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N A S A

b n04mNInzI-

T E CHN I C A L N O T E TN D-2395--

STRESS-INTENSITY FACTORS FOR ASINGLE-EDGE-NOTCH TENSION SPECIMENBY BOUNDARY COLLOCATIONOF A STRESS FUNCTIONby Bernard Gross, John E. Srawley,an d W il liam F. Brown, Jr.Lewis Research CenterCleueland, OhioN A T I O N A L A E R O NA U T IC S A N D S PA CE A D M I N I S T R A T I O N W A S H I N G T O N , D . C . A U G U S T 1 9 6 4

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TECH LIBRARY KAFB. NMIIIIIIIIlllIIIIIllllIll111

STRESS-INTENSITY FACTORS FOR A SINGLE-EDGE-NOTCH TENSIONSPECIMEN BY BOUNDARY COLLOCATION O F A STRESS FUNCTION

By Bernard Gross , John E . Srawley,and William F . Brown, J r .Lewis Research CenterCleveland, Ohio

NATIONAL AERONAUTICS AND SPACE ADMINISTRATIONFo r sa le b y the Offi ce of Tech nic al Services, Department of Commerce,

Washington, D.C. 20230 -- Pr ice $0.50

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STRESS-INTENSITY FACTORS FOR A S1NGL;E-EDGE-NOTCH TENSIONSPECIMEN BY BOUNDARY COLLOCATION OF A STRESS FUNCT I ON

by Bernard Gross, John E. Srawley,and W i l l i a m F. Brown, Jr.Lewis Research Center

SUMMARYA boundary va lue col loc a t ion procedure appl ied t o th e W i l l i a m s s t r e s s

func t ion w a s employed t o d e te rm in e t h e e l a s t i c s t r e s s d i s t r i b u t i o n i n t h e im -m ediate v i c i n i t y of t h e t i p of an edge c rack in a f ini te-width specimen sub-jec ted t o uni form t e n s i l e loading. This typ e of s ingle-edge-notch spec imen i sp a r t i c u l a r l y s u i t a b l e f o r de te rm in a ti on of p la n e s t r a i n f r a c t u r e to ug hn es sva lues . The ana ly t i ca l r e su l t s a r e expres sed i n such a way t h a t t h e s t r e s si n t e n s i t y fa c t o r may be determined from known c ond itio ns of specimen geometryand loading.A s t h e c rack leng th dec reased , t h e r e s u l t s ob ta ined by th e co l loca t ionprocedure approached those derived from a closed so lu t ion f o r an edge crack ina sem i - in f in i t e p l a t e . Over a range of r a t i o s of crack length t o specimenwidth between 0.15 and 0.40 t h e c o l l o c a t io n s o l u t i o n y ie ld e d r e s u l t s i n v e ry

good agreement with those derived from experimental compliance measurements.

INTRODUCTIONA method f o r c a l c u l a t in g t h e s t r e s s d i s t r i b u t i o n i n a t e s t specimen con-t a i n i n g a single-edge crack (sharp notch) and subjec ted t o a u ni fo rm t e n s i l eload i s d e sc r ib e d h e r ei n . The r e s u l t s a r e p a r t i c u l a r l y u s e f u l i n de te rm in in gs t r e s s i n t e n s i t y f a c to r s K f o r given con diti ons of load and geometry andth ere fo re permit t h e use of t h e s ingle-edge-notch specimen i n f ra c t ur e tough-n e s s t e s t i n g .The ASTM Sp ec ia l Committee on F ra c tu re T es ti ng of High St re ng th Me ta ll icM ate r i a l s i s sued a se r i e s of r epor t s desc r ib ing r ecen t developments i n f r a c t u r et ou gh ne ss t e s t i n g ( r e f s . 1 o 4 ) . It has been shown t h a t th e magnitude of t h ee l a s t i c s t r e s s f i e l d i n t h e i mmediate v i c i n i t y of a crack but beyond the crackt i p p l a s t i c zone may be chara c ter ize d by a single parameter K , t h e s t r e s si n t e n s i t y f a c t or ( r e f s . 1 and 5 ) . For any given material a c h a r a c t e r i s t i cva lue K,t o th e onset of r ap id f r ac tu re . L ike o the r m echanica l p rope r t i e s , K, i sof t h e s t r e s s i n t e n s i t y f a c t o r i s assumed t o ex is t t h a t corresponds

