Empirical Methodology to Estimate Yield Stress

8/3/2019 Empirical Methodology to Estimate Yield Stress

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A n E m p i r i c a l M e t h o d o l o g y t o E s t i m a t e a L o c a l Y i e l dS t r e s s in W o r k - h a r d e n e d S u r f a c e L a y e r sb y J .P . N o b r e , A . M . D i a s a n d M . K o r n m e i e rA B S T R A C T - - A m e t h o d o l o g y i s p r o p o s e d f o r es t im a t ing theloc a l y ie ld s t r es s i n wo rk -ha rde ned s u r fac e l ay e rs . I t i s bas edo n t h e c o n c e p t o f i n - d e p t h n o r m a l i z e d v a r i a t i o n o f h a r d n e s sa n d x - r a y d i ff ra c t io n p e a k w i d t h , b o t h o f w h i c h m e a s u r e t h es t ra in -ha rde n ing a t ta ined by the m ate r ia l s ' s u r fac e - t r ea tedl a y e r s d u e to , for ex am p le , s ho t -peen ing . I t s p r i nc ip le i s d i -r e c tl y f o u n d e d o n t h e c l a s si c a l h a r d n e s s t h e o r y . T o s tu d y t h ee v o l u t io n o f th o s e v a l u e s w i t h p l a s t ic d e f o r m a t i o n , s p e c i m e n so f f iv e s t e e l s w i t h d i f fe r e n t m e c h a n i c a l p r o p e r t i e s w e r e s u b -j e c t e d t o i n t e r r u p t e d t e n s i l e te s t s . T h e t e s t s w e r e p e r f o r m e da t s uc c es s i v e i nc rem ents o f p las t i c s t r a in , un t i l f r ac tu re o c -c u r r e d . T h e s p e c i m e n s w e r e l o a d e d a n d u n l o a d e d i n i n c r e -m e n t s o f a b o u t 2 % t r u e s t ra i n . A f t e r e a c h p l a s t ic s t r a in i n c r e -m e n t , h a r d n e s s a n d d i ff r a ct io n p e a k w i d t h w e r e m e a s u r e d . I tw a s o b s e r v e d t h a t t h e v a r ia t i o n s o f d i ff r a ct io n p e a k w i d t h a n dh a r d n e s s a r e r e l a t e d t o t h e m a t e r i a l' s s t ra i n - h a r d e n i n g , a n dt h e i r n o r m a l i z e d v a r i a t io n s c a n b e c o n s i d e r e d p r o p o r t i o n a l t othe no rm a l i z ed v a r ia t i on o f t he m ate r ia l ' s y ie ld s t r es s . Thus ,w h e r e t h e y i e l d s t r e s s o f t h e b u l k m a t e r i a l, i ts h a r d n e s s o r ac hara c te r i s t i c d i f fr ac t i on pe ak w id th v a lue , and the i r re la t i v ev a r i a t i o n s a l o n g t h e h a r d e n e d l a y e rs , a r e k n o w n , a n e m p i ri c a le x p r e s s i o n c o u l d b e u s e d t o e s t i m a t e t h e l o c a l y ie l d s t r e s s a sa f u n c t io n o f t h e t r e a t e d d e p t h .K E Y W O R D S - - M e c h a n i c a l b e h a v io r , s h o t -p e e n i n g , yi e lds t reng th , s t r a in -ha rden ing , m ic ro -ha rdn es s , x - ray d if f rac t i on

I n t r o d u c t i o nK n o w l e d g e o f l o c a l y i e l d s t r e n g t h i n a m a t e r i a l ' s h a r d -ened l aye rs , due to mechan ica l s u r face t r ea tmen t s , s uch a sfo r exam ple s ho t -peen ing , i s o f g rea t r e l evance in s eve ra l a r -e a s o f m e c h a n i c a l e n g i n e e r i n g s c i e n ce , l P r o b l e m s o f f a t ig u e ,c o n t a c t f a t ig u e o r w e a r n e e d v a l id b e h a v i o r l a w s f o r s u r f a c e -t rea ted ma te r i a l l aye rs . T ha t knowledge i s a l s o requ i red , fo rexam ple , i n the s tudy o f the s o -ca l l ed p la s t i c i ty e f fec t on there s u l t s o f the inc rem en ta l ho le -d r i l l i ng t e chn iqu e fo r r e s idua ls t re s s m e a s u r e m e n t s 2 - 4 o r f o r n u m e r i c a l p r e d i c t io n m o d e l s

o f re s idua l s t r e s s re l axa t ion due to d ynam ic load ing . 5 '6M echan ica l s u r face t r ea tmen t s , s uch a s s ho t -peen ing , u s el o c a l p l a st i c d e f o r m a t i o n ( n o t u n i f o r m p l a s t ic d e f o r m a t i o n )a s a wa y o f in t roduc ing co mp res s ive re s idua l s t r e s se s , and s oi m p r o v e t h e u se f u l l if e o f m e c h a n i c a l c o m p o n e n t s . A c c o r d -

J .P . N o b r e ( jo a o . n o b re@ d em . u c . p t) i s a n Au x i l ia r y Pr o fe s s o r a n d A . M . Dia sis a Ful l Professor, M echanic al Engineer in g Depar tm ent , Univers i ty o fCoimbra, P-3030, Coimbra, Por tugal . D r . M. K ornm eier is current ly work-ing a t the Europea n Pa tent Off ice , Munich , Germany.Or ig inal manuscr ip t submit ted: September 27 , 2002.Final manu scr ip t received: June 6 , 2003.DO I: 10.1177/0014485104039755

