i ty of the la tt ice . Th e d is locat ion requires appl ica t ion o f afo rce to o verco m e th i s ene rgy ba r ri e r and the mag n i tude o f th is fo rce is the Pe ie rl s -Nab arro s t ress , XpN. W hile m ovin g the d is locat ion wi l l a lso enc ounter o ther bar r iers suchas so lu te a toms , vac ancies , precip i ta tes , inc lus ions , bou ndar ies and o the r d is locations .Dif fere nt spacings and s ize scales characterize the se ad di t ional barr iers .

A t temp eratures o ther than absolu te zero , a toms v ibra te abo ut thei r la t t ice s i tes a t atem perature-de pend ent ampl i tude. Ad di t ions of thermal ene rgy ( i.e. , h ighertempera tu res ) ac t to inc rease the am pl i tude o f a tom v ib ra t ion , inc reas ing the p robab i l itythat an a tom wil l " jum p" f rom one equi l ibr ium s i te to another. This thermal en ergy thusac t s to dec rease the m agn i tude o f the energy p rov ided by the shor t r ange obs tac les tod i s loca tion mot io n a t any mo m ent in t ime , the reby dec reas ing the fo rce r equ ir ed to mov ethe d is locat ion f rom o ne equi l ibr ium pos i t ion to the next . Th e ef fec t o f s train ra te O . e . ,dis locat ion veloci ty ) has a s im i lar but op pos i te ef fec t to that of temperature . A s s t ra inra te increases less t im e is av ai lab le for the d is locat ion to ov erco m e the shor t- rangebar ri e rs p rov ided b y the l a t ti ce a toms . Th i s dec reases the e f f ec t o f the rmal ene rgy ,resul ting in an inc reased force required for d is locat ion mot ion .

Lon g-range o bs tac les d i f fer f rom shor t- range obs tac les be cau se changes in thermalenergy do no t g rea t ly a f f ec t the ab i l i ty o f d i s loca t ions to m ov e pas t them a t the s t ra in r a testypical ly exper ience d by c iv i l and mechan ical s t ructures that are not des igned fo r creepservice . This oc curs becau se a tomic v ibra t ion am pl i tude has l i t tle ef fec t on the s ize o f thelong-range ene rgy bar r ier presented to the d is locat ion , wh ich is on the order o f the inter -pa r t ic l e spac ing . Consequen t ly , the amoun t o f ene rgy r equ i r ed to mo ve a d i s loca t ion pas tthese l a rge obs tac les i s o rde rs o f magn i tude l a rge r than tha t p rov ide d by the inc reasedla t t ice v ibra t ion that resul ts f rom increases in temperature. In thei r work , Arm strong andZer i ll i conc luded f rom ca re fu lly ana ly ized s e t s o f Tay lo r exper imen ts tha t overcom ing thePe ie r ls -Nabar ro ba r r ie r s w as the p r inc ipa l the rmal ly ac t iva ted mechan i sm in BCCmaterials [27-30].

The fund am ental ly d i f ferent nature o f shor t- and long -range bar r iers to d is locat ionm ot ion mo t ivates a separa te t rea tment in d is locat ion-based cons t i tu t ive formulae .Consequen t ly , the f low s t res s o f a ma te ri a l i s typ ica l ly expres sed as a sum o f a thermaland thermal co m pon ents , associa ted wi th long-range and shor t - range bar r iers( respect ively) , as fo l low s:

o - = ( 7 )The f i r s t te rm occ urs due to the a thermal or long-range bar r iers to d is locat ion m ot ionwh i le the sec ond term descr ibe d the ac t ion o f therm al ly ac t ivated shor t- range barr iers .The shor t - range bar r iers includ e the Peier ls -Nabarro s t ress and d is locat ion forests .Peier ls-Nabarro s tresses ( lat t ice fr ict ion s tresses) are the controll ing short-range barr iersin BC C m etals wh i le d is locat ion fores t s tructures are the con tro l ling short -range bar r iersin FC C and H C P metals . This d i f ference is respon s ib le for the d i f ference in s train ra tesens i tiv i ty be tw een BC C and FC C meta l s. Focus ing a ttent ion on BC C m eta ls , the


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t empera ture depe nden ce of the probabi l i ty tha t a d i s loca t ion wi l l overcom e a shor t- r angeobstacle is give n by:

w here k is Bo l tzm an's cons tant , T is the absolute temp erature o f interes t , and A G is theact ivat ion ene rgy o f a barrier [31]. The s t rain rate ef fect shares a s im ilar form:

k = k o e x p ( A % T ) (9)Solv ing eq. (9) for A G gives

Equ at ion (10) clear ly shows that act ivat ion en ergy for dislocat ion m ot ion de creases wi thtemp erature and increases wi th s t rain rate .