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d ep en den t on t h e s t r a in r a t e , t h e t e mp e ra tur e , an d t h e t e s t i n g d i r e c t i o n . I nth e c a s e of s h e e t and p l a t e m a te r i a l s it i s a l so dependent on th e th ickne ss .The va lu e of may be dete rmi ned from t e s t s on specimens containing sharpnotches or c ra ck s, p ro vide d t h a t s u i t a b l e e x pr e s sio n s a re a v a i l a b l e t h a t g i vet h e s t r e s s i n t e n s i t y f a c t o r i n terms of t h e specimen geometry and ap pli ed lo ad sa t f r a c t u r e i n s t a b i l i t y .K,

Approximate solutions for K exi s t i n c l o s e d f o r m f o r a number of speci-men desi gns s ymm etrically notched wi th re spe ct to t h e t e n s i l e l o ad a x i s( r e f s . 5 t o 9 ) . The single-edge-notch tension specimen appears to be more ef-f i c ie n t , however, than symmetr ica l ly notched specimens with resp ect both to t h emater ia l and t o t h e l o a d in g c a p a c i t y re q u i re d ( r e f . 10). For h i s r ea so n, t h esingle-ed ge-notc h specimen may be of con side rabl e importance i n t h e determina-t i o n of K,s e c t i o n s a r e an inheren t requi rement of th e t e s t . Recen t, ve ry ca re fu l exper i -mental compliance measurements on th e s ingle-edg e-notch specimen (r e f . 10) pro-v i d e v a l u es of s t r a i n e n er gy r e l e a s e r a t e s as a f u n c t i o n of c r a ck l e n g th f romw'hich values of K may be derived. An a n a l y t i c a l s o l u t i o n i s desirable, how-ever, as an independent check on t h e exp erim enta l procedure; it a l s o h as t h eadvantage th a t t h e in f luence o f c e r ta in geomet r ica l pa ramete rs, such as t h er a t i o o f h e ig ht t o wi dt hte di ou s experimen tal measurements. Furthermore, t h e method of obta inin g ana n a l y t i c a l s o l u ti o n i s a p p l i c a b l e to a l l combinations of bending and tensiona p pl ie d t o a single-edge-notch specimen.

f o r pl a n e s t r a in c r a c k p r op a ga t io n where r e l a t i v e l y l a r g e c r o s s

V/W, may be r ap id ly determined without re so r t to

An a n a l y t i c a l s o l u t i o n t o t h e s t r e s s d i s t r i b u t i o n i n t h e s i ng le -e dg e-notch tens ion specimen i s obta ined here in by a boundary value col locat ion pro-c ed ur e a p p l i e d t o t h e W i l l i a m s s t r e s s f u nc t io n ( r e f . ll), which i s known tos a t i s f y t h e boundary con d i t ions a long an edge c rack .t h a t p e r m i ts e x pr e ss io n of s t r e s s i n t e n s i t y f a c t o r s i n terms of t h e measuredqu an t i t i es of load and specimen dimensions. In addi t ion, t h e inf lu enc e of t h eend e f f e c t on t h e s t r e s s i n t e n s i t y f a c t o r i s determined. The end ef fe ct de-r i v e s from t h e f i n i t e d i s t a n c e b etw een t h e c r a c k a nd t h e u ni fo rm ly l oa de dboundary, expres sed as V/W. A comparison i s made between t h e pres ent an al yt i-c a l r e su l t s and a c losed s o lu t io n ob ta ined by Wiggleswor th ( r e f . 1 2 ) f o r anedge c ra c k i n a s e m i - i n f i n i t e p l a t e .t er ms of t h e s t r e s s i n t e n s i t y f a c t o r i s compared with experimental results ob-t a i n e d b y o t h e r i n v e s t i g a t o r s f o r t h i s specimen w it h s t r a i n e ne rg y r e l e a s e r a t e(compliance measurement ) experiments.

