RS in Laminated Fiber Epoxy Composites

8/14/2019 RS in Laminated Fiber Epoxy Composites

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Res i dua l S ur face S t resses i n Lam ina ted C ross -p l yF i ber -epoxy Compos i te Mate r i a l sby H aro ld E . Gasco igne

A B S T RA CT - - Re s idu a l ( cu r ing ) s t r esses in a c r oss - p l ylaminated p la te are re la ted to the s t ra ins re leased whenind iv idua l p l ies are separa ted . Re leas ed d isp lacem ents a r edetermined us ing h igh-sens i t i v i ty mo i r6 in te r fe romet ry andl inea r i zed s t r a i n - d isp lacemen t equa t i ons a r e used t o de te r -mine res idua l s t ra ins . E las t ic o r tho t rop ic s t ress-s t ra in re la -t ions are used to ca lcu la te res idua l s t resses remote f romfree-edge s o f a [902o/02o/902o] g raph i te -epox y cross-p lypane l. T he m easu r ed s t r a i ns compa r e f avo r ab l y w i th t hosep r ed i c ted by l am ina ted p l a t e t heo r y . I n a second examp le ,t h e c i rcumferent ia l and rad ia l res idua l s t ra ins and s t ressesat the end-sect ion o f a th ick-wa l led cross-p ly g raph i te -ep-oxy cy l i nde r a r e de te r m ined .

IntroductionR es i dua l s t r e s s e s i n compos i t e l ami na t e s r e s u l t f r omvar i ous s ou r ces du r i ng m anuf ac t u r e bu t pa r t i cu l a r ly f r om

di f f e r ences i n t he r ma l con t r ac t i on o f ad j acen t p l ie s w h ena l ami na t e i s coo l ed f r om i t s cu r i ng t emper a t u r e .T hes er e s i dua l s t r es s e s depend o n l ami na t e cons t r uc t i on and ma-t e r ia l p r ope r t i e s a s w e l l a s t he f ab r i ca t i on p r oces s . R es i dua ls t r e s s e s a r e e s pec i a l l y i mpor t an t a t t he f r ee edge o f al ami na t e w he r e h i gh va l ues can c r ea t e ma t r i x c r ack i ng o rde l ami na t i on i n i t i a t i on . C u t - ou t s ( ho l e s ) i n l ami na t eds t r uc tu r e s c r ea t e f r ee edges w he r e t he bas i c i n - p l ane r e s id -ua l s t r e s s e s gene r a t e s t rong i n t e r l ami na r s hea r and no r m a lstresses9 O r gan i c , m e t a l, and c e r ami c ma t r i x l ami na t e s a r el ikely to p ossess res idual s t resses af te r fabr ica t ion9 Sinc efabr ica t ion res idual s t resses may be large , i t i s essent ia lt ha t t hey be un de r s t ood s o t ha t de s i gne r s can e s t i ma t e t he i ri n f l uence .

T w o r ecen t s t ud i e s i nvo l v i ng compos i t e ma t e r i a l haveus ed mo i r 6 in t e r f e r om e t r y t o meas u r e de f o r ma t i ons due t ore lease d res idual s t resses ; the f i r s t in a th ick cross-p ly r ing I

Ha roM E . G asco igne i s Pro fes sor o f Mechan ica l Eng ineering , Ca l ifo rn iaPo ly t echn ic S ta t e Un ivers i ty , San Lu i s Ob i spo , CA 9340 ZPap er was presen ted a t t he 1992 SEM Spr ing Con ference on Exper imen-t a l M e c h a n i c s h e l d i n L a s V e g a s N V o n J u n e 8 -1 1 .Or ig ina l manuscr ip t submi t t ed : Apr i l 31 , 1992 . F ina l manuscr ip t r e -ceived." M ay 27, 1993

and t he s ec ond i n a f l a t c r os s - p l y pane l . 2 N e i t he r o f t he s es t ud i e s p r e s en t s a un i f i ed expe r i men t a l p r ocedur e w h i chr e s u l ts i n de t e r mi na t i on o f t he r e s i dua l s t r es s e s a t p r e s e -l ec t ed l oca t ions . I t i s t he go a l o f t h i s pape r t o p r e s en t as igni f ican t s tep towar d such a uni f ied pro cedure 9 On lyc r os s - p l y l ami na t e s a r e cons i de r ed i n t h is pap e r i n w h i chr ep r e s en t a t i ve r e s i dua l s t r e s s e s a r e de t e r mi ned i n a f l a tpan el and in a cy l inder 9

ApproachD ur i ng coo l i ng f r om t he cu r i ng t emper a t u r e , c e r t a i np l i e s con t r ac t mor e t han t he i r ne ighbor s , l e ad i ng t o l ock ed-

in deform at ion s a t the am bient (ope ra t ing) t emperature 9W hen f r ee - edges o f a l ami na t e a r e p r e s en t, i n t e r lami na r a sw e l l a s i n - p l ane s t r e s s e s a r e c r ea ted . T h i s ' f r ee - edge p r ob-l em ' ha s been s t ud i ed ex t ens i ve l y in ca s e s o f mechan i ca l

9 3 . . . .l oad i ng . S i nce i n t e r rac i a l f o r ce s ( s t re s s e s ) ac ti ng be t w eenp l i e s t ha t d i f f e r en t ia l l y con t r ac t du r i ng co o l i ng f r om t hecur i ng t emp er a t u r e c r ea t e de f o r ma t i ons i n t he p l i e s , e l i m i -na t i on o f t he s e i n t e r f ac i a l f o r ce s ( s t r e s se s ) w i l l r e l i eve t heres idual deformat ions in the pl ies 9 A s s umi ng e l a s t i c r e -s pons e , i f a l l o f t he i n t e r f ace s t r es s e s a r e r emo ved , ea ch p l yr e t u r ns t o a s t r e s s - f r ee s t a t e t h r oughou t . I n t he p r e s en tappr oach , p l i e s a r e s epa r a t ed a t i n te r f aces us ing d i amon dcut t ing tools af ter repl ica t ing a cross- l ine di f f rac t ion gra t -i ng on t he s u r f ace w h e r e t he r e s i dua l s t r ai ns and s t r es s e sa r e t o be de t e r m i ned . T he r e l i eved d e f o r ma t i ons ( s t r a ins )a r e t he nega t i ve o f t he de f o r ma t i ons ( s t r a ins ) p r oduce d bythe res idual s t resses resul t ing f ro m coo l -dow n9 Th e re-l i eved d i s p l acemen t s a r e de t e r mi ned us i ng moi r 6 i n t e r -f e r o m e t r y a n d t h e c o r r e s p o n d i n g r e l i e v e d s t ra i n s a r ede t e r mi ned f r om t he l i nea r i zed s t r a i n - d i s p l acemen t r e l a -t ions 4

OU 1 ON~ OV 1 aNy~ X - - 0 X - - f 0X ~ Y - -- -0 y f O y ( 1 )

l ( a N ~ ~ N y(2 )