Us ing the se equat ions as the bas i s fo r the i r m odel , Zer i ll i and A rms t rong found tha tdis locations ov ercom ing Peierls -Nabarro barr iers are th e pr incipal therm al ly act ivatedm echan ism for deformat ion in BCC m eta ls [32]. The spac ing o f these obs tac les i s equa lto the la t tice spacing and thus is not a f fected b y pr ior plas tic s t rain as are the dis locat ionfores t s t ructures in FCC metals . Zer i l l i and A rm strong developed the fol lowingexpress ion for the thermal por t ion of the f low s t ress in eq. (7) (assuming a cons tantobs tacle spacing fo r BCC):

o-* = C 1 ex p( - f iT) (11)wh ere f l depen ds on s t rain and s t rain rate as fol lows:

f l = - C z + C 3 Ink (12)Com bin ing eqs. (11) and (12) wi th com m only accepted te rms descr ib ing Orow an- types t rengthening due to athermal barriers resul ts in th e fol low ing descr ipt ion o f the f lowbehav i o r o f B C C me t a ls :

cr f =c% +C 4s" + p_d- '/2 + C , -e x p [ - C zT + C3T. ln (~)] (13)w h e r e C] , C2 , C3 , C4 , t t , a nd n are m ater ial cons tants , and d is the grain diam eter . Th eform o fe q. (13) is cons is tent wi th dis locat ion-m echan ics based cons t i tutive m ode ls


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1 6 P R E D I C T I V EMATE RIAL MODELING

dev e lop ed b y o the r re sea rcher s [33 -34]. F rom the phys ica l under s tand ings tha t under l ieeq . (13) the e xpec ta t ion ar ises that the tem perature and s train ra te depen den ce ( i .e .,q-exp[-c27" + C3T. n(k)])s h o u l d b e co m m on to a l l fer ri t ic s tee ls . This i s bec ause the la tt ices t ructure is th e sa m e in a ll fer r it ic s tee ls i r respect ive o f thei r a l loying , heat t rea tment,deg ree o f co ld work , deg ree o f p r io r p las ti c s tr ain , and degree o f i rr ad ia t ion damage ;fac to r s w ho se in f luence a re desc r ibed by the te rms ~ + c48"+ ~/-1~. Fig. 5 dem onstra testhat exper im enta l data val idates th is expecta t ion . The se figures d isp lay an excel lentco r respondenc e be tw een the temp era tu re and s tra in r a te depende nce o f a w ide va r ie ty o fs tee l a l loys and the predic t ions of eq . (13) for pure i ron ( i .e ., C1 = 1033 M Pa, C2 =0.00698 / ~ and C3 = 0 .000415 / ~A Physical ly Ba sed Criterion fo r Cleavage Fracture

Cottre l l [35], Petch [36] , Sm ith [5-6] and ma ny o ther researchers hav e a t temp ted todesc r ibe c leavage f rac tu re m echan isms quan t it a tive ly . He re we exam ine the 1968 worko f T e t e l m a n , W i ls h a w , a nd R a n ( T W R ) [ 1 0]. T h e y d e sc r ib e t h e fo l l o w i n g se q u e n c e o fthree eve nts that es tabl ish the nec essary precursors for c leavage f rac ture :

Even t #1 .

Even t #2 .

Even t #3 .

C leavage f r ac tu re beg ins w i th mic ro -c rack nuc lea t ion , wh ich i s w e l ldocum en ted as r esu l ting f rom d i s loca tion m ot ion and accum ula t ion a tlong-range ob s tacles such as gra in boun dar ies and seco nd phase par ticles .The va lue o f s tr a in neces sa ry to nuc lea te the m ic ro -c rack quan t if ie s th isevent .Fo l lowing nuc lea t ion , the mic ro -c rack needs to p ropaga te th rough thegrain in wh ich i t w as nuc leated . A cr i t ica l value o f s tress t riax ia l ity isnee ded to keep the crack t ip sharp . Otherw ise d is locat ions wi l l be em it tedfrom the crac k t ip , there by b lunt ing i t, which s tops the c lea vag e f rac tureproc ess and prod uce s ins tead a non-propagat ing crack .The f ina l even t i s the sub sequen t p ropaga tion o f the m ic ro -c rack th roughthe boun dar ies that sur round the nucleat ing gra in . This ev ent occurs wh enthe open ing s tr es s a t the c rack t ip exceeds the m ic roscop ic c leavagefracture s t ress "ov er {a character is t ic d is tance of} a t leas t one gra ind iamete r in the p las t ic zo ne ahead o f the c rack . "

I n 1 9 73 , R K R [3 ] e x p a n de d o n T W R ' s h y p o t h e si s a n d d e v e l o p e d th e i r o w n"character is t ic d is tance" o f tw o gra in d iam eters a t lo we r t rans it ion reg ion temperatures .Ho we ver , the f a il ing o f the RK R m ode l , and a l l o f it s subsequen t p rogeny , has be en i tsexc lus ive focus o f T W R 's E ven t #3 , e f f ec t ive ly imply ing tha t Even ts 1 and 2 a re s a ti s fi edapriori . In spec i fi c , w e men t ioned ea r li e r in th is paper tha t the RK R m ode l does no tperform w el l a t temp eratures in the uppe r t rans it ion region . Ad di t ional ly , appl ica t ion o ft h e R K R - t y p e B E R E M I N m o d e l h a s b e e n u n s u c c e s sf u l i n t h e p re d i c ti o n o f f r a ct u re i nspecim ens that hav e an appl ied b iaxia l load . I t i s reasonab le to pos tu la te that bothfai lings re la te to R K R 's assum ed ins ignif icance o f Eve nt #2 (adequ ate s t ress tf i -ax ia li ty


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to ex tend the m ic ro -c rack to the boundar ies o f the g ra in in w h ich i t in it ia t ed ) fo r thefo l lowing r easons:

13 .Ik=,==l

!