ing to W ohl fah r t , 7 tha t de fo rm a t ion i s due to tw o m a in e f -fec t s : a d i rec t p l a s t i c e longa t ion a t t he s u r face , i nduced by thet a n g e n t ia l f o r c e s ( h a m m i n g e f f e c t ) , w h i c h i m p l i e s m a x i m u mcom pres s iv e re s idua l s t r e s s a t the s u r face ; and a p la s t ic s t ra inin the s ub - l aye rs , i nduced by the He r t z p re s s u re (He r t z e f -f e c t) , w h i c h i m p l i e s m a x i m u m c o m p r e s s i v e r e s i d u al s t r e s se swi th in the ma te r i a l , be low the s u r face . Fu r the rmo re , hea t d i s -s i p a ti o n c o u l d i m p l y p l a st i c c o m p r e s s i o n a n d t h e a p p e a r a n c eof t ens i l e r e s idua l s t r e s s e s a t t he s u r face . Depend ing o n them e c h a n i c a l c h a r a c te r i s ti c s o f t h e m a t e r ia l s , t h e c o m b i n a t i o nof the s e e f fec t s p roduces , by ove r l app ing , the f ina l i n -dep thcom pres s ive re s idua l s t r e s s p ro f il e .S h o t - p e e n e d l a y e r s a r e s u b j e c t ed t o c y c l i c p l a s t ic d e f o r m a -t io n , i n d u c e d b y t h e i m p a c t s o f t h e p e e n i n g m e d i u m , a t h ig hp la s t i c s t ra in ra t e s . T he re fo re , t he e l a s to -p la s t i c behav io r o ft h e m a t e r i a l n e a r t h e s u r f a c e c a n b e c o m p l e t e l y d i f f e r e n t f r o mtha t found fo r the bu lk ma te r i a l . 8 '9 G uech ich i 1~ p rop os eda n a n a l y t i ca l m o d e l , c o m p a r i n g t h e c a s e o f s h o t - p e e n i n g t oa s imp le cyc l i c load ing p roces s , a s a means o f p red ic t ingthe re s idua l s t r e s s e s induced by the d i f fe ren t t r e a tmen t con -d i t ions . In th i s mode l , two d i f fe ren t in -dep th reg ions we recons ide red . C los e to the s u r face , t he re i s a r eg ion ca l l ed the" a c c o m m o d a t e d z o n e " , c h a r a c t e r iz e d b y t h e s a t u r a te d v a l u eo f the re s idua l s t r e s s e s re l a t ed to the ma te r i a l ' s cyc l i c e l a s -t i c limi t . In th i s r eg ion , the ma te r i a l i s s ub jec ted to repea tedd e f o r m a t i o n , a n d t h e m a t e r i a l c a n t h e r e f o r e b e s u b j e c t e d t ocyc l i c ha rden ing o r s o f t en ing . As a re s u l t, t he s t ra in ha rden -ing behav io r s hou ld be mod i f i ed . Be low th i s r eg ion , the re i sa reg ion ca l l ed the " adap ted zone" , cha rac te r i zed by the fa s tdec l ine o f the re s idua l s t r e s s es . In th i s r eg ion the ma te r i a l i st a k e n t o b e s u b j e c t e d t o m o n o t o n i c p l a s ti c d e f o r m a t i o n .X - r a y s m a k e i t p o s s i b l e t o s t u d y t h e i n - d e p t h v a r i a t i o nof d i f f rac t ion peak wid th . T h i s va lue i s r e la t ed to the s qua rer o o t o f t h e m e a n q u a d r a t ic v a l u e o f t h e d i s t o rt i o n o f t h e c r y s -t a l l ine l a t t i c e 1 /2 , wh ich i s d i rec t ly p ropor t io na l to themic ros co p ic re s idua l s t r e s s ( th i rd o rder ). 11 ,12 Us ing s cann inge lec t ron ic m ic ros cop y , Ha ldm i 13 ana lyz ed the in -dep th va r i a -t i o n o f t h e d i s lo c a t i o n c o n f ig u r a t i o n in A F N O R 3 5 C D 4 s h o t -peen ed s t ee l . Hak im i found tha t , c lo s e to the s u r face , t he d i s -loca t ions a re s m a l l , f ine and in fo re s t . As the dep th inc rea s e s ,the d i s loca t ions t end to be longe r , r e c t i l i nea r and fewer . Hea l s o obs e rv ed tha t t he in -dep th d i s loca t ion den s i ty va r i ed ac -co rd ing to the va r i a t ion o f x - ray d i f f rac t ion peak w id th , i .e . ,t he h ighe r the d i f f rac t ion peak wid th , t he h ighe r the d i s loca -t i o n d e n si ty . M i c r o - h a r d n e s s m e a s u r e m e n t s m a d e a l o n g t h ec r o s s - s e c ti o n o f s h o t - p e e n e d s p e c i m e n s a r e a n e q u a l l y e f f i-c i e n t w a y o f c h a r a c t er i z i n g t h e m e c h a n i c a l b e h a v i o r o f t h ei r

76 9 VoL 44, No. 1, Feb rua ry 200 4 9 2004 Soc ie t y f o r Exper im enta l Mechan ics


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s u r face l aye rs , s ince ha rdnes s i s a d i rec t meas u re o f the ma -te r i a l ' s s t r eng th and c an be re l a t ed to i t s y i e ld s t reng th . 14S e v e r a l t e c h n i q u e s h a v e b e e n p r o p o s e d f o r e v a l u a t -

ing the mech an ica l p rope r t i e s o f m a te r i a l s u r face s . 5 '6 '8 '15D e s v i g n e s 8 d e te r m i n e d t h e y i e l d s t re n g t h o f A F N O R 3 5 C D 4s ho t -pee ned s t ee l a s a func t ion o f the t r ea ted dep th , ba s ed o nthe e l a s to -p la s t i c ana lys i s o f r e s idua l s t r e s s re l axa t ion du r ingfa t igue t e s t s , u s ing the e l a s to -p la s t i c c a l cu la t ion me thod o fZ a r ka a nd C as ie r . 16 Cao 9 us ed the in -dep th d i s t r ibu t ion o f x -r a y d i f f r a c ti o n p e a k w i d t h , o b t a i n e d e x p e r i m e n t a l l y t h r o u g h -ou t the s ho t -peened s u r face l aye rs , t o eva lua te the mechan-i ca l p rope r t i e s o f thos e l aye rs . C ao and Cas tex 15 us ed in -d e p t h x - r a y d i f f r a c t i o n p e a k w i d t h m e a s u r e m e n t s t o d e t e r -m i n e t h e s t r e s s - s t r a i n r e l a t i o n s h i p o f s h o t - p e e n e d s u r f a c elaye rs . Ba t i s t a and c o -w orke rs 6 ' 17 p ro pos e d an o the r me thodf o r e v a l u a t in g t h e m e c h a n i c a l p r o p e r t i e s o f s u r f a c e - t r ea t e ds tee l s ( s ho t -peen ing , n i t r id ing and ca rbon i t r id ing t rea tmen t s ) ,ba s ed on fou r -po in t bend ing t e s t s , u s ing the x - ray d i f f rac t ions i n 2 ~ m e t h o d f o r s t r e s s d e t e r m i n a t i o n a n d e l e c t r i c e x t e n -s om e te r s fo r s t r a in de te rmina t ion . V i rm oux e t a l . 5 , how eve r ,u s e d a n o t h e r p r o c e d u r e t o c h a r a c t e r i z e th e m e c h a n i c a l p r o p -e r t ie s o f t h e s u rf a c e l a y e rs o f t h e s te e l A F N O R Z 1 0 0 C D 1 7t rea ted by l a s e r s hock , ba s ed on the ana lys i s o f quas i - s t a t i cs phe r i ca l i nden ta t ions , compar ing expe r imen ta l i n -dep th in -den ta t ions p ro f i l e s (cha rac te r i zed by the rad ius and dep th ) ,u s ing d i f fe ren t loads , w i th thos e ob ta ined by f in i t e e l emen ts imula t ion . T h i s me thod i s ba s ed on the c l a s s i ca l ha rdnes stheo ry o f T abor 14 and the e l a s to -p la s t i c ind en ta t ion m ode l so f J o h n s o n 18 a n d S t u d m a n e t a l ) 9 O t h e r a u t h o r s u s e n a n o -i n d e n t a t i o n m e a s u r e m e n t s a n d c o m p r e s s i v e t e s t s a s a w a yto f ind the y ie ld s t reng th o f s pec imens s ub jec ted to the rmo-chem ica l s u r face t r ea tm en t s (e . g . , c eme n ta t ion ) . 2~ A n ew ap -p r o a c h f o r t h is p r o b l e m i s d e s c r i b e d b e l o w , w i t h s p e c i a l re f -e rence to the loca l in -dep th y ie ld s t reng th e s t ima t ion in the" a d a p t e d r e g i o n " o f t h e w o r k - h a r d e n e d s u r f a ce l a y e r s.D u e t o t h e c o m p l e x a n d l i m i t e d a p p l i c a t io n o f t h e m e t h -ods and t echn iques d e s c r ibed abov e , it was dec ided to s tudya n e w m e t h o d o l o g y f o r l o c a l y i e l d s t r e n g t h e s t i m a t i o n i nw o r k - h a r d e n e d s u r f a c e l a y e r s b y u s i n g m i c r o - h a r d n e s s i n -d e n t a ti o n s . T h e p r o p o s e d m e t h o d o l o g y , b a s e d o n t h e c l a s si c a lha rdnes s theo ry , 14 us e s the new conc ep t o f in -dep th no rm a l -i zed ha rdnes s va r i a t ion . T he s ame re s u l t s c an be ach ievedus ing x - ra y d i f f rac t ion peak wid th va lue s . Fo r th is , a c a l ib ra -t i o n p r o c e d u r e w a s c o n d u c t e d . T h i s w a s b a s e d o n t h e s t u d ya n d a n a l y s i s o f h o w m i c r o - h a r d n e s s i n d e n t at i o n s a n d x - r a yd i f f rac t ion peak wid th va lue s evo lve w i th the p la s t i c s t ra inimpos ed du r ing t ens i l e t e s t s . T ens i l e t e s t s we re ca r r i ed ou ta t s ucces s ive p la s t i c s t ra in inc remen t s un t i l f r ac tu re . Af te re a c h p l a s ti c s t r ai n i n c r e m e n t o f a b o u t 2 % , s a m p l e s w e r e u n -l o a d e d a n d h a r d n e s s a n d d i f f r a c t i o n p e a k w i d t h w e r e m e a -s u red . T h i s p rocedure p rov ided the ma te r i a l ' s s t r e s s - s t ra incu rve and , the re fo re , t he re l a t ion be tween the va r i a t ion o fh a r d n e s s a n d d i f f ra c t io n p e a k w i d t h a n d t h e v a r i a t i o n o f t h ey ie ld s t reng th due to the s t ra in -ha rden ing o f the ma te r i a l . T ov e r i f y t h e p r a c ti c a l a p p l i c a b i l it y o f t h e p r o p o s e d m e t h o d , t h ec a s e o f s h o t - p e e n i n g t r e a t m e n t w a s s e l ec t e d . A u n i f o r m s h o t -peen ing t rea tmen t was app l i ed to f ive s t ee l s w i th d i f fe ren tm e c h a n i c a l p r o p e r t i e s . F o r a l l s p e c i m e n s , i t w a s f o u n d t h a tb o t h h a r d n e s s a n d d i f f r a c t i o n p e a k w i d t h v a l u e s d i m i n i s h e ds tead i ly f rom the s u r face to the in te r io r , whe re the v a lue s inthe bu lk ma te r i a l we re reg i s t e red . T he re l a t ive va r i a t ions o fthe s e quan t i t i e s we re then com pare d to thos e ob ta ined in the