E x p e r i m e n t a l M e c h a n i c s 9 27


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9 (Y3 ----"~23"1~31"[12 J

-Q l! Qlz Q13 0 0 0Q12 Q22 Q23 0 0 0Q13 Q23 Q33 0 0 00 0 0 Q 4 4 0 00 0 0 0 Q5500 0 0 0 0 Q66

e ~ - o ~ I A T "e~ - ~2A TE~ - o~3AT

723

71z ( 3 )

( 4 )

w h e re th e c o o rd in a te a x e s c o in c id e w i th s y m m e t ry a x e s o fth e m a te r i a l . B e c a u s e th e g ra t in g i s a p p l i e d o n a s u r fa c ea f t e r c o o l -d o w n , th e r e s id u a l s t r a in s m e a s u re d a re th et e rm s ( e ~ - ~ IA T ) , e t c . S in c e th e s u r fa c e s o n w h ic h th eg ra t in g s a re r e p l i c a t e d a re f r e e o f s u r fa c e t r a ct io n s , e q (3 )s im p l i f i e s t o

~ 1 2 L 0 0 Q 6 6 [ ~ 1 2 J

( a )

w i th th e s u p e r s c r ip t R d e n o t in g r e s id u a l s t r a in o r s t r e s sw i th

Ez 1)12 E2Q , , - Q I ~ ---1 - ' 1 . )1 2 ' O 2 1 l - - 1 ) 1 2 1 ) 2 1

E2022 -- Q66 = G121 - ~ 1 2 'D 2 1 ( 5 )T h e n o rm a l s t r a in p e rp e n d ic u la r t o th e s u r fa c e m a y b ed e t e r m i n e d f r o m

'1-)13 1)23~3 = E~ - ~ 3 A T = - 7 G1 - - - (Y2E2 (6)T h e ' t e n s io n v a lu e ' o f P o i s s o n ' s r a t io , a )~ l, i s d e f in e d a s th ed e c re a s e in l e n g th o f a n e l e m e n t in th e o n e -d i r e c t io n d u eto a t e n s i l e fo rc e in th e tw o -d i re c t io n d iv id e d b y th ee x te n s io n in th e tw o -d i re c t io n . T h e c o m p re s s iv e v a lu e o fP o i s s o n ' s r a t io , v ~ l , i s c o r re s p o n d in g ly d e f in e d a s th ein c re a s e in l e n g th o f a n e l e m e n t in th e o n e -d i r e c t io n d u eto a c o m p re s s iv e fo rc e in th e tw o -d i re c t io n d iv id e d b y th ec o n t ra c t io n in th e tw o -d i re c tio n . D i f f e re n c e s b e tw e e n th eP o i s s o n ' s r a t io s in t e n s io n a n d c o m p re s s io n in c o m p o s i t e sh a s b e e n o b s e rv e d p re v io u s ly . 6 8

E x a m p l e s

F l a t L a m i n a t eA s a f i r s t e x a m p le , c o n s id e r a [9 0 d 0 d 9 0 n ] f l a t la m in a tes h o w n in F ig . 1 . T h e x , y , z , c o o rd in a te s y s t e m i s a l ig n e d

w i th th e p r in c ip a l m a te r i a l d i re c t io n s: x p e rp e n d ic u la r t oth e p l a n e o f t h e l a m in a te , y p a ra l l e l t o t h e 0 -d e g f ib e r s a n dz pa ra l le l to the 90-deg f ibe rs . Res idu a l s t ra ins and s t res sesa t l o c a t io n s a f f e c t e d b y a n d a t l o c a t io n s n o t a f f e c t e d b yf re e -e d g e s a re s o u g h t . T o d e te rm in e th e s t r a in s in th e p l a n eo f a f r e e -e d g e , s t r ip s a re i s o l a t e d f ro m a n e d g e a s s h o w nin F ig . 1 . I f the leng ths o f the s ides o f the p la te a res u f f i c i e n tly l a rg e , t h e e d g e e f f e c t s w i l l n o t b e f e l t s u f f i -c i e n t ly f a r f ro m th e e d g e a n d th e s t r a in s a n d s t r e s s e s w i l lb e in d e p e n d e n t o f y a n d z. N e a r th e e d g e s , i n t e r l a m in a rs h e a r s t r e s s e s a re r e q u i re d to e q u i l ib ra t e th e in -p la n es t r e s s es c re a t e d b y d i f f e re n t ia l t h e rm a l c o n t ra c t io n . In tu rn ,to s a t i s fy e q u i l ib r iu m , th e in t e r l a m in a r s h e a r s t r e s s e s r e -q u i re in t e r l a m in a r n o rm a l s t r e s s e s . T h e s e in t e r l a m in a rs t r e s s es h a v e s ig n s d e p e n d in g o n th e c o e f f i c i e n t s o f t h e r -m a l e x p a n s io n , c ~, o f t h e p l i e s . F ig u re 1 s h o w s th e s e n s e o fthe s t res ses deve loped fo r c~90 > a0 . Re l iev ing the in te r-l a m in a r s t r e s s e s w i l l r e l i e v e th e lo c k e d - in r e s id u a l s t r a in sin all plies 9T o d e m o n s t ra t e , i n a p a r t i c u la r c a s e , t h e r e l e a s e o fd e fo rm a t io n s th a t w e re lo c k e d - in d u r in g c o o l in g f ro m th ec u r in g t e m p e ra tu re , a 3 8 x 4 1 -m m (1 . 5 0 x 1 . 6 1- in .) c o u p o nw a s c u t f ro m a [9 0 2 d 0 2 d 9 0 2 0 ] ]M7 /8 5 5 1 -7 A (H e rc u le sMa te r i a l s C o . ) g ra p h i t e / e p o x y f l a t l a m in a te h a v in g a th i c k -n e s s o f 8 .3 8 m m (0 . 3 3 0 in .) . E a c h l a m in a w a s o n e - th i rd o fth e to t a l l a m in a te th i c k n e s s , i . e. , e a c h l a m in a h a d a th i c k -n e s s o f 2 . 7 9 m m (0 . 1 1 0 in .) . T h e p a n e l h a d b e e n s to re d inth e l a b o ra to ry in w h ic h s u b s e q u e n t m e a s u re m e n t s w e rem a d e f o r s e v e r a l w e e k s t o m i n i m i z e h y g r o s c o p i c e f f e ct s .A 1 2 0 0 g / r a m (3 0 4 8 0 g / in .) c ro s s - l in e re f l e c t iv e d i f f r a c-

w h e re U , V a re c o m p o n e n t s o f d i s p la c e m e n t in th e x a n d yd irec t ions , re spec t iv e ly ; N~ and Ny are f r in g e o rd e r s c o r re -s p o n d in g to U a n d V , r e s p e c t iv e ly ; a n d f i s t h e f r e q u e n c yo f th e r e fe re n c e g ra t in g w h ic h w a s 2 4 0 0 g /m m (6 0 , 9 6 0g / in .) t h ro u g h o u t . T h e c o r re s p o n d in g r e s id u a l s t r e s s e s a red e te rm in e d f ro m th e g e n e ra l c l a s s i c a l t h e rm o e la s t i c e q u a -t io n s w r i t t e n fo r t h e c r o s s -p ly l a m in a te a s 5