6 0 0

4 5 0

3 0 0

1 5 0

0

- 1 5 0

+ F o r g i n g / U n . l r r ~ F o r g i n g / P o w e r9 F o r g t n g / i ' e s t X P t a te , 'U r l - lr r= ' P l a t e / P o w e r 9 P l a t e / T e s tX WeldAJn-lrr A W e l d / P o w e rA W e l d / T e s t m , m , m P r e d i ct io n

0

A

- 3 0 0 , - ,- 2 0 0 0 2 0 0 4 0 0

T e m p e r a t u r e [ ~

6 O 0

4 5 0

~ oo1 5 0

? o-150-3O0

-200

z ~ H S L A Platea C - M n P l a t e & W e l dn Q & T Ma r t e ns i t ic P la te

P r e d i c t i o n

0 2 0 0 4 0 0

T e m p e r a t u r e [ ~4 0 0

~" 3 o 0rr 2 0 0~JIX . 100E

o< ~

r - 6 5 C D a t aI ~ - 1 2 C D a t al x 2 9 C D a t a

- 6 5 C P r e d i C t io n- 1 2 C P r e d i c t i o n

............ 29C Predicbon oO

4 0 0

11,~ ; 3 0 0(D

r . 2 0 0

n , , lOOE~ , o

r< ~> L i n k - P ~ a t e 14 Ai - -J f

O . 0 1 O , 1 0 1 0 0 0.1 0 o . l O 0

~ 0 O 0O l ~ o o 0 o lS t r a i n R a t e [ l/ s e c ]

C2 .45C DataZ x . ~ 0 C pDeadi~c~ n

- 9 0C P r e d i c t i o n f

R e f e r e n c e" C o n d i t i o nMinand S N 4 9 0 B

i io . o o l o . t l O lO O OS t r a i n R a t e [ l/ s e c ]

FIG. 5-E xp erim en tal data demonstrates an excellent correspondence between thetemperature an d strain rate dependence o f a wide va riety o f s teel alloys and thepredict ions o f eq. (13) o r pu re iron.


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In U pp er Trans it ion: In the low er t rans it ion region that R K R inves t igated , theassum ption that ad equate t r i -ax ia l i ty exis ts under a l l condi t ions is appropr ia tebe cau se the la t t ice f r ic tion s t ress , w hich is tem perature dep endent , i s h igh enou ghto p reven t s ign i fi cant d i s loca tion em is s ion f rom the t ip o f the m ic ro -c rack and ,consequ ent ly , i s h igh enou gh to preven t s ignif icant crack t ip b lunt ing [37-38] .H ow eve r , a t temperatures h igher in the t rans i t ion reg ion the f r ic t ion s t ressdecreases, a l lowing d is loca t ion emis s ion and mo t ion to occur more eas ily . Th i sleads to a greater poss ib i l i ty o f m icro-crack b lunt ing , and a c onseq uent fa i lure toach ieve Even t #2 . Thus , in upper t rans it ion the occur rence o f Even t #2 canno t beas sume d and mu s t be ch ecked fo r independen tly .I n th e P r es e n c e o f A p p l i e d B i a x i a l L o a d i n g : M o s t f r a ct u re m e c h a n i c s t e stspecim ens are s imilar in that the ap pl ied loading acts only to open the crack .W hile specim ens that have lo ad appl ied para l le l to the crack p lane are rare ,s t ructures that app ly load para l le l to the crack p lane are qui te comm on. Fo rexam ple , an y in ternal ly pressur ized vessel generates a b iaxia l s ta te o f s t ress in thevesse l wal l , and thermal s t resses are inherent ly two -or three d imen s ional in an yres t ra ined s t ructure . A rem otely appl ied mu l t i -axia l s t ress s ta te c lear ly m akes i teas ier to achieve E vent #2 and, conseq uent ly , eas ier for c leavag e frac ture to occur .Ho wev er , u s ing the max im um pr incipa l s t r es s a s ~7 in the B ER EM IN m ode l ( eq.(6) ) fa i ls to provid e a ny mea sure o f s t resses appl ied para l le l to the crack p lane,and thus can not be expe cted to predic t fa i lure for loading condi t ions o f th is type .

in c los ing i t is re levant to note that, in i t s current im plemen tat ion , c a lcula t ion o f theW eibu l l s tr e s s fo l lowing the a pproach sugges ted by the Berem in g roup invo lves anarb it ra ry s e lec t ion o f the pa ramete r ~ . f2 r ep resen t s the s i ze o f the "p roce s s zone" fo rc leavag e f rac ture , w hich is typical ly def ine d as e i ther the p las t ic zon e, o r as the regionwi th in wh ich the max im um pr incipa l s t r es s exceeds som e in teg ra l mu l t ip le (usua l ly 2 o r3) of the uniaxia l y ie ld s t ress . Thus , the current Berem in mo del includes an arb i t rary andnon-p hys ical ly base d def in i tion o f the proc ess zone for c leavage f rac ture . This arb i t raryse lec t ion can be r ep laced by appea l to T W R 's Even t #1 , wh ich sugges t s tha t the p roces szone for c leavage f rac ture is the region wi th in w hich suff ic ient s train has d eve lope d toinitiate a micro-crack.Sum ma ry and C onclus ions

In th i s paper we hav e r ev iew ed the deve lopm en t o f mode ls tha t a t temp t to p red ic t thet rans it ion f rac tu re behav io r o f BC C meta l s , beg inn ing w i th the l andmark paper pub l i shedby Ritch ie , Kno t t, and R ice (RK R) in 1973 and work ing fo rward to the B ER EM INm odel , which is the focus of current research ef for ts . In the pas t three dec adescons ide rab le p rogres s has been m ade in p red ic t ion accuracy . Never the les s , the bes tc leavage f rac ture m ode ls avai lable curren t ly st il l inc lude co eff ic ients back-ca lcula tedfrom f racture toug hness exp er iments , these being nec essary to incorporate in to the m ode lsthe tem perature and s tra in ra te depe nde ncy of fer ri tic s tee ls . M oreov er , exper imenta l


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evidence dem ons t ra ting tha t success fu l b l ind pred ic tions o f f a i lu re cannot be m ade evenfor the s imple case o f appl ied biaxial loading cal ls into q ues t ion the robustness o f thesem ode ls to the pred ict ion o f the fai lure condi t ions in eng ineer ing s tructures .