tens i l e t e st s , wh ich f ina l ly a l lowed the e s t im a t ion o f the loca ly ie ld s t re s s in the s ho t -p eened s u r face l aye rs .Descr iption of the Proposed Method

A c c o r d i n g t o T a b o r , 14 " a c o n v e n i e n t m e t h o d o f m e a s u r i n gthe e l a s t i c l imi t o f a ma te r i a l i s t o de te rmine i t s ha rdnes sus ing a py ramida l inden te r pos s e s s ing a l a rge apex ang le , a sin the V icke rs t e st . " H i s theo ry on the ha rdnes s o f me ta l s i sf o u n d e d o n e x p e r i m e n t a l e v i d en c e . H e o b s e r v e d t h a t w h e n ame ta l is s ub jec ted to h igh co ld work ing , i . e ., w i thou t c ap ac i tyf o r s u b s e q u e n t w o r k h a r d e n in g , p l a s t i c f lo w b e g i n s w h e n t h em e a n c o n t a c t p r e s s u r e i s g i v e n b y :

Pm = COy (1 )wh ere c i s a cons tan t o f abou t 3 and c ry can be cons id e red a sa " rep re s en ta t ive y ie ld s t re s s" . 14 Bas ed on th i s theo ry , T aborf o r m u l a t e d a n e q u a t i o n t o e s t i m a t e t h e u l ti m a t e s t r en g t h o fa m a t e r i a l. S e v e r a l a t t e m p t s h a v e b e e n m a d e t o e s t i m a t e t h e0 . 2% o f f s e t y i e ld s t reng th . Geor ge e t a l . 21 p ro pos e d a me thodu s i n g t h e R o c k w e l l h a r d n e s s d a t a . U s i n g V i c k e r s a n d M e y e rha rdnes s , C ahoo n e t a l . 22 foun d an e mp i r i ca l equa t ion tha ta l lows the ca lcu la t ion o f the y ie ld s t reng th , w i thou t need ingto know i t s s t r e s s - s t ra in cu rv e , g iven by :

H V 0C~y0,2 = - - ~ ( . l ) m-2 . (2 )In th is eq uat ion , C~y0,2 s the y ie ld s t re ngth , H V i s t he V ick -e r s ha rdnes s and m i s the M eye r exponen t ( the s t ra in ha rd -en ing coe f f i c i en t n can be re l a t ed to th i s expo nen t by n = m- 2 ). 14 T hey s how ed the eq ua t ion ' s va l id i ty fo r a 65S a lu -

m i n u m a l l o y an d 1 0 4 0 s t ee l , su b j e c t ed t o v a r i o u s a m o u n t s o fc o l d w o r k i n g . F o r a g i v e n M e y e r e x p o n e n t m , t h i s e m p i r i c a le q u a t i o n c l e a r l y s h o w s p r o p o r t i o n a l i t y b e t w e e n O y a n d H V .F o r a p e r f e c t l y p la s t ic m a t e r i a l , C a h o o n ' s e q u a t i o n i s re d u c e dt o T a b o r ' s e q u a t i o n , w h i c h a l l o w s t h e r e l at i o n s h i p b e t w e e nthe mean co n tac t p re s s u re and a rep re s en ta t ive y ie ld s t reng thob ta ined in a t ens ion o r compres s ion t e s t . Cons equen t ly , i fha rdnes s va r i e s to a ce r t a in deg ree , t he re s hou ld be a co r re -s pond ing va r i a t ion in y ie ld s t reng th .T h u s , b a s e d o n t h e p r o p o r t i o n a l i t y b e t w e e n h a r d n e s s a n dy ie ld s t reng th s t a t ed in T abor ' s and Cahoon ' s equa t ions , t heinc rea s e o f the loca l y i e ld s t reng th (Ac~y) , r e l a t ive to the y ie lds t reng th o f the bu lk ma te r i a l (C~y,0 ), c an b e ea s i ly e s t im a tedbas ed on the loca l inc rea s e o f ha rdnes s (A H V) , r e l a tive tot h e h a r d n e s s o f t h e b u l k m a t e r i a l ( H V y , o ) , us ing the fo l lowingf o r m u l a :

A t ~ y A H V- ( 3 )O y , 0 H V y , O

I f i t w e r e p o s s i b le t o f o r m u l a t e s o m e h y p o t h e s i s , a s i m -i l a r e x p r e s s i o n c o u l d b e t h e o r e t i c a l l y d e d u c e d w h e n h a r d -n e s s o r x - r a y d i f f r a c ti o n p e a k w i d t h a r e u s e d . F r o m p r e v i o u sworks , 5 ,6 whe re the ha rdnes s was cons ide red to va ry w i ththe p la s t i c s t ra in in acco rdance w i th a po ten t i a l l aw , s imi la rto the t rue s t re s s - s t ra in re l a t ions h ip fo unded in t ens i l e te s t s ,i t i s pos s ib le to cons ide r tha t bo th ha rdn es s (HV ) and d i f f rac -t i o n p e a k w i d t h ( H W V ) m i g h t v a r y w i t h t h e p l a s t ic s t r a i n ( e p )accord ing to the fo l lowing expre s s ions :

H V ---- H V y 0 -t- K H V S p H v' , I .u wv ( 4 )H W V = H W V y , o + a H W V S p 9

9 2004 Soc ie ty fo r Exper imen ta l Mechan ics Exper imen ta l Mechanics 9 77


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H e r e , n H V an d n H W V a re s t ra in -ha rden ing exponen t s andK H V an d K H W V a re s t reng th coe f f i c i en t s , co r re s pond ing toh a r d n e s s - s t r a i n a n d d i f f r a c t i o n p e a k w i d t h - s t r a i n r e l a t i o n -s h ips re s pec t ive ly , i den t i ca l t o the s t reng th c o e f f i c i e n t p r e -s en ted in we l l -kno wn t rue s t re s s -p la s t i c s t ra in re l a t ions h ips ,ob ta ine d du r ing s t anda rd t ens i l e t e s t s , a s , fo r exam ple , K y inthe fo l lowing equa t ion : 23

