Delamination in composite materials

8/13/2019 Delamination in composite materials

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a o ~ 6 1 ) . ~ 1IJ 00+ .00C 99 Pcrp _ _ Praa pic

MODELING OF DELAMINATION IN COMPOSITELAMINATES USING A LAYER-\VISE PLATE THEORY

E. J. BARBERot and J. N. REDDvtD e p a n m ~ n lof Engioring and Mechanics. Virginia Polytechnic Institute and

Stale Uni\crsity. Blackshurg. VA 2->610219. U.S.A.

R ~ r ~ n ~ d10 J u n ~1990: in r ~ f ; . f d urm 30 O l < - n , ~ , .1990

A b s t n d - The la) er-\\;se laminate theory or Reddy is extended to account for multiple delaminations between la)en. and the a s s o c i a t ~computational model is developed. DeJaminatior.sbct un lalers of composite plates are modeled by jump discontinuity conditions at the i n t e r r a ~ ,Geometric nonlinearity is included to capture la) cr buckling. The strain energy release rate dis-tribulion alonl the boundary of delaminations is computed by ~ novel aJlorithm. The computationalmodel prcKnled is \-alicbled through sa eral numerical examples. .

I 1 ~ T R O D U c n O N

The objective of [his study is to characterize deJaminations in laminated composite platesusing ala) er-\\ ise theory. \Ve wish to raise the quality of the analysis beyond that providedby conventional. equivalent single-layer laminate theories without resorting to a full threedimensional analysis. A computationai model based on the. layer-wise theory of Reddy(1987) is presented, and the model is used in the analysis of plates with delaminations.

The advantages of an equivalent single-layer theory over a3-0 analysis are many. Inthe application of J-O finite elements to bending of plates, the aspect ratio of the elementsm ~ t b ekept to a reasonable value in order to avoid shear locking. If the laminate ismodeled with 3-D elements. an excessively refined mesh in the plane of the plate needs tobe used because the thickness of an individual lamina dictates the aspect ratio of an element.On the other hand. a finite element model based on a laminate theory does not have thesame aspect ratio limitation because the thickness dimension is eliminated by integratingthrough the laminate thickness. HO\\ ever, the hypothesis commonly used in the conventional ie . both classical and shear deformation) laminate theories leads to a poorrepresentation of strains in cases of interest, namely. in thick composite laminates withdissimilar material layers.

A 2 -D laminate theory that provides a compromise between the 3-D theory and conventional plate theories is the layer-\\ise laminate theory of Reddy (1987), with layer-wisecontinuous representation of displacements through the thickness. Although this theory iscomputationally more expensive than the c onventional laminate theories, it pred:cts theinterlaminar stresses very accurately (Reddy t 01. 1989; Barbero t al 1990a,b). Furthermore, it has th e advantage of all plate theories in the sense that it is a two-dimensionaltheory, and does no t sutTer from aspect ratio limitations associated with 3-D finite e ~ e m e n tmodels_

The layer-\\ise representation of the displacements through the thickness .has provento be successful. Yu (1959) and Durocher and Solecki (1975) considered the case of a threelayerplatc. ~ 1 a u(1973), Sriniv3s (1973). Sun and \Vhitncy (1972), and Seide (1980) derived

theories for laycr-\\ ise linear dispJacenlcnts. Rcissner s mixed variational principle \1, 35 usedby t.1urakami 1986 and ToJedano and Murakami (1987) to include the interlaminarstresses as primary variables. Both continuous functions an d piece-wise linear furctions\vcre used. Reddys theory is chosen in this \\ ork hecause of the generality it o ~ e r sinmodeling delaminations.

Delaminations bet\\een laminae are common defects in laminates, usuaJly d e \ ~ J o p e d

t Present)) Assistant Professor. IXpartment of ~ f e c h a n i C 3 1and Aerospace Engineering. West Virginia l:niversity. t.forgantown. \VV U.S.A.