The r e s u l t s a r e i n a form

F i n a ll y , t h e c o l l o c a ti o n s o lu t i o n i n

SYMBOLSa crack len gth i n s ingle-edge-notch specimen, in .

c o e f f i c i e n t s o f W i l l i a m s s t r e s s f u nc t io ndnE Young' s modulus, p s i

s t r a in e ne rg y r e l e a s e r a t e w i th c r a c k e xt en sion ; o r c r a c k e x te n s ionforce , in . - lb / sq i n .99

2

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K

vWx,Y

s t r e s s i n t e n s i t y f a c to r of e l a s t i c s t r e s sf i e l d i n v i c i n i t y of b o rd er o f c r a ck ,p s iload pe r un i t t h ickness , l b / in .a n g ul a r p o s i t i o n c o o r d in a te s r e f e r r e d t o

c ra ck t i pdi s ta nc e (he igh t ) between crack p lane andloc a t io n of un iform s t r e s s , i n .specimen width, i n .coord inate axes wi th o r ig in a t c r ack t i p ,p a r a l l e l and pe rpend icu la r , r e spec t ive ly ,t o c ra ck p l an eu ni fo rm t e n s i l e s t r e s s a p p l i ed t o specimen,

p s is t r e s s i n x- and y-di rec t ions , p s i0yJ *xy

X s t r e s s f u n ct i on

uo - 10,oOO psi

o0 - 10,oOO psiFigure 1. - Specimen geometryand loading assumed forcollocation solution.

METHODThe method of ana lys i s c on s is t s i n f ind ing a s t r e s s f u n ct i on X s a t i s f y -

ing th e biharmonic equat ionnumber of s ta ti o n s along t h e boun darie s of t h e single -edge -notc h specimen showni n f i g u re 1. For t h e present purposes use i s made of t h e W i l l i a m s s t r e s s f un c-t i o n ( r e f . 11)w i t h t h e c o r r e c t i o n o f a t y p o g r a ph i c a l e r r o r i n t h a t r e f e re n c e :

$X = 0 and th e boundary condi t ions a t a f i n i t e

m

2n - 3[- os (n - ;)e + 2n + c os ( n + $4n=l, 2, . ..

+ ( -1) d 2nrn+1[- os (n - i cos(n +Because of symmetry (fig. 1) on ly even t e rm s o f t h e s t r e s s fun c t ion a r e con-s idered . The s t re ss es in te rms of X o b ta in e d b y p a r t i a l d i f f e r e n t i a t i o n a r eas fo l lows:

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2- a2x - a2x 2 a2x s i n e cos e + ax s i n ea y - - - - c o s 8 - 2 ZEF r z rax2 a,ax s i n e cos 0 a% s in20+ 2 -!---z r a02 r 2

The Wil l i ams s t r e s s f u n c t io n i s an Airy s t r e s s funct ion, which, bes id es s a t i s -fy i ng t h e biharmonic equation, a l s o s a t i s f i e s t h e bo un dar y c on d i t i o n s a lo ng t h ecrack surface , namely, th a t th e normal and shear ing s t re ss es be zero . Thus,when 8 = %T equa t ions (1)and ( 2 ) g iv eboundary requirements on th e s t r e s s func t ion f o r t h e specimen having th e geom-e t r y a nd t r a c t i o n s shown i n f i g u r e 1 a r e as fo l lows :

cry = 0, rxy = 0. The remaining

Along boundary A -,B:axx = o a = O

Along boundary B -,C:

Along boundary C -+ D:

Because of symmetry with respect t o t h e crack plane ( f i g . 1)o n ly h a l f t h especimen need be considered.For t h e purpose of determining t h e s t r e s s i n t e n s i t y f a c t o r as def ined in

d l of t h ere fe rence 13, which c h a r a c t e r i z e s t h e s t r e s s d i s t r i b u t i o n i n t h e imm ediateneighborhood of th e crack t i p ( r --* 0) , o n ly t h e f i r s t c o e f f i c i e n t4

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W i l l i a m s s t r e s s f un c ti on i s n e ce ss ar y, s i n c e t h i s t e rm i s dominant. A s shownl a t e r , d l i s p ro p or ti on al t o t h e s t r e s s i n t e n s i t y f a c t o r K. Values of d las w e l l as of t h e o th e r c o e f f i c i e n t s a r e o b t a in e d b y s a t i s f y in g t h e b ou nd arycondi t ions (eq .given boundary f o r a specimen with t h e geometry shown i n fi g u re 1 h a t i ss u b j e c te d t o a uniform s t r e s s of 10 ,000 p s i ac t i ng a t a d i s t a n c ecrack plane. Computat ions were made f o r sev era l r a t i o s of crack length t ospecimen wid th a / W between 0.04 and 0.5 and f o r va lues o f V/W ranging from0 .5 t o 1 . 5 .