? /cry = cry,O -'}- K y s p . ( 5 )T his way , i t i s pos s ib le to w r i t e

( H V - H V y , o ~ n / n H vC r y - - c r y , 0 = Ky \ KEy ]

( H W V - H W V y , o '~ n/ n H w vcry - cry,O = K y \ KH wv ] "(6 )

S ince a l l exponen t s n , n H V an d n H W V rep re s en t thema te r i a l ' s s t r a in ha rden ing p rope r ty , t hen n / n H V .~ 1,n / n N W V ~ 1 , and the y ie ld s t re ss cons equen t ly va r i e s l in -ea r ly w i th the ha rdnes s o r d i f f rac t ion peak wid th , w i th thev a r i a t io n o f t h e s e q u a n ti f ie s i t s e l f b e i n g a m e a s u r e o f t h e m a -te r i a l ' s s t r a in ha rden ing capac i ty . T hus , u s ing bu lk ma te r i a lr e f e r e n c e v a l u e s f o r n o r m a l i z a t io n p r o p o s e s , t h e f o l l o w i n ginc remen ta l r e l a t ion can be wr i t t en a s :

Ac ry A Hcry,O -= y Hy,O (7 )

w h e r e t h e l e t te r H c a n c o r r e s p o n d t o h a r d n e s s ( H V ) o r d i f fr a c -t ion pe ak w id th (HWV), C ry ,0 and Hy , 0 co r re s p ond to the y ie lds t re s s a n d h a r d n e s s o r d i f f ra c t i o n p e a k w i d t h o f t h e b u l k m a -te r i a l , r e s pec t ive ly , and y i s a cons tan t ( s ca le fac to r ) g ivenb y :K y H y , Oy - ( 8 )K H cry,O

T h e d e d u c e d e q ( 7 ) i s t h e b a s is o f t h e p r o p o s e d m e a s u r e -m e n t m e t h o d a n d r e l a t e s t h e l i n ea r i ty b e t w e e n r e l a t i v e i n c re -me n t s o f ha rdnes s (d i f f rac t ion peak wid th ) an d y ie ld s t re s s .T h a t e q u a t i o n c a n h e w r i tt e n i n a v e r y c o n v e n i e n t f o r m , a l l o w -ing the y ie ld s t re s s to be e s t ima ted , fo r e ach l eve l o f p l a s t i cd e f o r m a t i o n , i n t e r m s o f t h e n o r m a l i z e d v a r i a t i o n o f h a r d n e s so r d i f f rac t ion peak wid th , a s fo l lows :

Oy ~--- cry,O 1 + y (9)T he d i f fe ren ce be tween eqs (3 ) and (7 ) i s t he appea ra nce o ft h e c o n s t a n t , / i n t h e l at te r , T h e e x p e r i m e n t a l s t u d y p r e s e n t e db e l o w w a s c o n d u c t e d t o v e r i f y t h e v a l id i t y o f t h e l i n e a r it yhypo th es i s and the app l i cab i l i t y o f thos e equa t ions fo r loca l

y i e l d s t r e n g t h e s t i m a t i o n i n w o r k - h a r d e n e d s u r f a c e l a y e r s .T h e d e p e n d e n c y o f t h e c o n s t an t y a c c o r d i n g t o t h e m a t e r i a l sus ed in th i s s tudy i s a l s o ana lyzed .M a t e r ia l s a n d E x p e r i m e n t a l P r o c e d u r e

T h e e x p e r i m e n t a l s t u d y w a s d i v i d e d i n to t w o p a r t s u s i n gf ive s t ee l s w i th d i f fe ren t mechan ica l p rope r t i e s . In the f i r s tpa r t , i n o rde r to ve r i fy the hypo thes i s s ugges ted above andthe va l id i ty o f the dedu ced eq ua t ions , a s e r i e s o f in t e r rup tedtens i l e t e s t s was pe r fo rmed . T hes e t e s t s a l lowed the s tudy

a n d a n a l y s i s o f h a r d n e s s v a r i a t i o n a n d x - r a y d i f f r a c t i o n p e a kwid th va r i a t ion w i th p la s t i c s t ra in . In a s econd phas e , s e -l ec t ing the ca s e o f the s ho t -peen ing t rea tm en t , t he me thodd e s c r i b e d a b o v e w a s a p p l i ed t o d e t e r m i n e l o c a l y i e l d s t re s sa s a func t ion o f dep th , i n work -ha rd ened s u r face l aye rs . T h i sp r o c e d u r e w a s c a r r i e d o u t , c o m p a r i n g t h e r e su l t s o f t h e n o r -m a l i z e d v a r i a t i o n o f t h e h a r d n e s s a n d t h e d i f f r a c t i o n p e a kw i d t h d e t e r m i n e d i n t h o s e l a y e r s.T a b l e 1 s h o w s m e c h a n i c a l p r o p e r t i e s a n d c h e m i c a l c o m -pos i t ion o f the ma te r i a l s u s ed , l i s t ed by y ie ld s t reng th , i ninc rea s ing o rde r .T w o t y p e s o f s p e c i m e n w e r e m a c h i n e d f r o m e a c h a s -rece ived p la t e . Fo r t ens i l e t e s t s , p ropor t iona l s pec imens , a c -c o r d i n g t o A S T M E 8 , w i t h r e c t a n g u l a r c r o s s -s e c t i o n a l a r e a s ,w e r e u s e d . F o r s h o t - p e e n in g t r e a t m e n t , s p e c i m e n s w e r e m a -ch ined in p lane p iece s , 12 mm th ick . A l l s pec imens , a f t e rg r ind ing , we re s ub jec ted to an e l ec t ro ly t i c po l i s h ing to re -m o v e t h e l a y e r a f f e c te d b y t h e m a c h i n i n g . T h i s p r o c e d u r e w a sc o n t r o l le d b y m e a n s o f x - r a y d i f f r a c ti o n , a n a l y z i n g t h e e v o l u -t i o n o f d i f fr a c t io n p e a k w i d t h w i t h t h e d e p t h o f t h e r e m o v e dma te r i a l s u r face l aye r . I t was obs e rved tha t , a f t e r a c e r t a ind e p t h , t h e v a l u e o f t h e d i f f r a c ti o n p e a k w i d t h r e m a i n e d a l -m o s t c o n s ta n t . T h e e s t a b l i sh e d m e a n v a l u e w a s c o n s i d e r e d t ob e t h e r e f e r e n c e v a l u e o f t h e b u l k m a t e r ia l . I n t h e c a s e o f t e n -s i le s p e c i m e n s , t o o b t a in h o m o g e n e o u s a n d p l a n e s u r f a c e s,the e l ec t ro ly t i c po l i s h ing was ca r r i ed ou t in an e l ec t ro ly teb a t h a g i t a te d i n a r o t a t io n a l m o v e m e n t i m p o s e d b y a m a g -n e t ic f i e ld , to a l l o w t h e g r a d u a l a n d h o m o g e n e o u s r e m o v a lo f m a t e r i a l f r o m t h e s p e c i m e n s u r f a c e s . A s a m e a n v a l u e ,a b o u t 2 0 0 I x m o f m a t e r i a l w a s r e m o v e d f r o m e a c h s p e c i m e ns ur face . Af te r th i s p r io r t r e a tmen t , t he s pec imens we re s ub -j ec ted to ax ia l t ens i l e t e s t s . T he s pec imens we re loaded andu n l o a d e d i n i n c r e m e n t s o f a b o u t 2 % t r u e t o ta l s t ra i n. A f t e reach s t ra in inc remen t , t he s pec imens we re s ub jec ted to ha rd -nes s t e s t s and x - ray d i f f rac t ion . T h i s p rocedure was ca r r i edo u t u n t il t h e m a x i m u m l o a d w a s a t ta i n ed . A f t e r t h i s p o in t ,s p e c i m e n s w e r e t e n s i o n e d u n t i l f r a ct u r e . A l t h o u g h t h e r e g i o nf r o m t h e b e g i n n i n g o f t h e n e c k i n g z o n e a n d t h e f r a c t u r e p o i n tc o u l d g i v e m o r e i n f o r m a t i o n , t h i s w a s n o t m e a s u r e d d u e t othe d i f f i cu l ty in t ak ing ha rdnes s and d i f f rac t ion peak wid thm e a s u r e m e n t s i n t h a t r e g i o n . T h e f i n al c r o s s - s ec t i o n a r e a s o ft h e s p e c i m e n s w e r e m e a s u r e d u s i n g o p t i c a l m i c r o s c o p y a n dt h e f i n al h a r d n e ss a n d x - r a y m e a s u r e m e n t s w e r e t a k e n .T heore t i c a l ly , due to the geo me t r i c s imi la r i ty obs e rved inVicke rs ha rdnes s inden ta t ion , on a mac ros cop ic s ca le , t rueh a r d n e s s i s a l m o s t i n d e p e n d e n t o f t h e l o a d a p p l i e d a n d i sc o n s i d e r e d t o b e a c o n s t a n t o f t h e m a t e r ia l . H o w e v e r , o n amic ros cop ic s ca le , due to s o -ca l l ed inden ta t ion s i ze e f fec t( ISE ) , appa ren t ha rdne s s can va ry w i th the inden te r s i z e , thedep th o f the inden ta t ion o r the t e s t l oad ap p l i ed . 24,25 Gen e r -a l ly , depend ing on the inden te r s i z e and inden ted ma te r i a l ,t h e a p p a r e n t h a r d n e s s v a l u e d e c r e a s e s a s y m p t o t i c a l l y w i t hinc rea s ing load , un t i l t rue ha rdnes s i s a t ta ined , and tha t va r i -a t ion cea s e s . T h i s i s why , in the t ens i l e te s t s s pec im ens , ha rd -n e s s i n d e n ta t i o n s w e r e m a d e u s i n g t w o l o a d s c o r r e s p o n d i n gto 100 and 300 g . In gene ra l , i t was fou nd tha t ha rdnes s va lue sco r re s pond ing to a load o f 100 g a re s l igh t ly h ighe r than fo rt h o s e o f 3 0 0 g . H o w e v e r , t h e m e t h o d o l o g y p r o p o s e d i n t h isw o r k u s e s n o r m a l i z e d h a r d n e s s v a r i a t i o n v a l u e s w h i c h a v o i dtha t d i f fe rence in the f ina l r e s u l t s when one o r o the r load i sus ed .S h o t - p e e n i n g s p e c i m e n s w e r e t r e a te d u s i n g i d e n t i ca l p a -rame te r s , s hown in T ab le 2 . T o s tudy the in -dep th evo lu t ion