I - o - o - o, ~ ~ vo ol _ _-- ,F

I - I o o- ~ 1 7 6 1 7 6o o o . ~ , . . r ' l , . . ~ 9 0o oo 9 o 9 o 9r , - ' ~ 1 7 6 ! io - v uo o o / io oo o oo o o . . ~ . - - n , . - ~ 9 0o n o n o n J

(~90~ ~ 0( b )

F i g, 1 - - G r o s s - p l y l a m i n a t e . (a ) L a m i n a t ec o n s t r u c t io n a n d c o o r d i n a t e s y s t e m . ( b)F o r m a t i o n o f r e s i d u a l s tr e s s e s u p o n c o o l -i n g f r o m c u r i n g t e m p e r a t u r e

2 8 9 March 1994


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Ny Nz

Fig. 2-Ny and Nz disp lacem ent fri nges for sur face ply of [902o/0zd90~0]cross-ply laminate. (a) Initial2.16-mm (0.085-in.) layer thickness removed. (b) 0.50-mm (0.020-in.) thickness layer removedfrom original 2.16-ram layer

tion grating was replicated on one of the lateral surfacesusing PC-10 adhesive (Measurements Group Inc.). Thethickness of the adhesive layer in the replication processis typically less than 25 p-m (0.001 in.). The coupon w iththe replicated diffraction grating was placed into the field

- 4of the mterferometer and the null-field obtained in theprojection plane for both the y- and z-direction displace-ment fields, V and W, respectively. The specimen wasremoved f rom the interferometer . The mold f rom whichthe grating was taken was then placed in the interferometerto verify that the sam e null-field was obtained. Thereafter,the interferometer could be brought to null-field immedi-ately prior to determining the residual displacements in theplane of the grating using the grating mold as the referencefor released deformation. This was a critical step in theprocedure since several hours could elapse between plyseparation and obtaining the final displacem ent interfero-grams. The outer (90-deg) ply was separated from thecenter (O-deg) ply using a diamo nd-wafering saw cutting

parallel to the interface between the 90-deg and 0-degplies. The thickness of the layer removed was 2.16 mm(0.085 in.) . The relieved displacements in the plane o f thesurface are shown in Fig, 2. The effect of the edges isclearly evident extending inward from each edge approxi-mately one-third of the length of the side of the coupon.Thus, approximately the central one-third of the length ofthe edge is free of edge-effect. The representative strainreleased perpendicular to the fibers was -0.37 percent(residual strain +0.37 percent) and representative strainreleased parallel to the fibers was +0.014 percent (residualstrain -0.014 percent). Fig ure 2 also shows the correspond-ing displac eme nt fields after a 0.5 l-ra m (0.020-in.) surfacelayer was removed from the 2.16 -mm thick original layer.No significant change in response was observed indicatingthat the locked-in deformations were fully released withthe first cut at the interface and, thus, substantiated thepremise that remov al of stresses at the interface returns thelamina to a stress-free state.

Exper imen ta l Mechan ics 9 29


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T A B L E 1 - - M A T E R I A L P R O P E R T I E S *ii IProper ty IM7/8551-7A AS4/3501-6E l (t e n .) 1 6 5 x 1 0 S ( 2 4 x 1 0 a) 1 4 5 x 1 0 3 ( 2 1 x 1 0 a)E 1 ( c om . ) 148 x 103 ( 21 . 5 x 103 ) 143 x 103 ( 20 . 7 x 103 )E 2 (t en .) 7 . 8 x 1 0 a ( 1 . 1 4 x 1 0 a) 9 . 3 x 1 0 3 ( 1 .3 5 x 1 0 3 )E 3 *( te n .) 7 . 8 x 1 0 a ( 1 .1 4 x 1 0 3 ) 9 . 3 x 1 0 a ( 1 . 3 5 x 1 0 a)Su~ ( t e n .) 2 4 8 0 ( 3 6 0 ) USu~ ( c o m . ) 1 6 5 0 ( 2 4 0 ) 1 7 2 0 ( 2 5 0 )Su2 ( ten . ) 53 (7 .7) Uv12 0 . 3 0 . 3v 2 (" 0 . 01 4 0 . 020v2a U Uc ~r + 21 . 6 x 106 / ~ + 21 . 6 x 108 / ~~_ -0.4 x 10~ / ~ - 0 .1 x 108 / ~

* H e r c u l e s M a t e r i a l s C o . d a t a b a s et S t if f n e s s a n d s t r e n g t h i n M P a ( k s i) a t 2 0 ~T = T r a n s v e r s e t o f i b e r sL = A l o n g f i b e r s* * C a l c u l a t e d f r o m r e c i p r o ic i t y a t 2 0 ~U = U n k n o w n* P r o p e rt y n o t a v a i l a b l e - - a s s u m e d t o b e a p p r o x i m a t e l y E 2

I t i s in teres t ing to compare the s t rains released in theex p er imen t wi th t h e r es id u a l s t r a i n s p red i c t ed b y l ami -n a t ed p l a t e t h eo ry (LP T) . Th e ap p en d ix g iv es t h e LP Tan a ly si s co r r esp o n d in g t o t h e p resen t ex amp le . W i th u n -cer tain ty in mater ial p roper t ies , p r incipal ly the values o ft h e co ef f i c i en t s o f t h e rmal ex p an s io n , t h e m easu red v a lu esof the in -p lane s t rains on the lateral surfaces away f rom theed g es an d t h o se p red i c t ed b y t h e LP T ag ree q u i te f av o r -ab ly . Tab le 2 g ives thes e resu l t s .Using the mater ial p roper t ies g iven in Table 1 , res iduals t resses in the 90-deg layer are (~y= 28 MP a (4 .1 ksi ) , whichi s ap p ro x im ate ly 5 0 p e rc en t o f t h e t r an sv er se t en s il es t reng th , and (~z = -20 M Pa ( -2 .9 ksi ) o r about o ne perc en to f t h e l o n g i tu d in a l co m p ress iv e s t r en g th.To p o r t r ay t h e i n f l u en ce o f cu t -o u ts (h o l es ) o n t h e s t a teof res idual s t rain (s t ress) near the in tersect ion of the cu t -ou tan d t h e su r f ace o f t h e l amin a t e , a 9 .3 8 -mm (0 .3 7 5 - in . )d i amete r h o l e was mach in ed t h ro u g h t h e t h i ck n ess o f aco u p o n t ak en f ro m th e sam e p an e l. A d i amo n d -co re d r i llwas u sed w i th l i b e ra l am o u n t s o f co o l an t an d t h e cu t t i n gsp eed r ed u ced t o min imize cu t t i n g d amag e an d i n t rod u c-t ion of machin in g s t resses . A cross- l ine g rat ing as p re-