To address these shor tcomings and progress tow ard fu l ly pred ic t ive models twoenhancem ents a re nee ded to the BER EM IN mod el . F i rs t, i t needs to incorpora te ad i s loca t ion-mechanics based cons t itu tive mod el such as tha t p rop osed by Z er il li andArms t rong . Such mo dels account for the phys ica l phe no m ena respons ib le fortemp erature and s t rain rate d epe nde ncy in fer r i tic s teels so that th ese t rends do n ' t h ave tobe back ca lcu la ted f rom toughne ss da ta ; ins tead the y becom e in tegra l part s o f the model .Secondly , a phys ica lly -based c r it e rion for c leavage f rac ture i s need ed so tha t the m ode lcan b e appl ied wi th conf idence to any cond i tion o f in te res t w i th in the t rans it ion f rac turereg ime. To th is end the use of a m odel p roposed by T ete lman, Wi lshaw, and Ran (TW R)is sugges ted. W e dem onstrate that the TW R 3 -event c leava ge f racture cr i ter ion is in facta m od e gen eral express ion o f the cr i tical st ress / cr i t ical dis tance cr i ter ia popu lar ized byRK R, and that this increased general i ty is exp ected to recon ci le recog nized def iciencies inRKR, and RKR - l ike c leavage f rac ture mo dels (such as the BER EM IN m odel ).Reference s

[1 ]

[2 ]

[ 3 ]

[ 4 ][ 5 ][ 6 ][ 7 ]

[ 8 ]

W atanabe, J ., Iwa date, T., Tanaka, Y. , Yo kob oro, T. , and A nd o, K. , "Frac tureToughness in the Trans it ion Region , " Engineering Fracture Mechanics, 28, pp.589-600, 1987.Kno t t , J .F . an d Boceaceini , A.R. , "The Fracture M echa nics-M icros t rucm reLinka ge," M ech anics and Materials: Fun dam entals and Linkages , M eyers , M .A. ,et al., Eds., W ile y & Sons, pp. 399-424, 1999.Ritchie, R. , Kn ott , J.F., and Rice, R. , "O n the R elat ionship B etw een Crit icalTen si le Stress and Fracture Stress in M ild S teels , " JM ech Phys So l, 21, pp. 395-410, 1973.Orow an, E., Transactions o f the Institute o f Eng ineers a nd Shipbuilders inScotland, 89, p. 16 5, 1945.Sm ith, E., P hysical Bas is o f Yield and Fracture, Co nferen ce Proceedings , Ins ti tuteo f Physics and th e Physical Society, Lond on, p . 36, 1966.Sm ith, E., "C leav ag e Frac ture in M ild Steels ,'" Int. J. Fra ct. M ech., Vo l. 4, pp.131-145, 1968.Rice, J .R. , and Ro sengren, G.F., "Plane Strain Deform at ion Ne ar a Crack Tip in aPow er -Law Hardening M ateria l, " Journal o f Mechanics a nd Physics o f Solids, 16,pp. 1-12, 1968.Hu tchinson, J.W. , "Singular Behavior at the E nd o f a Ten s i le Crack in aHardening M ater ia l , " Journal o f M echanics and P hysics o f Solids, 16, pp. 13-31,1968.


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20 PR ED IC TIVEMATERIAL MODELING

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pa r am e te r - I . S t r uc tu r e o f f ie lds , " Journal o f Mechanics and Physics o f Solids,39, pp. 989-1015, 1991.[ 15 ] H a n c o c k , J . W . , R e u t e r , W . G . , a n d P a r k s , D . M . " C o n s t r a in t a n d T o u g h n e s sP a r a m e t e r i z e d b y T," Constraint Effects in Fracture, AS TM S TP 1171, E . M .H a c k e t t , K . - H . S c h w a l b e , a n d R . H . D o d d s , E d s . , A m e r i c a n S o c i e t y f o r T e s ti n gan d M a te r ia l s , Ph i l ade lph ia , pp . 21-41 ,1993 .[ 16] Ch ao , Y J . , Yang , S ., and Su t ton , M . A . , " O n t h e F r a c tu r e o f S o l i d s C h a r a c t e ri z e d

b y O n e o r T w o P a r a m e t e rs : T h e o r y a n d P r a c t ic e , " Journal o f the Mechanics a ndPhysics o f Solids, 42(4) , pp.629-647, 1994.[ 17] N ew m an, J .C . J r, , Cr ew s , J .H . , B ig e low , C .A . , and D aw icke , D .S . , "Va r i a t ions o fa G l o b a l C o n s t ra i n t F a c t o r in C r a c k e d B o d i e s U n d e r T e n s i o n a n d B e n d i n gL o a d s , " Co nstraint Effects in Fra cture, The ory an d Applications." Seco nd Volume,A S T M S T P 1 2 44 , M a r k K i r k a n d A d B a k k e r E d s. , A m e r i c a n S o c i e ty f o r T e s ti n gand M ater ia l s , Ph i lade lph ia , pp. 21-42, 1995.[ 18] O 'Do w d, N .P . , Sh ih , C .F ., "Fam i ly o f c r ack- tip f i e lds cha r ac t e r i zed by a t r iax i a l i typa r am e te r - I I. F r ac tu r e Appl i ca t ions , " Journal o f Mechanics an d Physics o fSolids, 40, pp . 939-963, 1992.[ 19] D odd s , R .H . , A nde r son , T .L . , and Ki r k , M .T . , "A Fr am ew or k to Co r r e l a te a /WRat io E f f ec ts on E la s t i c - P la s t i c F r ac tu r e Tou ghn ess (Jc)," International Journal o fFracture, 48, 1991, pp. 1-22,[ 20] K i r k , M .T . , Ko ppen hoe f e r , K.C., and S h ih , C .F ., "E f f ec t o f Co ns t r a in t onS p e c i m e n D i m e n s i o n s N e e d e d to O b t a i n S t r u c tu r a ll y R e l e v a n t T o u g h n e s s