78 9 VoL 44, No. 1, Feb rua ry 200 4 9 2 0 0 4 S o c i e t y f o r E xp e r i m e n t a l M e ch a n i cs


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T A BLE 1 - - M A T E R I A LS , M E C H A N I C A L P R O P E R T I E S A N D C H E M I C A L C O M P O S I T IO NMaterial cry0.2 Om ** Che mical com posi t ion (% weight)Nr. -A ISI (MPa) (MPa) n * t t V C S i Mn Cr Mo Ni V1 - (4 20 ) 350 630 0 .24 200 0 .38 0 .9 0 .5 13 .62 - 1045 440 720 0 .18 220 0 .45


5/9

1 1 0 0 .1 0 0 0 1

9 0 0 1- 8 0 o iE 7oo i

6 0 0 15 o o i

. E 4 0 0z 3 0 0

2 0 0 11 0 0 1

0

A I S I ( 4 2 0 )

, /i I . ~ . . . . . i . . . . . . . . . I . . . .

A I S I 1 0 4 5

1 -t A I S I 3 4 1 54.4

0 5 1 0 1 5 2 0 2 5 3 0 0 5 1 0 1 5 2 0 2 5 3 0 0 5 1 0 1 5 2 0 2 5N o m i n a l S r a i n [ % ] N o m i n a l S t r a i n [ % ] N o m i n a l S t r a in [ % ]

F i g . 1 - - E x a m p l e s o f n o m i n a l s t r e s s - s t r a i n c u r v e s o b t a i n e d i n t h e i n t e r r u p t e d t e n s i l e t e s t s3 0

A I S I 4 3 3 7\

, , . , . . . . , . . . . . . . . . . . . . .5 1 0 1 5 2 0 2 5 3 0N o m i n a l S t r a in [ % ]

1 4 0 0 ,

1 2 0 0

1 0 0 0

8 0 0

6 0 0 .

4 0 0 ,

2 0 0 I

A I S I ( 4 2 0 )

/A I S 1 1 0 4 5

y = 52 4.75x 01,892 = .

YA I S 1 3 4 1 5

4-I

1

y = 98 2.51 xa o591

A I S I 4 3 3 7

,i, , , , , , , , . . . . . . . , . . . . . . . . . , . . . . . . . , . . . . . . . . . . , . , , , . . . . . . . . . . , , , , , , , , , , , . , . ,

2 0 4 0 6 0 8 0 1 0 0 2 0 4 0 6 0 8 0 1 0 0 0 2 0 4 0 6 0 8 0 1 0 0 0 2 0 4 0 6 0 8 0 1 0 0T r u e S t r a i n E p la s t % ] T r u e S t r a i n P--*plast % ] T r u e S t r a i n S p la s [ % ] T r u e S t r a i n s p l a= [ % ]

F ig , 2 - - C o r r e s p o n d i n g t r u e s t r e s s - p la s t i c s tr a in c u r v e s ; t h e m a r k e d p o in t c o r r e s p o n d s t o m a t e r ia l f r a c tu r e

,P .

" E~ 2

" 1

X

A I S I ( 4 2 0 )

AhardnessSdiffraction pea k width

, , , , . . . . . . . , , , , , , , ,2 0 4 0 6 0 0 0 1 0 0

T r u e S t r a i n ~ p l a s t % ]

A I S I 1 0 4 5

O diffractionpeak width&hardness

, , , , , , , , , , , , , , , , , ,2 0 4 0 6 0 8 0 1 0 0

True S tra i n F - . . p l as [ % ]

A I S I 3 4 1 5

Odiffractiortpeak width&hardness

9 , 9 , , 9 9 i 9 9 9 , 9 , , , , 9 92 0 4 0 6 0 8 0 1 0 0

T r u e S t r a i n S p la st % ]F i g . 3 - - - H a r d n e s s a n d d i ff ra c t io n p e a k w i d t h a s a f u n c ti o n o f t r u e p l a s ti c s t r a in

4 5 0A I S I 4 3 3 7

:1 4003 5 0

3 0 0 ~ '

" 2 5 0 " ~t 2o0- 1 1 5 0 =.i

1 0 09 diffractionpeak w i d th 30&hardness-i' , , , , , , , , . . . . . . . , , , , 0

0 2 0 4 0 6 0 8 0 1 0 0T r u e S t r a i n S p l a s t % ]

variat ion . That i s , the interpolat ion o f experim ental dataseems to indicate a d irect proport ional i ty between the rela-t ive increments o f d i ffract ion peak width and yie ld stress . Thejust i f icat ion for th is d i f ference between the measuring meth-ods could be related to the ef fect of the p last ic deformationinduced b y the hardness indentat ion i t se lf . E quivalent p last ic

strain , induced by a pyramidal indenter, i s c lose to 8%, anda relat ion of about 3 i s found between the yie ld stress andthe hardness o f the material. 14,18 This was also co nf irmed b your experimental data. In the case of x-rays , the material i snot d isturbed by the measurement process i t se l f , and there-fore there exists a d irect proport ional i ty betw een n orm al ized

8 0 9 Vol. 44, No. 1, February 2004 9 2004 Soc ie ty for Exper imenta l Mechan ics