: Oiflon C. Garvin Professor.

373


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either during manufacturingor during opcrationallife of the laminate (c.g., fatigue. inlpact).Delaminations may buckle and grow in panels subjected to in-plane compressive loads.Delaminated panels have reduced load


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k b m i ~ u ~ nin conlrositc bminatet 375

The laycr-\\ isc theory of Reddy (1987) is extended here to model the kinematics ofmultiple c J e l ~ m i n a t i o n s oThe theory is applied to embedded delaminations that are entirelyseparated from t h ~ base laminate after buckling. Numerical results are presented for anumber of problems and the results are compared to existing solutions.

~ . TilE lAYER\VlSE L ~ I I S TPLATE THEORY

Increased use of laminated composite plates has motivated the development of refined

plate theories to overcome certain shortcomings of the classical laminate theory_ The firsto r d c r n ~higher-order shear deformation theories (see Reddy, 1984. J989, 1990 yieldimproved global response, such as maximum deflections, natural rrequencies and criticalbuckling loads. Conventional theories based on a single continuous and smooth displacement field through the thickness of a composite laminate give poor estimation of theinterlaminar stresses. Since important modes of failure are related to interlaminar stresses,refined plate theories that can model the local behavior or the plate mort accurately arerequired. The layer-\\Oise plate theory is shown to provide excellent predictions of the localresponse. i.e., intcrlaminar stresses, in-plane displacements and stresses, etc. (Barbero,1989 . This is due to the refined representation of the laminated nature of composite platesprovided by the theory and to the consideration of shear deformation effects_ Before wepresent the theory for delamination modeling, a revie\v of the basic elements of the theoryis first presented.

Consider a laminated plate composed of N orthotropic laminae, each being orientedarbitrarily with respect to the laminate . ~ , - c o o r d i n t ~which are taken to be in themidplane of the laminate. The displacements (II 2 u) at a point . : ~y z in the laminateare assumed to be of the form see Reddy, 1987), .

u,(.t ,) , :) = u(.t, y) + Vex, ) , =)u ~ x .y :) = L- .t, J + V(x, } , :u)(:c,) = = ,r(.t. ) ), (1)

\\ here (l I l \ Ir ) are the displacements of a point (x, ) ,0) on the reference plane of thelaminate. an d U and V are functions \\ hich vanish on the reference plane:

U(x,y,O) = V(x,y,O) = (2)

In order to reduce the three-dimensional theory to a tYlo-dimensional one, Reddy 1987 suggested layer-wise approximation of the variation of U and V with respect to thethickness coordinate, =:

U (x, ) = = L uJ (x, y) t (=)

I

ex y,:) = L I J (x, y)tjJi : ) .J I

(3)

\ \ h ~ r e an d l arc undetermined coefficients and are any piece-\vise continuous functionsthat ~ l [ i s f ythe condition

t 'i(O) = 0 rorall j It2, ~ n . (4)

The approximation in eqn 3 can also be vie\\ ed as the global semi-discrete finiteelement approximations Reddy, 1 9 8 ~ .through the thickness. In that case p denote thegloOJI interpolation functions. and II i and IJ are the global nodal values of U and V (andpossihly their derivatives) at the nodes through the thickness of the laminate.


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.. ::. ~ ~ 1 .

376 E. J. B A I tI ItO ~ n d J. S . R EU U Y

J. A ~ t O DE L FOR THE S T l:D Y O F D l : \ ~ l 1 N A T 1 0 S S

r 3.1. IntroductionD elam ina tion buck ling in lam ina te d pl at es sub je ct ed to in -p la ne co m p res si v e lo ads isw l l r o~ nized as a lim iting fac to r on the p er fo rm anc e o f co ni po si te stru c t ure s. While theaccu racy ,f the an al ys is is o f pa ra m oun t im p or ta nc e to the co rr ect e v al ua ti on o f da m agein co m pos Ite s, the cost o f anal ysis precludes the use o f th ree -d im ens io n al models, T hissection de ls with the fOrJlluhuion of a lan lin ;l tcd p lat e th eo ry th at ~ ha n d le nl ul tip lcde la m in at io ns in com po si te pla tes .