v iously descr ibe d was rep l icated on one lateral surface. A0 .5 l - r am (0 .0 2 0- in .) su r f ace l ay er was r em o v ed u s in g ad i amo n d -wafe r i n g saw. Th u s , t h e i n t e r l amin ar t r ac t i o n sac t i n g a t t h e p l an e o f t h e cu t were e l imin a t ed an d t h esurface s l ice relaxed to i t s s t ress- rel iev ed s tate . Th e resu l t -i n g Va n d W d i sp l acemen t f ri n g es a r e sh o wn in F ig . 3 . Th es ig n o f t h e f r i n g e g rad i en t was d e t e rmin ed u s in g ca r r i e rf r inges o f ex tension .9 Th e ex ce l l en t f r in g e c l a r i ty an d h ig hf r i n g e d en s i t y p e rmi t r e l i ab l e d e t e rmin a t i o n o f t h e f r i n g egradie nts an d strain. Th e residu al (tensile ) strain, ey, per-p en d i cu l a r t o t h e f i b e r d i r ec t io n was +0 .4 5 p ercen t . Th ecorres ponding res idual tensi le s t ress was ~y = 35 M Pa (5 .1k s i ). A t o th e r l o ca t i o n s o n t h e b o u n d ary o f t h e h o l e , r e s i d-ual shear s t resses may reach s ign i f ican t levels which , inco mb in a t i o n wi th n o rmal s t r es ses , may cau se i n -p l an emat r i x c r ack in g .Th e r es id u a l d e fo rmat io n s i n t h e p l an e o f t h e f r ee -ed g eare d e t e rmin ed i n a s imi l a r way . A c ro ss - l in e g ra t in g wasrep l i ca ted o n t h e ed g e o f t h e co u p o n p erp en d i cu l a r t o t h ez ax i s. Tw o cu t s w ere m ad e n ear ly p a ra l l e l t o t h e i n t e r f aceb e tween t h e 0 -d eg an d 9 0 -d eg p l i es u s in g a 0 .3 5 -mm(0.014- in .) d iam ond s l i t t ing saw de noted as Cut I and CutI I in F ig . 4 . These cu ts were made at a s l igh t ang le wi thresp ec t t o t h e p ly i n t e r f aces - - t h e i n t e r f aces a r e i n d i ca tedb y th e s ym b o l 0 . . . . . 0 . T h e p u r p o s e o f t h e se cu t s w a sto sep ara t e t h e cen t r a l 0 -d eg l amin a f ro m th e 9 0 -d eg l am-ina, thereby remo ving the in ter facial s t resses . The purpo seo f t h e s l i g h t an g l e o f t h e cu t r e l a t i v e t o t h e i n t e r f ace wasto r ev ea l t h e s l o p e o f b o th t h e U an d V f i e l d d i sp laceme n tf r inges near and a t the in ter face as seen in F ig . 4. Nex t , CutI I I (F ig . 3 ) was mad e b y p ar t i n g o f f a l ay er o f 0 .8 0 mm(0 .032 in . ) paral lel to the g rat ing . Three separate s t r ipswere t h u s c r ea t ed ; o n e f ro m th e cen t r a l l amin a an d o n eeach fo r t h e two o u t e r lamin ae . Each s t ri p was h e ld in p l aceon a f la t g lass p late using two-s ided adhe sive tape, and theg lass p late was inse r ted in to the in ter ferom eter af ter tak ingg rea t ca re t o ca re fu l l y a l ig n t h e i n t e r f e ro mete r fo r n u l l-f ie ld using the specim en grat ing mold . The U (x-d i rect ion)and V (y-d i rect ion) d isp lace me nt f r inges are Shown in F ig .4 . On e en t ir e i n t e r f ace i s v i s i b le a lo n g 0 . . . . . . 0 ad j acen tt o t h e ch ev ro n sh ap ed (v o r ) f r i n g es in t h e U- f i eld . Th esech ev ro n f r i n g es i n d i ca t e a ch an g e i n s i g n o f t h e t h ro u g h -th e - t hi ck n ess n o rmal r es id u a l - s tr a in co m p o n en t ~ c ro ss -i n g t h e i n t e r f ace f ro m p o s i t i v e i n t h e cen t r a l 0 -d eg l amin ato negat ive in the ou ter 90-deg lamina. This i s s imi lar toth e r ev er sa l i n s i g n o f t h e t h ro u g h - th e - th i ck n ess n o rmals t r es s v e ry n ear t h e i n t e r f ace o f a mech an i ca l l y l o ad edcro ss -p ly l amin a t e d esc r i b ed i n Ref . 3 (p ag e 2 0 9 ) . Th eres id u a l s t ra i n s i n t h e u n i fo r mly resp o n d in g cen t e r o n e-th i rd o f the leng th of the edge are: ex= +0 .50 perce n t in thecen te r lamina and -0 .22 p ercen t in the ou ter lam ina, ey =

T A B L E 2 -- C O M P A R I S O N O F R E S I D U A L S T R A IN S D E T E R M I N E D B Y E X P E R IM E N T A N D L A M I N A T E D P L A T ET H E O R YI I I I I I I I I I I I I0 - d e g L a m i n a 9 0 - d e g L a m i n a

] I t o F i be r s ( pe r c en t ) _L. o F i be r s ( pe r c en t ) [ I t o F i be r s ( pe r c en t ) .1_ o F i be r s ( pe r c e n t )M e a s u r e d N o t M e a s u r e d N o t M e a s u r e d - 0 .0 1 4 + 0 . 3 6T h e o r y - 0 . 0 3 7 + 0 . 3 4 - 0 . 0 1 6 + 0 . 3 2

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-0.04 percent in the center lamina and +0.33 percent in theouter lamina. An elemental s tr ip shown in Fig. 4 a t theintersect ion o f the f ree-edge and the la teral sur face o f thelaminate exper iences uniaxial loading; a special case ofplane stress. The values of residual strains at this location

41(1.61)

Dimensions: ram(in]

9 C u t I I ID 0.8010.032)

' Vi I ,,,,,u.ii

/ ~ 0 . 3 5 (0 . 0 1 4 )

3301

O)

are ex = -0.22 percent, and ey = +0 .33 perce nt. Th is me antthat when the res idual s tresses were released, the elementalstr ip respon ded as if a co mp ressi ve stress, Cry, acted a loneresult ing in a Poisso n' s ra t io v~ = 0.22/0.33 = 0.67. Thisseemingly large value of Poisson' s ra t io in compress ionhas been observed in s imilar laminae. 8Thick-wa l led Cy l inder

Thick-walled cross-ply laminated cylinders are of cur -rent interest for use in submersible structures. The influ-ence of cur ing s tresses has been s tudied theoret ical ly underthe assumptions of l inear -elas t ic response and tempera-ture- independent proper t ies . 1~ The same metho dolog y asdescr ibed for the f la t laminate can be used to determine theresidual strains on surfaces of a cross-ply cylinder. Apor t ion of a cylinder near i ts end sect ion is shown in Fig.5. At an end sect ion of the cylinder , there wil l be af ree-edge ef fect with the s trains and s tresses varying in theaxial (z) direction. Th e r-0 pla ne is similar to the x-y planeat the f ree-edge of the laminate as shown in Fig. 4. Like-wise, there will be residual strains and stresses in the r-0plane. The representat ive s trains released in the plane o fthe f ree sur face at the end of the cylinder due to the remov alof the circumfere ntial and radial stresses, co and crrrespec-tively, (at the free surface 6z = 0) will be indep enden t o f 0

V - Field

W - Field(b)