6/9

A I S I ( 4 2 0 ) A I S 1 1 0 4 5 A I S I 3 4 1 5 A I S I 9 2 5 5

y = 2 .9908xR 2 = 0 .9905

y . . . . . . . . . , . . . . . . . 9 I : I/ / I t "= 2 .9346x

y = 1 A 8 1 6 x R 2 = 0 . 9 6 2 6 y = 0 , 9 0 3 8 x~ //;=;o ..... I t > z .......' I / l J / / : : ...... o....1 . . . . 1 / , . ,d. , , . . . . ( .v>

/ / a, .... HV ) ~ y . .. . .. . ?:ra:tio:ea:wtdt::W:) U . . . . . . . 7:ff:acii:p ....... (HWV)0 ~ . . . . . . . . . . . . . . . . . . .1 2 0 1 2 0 1 2 0 1

N o r m a l i z e d V a r i a t io n o f H V o r H W V N o r m a l i z e d V a r i a ti o n o f H V o r H W V N o r m a l i z e d V a r i a t io n o f H V o r H W V N o r m a l i z e d V a r i a t io n o f H V o r H W VFi9 . 4 - -N orm a l i zed va r i a t ions o f y i e ld s t re s s v e rs us ha r dne s s (HV) (o r d if frac tion pea k wid th (HWV))

9 I A o l O ~ = 2 " 8 2 6 2 * ( / ~ " IV / H V ~ I 9

~ 1.5 ~

iN # oo .,o ,..,L I~ H W V / H W V 0 ), . _ - o . . .Z

0 i . . . . , . . . . , . . . . , . . . .0 0 . 5 1 1 . 5 2

N o r m a l i z e d V a r i a t io n o f H V o r H W VF ig . 5 - - N o r m a l i z e d v a r i a t io n o f y i e l d s t r e s s v e r s u s h a r d n e s s(HV) (or d iffrac tion pe ak width (HWV)) for mate r ia ls use d int h i s s t u d y

y i e l d s t r e ss v a r i a t io n a n d n o r m a l i z e d d i f f r ac t i o n p e a k w i d t hva r i a t ion . In add i t ion , t h i s imp l i e s a d i f fe rence in the s ens i -t i v it y o f t h e m e a s u r i n g m e t h o d s . T h e d i f f r a c t io n m e t h o d i smo re s ens i t ive , bu t mo re d i f f i cu l t t o pe r fo rm , wh i l e the ha rd -n e s s m e t h o d i s l e ss s e n s i t iv e b u t m u c h e a s i e r t o p e r f o r m .For the s t ee l s s tud ied , the p ropor t ion a l i ty cons tan t 7 s eem st o b e i n d e p e n d e n t o f t h e m a t e r i a l s ' s t r a in - h a r d e n i n g b e h a v i o r .T h a t b e h a v i o r c o u l d t hu s o n l y b e d e t e r m i n e d b y k n o w i n g t h en o r m a l i z e d v a r i a t io n o f h a r d n e s s o r d i f f r a ct i o n p e a k w i d th .T h i s s tudy s hou ld , howeve r , be deve loped in the fu tu re ande x t e n d e d t o a n o t h e r m e t a l s a n d a l lo y s .A p p l i c a ti o n o f th e P r o p o s e d M e t h o d o l o g y : E s t im a t i o no f a L o c a l Y i e l d S t r e s s i n S h o t - P e e n e d S u r f a c e L a y e r s

T h e e m p i r i c a l m e t h o d o l o g y d e s c r i b e d a b o v e w i l l s u b s e -quen t ly be us ed to eva lua te the s t ra in -ha rden ing cu rves o fma te r i a l s and to e s t ima te a loca l y i e ld s t re s s , a s a func t ionof the dep th , i n the s u r face l aye rs a f fec ted by a s im i la r s ho t -p e e n i n g t r e a tm e n t , w h i c h w a s a p p l i e d t o s e v e r a l o f t h e s t e e l

s pec imens l i s t ed in T ab le l . T he cu rves p re s en ted in F ig . 4wi l l be us ed a s an expe r imen ta l c a l ib ra t ion , l i nk ing the no r -ma l i zed va r i a t ions ob ta ined du r ing t ens i l e t e s t s w i th thos eob ta ined in s ho t -peened s u r face l aye rs . T h i s p rocedure w i l lb e u s e d t o e s t i m a t e t h e y i e l d s t r e s s t h r o u g h o u t t h e w h o l et rea ted dep th .S o m e o b s e r v a t i o n s s h o ul d , h o w e v e r , b e m e n t i o n e d b e f o r et h e p r o p o s e d m e t h o d o l o g y i s a p p l i e d t o l o c a l y i e l d s t r e s se s t ima t ion in s ho t -p eened s u r face l aye rs . F i r s t o f a ll , t he ca l i -b ra t ion p roce dure was ob ta ined f ro m un iax ia l t ens i le t es t s . I ti s imp l i c i t ly acce p ted tha t t he ma te r i a l ha s a s imi la r mech an-i ca l beh av io r in t ens ion a s i t does in the ca s e o f com pres s ionl o a d in g . F o r m a t e r i a l s w h i c h p r e s e n t d e f o r m a t i o n a s y m m e t r yi n t e n s io n l o a d i n g , c o m p a r e d w i t h c o m p r e s s i o n l o a d i n g , a s i nt h e c a s e o f m a g n e s i u m a l l o y s, 27 f o r e x a m p l e , t h is p r o c e d u r erequ i re s cau t ion , s ince y ie ld s t reng th in t ens ion and com pres -s ion i s ve ry d i f fe ren t . Howeve r , r e l a t ive to s t ra in -ha rden ingbehav io r , i f d i f fe rences a r i s e , t hey s hou ld b e re f l ec ted in then o r m a l i z e d h a r d n e s s o r d i f f r a c t io n p e a k w i d t h v a r i a t i o n v a l -ue s ob ta ined . On the o the r hand , i t i s we l l know n tha t mac ro -ha rdnes s o r t rue ha rdnes s va lue i s a ma te r i a l ' s cons tan t andi s t r ad i tiona l ly re l a t ed to the y ie ld s t re s s o f the m a te r i a l un -de r t e s ting . T he m ic ro -ha rd nes s va lue , howeve r , is no longe ra ma te r i a l ' s cons tan t , bu t depends on the inden te r s i z e , l oada n d d e p t h o f i n d e n t a ti o n . 24 T h i s I S E i s n o t w e l l u n d e r s t o o dy e t a n d i t h a s b e e n e x p l a i n e d u s i n g v a r i o u s m e c h a n i s m s . 25For the s t ra in -ha rden ing cha rac te r i za t ion o f s ho t -peened s u r -f a c e l a y e rs , th e l o w e s t p o s s i b l e l o a d s w o u l d b e u s e d f o r m o r el o c a l i ze d V i c k e r s m i c r o - h a r d n e s s r e a d i n g s a n d t h e I S E c a nthe re fo re in f luence appa ren t ha rdnes s va lue s . Howeve r , fo rt h e m e t h o d o l o g y p r o p o s e d h e r e , t h e u s e o f t h e s a m e l o a dth roughou t the who le t r ea ted dep th , r ega rd le s s o f i t s va lue ,and the co r re s pon d ing re l a t ive ha rdnes s va r i a t ion va lue s , c anm a k e t h e I S E n e g l ig i b l e, s i n c e t h e n o r m a l i z e d h a r d n e s s v a r i a -t ion i t s e l f i s a me as u re o f the s t ra in -ha rde n ing a t t a ined b y thet rea ted m a te r i a l , a s s een ea r l i e r . F ina lly , t he d i rec t e f fec t o fthe re s idua l s t r e s s s t a t e on mic ro -ha rdnes s read ings s hou lda l s o be t aken in to accoun t . In the ca s e o f t ens i l e t e s t s , noma c ro re s idua l s t r e s s e s a r i se a f t e r the s pec im en i s un loaded ,s i n c e th e i m p o s e d p l a s t ic d e f o r m a t i o n c a n b e c o n s i d e r e d u n i -f o r m a l o n g t h e c r o s s - s e c t io n s o f t h e s p e c i m e n s . I n t h i s c a s e ,m i c r o - h a r d n e s s r e a d i n g s a r e n o t a f f e c t e d b y m a c r o r e s i d u a ls t re s se s , and the re fo re the i r va r i a t ions , i n a f i r s t app roac h , c anbe a t t r ibu ted to the s t ra in -ha rden ing e f fec t . Howeve r , h ighc o m p r e s s i v e r e si d u a l s t r e s se s i n d u c e d b y t h e s h o t - p e e n i n g