3.2. Fornlul,:1 iO I l o f th e h e o r yM odelin g o f de la m in a tio ns in la n lin at cd c o m p o s ite p late s requi re s a n ap p ro pr ia tekin em ati ca l de sc ri pt io n to allo w for se pa ra ti on a n d sl ipp in g . T his can be in co rp o ra te d in tothe layer-wise th e o ry by pr op er m odifi ca tio n o f th e exp an sio n o f the disp la cem en ts I)

thr ou gh the th ick nes s. The layer-w ise the ory can be ex te nd ed to m ode l the ki ne m at ic s o f alay ered pl ~ t w ith pro visi on for delam ina tion s, by usi ng the fol lo w in g ex p an si on o f thedispl ac em ent s th ro ug h the th ic kn ess o f the p la te :

N Du . x , ) ,= ) = u .t , + L t / li :) U J{ .t . y ) + L H :) U / x , y )J-. i

,\ DU2 X, Y z = vex Y) L ; i : ) LJ (x , y ) L H i( :) Vi (x , y )

I I

D

U l( X , ) 1 , z = , v x . ) ) L H i z ) Jy i( . 1 :,) )J - a

(5 )

where the ste p fu nct ion s H I ar e com pu te d in term s o f th e H ea vi si de ste p fu nc ti on s fI as:(

,- -

H i : ) = H : - : J = 1 fo r Z ~ Z jH I : ) = H : - : / = 0 for : < : 1 (6)

In eq n 5 41 z ar c lin ear L ag ra ng e in te rp ol at io n fu nc ti on s , N is th e nu m b e r o f lay er s usedto m od el the la m in ate an d D is th e nu m ber o f d el am ina ti on s. T he jU lnp s in the di sp la ce m en tsat t h e jt h de la m in at ed int er fa ce are given by VI, Vi an d IV i U sin g the ste p fu n ction s H J( z ),w ec an m odel an y n un lb er o f delamination s th ro u g h th e th ic k ne ss ; the nU lllber o f add it io nalvar ia bl es is equ al to the num ber o f dc la m in at io ns con sid e red . At d e la m ina te d int erfac es,the dis pl acem ent s o n ad ja ce nt layers remain in d ep en d en t, allo\ \ in g for sepa ra ti o n andslip pin g.

A lth o u g h no n lin ear effects are im p o rta n t, ro ta tio n s a n d di sp la ce m en ts a re n ot exp ec tedto be so la rg e as to re qu ir e a full non linear ana ly si s. O n ly th e von K arm an nonl in ea rit yin the kin em ati c eq u a ti on s needs to be co nsid ere d.T h e li near s tra in s of the th eor y ar e

I ell ev J;\ ClI i CI J ) I D CUi i: i = - - + - f /J ~ H - 1 2 OJ ex 27 cy ex 2 cy c.

cHJ : = 0 be ca use ~ : : : 0

o

(7)


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D el am in a tio n in co m posite lam ina tes

I [ Cl ~ t; c ~ D aWl] , _ = ~ - v H J -. 2 C c: 7 0

77

8 )

w he re the un der lin ed te rm s a r e du e t o t h e in tr o d u ct io n o fthe de la m inati on va r ia bl e s V I,VI, W I.