Fig. 3-Released residual displacement fieldsfor cross-ply laminate with through-the-thick-ness hole. (a) Coupon dimensions. (b) Vand Wdisplacement fields (perpendicularand parallel to fiber direction at surface) Fig. 4 - -U and V residual displacement fringes on theedge of a [902o/020/902o]cross-ply laminate

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Fig. 5-- Sur face slice removal for thick-walled cross-ply cylinder

because of axial symme try. Anomalies in mater ial proper-t ies and fabr icat ion wil l, o f course, give local var iat ionsf rom average ( representat ive) values . However , thesestrains and stresses may depend on axial (z) position.The fol lowing procedural s teps are suggested to revealthe important characteristics of the residual stresses atsur face points o f the cylinder . (1) To determine the axialdepen denc e o f e0 and ez on, say, the o uter surface, a sma llf la t sur face can be produced by careful ly machining orgr inding us ing diam ond tools . The depth o f this f la t can be

made, say, one- third of the thickness of the outer plywithout causing an appreciable force release in the outerply. This f ia t is des irable s ince replication on curved sur -faces is very difficult. Next, a grating G~ is replicated onthis f lat and, subsequently, a thin layer containing thegrat ing is removed with the thickness of the layer conf inedto the outer ply. The releas ed strains will indicate the natureof the axial dependence in the outer ply. The sam e proce-dure could be used to determine the axial dependence ofthe sur face s trains on the ins ide sur face but with great lyincreased dif f iculty due to the concave curvature of thesurface. To determin e the released strains in the r-0 plane,a grat ing G2 is replicated on the end sect ion of the cylinderand a thin s l ice paral le l to the sur face is removed. Thispiece can be cut into two por t ions f rom which informationcan be extracted for the circumferentia l pl ies f rom onepor t ion and fo r the axial pl ies f rom the oth er por t ion (seeFig. 6). Stresses ~0 are remov ed by radial cuts and stresses(~r are remo ved by making shor t, s tra ight cuts approxi-mate ly parallel to the plies.

As an examp le of these s teps , a wafer containing G2 wastaken f rom a AS4/3501-6 graphite/epoxy cylinder with([903/0]20/903) stacking sequence design ated 89-1A whic hwas used in a pre viou s study. 11 The cured p lies had athickness of 0.18 mm (0.0072 in. ) giving a representat ivethickness of the three-ply circumferentia l layers of 0.55mm (0.0217 in.) . The cylinder had an ins ide diameter of17.78 cm (7.00 in.) and a nominal wail- thickness of 1.58cm (0.62 in. ) . F igure 6 shows the displacement f r inges inthe r -0 plane n ear the outs ide diameter result ing f rom therelease o f both 60 and ~Yr The circumfe rential and radia ls tresses for the outer four c ircumferentia l pl ies which aregiven in Fig. 6 were determined f rom eq (4) with theprincipal ma terial directions for the 903 plies taken as 1 =0, 2 = z, 3 = r . The exact locations of calculated stressesare shown for each ply in Fig 6. The use of carder f r inges

Fig. 6--C ircu mfe ren tial and radial residual displac eme nt fringes at end section of a thick-walled cross -ply cylinder.(a) Microphotograph showing cuts that release ~o and ~r. (b), (c), (d) are Vdispla cemen t fringes with different carrierfringe s of rotation. (e), (f), (g) are U displacem ent fr inges with different c arrier fringes o f rotation

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o f ro t a t i o n were i n d i sp en sab l e i n accu ra t e ly d e te rmin in gthe re leased s t rains . 12 Av erag ing o f f r inge spacings andf r i n g e an g les o v er t h e a r eas sh o wn to g e th er wi th s t an d ardd ev i a t i o n s a r e g iv en . S in ce t h e o u t e r c i r cu m feren t i a l p l i e sh ad co mp ress iv e c i r cu mferen t i a l re s i d u a l s t ra i n s, t h e v a lu eof v+21 was used in calcu lat ing the res idual s t resses usingeq (4) . Som e smal l unce r tain ty ex is ts in the res idual s t rainsd e t e rmin ed a t t h e b o u n d a ry d u e t o m in o r d i s t o rt i o n o f t h eg ra t i n g cau sed b y ep o x y sh r in k ag e . Th i s u n cer t a in ty i sd i scu ssed i n th e fo l l o win g sec to n . Th e v a lu e o f ~0 = -1 4 0M P a ( -2 0.3 k s i ) i n t h e o u t e r p ly is i n f a i r ag reemen t wi thth e v a lu e o f -1 5 k s i d e t e rmin ed u s in g a l ay er - r emo v a lme th o d fo r r i n g s cu t f ro m th e sam e cy l i n d er .1 3 Th e v a lu eso f ~ r ca l cu l a t ed i n t h e o u t e r fo u r c i r cu m feren t i a l p l i e s i sco n s id e rab ly l a rg e r ( ap p ro x imate ly t en t imes l a rg e r ) th anth o se d e t e rmin ed b y e x p er ime n t in Ref . 1 3 o r ca l cu l a t edfor a s imi lar cy l inder in Ref . 10 . I t i s in teres t ing to no tethat the rad ial res idual s t ress , o r , increases very qu ick lyf ro m zero a t t h e f r ee su r face t o +1 4 .9 M P a (2 .2 k s i ) n ea rt h e i n n er su r f ace o f t h e o u t e r p ly as sh o wn in F ig . 6 .E x p e r i m e n t a l E r r o r S o u r c e s

As in an y ex p er imen t , Un cer t a in t i e s o ccu r t h a t mayd e t e r i o ra t e th e a ccu racy o f t h e r esu lt s . Th e fo l l o win g u n -ce r t a in t i e s an d so u rces o f e r ro r a r e co n s id e red .(1 ) Un cer t a in ty o f t h e sp ec im en -g ra t i n g fr eq u en cy . Th eme th o d b y wh ich t h e mas t e r g r a ti n g ( f ro m wh ich sp ec imenrep l i ca t e co p i es u sed i n t h e p resen t wo rk were m ad e) was