M e a s u r e s " , Constraint Effects in Fracture, AS TM S TP 1171, E.M . Hacke t t , K . - H .S c h w a l b e , a n d R . H . D o d d s , E d s . , A m e r i c a n S o c i e ty f o r T e s t in g a n d M a te r ia l s,Ph i lade lph ia , pp. 79-103, 19 93.


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K IR K E T A L . O N M O D E L IN G O F S T E E L S S I N C E R K R 21

[21 ] W allin, K ., "T he Size Effect in Kzc Results," Eng ineering Fracture M echanics,22, pp. 149-163, 1985.[2 2] Berernin, F.M., "A Local Criterion for Cleavage Fracture o f a N uclea r Pressure

Vessel Steel," M etallurgical Transactions, 14A , pp. 2277-2287, 1991.[2 3] Griff ith, A, "The Pheno me na o f Rupture and Flow in Solids," PhilosophicalTransa ctions, Series A, 22 1, pp. 163-198, 1920.[2 4 ] Gao , X ., Do dds, R.H., Tregoning, R.L., Joyee, J.A., and Link, R.E., "Effects o fLoading Rate o n the W eibull Stress M odel for Simulation o f Cleavage Fracture inFerd tic Steels," Proc. o f the 2000 A SM E Pressure Vessel and Pipin g Conference,ASM E, July 2000.[2 5] Gao, X ., Dodds, R.H., Tregoning, R.L. , and Joyce, LA. , "Cleavage Fracture in

Surface Crac k Plates: Experim ents an d Nu merical Predictions," Proc. o f the 1999AS M E Pressure Vessel and Piping Conference, ASM E, July 1999.[2 6 ] W illiams, P.T., Bass, B.R., and M cA fee, W .J. , "A pp lication of the W eibullM ethodology to a Shallow-Flaw Cruciform B end Specimen Un der BiaxialLoading C onditions," Fracture Mechan ics, 31st Volume, A S T M STP 1389, G. R.Halford and J. G allagher, Eds., Am erica n Society for Testing and M aterials, 2000.[27] F.J. Zerilli and R.W . Arm strong, s AppL Phys., 61, (1987) 1816.[28] F.J. Zerilli and R.W . Arm strong, Shock and Com pression o f Cond ensed M atter,

1989, eds. S.C. Schm idt, J.N. Johnson, an d L.W . Dav ison, Elsevier, Am sterdam,1990, p357.[29] F.J. Zerilli and R.W . Arm strong, s AppL Phys., 68, (1990) 1580.[30] F.L Zerilli an d R.W . Arm strong, or. AppL P hys., 40, (1992) i803.[ 3 1 ] Ko cks, U .F., A.S . A rgon, and M .F. Ashby, Progr. M ater. Sci., 19, p. 1, 1975.[3 2 ] Zerilli, F. J. and R. W . Arm strong, "Dislocation-mec hanics-based constitutiverelations for material dynam ics calculations," s AppL Physics, Vol. 61, No. 5, 1March 1987.[3 3 ] Johnson, G .R., and W. H C ook, Proc. 7 h Int. Syrup. Ballistics, A m . Def. Prep.Org. (ADP A), Th e Netherlands, p. 54 1, 1983[3 4 ] Follansbee, P.S. and Kooks, U.F. , Ac ts Met ., 36, p.81, 1988.[3 5 ] Cottrell, A.H., TheoreticalAspects o f Fracture, Fracture , A verbac h, B .L, et al.,.Eds. , W iley & Sons, p. 20 -44, 1959.[3 6 ] Petch N.J., "T he Du ctile-Cleavage Transition in Alpha Iron ," Fracture, Averbach ,B.L, et al ., Eds. , W iley & Son s, p. 5 4-64 , 1959.[3 7 ] Rice, J.R. and Thomson , R. "Ductile versus brittle behaviour o f crystals," Phi l

Mag, 1974:29(1):73-97.[3 8 ] W eertman, J., Dislocation Based Fracture M echanics, Singapore: W orldScientific, 1996.