T h e non line ar po rt io n o f th e st ra in s a re :

I OlV J t ~ ~ H H J a W a l ~ ,aW 1 = - - - ~ - L H - 2 i I ax I

Ix: = 1 ,: = I: = o

T h e vir tu al str ai n energ y is n o w g iv e n by

lO a )

w he re

d U cL is the c o nt ri b u tio n o f th e c la s s ic a l line ar t e rm sdU C L is the co nt ri bu tion o f th e vo n K ar n t an classic al n on lin ear term slJU DL is the con tr ib ut io n o f th e new li near ter m s [u nderl in ed in eqn 8 )]

dU D Y L is the co n tr ibut io n o f th e new no n li nea r ter m s [u nderlin ed in eqn 9)] . lO b)

The co n trib ut io n o f the co n ve n ti on a l disp la cem en ts to lin ear te rm s is:

I { u N I cbui olJu N e d t ~u o~v )U C L = r N - - ; : - r ; ; r ; N~J ; ; -; -u t I U . . u I u:c t u XN cui at ) cc5U N CO V N }+ L N ~ Q z a L Q~ u l QJ ~ L Qf .vl - q O ~ d A 11 )- I J u X .T j _ I Y j _ I

T he co n tr ib u tio n o f the von K a rm a n n o n li n ea r te rm s is

1 2 )


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31 8 E. J. B A U JER O a n d J. N. R w u y

T h e c o n tr ib u tio n o f th e new linear term s is

, D f.{. CcSW; ... cc5 Vi elJU cdVJ 0 6U 1 C6J1 )}UDL = L Q~ ~ + Q~ ~ M~ ~ ~ ~ + M q -a - + ~ d x d y_ v.. t ly (J .t I) Y IX

(13a)

w h e re

f ,.-

T h e c o n tr ib u tio n o f the new n o n lin ea r te rm s is

i { [ ; au, ~ i ccSu- aWl ; 0. . a ~w 06K aWl UD NL = L M x ~ + - ~ - + M ,. ~ + i v X v X u:C (J. t: u y uY lJ V ); lVOb Wi

a ~u

aWi u e W i C~K aWI ] D D I M z, -a a ~ ~ ; ~ a ; + L L iM l Y IX uY U ) I. C Y u X I JX aw; a ~wJ + a ~W OWJ + iMii OWi 006Wi + C~W aWJ ~ ~ h ~ , ~ ~ ~

OW; a ~ wJ aWi CcSW } ~ n+ M J ~1 0) O O C 0)

w here

T h e b o un d a ry co n d itio n s o f th e th eo ry a re g iv en b e lo w :

(1 3 b )

(1 4 a )

(1 4 b )

1 .

G e o m e tr ic F o rc eu Nxnx+ NZTn,v N z,n x+ N n ,w Q .,nx+ Q ,.n,.II N~ n x N~ .r n xI J ~

n .~ t ~ n ,.

V j A f ~n ~+ I \ f ~. , .n .rVi ~f ~J nx + 4 \ f~ . IJ , .W i Q~ \ : I1 .1 + Q. ~: n .r . I S

T h e l a n li n a te c o n s t i tu t iv e e q u a t io n s c a n b e o b ta in e d in th e usual m a n ne r see R eddy,19 8 8 ; B a r b e r o , 198 9).

3 .3 . F r a c tu r e 111eclzanics an a l) si sT h e d e fo rm a t io n field o b ta in e d from the laycr-\1r ise t h e o ry is used to com pute the stra ine n e rg y re le ase rates a lo n g th e b o u n d a r y o f the d e la m in a ti o n . D elm inatio ns usually exhib itp l a n a r g ro \ \ t h , i.e., th e c ra c k g ro w s in its o rig ina l p la n e . H o w e v e r, th e shape o f the crack


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D cl am in ~t i o n in c om p os ite lam in at es 379rnay v a ry ,, ith tim e . F o r exam ple. a crack sha pe in it ia lly elli p ti c u sually grow s with varia b lea l =ct ra lio . T h c rc r o r ~ . the ~ r ac k gro\\-th is no t self -sim ilar. Ho \ \ e \ ~ r a self-sim ilar v irtu alc r a ~k cX h:nsion \\-ill assum ed in o rd e r to co m p u te the d is tr ib u tio n o f the strain energyr c : J ~a s~ ra tc G (s ) a lo n g the bo undary o f th e d e la m in a tio n .