14mad e p rec lu d es an u n cer t a in ty o f mo re t h an 1 f r i n g e / r amin t h e f r eq u en cy . Th i s u n ce r t a in ty wo u ld l ead t o a n eg l i -g ib l e e r ro r o f 0 .0 4 p e rcen t i n t h e s t ra i n s u s in g eq (1 ) an deq (2) .(2 ) Ou t -o f -p l an e warp ag e o f t h e sp ec imen g ra t i n g d u e t or e l ease o f r es id u a l s t r es ses. I f t h e o u t -o f -p l an e warp ag ecau ses a ro t a t io n ab o u t an ax i s p e rp en d i cu l a r t o t h e r e f e r -en ce g ra t i n g l i n es, n o e r ro r i s i n tro d u ced . I f warp ag e c au sesa ro tat ion of the g rat ing about a l ine paral lel to the referenc eg ra t in g l i n es , an ex t r an eo u s mo i r6 f r e q u en cy F e = - f W2/2i s cau sed wh ere W i s t h e ro t a t i o n an g l e ( as su med t o b esmal l ) an d f i s t h e f r eq u en cy o f t h e r e f e r en ce g ra t in g ( seeRef . 4 ) For a ro tat ion W = 0 .01 rad , the ex t raneous s t raina, = -50 p ,m/m o r -0 .005 percen t . I t i s h ig h ly un l ike ly thatt h e d e fo rmat io n s o f t h e r emo v ed su r f ace l ay er s i n t h ep resen t s t u d y w o u ld b e t h i s l a rg e . Th u s , t h e e r ro r d u e t oo u t -o f -p l an e warp ag e i s j u d g ed t o b e i n s ign i f ican t .(3 ) In t ro d u c t i o n o f ca r r i e r f r i n g es o f ro t a t io n i n t ro d u ceex t r an eo u s f r i n g es wi th f r eq u en cy F e = - f 02/2 wh ere 0 i st h e an g l e b e tw een t h e i n i t i al sp ec im en g ra t i n g l i n es an d t h ere fe r en ce g ra ti n g l in es an d f i s t h e f r eq u en cy o f t h e r e f e r -en ce g ra t i n g ( see R ef . 1 2) . Th e ca r r i e r f r in g es a r e r e l a t edto 0 by F c = f O . T h e l a rg es t ca r t i e r f r eq u en cy u sed i n t h ep resen t s t u d y was 3 5 f r i n g es / ram fo r t h e o u t e r c i r cu mfer -en t ial p ly shown in F ig . 6 . The ex t raneous s t rain in th iscase i s - 1 0 6 g m/m o r ap p ro x im ate ly -0 .01 p e rcen t , mak in gth e ap p aren t v a lu e o f 8 r at t h is l o ca t i o n 0 .0 2 8 p ercen t w h ichi s ap p ro x ima te ly 2 8 -p ercen t l o w. At o th e r l o ca t io n s i n F ig .6 t h i s e r ro r is sm al l en o u g h t o b e n eg l ec t ed .(4 ) R ig id -b o d y ro t a t i o n s o f p o r t i o n s o f l ay er s o cc u r a f t e rr e l ease o f r es id u a l s t r es ses . S in ce a fo u r -b eam o p t i ca lsy s t em w as u sed , r i g id -b o d y mo t io n s o f t h e g ra t in g d o n o t

affect calcu lat ion of shear s t rains (Ref . 4 , page 345) . No r-mal s t ra i n ca l cu l a t io n s a r e n o t a f f ec t ed b y r i g id -b o d y ro ta -t i o ns r eg ard l es s o f t h e t y p e o f o p t i ca l sy s t em u sed .(5 ) S h ear l ag i n t h e sp ec ime n g ra t in g . Bec au se t h e d i f -f r ac t i o n g ra t i n g i s s ep ara t ed f ro m th e co m p o s i t e su r f ace b ya t h in l ay er o f ep o x y ap p ro x imate ly 2 5 -g m (0 .0 0 1 - in . )th ick , abrup t changes in s t rain are no t transm i t ted prec iselyto t h e ex t e rn a l su r f ace o f t h e g ra t i n g . H ig h - s t r a in g rad i en tzo n es o n t h e co mp o s i t e su r f ace a r e en l a rg ed o n t h e g ra t i ngsu r f ace an d p eak s t r a in s a r e r ed u ced . Va lu es o f s t r a i nd e t e rmin ed a t p ly i n t e r f aces wh ere r es in - r i ch zo n es ma yex i s t w i l l t h e re fo re b e l e s s accu ra t e t h an a t l o ca t i o n s awayf ro m su ch h ig h - s t ra in g rad i en t r eg io n s. S in ce a l l n u mer i ca lr esu l ts p r esen t ed i n t h i s p ap er a r e fo r l o ca t io n s aw ay f ro mp ly i n t e r f aces , sh ear -l ag e r ro r i s j u d g ed t o b e u n imp o r t an t .(6 ) Dis to r t ion of the g rat ing at saw cu t o r f ree edge. I fex t r eme ca re i s u sed i n cu t t i n g wi th d i amo n d wafe r i n gb l ad es, n o d amag e t o t h e g ra ti n g o ccu r s . S lo w sp eed s a r er eco mmen d ed wi th t h e u se o f l i b e ra l amo u n t s o f cu t t i n gf lu id t o k eep t h e ed g e b e in g cu t co o l an d f r ee o f d eb r i swh ich can d am ag e t h e cu t ed g e . Th e b l ad e r emo v es m ate -r i a l wh ich i s n o t av a i l ab l e fo r an a ly s is . Th e wid th o f t h esaw b l ad e can o f t en b e se l ec t ed t o c lo se ly match t h eth i ck n ess o f a p ly so t h a t co m p le t e p li e s may b e r em o v edwi th a s i n g l e cut . Wafer i n g b l ad es m ay b e o b t a in ed wh ichremo v e as l i t t l e a s 0 .2 0 mm (0 .0 0 8 - in . ) . A mo re se r i o u se r ro r may b e i n t ro d u ced n ear a b o u n d ary d u e t o sh r i n k agein t h e e p o x y u sed t o r ep l i ca t e th e g ra t i n g. Th i s sh r i n k ag en ear t h e b o u n d a ry i s m an i f es t ed b y d i s t o r t io n o f t h e g ra t i n gand the generat ion o f 'parasi t ic ' f r inges s imi lar to those inphotoelas t ic coat ings . This c auses er ror p r incipal ly in nor-mal s t ra i n s p e rp en d i cu l a r t o t h e b o u n d ary . Th ese p a ras i t icf r i n g es can b e r ead i l y i d en t i f i ed b y i n t ro d u c in g ca r r i e rf r i n ges o f ro t a t i o n an d o b se rv in g an y cu rv a tu re o r ' h o o k -in g ' o f t h e f r i n ges n ear t h e b o u n d ary . Th ese f r i n g es can b emin imized , an d i n so me cases co mp le t e ly e l imin a t ed , b ycare fu l l y c l ean in g an y ex cess u n cu red ep o x y f ro m th ein t e r sec ti o n o f t h e g ra t i n g p l an e an d t h e b o u n d ary d u r in grep l icat ion .

(7 ) Un cer t a in ty i n m ate r ia l p ro p er t i e s. Ex cep t a t r e s i n -r ich or res in -s tarved locat ions in the mater ial , the repre-sen t a t i ve v a lu es g iv en i n T ab l e 1 a r e f e l t t o b e accu ra t ewi th in +2 -3 p e rcen t fo r mo d u l i i an d w i th in +3 -5 p e rcen tfo r t h e co ef f i c i en t s o f t h e rmal ex p an s io n . Th u s , i n mo s tcases , the g reates t er rors in the calcu lated va lues o f s t rainand s t ress are caused by uncer tain t ies in mater ial p roper-ties.

Conc lus ions(1 ) Th e d e t e rmin a t i o n o f r es id u a l su r f ace s t r a i n s an ds t r esses fo r c ro ss -p ly l amin a tes as d esc r i b ed i n t h i s p ap eris a s t raigh t forward bu t ted ious p rocess .(2 ) Ex t r eme ca re m u s t b e t ak en , i n t h e meth o d d esc r i b ed ,

in releasing in ter -p ly t ract ions using d iamon d-cu t t ing too lsso as n o t t o d amag e t h e ex c i sed sec t i o n o r t h e d i f f r ac ti o ngrat ing .