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t r e a t m e n t c o u l d h a v e s o m e i n f l u en c e o n m i c r o - h a r d n e s s m e a -s u reme n t s . In a recen t s tudy on th i s i s s ue , 28 a c l ea r in f luenceof re s idua l s t r e s s e s on ha rdnes s read ing s wa s conf i rmed . Intha t s tudy , a bend ing ba r , s ub jec ted to fou r -po in t bend ing ,w a s p l a s t i c a l ly d e f o r m e d u p t o 2 % p l a s t ic s t r ai n . M e a s u r e dh a r d n e s s v a l u e s , o b t a i n e d d u r in g s p e c i m e n ' s l o a d i n g , c l e a r lys h o w d e p e n d e n c y o n t h e b e n d i n g l o a d . D e c r e a s i n g h a r d n e s sva lues , r e l a t ive to the bu lk m a te r i a l va lue , a re obs e rve d on thetens i l e s ide and inc rea s ing ha rdnes s va lue s a re obs e rved onthe com pres s iv e s ide o f the ba r . Af te r un load ing , the re s idua ls t re s s s t a t e induces a s imi la r behav io r in ha rdnes s read ings .T h i s e f fec t i s , howeve r , more p ronounced in the t ens i l e r e -g ion . In the com pres s iv e reg ion , the e f fec t o f the re s idua ls t r es s e s o n t h e h a r d n e s s r e a d i n g s w a s n o t c o m p l e t e l y c la r -i f i ed and fu r the r s tud ie s s hou ld be pe r fo rmed , e s pec ia l ly toc l a r i fy t h e c o m b i n e d e f f e c t o f m a c r o / m i c r o r e s i d u a l s t r e s s e sand the ma te r i a l ' s s t r a in -ha rden ing . Nev e r the le s s , i t s hou ld ber e m e m b e r e d t h a t t h e p r o p o s e d m e t h o d o l o g y w o r k s w i t h r e l -a t i v e i n c r e m e n t s o f h a r d n e s s , w h i c h s e e m t o b e a d i r e c t m e a -s u re o f the s t ra in -ha rden ing e f fec t . Fu r the rmo re , s ys t em a t i ce r ro rs in abs o lu te ha rdnes s read ings can a l s o be reduced byu s i n g r e l a ti v e i n c r e m e n t v a l u e s . T h e i m p o r t a n c e o f w o r k i n gw i t h n o r m a l i z e d h a r d n e s s v a r i a t i o n v a l u e s s e e m s , t h e r e fo r e ,to be o f g rea t i n t e re s t .F i g u r e 6 s h o w s t h e i n - d e p t h e v o l u t i o n o f h a r d n e ss a n dd i f f r ac t i o n p e a k w i d t h a l o n g t h e s h o t - p e e n e d s u r f a c e l ay e r s .F o r a l l c a s e s, b o t h d e c r e a s e c o n t i n u o u s l y f r o m t h e s u r f a c e t othe in t e r io r o f the ma te r i a l , and the re fo re no s igns o f ma te r i a ls o f t e n i n g a t t h e s u r f a c e s e e m t o h a v e o c c u r r e d b e c a u s e o ft h e s h o t - p e e n i n g t r e a t m e n t , e v e n i n t h e c a s e o f t h e q u e n c h e da n d t e m p e r e d s t e e l ( A I S 1 4 3 3 2 ) . O n t h e o t h e r h a n d , i t i s c le a rtha t t he g rea te r s t r a in -ha rden ing ca pac i ty o f the ma te r i a l , t heh ighe r a re the no rma l i zed va r i a t ion va lue s a t t a ined , wh ichc o n f i r m s a g a i n t h a t t h e n o r m a l i z e d v a r i a t i o n s o f h a r d n e s sa n d d i f f r a c ti o n p e a k w i d t h a r e a m e a s u r e o f t h e m a t e r i a l ' ss t ra in -ha rden ing behav io r .

Fo r a l l c a s e s , t he ma te r i a l work -ha rdens a t t he s u r face .T h i s b e i n g s o , th e n o r m a l i z e d v a r i a t i o n o f h a r d n e s s o r d i f f r a c -t i o n p e a k w i d t h , o b t a i n e d a l o n g t h e s h o t - p e e n e d l a y e r s , c a nb e c o m p a r e d w i t h t h a t o b t a in e d f r o m p l a s t ic d e f o r m a t i o n i m -pos ed in the t ens i l e t e s t s . T hus , t ak ing in to accoun t the ex -

pe r imen ta l r e s u l t s p re s en ted in F igs . 4 and 5 , wh ich conf i rmthe va l id i ty o f eq (9 ) , a loca l y i e ld s t re s s a s a func t ion o f thet rea ted dep th can be e s t ima ted . F igu re 7 p re s en t s the re s u l t sob ta ined , s how ing tha t s imi l a r r e s u l t s a re ob ta ined no m a t t e rw h e t h e r h a r d n e s s o r d i f f r a ct i o n p e a k w i d t h d a t a a r e u s e d .

I t s hou ld neve r the le s s be n o ted tha t , i n the ca s e o f ha rdnes sda ta , i f a d i rec t c a l ib ra t ion i s pe r fo rmed us ing the abs o lu teva lues f rom F igs . 2 , 3 and 6 , d i s c repanc ie s occu r be tweenthe ha rdnes s an d d i f f rac t ion peak wid th da ta , u s ed fo r the lo -ca l y i e ld s t re s s e s t ima t ion , and e r roneous re s u l t s may a r i s e .T h i s i s e x p l a i n e d b e c a u s e s p e c i m e n p r e p a r a t i o n p r o c e d u r e s(cu t t ing , mec han ica l po l i s h ing ve rs us e l ec t ro ly t i c po l i s h ing ) ,u s e d f o r h a r d n e s s i n d e n t a ti o n m e a s u r e m e n t s , a f f e c t th e f i n a lr e su l t s. A c o m p l e m e n t a r y s t u d y w a s c a r r i e d o u t t o e v a l u a t et h e e f f e c t o f th e c u t t in g p r o c e d u r e a n d m e c h a n i c a l p o l i s h i n gu s e d f o r s p e c i m e n p r e p a r a t i o n . D e p e n d i n g o n t h e m a t e r i a l ,h i g h e r h a r d n e s s v a l u e s , b e t w e e n 1 5 a n d 2 0 HV, w e r e o b t a i n e df o r s p e c i m e n s s u b j e c te d t o c u t t i n g a n d m e c h a n i c a l p o l i s h i n gc o m p a r e d w i t h t h o s e o b t a i n e d f o r t h e s p e c i m e n s s u b j e c t e dto e l ec t ro ly t i c po l i s h ing a lone . T h us , s ince abs o lu te ha rdnes sread ings in the s ho t -peened s u r face s a re s l igh t ly h ighe r thanthos e ob ta ine d in the t ens i l e s pec im en s u r face s , due to the s u r -f a c e p r e p a r a t io n , a c o n s e q u e n t o v e r e s t i m a t i o n o f e s t i m a t e dloca l y i e ld s t re s s i s obs e rved , compared w i th tha t ob ta inedb y u s i n g t h e d i f f r a c ti o n p e a k w i d t h m e t h o d . H o w e v e r , u s i n gre la t ive va r i a t ion da ta , t h i s e f fec t i s avo ided and s imi la r r e -s u l t s a re ob ta ined when e i the r ha rdnes s o r d i f f rac t ion peakwid th d a ta a re u s ed , a s F ig . 7 s hows . Fo r the ma te r i a l s s tud ied ,s imi la r loca l y i e ld s t res s va lue s we re de te rmin ed in the s ho t -p e e n e d l a y e r s u s in g b o t h m e t h o d s . T h e r e f o r e , t h e p o s s i b l e i n -f l u e n ce o f r es i d u a l s t r e s s es a n d / o r I S E o n t h e m i c r o - h a r d n e s sr e a d i n g s a l s o s e e m s t o h a v e b e e n a v o i d e d s o m e h o w w h e nnorma l i zed ha rdnes s va r i a t ion da ta a re u s ed . Conve rs e ly , i tc o u l d o n l y b e t r u e i f n o r m a l i z e d h a r d n e s s v a r i a t io n d e p e n d son ly on the ma te r i a l s t r a in -ha rden ing e f fec t , wh ich s eems tob e c o n f i r m e d b y o u r e x p e r i m e n t a l r es u l ts .In F ig . 7 , i t c an a l s o be obs e r ved tha t t he y ie ld s t re s s c lo s eto the s u r face a t t a in s va lue s c los e to the ma te r i a l ' s f r ac tu res t re s s, w h i c h c o n f i r m s t h a t p l a st i c d e f o r m a t i o n w a s v e r y h i g h .M o reove r , i n the l aye rs nea r the s u r face , t he ma te r i a l i s sub -j e c t e d t o f e w c y c l e s o f o l i g o c y c l i c f a t i g u e d u e t o t h e i m p a c t s