( 3 ) ) a t a f i n i t e number o f s t a t io n s equa l ly spaced a long aV from the

The co l loca t ion p rocedure req u i res a matr ix so l u t io n of twice as manyequat ions as t h e number of boundary s ta t i o n s se le cte d f o r each combination oft h e independent va r ia ble s . This problem w a s programed f o r a d i g i t a l c om puterw i th t h e u s e of double p rec i s ion a r i thmet ic (16 s i g n i f i c a n t f i g u r e s ) .I n t h i s sol ut io n, t h e number of boundary s ta t i o n s i s i nc re a se d u n t i l t h e

f i r s t m a t r ix c o e f f i c i e n t d l c on ve rg es t o a s u f f i c i e n t l y s t a b l e v al ue . F ig -u r e 2, f o r example, shows t h e f i r s t m a t r ix c o e f f i c i e n t as a f u n c t i o n of t h e

Ratio of height to width atwhich uniform loadingis assumed to occur,$1..6059~ 23

Number of boundary stationsFigure 2. - First matrix coefficient as function of number of boundary stations.Tensile stress applied to specimen, 10,OOO psi: specimen width, 1 nch; ratioof crack length to specimen width, 1/3.

number of boundary s t a -t o n s f o r c o n f ig u r a t io n sw i th s e v e r a l V/W r a t i o sa t a va lue o f a/w of1/3. T h e v a r i a t i o n i nt h e f i r s t m a t r ix c o e f f i -c i e n t i s not more than+1 pe rc en t when t h e num-ber of boundary s ta t ionsi s inc reased from11 t o 23.

The s t ress - func t ionva lues a t 50 s t a t i o n salong t h e boundary werecomputed f o r se ve ra lgeometries with a l l dncoef f ic ien t s . Theseval ues were i n goodagreement wi t h th e pre-scr ibe d value s . Thes t r e s s - f u n c t i o n d e r i v a -t i v e no rm al t o t h e bound-ar y, however, showed pe r-t u r b a t i o n s n e a r t h ecor ner s of th e specimen.The effect of t h i s v a r i a t i o n i n t er ms o f s t r e s s d i s t r i b u t i o n t hr ou gh ou t t h especimen could be determined only by ad di t i on al computat ion. This add i t i on ale f f o r t d i d no t a pp ear j u s t i f i e d s i n c e t h e f i r s t m a t r ix c o e f f i c i e n t w a s , f o rp r a c t i c a l p urp oses, i n s e n s i t i v e t o t h e s e p e r t u r b at i o n s .

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RESULTSS t r e s s I n t e n s i t y F a c to r s

The s t r e s s i n t e n s i t y f a c t o re f f i c i e n t of t h e W i l l i a m s s t r e s s f u n c ti o n . The e x pr e ss i on f o r t h e s t r e s s i nt h e y - d i r e c t i o n i n t h e im mediate v i c i n i t y of t h e c ra c k t i p i s obta ined f romt h e dominant term as Tollows:

K may be der ived i n terms of t h e f i r s t co-

82+ s i n - s i n

The expr ess ion f o r ISs t r e s s a n a l y s i s , i s as f o l l o w s :

g ive n i n r e f e r e n c e 1, based on t h e Westergaard crackY

2 2 30)8cos (1 + s i n - s i n -IS = -JS

(4 )

Thus

Ratio of crack length to specimen width, alWFigure 3. - Collocation results of a plot of dimensionlessparameter against ratio of crack length to specimenwidth in single-edge-notch specimen.

A s shown i n f i g u re 2, s m a l l o s c i l l a t i o n ssometimes occur i n th e f i r s t matr ix co-e f f i c i e n t . For t h i s r ea so n s t r e s sin te ns i t y fa c t o r s were computed wi thva lues of dl averaged from 1 1 t o 23boundary points .For purposes o f f r ac tu re toughnesst e s t i n g t h e r e s u l t s o f t h e c o l lo c a t io n

procedure are convenient ly expressed int h e form of a dimensionless parameterinvolving K and th e measured qu an t i t i e sa s a fu nc ti on of a/W ( r e f . 10). Thus,

where P i s t h e l o ad p er u n i t t h i c k n e s s .A s d i sc u ss e d l a t e r , t h i s form i s u s e f u lf o r a comparison o f t h e an a l y t ic a l r e -s u l t s with exper imenta l compliance cal i -b r a t i o n d a t a . A curve der ived f romequa t ion (3 ) r e l a t i n g K ~ W / P ~ o a/wi s g i ve n i n f i g u r e 3.t o any V/W v a lu e g r e a t e r t h a n a b o u t0.8 ( s e e f i g . 4 ) .