(3) F or the flat [9020/020/9020] gra phi te-e pox y lamin ate,g o o d ag reemen t was fo u n d b e tween t h e ex p er imen ta l l ydetermined in -p lane res idual surface s t rains and corre-

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sponding res idual surface s t rains p red icted by laminatedplate theory ,(4) Residual s urface s t rains and s t resses at the end sec-t ion ne ar the ou ter surface o f a th ick -wal led ( [903/0]2o/903)g rap h i t e - ep o x y cy l i n d er were d e t e rmin ed . Th e i n t ro d u c-t i o n o f ca r r i e r f r i ng es i n mo i r6 i n t e r f e ro met ry i s an imp o r -t an t t o o l i n i n c reas in g t h e accu racy o f t h e r esu lt s wh e re t h eload- indu ced f r inges are i r regu lar o r sparse.

(5 ) Mater ial p roper t ies used in const i tu t ive equat ionssh o u ld b e accu ra t e ly d e t e rmin ed t o i n su re t h a t t h e ca l cu -lated res idual s t resses are real i s t ic .

AcknowledgmentsAp p rec i a t i o n i s ex p ressed to t h e P h o to mech a n i cs Gro u p

in t h e Dep ar tmen t o f En g in eer in g S c i en ce an d M ech a n i csat Virg in ia Po ly technic Inst i tu te and State Univers i ty fo rt h e i r a s s i s t an ce d u r in g 1 9 8 9 -1 9 9 0 wh en t h e wo rk d e -scr ibed herein was in i t ia ted . Special thanks are ex tendedto P ro fes so r Dan i e l P o s t fo r h is ad v i ce an d en co u rag emen t .Th e au th o r wi sh es t o t h an k t h e Hercu l es M ate r i a ls Co m-p an y fo r su p p ly in g t h e sp ec imen s u sed i n t h e s t u d y .

References1. Lee, J., Czarnek, R. and Guo, Y., "Interferometric Study o f ResidualStrains in Thick Composites," Proc. 1989 SEM Sp ring Conf. on Exp.Mech., 356-364 (1989).2. Lee, J. and Czarnek, R., "Measuring Res idual Strains in CompositePan els Using Mo ird Interferometry, '" Proc. 1991 S EM Spring Conf. onExp. Mech., 405-415 (1991).3 . Herakovich , C .T. , "Fre e Edge Ef fec ts in Lam inated Composi tes ,"Handbook of Composites, ed. C.T. Herakovich a nd EM . Tarnopolskii,Else vier Science Publishers, 2, C hap. 4 (1989).4 . Post, D. , "Moir~ ln terferometry ," Handbook o f Experimenta l Me-chanics, ed. A. S. Kobayashi, P rentice Hall, E nglew ood Cliffs, N J, Chap.7 (1987).5. Agarwal, B .D. and Broutman, L.J. , Ana lysis and Performance o fFib er Composites, John W iley and Sons, Inc., 148-155 (1990).6 . Halp in , J .C . , "Prim er on Composi te Materia ls ," Technomic P ub-lishing Co., Lancaster, PA , 168-172 (1984).7 . Bert , C . W. , "M odels or Fiber Com posites w i th Di f feren t Propert iesin Tension and Compression," J. Eng. Mat. Tech., 344-349 (1977).8. Gascoigne, H.E. a nd Abdallah, M. G., "Strain Analysis o f a Bonded,Dissimi lar, Composi te Mater ia l T-Jo in t Using Moir~ ln terferome try ,"Opt . and Lase rs in Eng . , 13 , 155-165 (1990).9 . Han, B. and Post , D. , "The Ti l ted Pla te Me thod for In troducingCarrier Fringes o f Extension in Moirg In terferometry ," EXPERIMENTAL

TEC~Qt;ES, 13 (7), 25-29 (1989).10 . Hyer, M. W., "Hydrosta tic Response o f Th ick Laminated Compos-ite Cylinders, " J. Reinfo rced PIastics and Com posites, 7, 321-340 (1988).11. Abdallah, M . G., Cairns, D.S. and Gascoigne, H.E., "ExperimentalInvestigation of Thick-Walled Graphite~Epoxy Composite Ring UnderExternal Hydrosta tic Compressive L oading ," Proc. 1991 SE M SpringConf. on Exp. Mech., M ilwaukee, 626-631 (1991).12 . Guo, Y. , Post, D. , and Czarnek, R . , "The Mag ic o f Carrier Fringesi n M o i r ( l n t e r f e r o m e t r y , " EXPERtMEN'rALMECHANtCS, 29, 169-173(1989).13. Abdallah, M .G., "Residual Stresse s in Thick -Wa lled CompositeRings," Proc. 1992 SEM Spring Conf . on Exp . Mech . , Las Vegas,1063-1070 (1992).14. Post, D., MeKelvie, J. , Tu, M. and Dai, F., "Fabrication o fHolographic Grat ings Using a Moving Source ," Appl. Opt . , 28 (15) ,3494-3497 (1989) .

A P P E N D I X

Laminated Plate AnalysisIn c l as s i ca l l amin a t ed p l a te t h eo ry (L P T) 5 th e fo l l o win gassu mp t io n s a r e mad e . (1 ) Di sp l acemen t s a r e co n t i n u o u sacro ss t h e i n t e r f ace b e tween two l amin a . (2 ) Th e i n -p l an e

d i sp l acemen t s a t an y p o in t a r e r e l a t ed t o t h e d i sp l acemen t so f t h e g eo met r i c mid p l an e . (3 ) P l an e sec t i o n s r emainp lane. (4 ) In -p lane s t rains are related to d isp lacementgrad ien ts at the geom etr ic midplane. (5 ) S t rains vary l ine-ar ly (at most ) across laminate th ickness . (6 ) Through- the-th ickness normal s t rains are neg lected .F ig u re 7 sh o ws t h e co o rd in a t e sy s t em an d d efo rmat io n sin the x - y p lan e . F ro m g eo met ryv = Vo - xo~ (A1 )

wi th~U oOY (A2)

A l s o

W = W o - - X ~z (A3 )Th e l i n ea r i zed s t r a in -d i sp l acemen t r e l a t io n s g iv e

Ov OVo O2Uo- ~ y 3 y x O y2

Ow 3Wo b2Uo~ Z - ~ z ~ z - X ~ z ~

by 3w bV o 3Wo 32Uo~(y~= -q -- + - -V- = - - + - - -2 xo z o y O z 3 y ~ y O z (A4 )Eq u a t i o n (A4 ) can b e wr i t t en i n mat r i x fo rm as

, , zL v y z J L ~ y z J (A5 )

o ow her e e y, and ~ z, and " ~ ~ z a r e t h e midplane strains and ~:y,~c~, and ~Cyza r e t h e plate curvatures . The s t ress-s t rain rela-t ions fo r the k th lam ina are

f F ~ 1 7 6 1 l f tQ , 6 Q 2 0 0 6 J L y J ( s a m e ) , ,,(A6 )

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. v , I"po g g g ~7 d O g ,) l o t

P8"6"6"6"8"6"8"~o o o o o o o o o o o) o o o o o o o o o o ro o o o o o o o o o oi o o o o o o o o o o () o o o o o o o o o oo o o o o o o o o o (o o o o o o o o o o

o o x o o o o o o () o r ~ 3000000O0 )00000(000000)0000000000r00000000000~ o o o o o o o o n o r