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Fig. 7--1n-depth estimation of local yield stress in shot-peened surface layers. Comparison between the results obtained usingthe normalize d variations of hardness and x-ray diffraction peak width

of the peening medium . Never theless , i f s tress-strain curvescorresponding to the mater ial near the surface (accommo-dated zone) , exhibit a s train-hardening exponen t close to thatobtained for the bulk m ater ial , as suggested in a f irst approachin the meth od o f Cao an d Castex, 15 the propose d meth odol-ogy ca n give reasonable est imates of the local yield s trengththroughout the whole shot-peened surface layer .Conclusions

The exper ime ntal data obtained in f ive s teels with dif fer -ing mechanical proper t ies , subjected to interrupted tensi letests, show that both the hardness var iat ion and x-ray dif f rac-t ion peak width var iat ion, with the plast ic deformation, arerelated to the s train-hardening capacity of a mater ial , withtheir normalized var iat ions being propor t ional to the norma l-ized var iat ion of the mate r ial 's yield s tress. This exp er imentalobservation al lows the fol lowing em pir ical equation to be ob-tained, which en ables the yield s tress of a mater ial subjectedto a given amount of plast ic deformation to be determined,when the relat ive var iat ion of hardness or dif f ract ion peakwidth and the yield s tress of the bulk mater ial are known:

O'y = O'y,O 1 + y

Here, the let ter H can corresp ond to hardness (HV) or todiffraction p ea k width (HWV), Cry,0 and Hy,0 corresp ond tothe elast ic l imit and hardness (p eak width) of the bulk mate-rial, respectively, and y is a constant. For the steels studied,the exper imental results have shown that the value of thisconstant seems to be independent of the mater ial 's s train-hardening, and a value close to 1 is obtained if the relat ivevar iat ion of diff ract ion peak width is used, while a value closeto 2.8 is obtained, i f the relat ive var iat ion of hardness is used.Taking into account the exper im ental results , an empir icalmethodology, based on the equation above, is proposed forthe est imation of a mean local yield s tress , as a function ofthe depth, in work-hardened surface layers . The est imationof the local yield s tress , using the normalized var iat ions ofhardness and dif f ract ion peak width, obtained in shot-peenedsurface layers , gives identical results, w hich seems to conf irmthe validi ty of the propose d methodology.

AcknowledgmentsThe authors gratefully acknow ledge the f inancial suppor tof Funda9~o para a Ci~ncia e Tecnologia, FCT-Portugal ,

p ro jec t POCTI /35953/EME/2000 , which made th i s workpossible.References

1. Scholtes, B., "Eig enspannu ngen in Mech anisch Ran dschichtve r-for mte n Werkstoffzust?inden: Ursachen, Ermi ttlung und Bewer tung,"Deutsche Gesellschafi fu r Mate rialku nde--D GM lnformationsgesellschaftmbH, Frankfurt (1991).2. Nobre, J.P, Kornmeier, M ., Dias, A.M., and Scholtes, B., "Use of theHole-Drilling Meth od fo r Measuring Residual Stresses in Highly StressedShot-peened Surfaces," EX PERIMENT AL MECHANICS, 40 (3), 289-297(2ooo).3. Nobre, J.P., Kornmeier, M., Gibmeier, J., Dias, A.M,, and Scholtes, B.,"Plasticity Effect On Residual Stress Results U sing Different Hole-DrillingEvaluation Methods," Proceedings of the 6th International Conference onResid ual Stresses ( CR S 6), Oxford, 10-1 2 July, Vol. 2, lO M communi cations,1188-1195 (2000).4. Gibmeier, J., Kornmeier, M., and Scholtes, B., "'Plastic Deformationduring Application o f the Hole-Drilling Method," 5th European Conferenceon Residual Stresses ( ECRS 5), Delft, 28-3 0 September, A.J. Bottger et aL,editors, VoL 1, Tran s Tech Publishers, 13 1-1 36 (1999).5. Virmoux, P., Inglebert, G., and Gras, R, "Charact erisation of Elasto-Plastic Behaviour fo r Contact Purposes on Surface Hardened Materials,"Dissipative Processes in Tribology, D. Downson et al., editors, ElsevierScience, Amsterdam, 287-301 (1994).6. Batista, A. , Dias, A., Virmoux, P.., Inglebert, G., Le Flout', J., and Le-brun, J., "Characterisation of Mechanical Properties and Residual Stressesin Hardened Materials," Recent Advances in Experimental M echanics, Pro-ceedings o f the l Oth Interna tional Conference on Experime ntal Mecha nics,Lisbon, 18-22 July, J. Gomes , editor, A. Balkema, 739 -74 4 (1994).7. Wohlfahrt, H., "The Influence of Peening Co nditions on the Result-ing Distribution of Residual Stress," Proceedings of the 2nd InternationalConference on Shot Peening, Chicago, M ay 1984, SEM Ed, 316-331 (1984).8. Desvigne s, M., "Influence du Grenaillage de Pr~contrainte sur laTenue en Fatigue de l'Acier 35CD4," Thesis: Docteur en G~nie M~canique,ENSAM, Paris (1987).9. Cao, W., "Adoucissement Cyclique des Aciers et Relaxation des Con-traintes Residuelle s de Grenaillage en Fatigue," Thesis: Doct eur en G~nieM~canique, ENSA M, Paris (1989).10. Guechichi, H., "PrEvision de s Contrainte s R~siduelles du es auGrenaillage de Prdcontrainte," Thesis: Do cteur en Gin ie Mdcanique, EN-SAM, Paris (1986).11. Ji, N., "'Application de l'Ana lyse d es Profils de Raies de Dif fractionX pour Caractdriser L'Etat Microstructural et MEcanique des Mat~riauxM(ta lliqu es," Thesis: Docteu r en G(n ie M~canique, ENSA M, Paris (1989).12. Maeder, G., Lebrun, J.L.,an d Sprauel, J.M., "D~termination parDiffraction des Rayons X des Contraintes Internes: Aspects Macro e

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Empirical Methodology to Estimate Yield Stress

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