T he v irtu a l c ra c k ex tension m ethod po s tu la te s th a t the s tra in energy release ra te canbe com p u ted from t h ~ s tra in energy t ~ an d the rep resen ta tiv e c rac k leng th Q as:

G (s) = d U /d a . (16)

In the ac tu a l im p le m e n ta t io n o r the v irtua l crack ex tension m eth o d , how ever, eqn (16) isa p p ro x im a te d by a q u o tie n t 6 U It a that in the lim it ap p ro x im ates th e vaN e o f the str ai nenergy release ra te G. In the Jacobia n deriv a tiv e m ethod (B arbero and R eddy, 1990), theG ( s ) is c o m p u te d \\ ithout ap p rox im ating th e deriva tive , s o it d o e s n o t require th a t wecho ose th e m a g n i tu d e o f th e virtu al cra ck extensio n l a .

T h e b o u n d a ry o f a de lam ina tio n is m odeled as b o u n d a ry co nd iti on s on the delam in a tio n v ariab le s U i, Vj, an d Iy i, by se tti ng them to zero. A self-sim il ar virtual c rac kex tensio n o f th e c r a c k (delam inatio n ) is specified fo r th e nodes o n th e b o u n d a ry o f thed e lam in a tio n . i.e . th e t\\ o com ponen ts o f the n o rm al to the de lam in a tio n boundary a respecified for each n o d e o n the boundary . T h e J O M is th en used a t each configu ration (o rlo ad ste p) to c o m p u t e G ( s ) from the disp la cem en t field.

3. 4. F inite -e le ,,,eIJ nlolJelT h e generaliz ed displacem ents (u , D , lV , II, 01, Ul, VI, WI) a r e expressed over each

elem en t as a l in e a r c o m b in a t io n o f the I\v o-d im ensional in te rpo la tio n functio ns pi and th en od a l v ~l l u c s II,. L u , .II:.l . Vi, VI, n l) a s ro n ow s:

(li t .. u l , r J V i, Vi, U i) = L U i , V i , i u vi U1 Vi Wnt/li (1 7 )

; I

W here is th e n u m b e r o f nodes per elem ent. U sing eqn (17) in eqn (lO a), we obta in th efoJlo\ving finite ele m en t e q u a ti o n s :

k k:: k l 2 3 kkJ {A} {q}vki kii k~; . k: ) k 2 J {AI} {ql} 1 0

k ~ k_ ~ k ~. ; k : ) k ) {AN} = {( } (18a) ,\ , Dk: ~ k J 2 k J ) {3 1 } { } IN II ID

k ~ kJ ~ { D } {qD}1 D I DD

\\ h e re

{6 }T = {li t :. . t . , u ., t . , K }

{11 } T - { j j j J }I I, . L t II 1 1ft

6 ,} T - V I J j J ~/ i VJ V i W I} (18b)- \ I I I II

a n d ~ s u b n la t ri ce s [ k ] . [k) :] , [k} ). (k;: ] , [k;)l, [ k~ ] t [k /l W ith i j 1, . . . , N andr , s= I, . . . ,D a re g iv en in B arbero (1989). T h e load vecto rs {q}, {ql} . . . { ~ } and{ q l l , { j n} a re a n a l o g o u s to {6}, { ~ I} . . . , { ~ N} a n d {A I}, . .. , { ~D} respectively [seeeqn (18b)]. T h e n o n l in e a r algebraic system is solved by th e N ew to n -R a p h so ri alg orithm .T h e c o m p o n e n t s o f th e Ja c o b ia n m atr ix a re a ls o given in B arbero (1989).

T h e n o n li n e ar e q u a t i o n s are lin ea rize d to fo rm u la te the e ig en v a lu e problem associated\\ jth h ifu r c a t io n b u c k lin g an a ly si s :


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380 E. J. 8 A I t8 I t O an d J. S . R 1. UUY

[KDJ - i. K c ; cJ = 0 1 9 . w he re KD l is the lin ea r p ar t o f th e d ir ec l stiffn ess matrix and KG is th e ge o m etri c st if fn es s

m atrix .