D e f o r m e d U n d e f o r m e d

Fig. 7- -Def in i t ion of the undeformed and deformedcross sect ion of a lam inated p late

1w h e re ~ 0 a re th e t r a n s fo rm e d s t i f fn e s s e s w h ic h a re re l a t e dto the s t i f fness in the p r inc ipa l ma te r ia l d i rec t ions , Qu , byeq 5 .61 o f Ref . 5 .In ' t h e c a s e o f t h e rm a l s t r a in s n o w c o n s id e re d , t h e m e -c h a n ic a l s t r a in s a re g iv e n b y

"~y~J tYy zJ [ ~yzj ( A 7 )

w h e re { e} a re th e to t a l l a m in a s t r a in s g iv e n b y

I y 1 , 1 f t[ ' Y y z J [ ~ y z J ](]yz (A 8 )w i th th e th e rm a l s t r a in s g iv e n b y

V ; y z J L ~ y ~ A r J (A 9 )

w h e r e ~ y , o ~ a n d ~ y z a r e t h e ( t r a n s fo rm e d ) c o e f f i c i e n t s o fth e rm a l e x p a n s io n . T h e th e rm a l s t r e s s e s fo r t h e k th l a m in aa re

I 'y /~ I O l l O ] 2 # 1 6 1 I ~ y - - 0 ( ,y A T ]o T I = 1 a 1 2 0 % 2 Q G 6 | ~ E z - O l ,A T,.[T T - - _ _ _ _[ r z J k [ O , 6 Q z 6 0 6 6 J , [ y y z - ~ z A T J (AlO)T h e m id p la n e s t r a in s a n d c u rv a tu re s a re r e l a t e d to th et h e r m a l f o r ce ( N ) a n d t h e rm a l m o m e n t ( M r ) r e s u l t a n ts b y

f ! o t ; r A " "_ _L B , I D ' J ( A l l )w h e re , fo r t h e c a s e o f c ro s s -p ly l a m in a te s y m m e t r i c w i thre s p e c t t o th e m id p la n e ,

[ A ' ] = [ A ] - '[ B ' ] = [ 0 ]

{ M T } = { 0 } (A 1 2 )w i th

n[A]~ = [Au]~ = ~ ( Q i j ) k ( hi : - h k - , )

k = l

w h e re h k i s t h e th i c k n e s s o f t h e k th l a m in a . H e n c e , a se x p e c te d , t h e p l a t e c u rv a tu re s a re z e ro fo r t h e s y m m e t r i cc ro s s -p ly l a m in a te .

T h e th e r m a l fo rc e r e s u l ta n t fo r t h e p re s e n t c a s e i sI N I V ! I [ Q 2 2 [ f f ' T ' 4 ; - Q 1 2 0 ~ L ] [ Q l l l ~ L n t ' Q 1 2 0 [ , T

( A 1 3 )Fro m e q (A 12) th e inver se o f [A], [A] -~ , g ives th e e lem entso f [A ' ] a s

A'I 1 -' - A22A u A 2 2 - A 2 2- A l zAtl 2 _--A u A 2 2 - A ~ 2A uA ' 2 2 - - A uA22 - A~2

wh ereAli = ( 2Q22 + Q u ) hAm = ( 3Qm ) h

A22 = ( 2Q u + Q2z ) hw i th

ELO l I - - 1 - 19LragrL~Lr ELQ 1 2 - -- 1 - ~Lra~rc

E rQ 2 2 - - '- 1 - ~Lra&L (A 1 4 )

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C o m p u t i n g t h e c o m p o n e n t s o f t h e m i d p l a n e s t r a in s g i v e s( 2 Q l l + Q 2 2 ) [ (Qn+2Q,2)o~L+(2Q22+Q12)o~r] - 3Q12 [ (2Qn+Q,2)o~L+(Q22+2Q,2)o~r]4= ( 2 Q 2 2 + Q l l ) ( 2 Q l l + Q 2 2 ) - 9Q22(2Q22+Q11)[(2Q11+Q12)o~L+(Q22+2Q12)otr] 3Q12 (Q1,+2Q12)o~L+(2Q22+Q12)o~r]

( 2 Q 2 2 + Q l l ) ( 2 Q l l + Q 2 2 ) - 9 Q ~ 2

S i n c e v 1 2 v2 1 i s v e r y s m a l l c o m p a r e d t o u n i t y , i t w i l l b e n e g l e c t e d s o t h a tQl l -~ E l , Q22 = E2, Q12 -~ v12E2

D e f i n i n g m = E f f E 1 , r = V 12 m a n d r e w r i t i n g ( A 1 5 ) g i v e se ~ = ( 2 + m ) [ ( l+2r ) aL+( 2m +r ) O~r ] - 3r [ ( 2+r ) o~L+( m+2r ) o~r ] A T

( 2 m + l ) ( 2 + m ) - 9 r 2= ( - 3 r [ (l + 2 r ) a L + ( 2 m + r ) O ~ r ] + ( 2 m + l ) [ ( 2 + r ) o t L + ( m + 2 r ) ~ r ] A T

( 2 m + l ) ( 2 + m ) - 9 r z

A T

A T ( A 1 5 )

( A 1 6 )

Num erical Exam pleU s i n g a v e r a g e v a l u e s o v e r t h e t e m p e r a t u r e r a n g e o f A T

= - 1 6 0 ~ t o b e (X L = - 0 . 4 X 1 0 -6 /~ C ~r = - 2 1 . 6 x ' 1 0 - 6 / ~g i v e s ey ~ = - 3 0 4 x 1 0 6 m / m . H e n c e , i n t h e 0 - d e g l a y e r , th em e c h a n i c a l s t r a i n p a r a l l e l t o t h e f i b e r i s~Mt0deg]~ - -- ~ y - - ~ L A T = - 3 6 8 x 10 -6 m J m o r -0 . 0 3 7 p e r c e n t

a n d , in t h e 9 0 - d e g l a y e r , th e m e c h a n i c a l s t ra i n p e r p e n d i c u -l a r t o t h e f i b e r i s

~ [ 90 deg] = ey - o~r A T = +3 15 2 x 10 -6 m / m o r + 0 .3 2 p e r c e n t

T h e c o r r e s p o n d i n g t o t a l m i d p l a n e s t r a i n p e r p e n d i c u l a r t ot h e f i b e r i s e z~ = - 9 3 x 10 -6 m / m a n d t h e m e c h a n i c a l s t ra i ni n t h e 0 - d e g l a y e r p e r p e n d i c u l a r t o t h e f i b e r is

~M[0deg]= ~ o - - [~ T A T = + 3 3 6 3 x 10 -6 m / m o r + 0 .3 4 p e r c e n t

T h e m e c h a n i c a l s t r a i n p a r a l l e l t o t h e f i b e r i n t h e 9 0 - d e gl a y e r i s

~zM[90 eg] -- - 1 5 7 10 -6 m / m o r - 0 . 0 1 6 p e r c e n t

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RS in Laminated Fiber Epoxy Composites

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