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Anssi Laukkanen,~Kim Wallin,2 Pekka Nevasmaa, t and Seppo TLlatinen1T r a n s f e r a b i l i ty P r o p e r t i e s o f L o c a l A p p r o a c h M o d e l i n g i n t h e D u c t i l e to B r i t tl eT r a n s i t i o n R e g i o n

R e f e r e n c e : Laukkanen, A., Wallin, K., Nevasmaa, P., and Tghtinen, S., "Trans ferab i l i tyP r o p e r t ie s o f L o c a l A p p r o a c h M o d e l i n g i n t h e D u c t i le t o B r i t t le T r a n s i t io n R e g i o n ,"Predict ive M ateria l Modeling: Combining Fu nda me ntal Ph ysics Understanding,Computational Me thods an d Empirically Observed Behavior, A S T M STP 1429, M . T.Kirk and M. Erickson Natishan, Eds., ASTM International, West Conshohocken, PA,2003.Abstract : Qualitative descriptive potential of local approach models for upper shelf andtransition region has been well established. Models such as the Gurson-Tvergaard-Needleman and the Beremin model have been demonstrated to enable characterization offracture phenomena over the entire transition regime. The foundation for consistent localapproach characterization of material failure has been set, but quantitative application isstill at developing stages. Current work addresses properties of a modified Bereminmodel in terms of material parameter consistency and transferability, which are taken asmeasures of performance with respect to quantitative applicability. Three-point bendspecimen fracture toughness data of sizes 3x4x27 ram, 5x5x27 ram, 5x10x55 mm and10x 10x55 mm are simulated in the ductile to brittle transition region using finite elementmethods and inference of local approach parameters is performed using pointwisecollocation and stochastic methods. The suitability of the calibration methods and overallmodel performance are evaluated and demonstrated.K e y w o r d s : Brittle Fracture, Beremin model, Miniature Specimens, Local Approach,Transferability.I n t r o d u c t i o n

The use of fracture mechanics in design and failure assessment is in some practicesimpeded by the difficulties in quantifying the structure related constraint andtransferability properties of experimental test data. It is well known that specimen size,crack depth and loading conditions may effect the materials fracture toughness.Transferability of small specimen toughness data to real structures has long been the keyissue in fracture mechanics research. Methods based on the Weibull statistic, such as the"Master Curve" methodology and local approach methods of fxacture have beendeveloped and are able to characterize the scatter of fracture toughness test results and theeffects of specimen dimensions on the data distribution, making it possible to defineJ Research Scientists and 2Research Professor, VTT Indusl~ialSystems,P.O. Box 1704, FIN-02044VTT,Finland.

22Copyright92004 by ASTM International www.astm.org


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LAUKKANE N ET AL. ON TRANSFER ABILITY OF PROP ERTIES 2 3

f rac tu re toughness pa ramete r s fo r a g iven p ro bab i l i ty o f f a ilu re and de ve lop s ea l ingm od els fo r tough nes s transferabili ty.

Ongo ing dev e lopm en t o f loca l approach m ethod s fo r c leavage f rac tu re has l ed to thein troduc tion o f s evera l d i f fe ren t types o f ma te r ia l m ode l s and pa ram ete r ca l ib ra t ionprocedures . The f eas ib i li ty o f these method s to p rac t i ca l pu rposes has usua l ly beendemons t r a ted b y pe r fo rming pa ramete r in fe rence on the bas i s o f a t ta inab le exper imen ta lda ta and in man y cases i t can be a rgued tha t eve n l imi ted genera l i ty o f the used m ethodsis doubtfu l . I t can be s ta ted that qual i ta t ive desc r ip t ive potent ia l of local approach anddam age m echan ics m ode ls has been we l l au then t ica ted , bu t quan t i ta t ive p roper t i e s a re acom plete ly d i f ferent mat ter .

The cu r ren t wo rk con t r ibu tes to the mat te r by ana lyz ing a r e la t ive ly ex tens ive f r actu retoughness dataset us ing a m odif ied Bere m in m od el [1 ,2 ,3]. This co l lec t ion of frac turetough ness data in the du ct i le to br i tt le trans i t ion region cons is ts o f bo th i r radia ted andre fe rence da ta o fA 53 3B C1. 1 ( JRQ) ov er a w ide t em pera tu re r ange and w i th d i f f e ren ts ize bend type f rac tu re mechan ics spec imens . T he exper imen ta l r e su l ts a re p roces sedus ing the M as te r Curve m ethod and tw o fundam en ta l ly d if f e ring loca l approachparam eter ca libra t ion m ethod s are appl ied on the bas is o f the dataset. De pen den cies in thecal ibra ted param eters are desc r ibes and d iscussed , and on the bas is o f the resul ts , aca l ib ra tion m ethod o logy p roduc ing the w ides t r ange o f p roper ty t rans fe rab i l ity i spresented .T e s t i n g a n d m a t e r i a l

Th e material and experimental proce dures are de scribed in detail in [4] , w he re thef racture toughness data is appl ied to dem ons tra te per forma nce o f smal l f rac ture mech anicsspecim ens in the du cti le to brit tle transit ion region. A 53 3B C1.1 (JRQ ) in T-L orientationhas a f ie ld s t rength o f 486 M Pa and a tens i le s trength o f 620 MPa. I rradia tion o f the s tee lwa s per formed approxim ately 15 years ago a t a temperat~xre o f 265~ with a neutronf luency of approximately 1 .5 x 1019n/era2 (E > 1 M eV ). At the irradiated cond ition theyie ld s trength is 687 MP a and tens i le s trength 815 M Pa. T he T28J transit ion temperaturesare -29~ and +84 ~ and upp er she lf Ch arpy-V energies are 210 J and 129 J , respectively.