This curve app l ies

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Infl uen ce of End Ef fe ct sA s an a i d t o opt imizing t h e specimen des ign , ca lc ula t ion s were made t oshow how cl os e t h e assumed po si ti on of t h e uniformly loaded boundary could bet o t h e c ra c k p la n e w ith ou t a f f e c t in g t h e s t r e s s i n t e n s i t y f a c t o r .va lue of t h e pos i t io n , of course , i s a funct ion of the specimen width,and express ing the f i r s t m a t r i x c o e f f i c i e n t as a f u n c t io n of t h e r a t i o ofh e ig h t t o w id th ( s e e f i g . 1, p. 3 ) i s convenient f o r var iou s va lues of a/w.According t o f ig u r e 4, t h e f i r s t matrix c o e f f i c i e n t a t0.15 and 0 .5 i s e s s e n t i a l l y c o ns ta nt f o r r a t i o s of h ei gh t t o w id thg r e a t e r t h a n a bo ut 0.8.

For a givena / W

a / W ra t ios be tweenV/W

-3, 000.5 . 7 .9 1.1 1.3Ratio of height to width, V IW

Figure 4. - First ma trix coefficient as function of ratio of height to width for variousratios of crack length to specimen width. Tensile stress applied to specimen,10,OOO psi: specimen width, 1 nch.

I n a p ra c t ic a l t e s t specimen t h e means by which load i s app l i ed in t roducesnonuniformly s t r es se d regions a t t h e ends of t h e specimen. The actual specimenle ng th must t he re fo re exceed th e minimum len gt h determined from fig ur e 4.t h e c a s e of a pin-loaded specimen, f o r example, t h e optimum r a t i o of t o t a ll e n g t h t o w id th i s about 4 ( re f . 10).I n

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DISCUSSION OF RESULTSComparison with Wigglesworth Solution .

A check on t h e va l i d i t y o f t h e p resen t so lu t ion can be ob ta ined by com-par i son w i th th a t r epor ted by Wigglesworth ( re f . 1 2 ) f o r an edge c rack i n asem i- i nf in i te p la t e . These two sol ut i on s should converge as t h e c r a c k l e n g t ht e n d s t o z ero . I n t h e M e d i a t e v i c i n i t y of t h e c ra ck t h e f i r s t t e rm of th eWigglesworth so lu ti on predominates, and 5 may be ex pre ssed i n terms of thecoordin ate sys tem shown i n f ig u re 1 (p . 3 ) as fo l lows:

This equation may be compared with t h e corresponding express ion obt aine d by th epr es en t met hod:

+ s i n - s i n -2e2 25 = Jr (4 )where the f i r s t m a t r i x c o e f f i c i e n t dl depends on 50 and t h e specimen dimen-s io n s . The r e s p e c t i v e v a lu e s of i n a ny d i r e c t i o n n e ar t h e c r a c k t i p maybe compared by cons ider ing a single-ed ge-notch specimen of un it width. For t h esame uniform stress i n b o th ca se s. t h e r a t i o of 577 obta ined f rom theWigglesworth solution (eq. ( 8 ) ) t o t h a t computed f$om equ ati on ( 4 ) should ap-

\\

8

\\

\\

\\

8Crack length, .Figure 5. - Comparison of stress ratios in immediate vicinity of crack tip obtained by collocation

solution and solution of reference 12.proach 1 as t h e c r a c k l e n g th d e c r e a s e s .i n accordance wi th th i s behav ior . The r e s u l t s p r e se n t e d i n f i g u r e 5 a r e

Comparison with Experimental ResultsTwo experimental compliance ca li br at io ns of sin gle-edg e-notch specimens

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11111111111111111111II I 11 I II II I I I I II I I 1111 I 1111111111~11111~11

l oa de d i n t e n s io n a r e a v a i l a b l e f o r c om pariso n, t h e e a r l i e r by S u l l i v a n( r e f . 1 4 ) and a more re ce nt one by Srawley, Jo ne s, and Gross ( r e f . 1 0 ) . Thedesign of t h e specimen of r efe ren ce 1 4 w a s loaded through pins separa ted by adis tance le s s than twice th e width, which in t roduced la r ge end e f f ec t s t ha ta r e not accounted f o r i n t h e ana l y t i ca l so lu t ion . The spec imen used in re fe r -ence 10 w a s of s u f f i c i e n t l e n g th t h a t end e f f e c t s were n e g l ig ib l e , s i n c e t h ecompliance measurements were made over a gage length of 8 inches on a specimen3 inches wide that w a s loaded through pins 10 inches ap ar t . The suf f ic ien cy oft h i s gage length w a s es t abl ish ed by pre l imina ry exper iments . For t h i s reasonb e t t e r agreem ent w i th t h e a n a ly t i c a l r e s u l t s i s t o be expec ted from t he exper i -ments of r e f e r e n c e 10 than f rom tho se of refer ence 1 4 . Furthermore, as d i s -cussed in refe renc e 10, t h o s e d a t a a r e e x pe ct ed t o b e more p r e c i s e t h a n t h eda ta of re fe ren ce 1 4 because of di ff er en ce s i n specimen s i z e and measurementtechniques .

The experimental compliance procedure gives r e s u l ts i n terms of t h e s t r a i ne n e r g y r e l e a s e r a t e g.9For t h e purposes of comparing th e an a ly t ic a l wi th th e exper imenta l r e su l t s th emost reasona ble procedure appears t o be a conversion on t h e p la ne s t r e s s b a s i s .Thus,

The corre ct procedure fo r conver t ing th es e values oft o s t r e s s i n t e n s i t y f a c t o r s i s n ot y et c om p le te ly s e t t l e d ( s e e r e f . 10).

o r i n terms of exper imenta l ly measured qua nt i t ie s ,

I L

where i s determined by experimental compliance procedures and P i s t h eload per unit specimen thickness.Rat io o fc r ack l en g tht o specimenwidth,

a/wI

0.05.10.1 5.20. 25.3 0.35. 40. 4 5.50

Dimens o n l e s paramet e r,K2W/P2

Fxperiment a1r e s u l t sRef. 10

0.314.556.816

1.1801 . 7 3 52 . 5 7 13.7755.4367 . 6 4 110.477

Ref. 1 40.35

. 651.001 . 4 01 . 9 72 .804 . 2 06 . 1 88 . 9 0

1 2 . 5 0

Col loca t ionr e s u l t s

0.204.445.7581.180

1.7682.6033.8135.596

12.399a. 276

The comparison between e xp er i-mental and ana ly t ica l r e s u l t s i sshown i n t h e t ab le . A s might beexpected from the foregoing discus-s io n of th e compliance ca l i br a t io nexper iments, th e re su l t s g iven i nre fe rence 1 4 are cons i s ten t ly h igh-e r t h a n e i t h e r t h o s e o b ta ine d b yth e p r e se n t c o l l o c a t i o n s o lu t i o n o rt h o s e r e p o r t e d i n r e f e r e n ce 10. I nco nt ra st , ve ry good agreement be-tw een t h e a n a ly t i c a l s o lu t i o n a ndt h e da ta of re fe rence 10 i s notedf o r v a l u es of a / W between 0.15and 0.40. The differences betweenth e s e two s e t s of r e s u l t s a t t h elower values of a / W are probablya s s oc i a te d w i th u n c e r t a in t i e s i n

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I I1 I

t h e lower range of th e exper imenta l data. For h e a / W values above 0.4, d i f -fe rences due to bending of t h e ex peri men tal compliance specimen, which are nottaken in to accoun t by th e an a l y t ic a l so lu t ion , become impor tan t . Th is bend ingd e c r e a s e s t h e e c c e n t r i c i t y o f l o a d in g w i th r e s p e c t to t h e uncracked sec t ion ,and th e compliance f o r a given slot l e n g th i s t h e re f o re s l i g h t l y l e s s t h a n i fno bending took place.