The inves tiga ted spec imen geom et rie s were o f th ree -po in t bend type (3PB ) w i th s izes10-10-55 ram, 5-10-55 rnm, 5-5-27 mm and 3-4-27 m m , respect ively . The spec im ens arep resen ted s chemat ica l ly in F ig . 1. Refe rence o r ien ta t ions were that o f T-L whi le thei rr ad ia ted w ere L-T . The spec imens were t e s ted in th ree po in t bend ing w i th a span tospec imen w id th r a tio o f 4 . /n i t i a l c r ack l eng th to spec imen w id th was 0 .5 and thespec im ens w ere s ide -g rooved 10% on each s ide . The t e s ting and ana lyses rou t inesf o l lo w e d A S T M sta nd ard E 1 8 2 0 -1 9 9 9a ( T e s t M e t h o d f o r M e a s ~ e m e n t o f F r a c tu r eToughn ess ) and E 1921-2002 (S tandard T es t Me thod fo r De te rmina tion o f Re fe renceTem perature , To, for Ferr it ic S teels in the Trans i t ion Range) .


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Fig. 1--T hree -poin t b end specimens used in material characterization and numericalanalysis.

Master C urve AnalysisThe Master Curve analys is fo l lowed the AST M Test Method for Determinat ion of

Refe rence T emp erature, To, for Ferritic Steels in th e Tra nsition R ang e (E 1921-2002).Tw o levels of censoring were4 applied. First, for all data referring to "n on-cleav age"(ductile end o f test) i t was prescribed that 6i = 0. Second, all data violating the specimensize validity criterion were assigned the toughness valu e corresponding to the v aliditycriterion with 8i = 0.

A plastic q-factor of 2 was use d instead of 1.9 as prescribed in E1921. T he reason forthis was that the testing and J-analysis followed E1820. The effect of using the q-factor o f2 instead of 1.9 is a 1-2~ bias of the To values towards lower temperatures.

For the comp arison o f different size specimen data, and for the calculation of theMaster Curve transit ion temperature To, all data were thickness-adjusted to the referenceflaw length (thickness) B0 = 25 m m with Eq. 1 as prescribed in E1921:

.( (1 )


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LAUKKANE N ET AL. ON TRANSFER ABILITY OF PROPER TIES 2 5

T h e t h ic k n e s s , B , r e f e r s to t h e n o m i n a l t h i c k n e s s , r e g a r d l e ss o f s id e - g r o o v i n g .F o r a l l d a t a s e ts , T o w a s e s t i m a t e d f r o m t h e s i z e - a d ju s t e d K se d a t a u s i n g a m u l t i -

t e m p e r a tu r e r a n d o m l y c e n s o r e d m a x i m u m l ik e l i h o o d e x p r e s si o n ( E q. 2 ) [ 5] :I1~ 1 .1 1 8 i . ex p {0 .0 1 9 [T -T o ] } ( K , q - 2 0) 4 .e xp {0 .0 19 [T i - T o ] }= + 7 7 " e x p { 0 . 0 1 9 [ T i - T 0 ] } s ( l l + 7 7 - e x p { 0 . 0 1 9 [ T i - T o ] } ) s = 0 ( 2 )

T h e t r a n s it io n t e m p e r a t u r e , T o , w a s s o l v e d b y i t e r a ti o n f r o m E q . 2 . Y i e l d s t r e n g tht e m p e r a tu r e d e p e n d e n c y w a s e v a l u a te d f o l l o w i n g t h e S I N T A P - p r o c e d u re a s

r = o _ ~ r + l e5 / (4 9 1 + 1.8T)-189MPa, w h e r e R T d e n o te s r o o m t em p e r a tu r e a n d' yt e m p e r a t u r e i s g i v e n i n d e g r e e s C e l si u s .

I d e a ll y , t h e M a s t e r C u r v e s h o u l d n o t b e f i t te d t o d a t a b e l o w T o - 5 0 ~ i .e . d a t a c l o se too r o n t h e l o w e r s h e lf , w h e r e a t e m p e r a t u r e f it b e c o m e s h i g h l y in a c c u r a te a n d w h e r ed e v i a ti o n in t h e l o w e r s h e l f t o u g h n e s s fr o m t h e M a s t e r C u r v e a s s u m p t io n m a y b i as t h er e su l ts . A l s o , t h e e f f e c t o f e x t e n s i v e d u c t il e t e a r i n g s h o u l d b e a v o i d e d . T h e r e f o r e , in t h ed e t e r m i n a t i o n o f T 0 t h e d a t a w e r e l i m i t e d t o - 50 ~ 17 6 T h i s i s i n h a r m o n y w i t ht h e l at e st r e v i s i o n o f th e M a s t e r C u r v e s t a n d a r d , A S T M E 1 9 2 1 - 2 0 0 2 .

I n o r d e r t o p e r f o r m l o c a l a p p r o a c h p a r a m e t e r c a l ib r a t io n o n t h e b a s i s o f M a s t e r C u r v es ca t te r , t h e n o rm a l i za t i o n f r ac tu r e t o u g h n es s , K o , w as e s t i ma t ed . T h e M as t e r C u rv e s ca t te ri s d e sc r ib e d b y E q . 3 a c c o r d in g t o A S T M E 1 9 2 1 - 2 0 0 2 :

PI [ ~ ,K o_K~ (3 )

T h e e s t i m a t i o n o f t h e n o r m a l i z a t io n f r a c tu r e t o u g h n e s s K 0 , c o r r e s p o n d i n g to a 6 3 . 2 %f a il u r e p r o b a b i l i t y w a s c a r ri e d o u t o n t h e b a s i s o f r a n d o m l y c e n s o r e d M a x i m u mL i k e l i h o o d , E q . 4 [6] :

+x 11 , (4 )

w h e r e t h e c e n s o r i n g p a r a m e t e r 6 i i s 1 f o r u n c e n s o r e d a n d 0 f o r c e n s o r e d d a ta , t h el i m i t i n g f r a c tu r e t o u g h n e s s h a v i n g a f i x e d v a l u e o f 2 0 M Pa ~ m .N u m e r i c a l a n a ly s is