SUMMARY OF RESULTSThe r e s u l t s o f a n a n a l y t i c a l in v e s t i g a t i o n of t h e s t ress i n t e n s i t y f ac -

tors for a s ingle-edge-notch te ns io n specimen obta ine d by a boundary v alu eco l loca t ion p rocedure app l ied to t h e W i l l i a m s s t ress f u n c t i o n are as fo l lows:1. The v a lu e s o f t h e s t r e s s i n t e n s i t y f a c to r were i nd ep en de nt o f t h e d i s -tan ce between th e uniformly loaded cross sec t io n and th e no tch p lane p rov idedt h a t t h i s d i s ta n ce was g r e a t e r t h a n 8 0 p e r ce n t of the wid th .2. A t s m a l l r a t i o s o f c r a ck l e n g th to specimen wid th th e p resen t re su l t swere i n good agreement w it h a c losed so lu t ion ob ta ined fo r an edge c rack in as e m i - i n f i n i t e p l a t e .3. When t h e a na ly t i ca l r e su l t s were expressed i n app ropr ia te d imension-l e s s form, ve ry good agreement was obta ined wi th comparab le re su l t s ob ta inedfrom a h ig h ly a c c u r a t e e x p e rim en tal s t r a i n e ne rg y r e l e a s e r a t e de te rmina t ion .

L e w i s Research CenterNat iona l Aerona utics and Space Administrat ionCleveland, Ohio, May 5, 1964

REFERENCES1. ASTM Sp ec ia l Committee: Fr ac tu re Te st in g of High-Stren gth Sheet M ate ria ls,p t . 11. ASTM Bull . 244, Feb. 1960, pp. 18-28.2. ASTM S p e c i a l Committee: The Slow Growth and Rapid Prop agat ion of Cracks.M ate ria ls Res. and Standards, v ol. 1, no. 5, May 1961, pp, 389-393.3. ASTM S p e c i a l Committee: F r a c tu r e T e st in g of High-Strength Sheet Materials.Materials Res. and Standard s, v ol . 1, no. 11, Nov. 1961, pp. 877-885.4. ASTM Sp eci a l Co mi t te e : Progress i n Measur ing Fra ctu re Toughness and UsingFr ac tu re Mechanics. Ma ter ial s Res. and Standards, vol. 4, no. 3,Mar. 1964, pp. 107-119.5. Irwin, G. R.: Analysis of St re ss es and Str a i ns Near th e End of a CrackTravers ing a P la te . Jo ur . Appl. Mech., v ol . 24, no. 3, Sept. 1957,pp. 361-364.

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6. Sneddon, I. N.: The Dis t r ibut ion of St re ss i n t h e Neighbourhood of a Cracki n an E la st ic So lid . Proc. Roy. SOC. (London), Ser. A, vol . 187,no. 1099, Oct. 22, 1946, pp. 229-260.7. Mendelson, Alexander, and Spero, Samuel W. : E l a s t i c S t r e s s D i s t r i b u ti o n i na Fini te-Width Orthotro pic Pl a t e Conta ining a Crack. NASA TN D-2260,1964.8. Bowie, Oscar L.: Rectangular Tensile Sheet with Symmetric Edge Cracks.TR-63-22, Army M a te r ia l s R e s . Agency, Oct. 1963.9. Irwin, G. R . : Crack-Ekdension Force for a Part-Through Crack i n a P la t e .. Jour. Appl. Mech. (T rans . ASME), ser. E, vol. 29, no. 4, Dec. 1962,pp. 651-654.

10. Srawley, John E. , Jones, Melvin H., and Gross, Bernard: ExperimentalDetermination of t h e Dependence of Crack Exte nsio n Force on Crack Lengthfor a Single-Edge-Notch Te ns io n Specimen. NASA TN D-2396, 1964.

11. W i l l i a m s , M. L. : On t h e S t re ss Dis t r ibu t ion a t the Base of a Sta t ionaryCrack. Jo ur . Appl. Mech., vo l . 24, no. 1, M a r . 1957, pp. 109-114.1 2 . Wigglesworth, L. A. : S t r e s s D i s tr i b u t i o n i n a Notched P la te . Mathematika,

V O ~ . 4, 1957, pp. 76-96.13. Irwin, G. R . : Fracture . Vol . V I - E l a s t i c i t y and P l a s t i c i t y . E ncyclo -pedia of Phys., S. Fliigge, ed. , Springer-Verlag (Berlin), 1958,pp. 551-590.14. Sul l ivan, A. M . : N ew Specimen Design f o r Pl ane -St rai n Fr ac tu re ToughnessTes t s . Mate r ia l s R e s . and Standards, vo l. 4, no. 1, Jan. 1964,pp. 20-24.

NASA-Langley, 1964 E-2488 11

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