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26 PR ED IC TIVEMATER IAL MODELING

Num er ica l ana lyses a t tempera tu res co r respond ing to exper imen ta l f r ac tu re toughnessd a t a w e r e c a rr ie d o u t u s i n g th e W A R P 3 D r e se a r ch c o d e v e r s i o n 1 3.9 d e v e l o p e d b yUn iver s i ty o f I ll ino i s [2] . The com puta t ions p resen ted in the cu r ren t paper we re ca r ri edou t in 2D p lane s tr a in cond i t ions. T he e last ic -p las ti c ma te ria l behav io r wa s mo de le d w i than inc rem en ta l i so t rop ic ha rden ing fo rmula tion . T he de fo rm at ion desc r ip t ion waspresen ted in a f in it e s tr a in Lagrang ian f r amework . M eshes we re genera ted fo r 10 10x55m m , 5 x 1 0 x 5 5 r a m , 5 x 5 x 2 7 m m a n d 3 x 4 x 2 7 m m s m a ll 3 P B s p e ci m e n s . 8 n o d e b b a r-s tab i li zed 3D so l id e lemen ts were used in the computa t ions . A m esh fo r a CV N s izespec im en i s p resen ted in F ig . 2 .

Fig. 2 - - M e s h u s e d o r C V N s iz e s pe cim e n .The tem pera tu re depend ency o f y ie ld s tr eng th w as eva lua ted in accordance w i th the

M as te r Cu rve method . S t ra in ha rden ing exponen t was eva lua ted on the bas i s o f y ie ld totens i le s t rength ra t io us ing an exp ress ion deve lope d b y Au erkar i [7]:

(Cro/cr~ = ( 3 0 " I / N ) - 1 / N ) , w h e r e c ro is the yie ld strength, crt~ the ten sile stre ngth andN the s t ra in hardening expo nent .O n the bas is o f num er ical a nd e xper im enta l f rac ture toughne ss resul ts for the t ransi tion

reg ion the pa ram ete r s o f a th ree -pa ramete r W eibu l l d i st r ibu t ion were f i tt ed us ing am a x i m u m l ik e l ih o o d (M M L ) s c h e m e . T h i s w a s p e r f o r m e d u si n g th e W S T R E S S [3 ] c o d eand Ma tLab bu i l t eva lua t ion rout ines . The th ree pa ramete r W eibu l l mode l /m od i f i edBerem in m ode l fo r c leavage in it ia t ion i s p resen ted as

P / = 1 - e x p [ - ( ~ - c r th m ] ,L t ~ 1 7 6 J

( 5 )


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LAUKKANE N ET AL. ON TRANSFER ABILITY OF PROPER TIES 2 '7

w h e r e o -W s t he W eibu l l s tr e s s, o -, t he s ca l e pa ram ete r , m the shape pa ram ete r and o -~the t h resho ld s t r e ss . The W eibu l l s t r e s s is p resen t ed as

1

t v 0 g J ( 6 )w he re o-~ is the f i rs t p r i n c i p a l s t r e s s , Vo i s a r e f e r e n c e v o l u m e s e t t o u n i t y a n d f 2 i s th ef rac tu re p roces s zone . T he p roces s zo ne wa s d e f ine d as f ] : o " > )L. o -0 , w her e two va lueso f ~ we re used , 1 and 2 , t o eva lua t e r e su l t s p roces s zo ne depe nden cy , o -,h wa s iden t i f i edw i t h K ~ , , i .e . i t c o r r e sp o n d s t o a W e i b u l l st re s s a t K = K ~ , .

T h e c a l i b ra t io n o f th e s h a p e a n d s c a l e p a r a m e t e r s w a s c a r r i e d o u t u s i n g a M M Lr o u ti n e. T h e f o r m o f u s e d M M L f o r a W e i b u l l p r o b a b i li ty d e n s it y f u n c t io n o f E q . 5 w a s

L ( m , o - , ) = m e t I Io -w l e x p - o r E c r , e x p - ( n - r ) c r , c r w c ,i = l / i M

( 7 )

w h e r e n - r i s t h e n u m b e r o f c e n s o r e d s a m p l e s . T h e p a r a m e t e r s w e r e d e t e r m i n e d b yi t e ra t i v e l y f i n d i n g th e m a x i m a o f E q . 7 .

T h e c a l ib r a t io n p r o c e s s c o n s i s ts o f d e t e r m i n i n g t h e v a r i a ti o n o f th e W e i b u l l s tr e ss w i t hd i f f e r e n t v a l u e s o f t h e s h a p e p a r a m e t e r to f i n d t h e m a x i m u m o f E q . 7 b y i n p u t ti n g t h ee x p e r i m e n t a l f r a c tu r e t o u g h n e s s r e su l ts . T w o d i f f e re n t p h i l o s o p h i e s w e r e a d o p t e d f o r t h i sp roces s . F i r st , a d i rec t app roach by us ing t he ac tua l exper imen ta l s amples a t a f i xedt e m p e r a t u r e w a s u t il iz e d . S e c o n d , th e M a s t e r C u r v e n o r m a l i z a t i o n t o u g h n e s s w a sd e t e r m i n e d a n d t h e r e s u l ti n g d i s tr i b u ti o n w a s u s e d t o s t o c h a s t i c a l l y g e n e r a t e t h e s a m p l efo r pa ra m ete r i n fe rence . In t h i s case Mo n te -C ar lo s am pl ing

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