4. ~ L ~ tE R I C AL RESULTS

Several exa::-.ples ; re p re se n te d in or d er to validate the prop ose d fo rm u la li on . A n aly tica l so lu tions ~ be deve loped fo r si m p le cases and lhey are used for c o n lp a r i s o n w ithth e m ore gene ra l J pp ro ~ c h p res en ted here .

4 1 Square thin fil n l de / Jj Jli na tionA th in layer d el am i l at ed fr om a n is o tr op ic squa re plate flex ura l ri gidi ty , by acon cen tr at ed load P a t its c e nt e r is con si d er ed . The dela m in at io n is als o a ss u m ed to besq u a re w ith side 2a T ht: base la m in a te is ass um ed to be rigid \\ i th re sp ect t o t h e th ind e lam ina te d layer. A n ar. aly tic al s o lu ti o n fo r th e linear den ec tion an d st ra in e n e r g y re le asera t e ca n be der iv ed assum in g th a t th e dela m in at ed la ye r is clam ped to t h e ri gi d b a s ela m ina te . D ue to th e biaxual sym m et ry , a qu a d ra nt o f th e squ are de la m in a ti on is a n a ly z e dusi ng 2 x 2 a n d S x 5 me s h ~ s o f n in e -n o d e ele m ent s. E ithe r the c lam ped b o u n d a r y c o nd i t io nis im pos ed o n the bo un dar y o f th e d el am in at io n o r an add itio nal ban d o f el em en ts w ith ac lo se d del am ination is p la ced a ro u nd th e de la m in ate d ar ea to sim ul at e the no n d e lam in a ledreg io n . B ot h m oe els p ro du ce co n si sten t re sul ts for tran sv erse defle ctio ns an d a ve ra g e st ra inener gy re lea se r a :~ s A fine m es h is nece ssary to ob ta in a sm ooth d is tr ib u tio n o f th e st ra inene rgy relea se ra t: G a lo n g th

e b o u n d a r y o f a squa re delam in at io n. T he lin ear s o l u ti o n fo rP = 10 co m par es well w ith th e a n al yt ic a l so lu ti on see Fig . I) . The li n ea r a n d n on li n earm ax im u m d el am inat io n op en in g W a n d ave ra ge strain energ y re leas e ra te Gilt a s a functiono f th e ap pli ed loa d P are show n in F ig . 2. I t is evide nt tha t the m em br an e str es se s c au se d

1 0 . 0 . . . . . . .... U n e o r P .1 0 N o nl in e ar p .S O_ _ _ _ N on li ne or P 0 0

C lo s e d fo r m P . 1 01 .0

2 .0

aa 1.0 od 4 -0

0 .0 +-------------,,..=11I-...0 0 . , 2 O .J 0.. . o ~x / a

Fig. I. i st r ib ~ l o n o r the str a in en erg ), rele35e ra te along the bou nd ary or a squ3re d e lam in a tio nd ue to c o n c en tr a te d J03d.

t f

7. 0

....

1. 0 - O /o P

r

N o n J ir .e o r- - w n e c r

0. 0 +----r-- I..---.--a - ..--, OO

1

.0

~ o

w O / c p4 .0

GAvO/p2 J .O

p

Fi g . 2. ~ 1 a~ im L ~ d ela m in at ion o p e n in g U a n u 3\C ra g\. slr J in ene rg y r c ~a S4: ra te ac co rd in g ton on li n ea r an al ys is o f a sq u a re c :l J rn in Jl ion .


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. D c I m l n ~ l i o nin comrosite LlminaltS 381

by the geometric nonlinearity reduce the m3gnitude of the a\ erage strain energy releaserate considerably (\Vhitcomb and Shivakumar. 1987).

4.2. ill fi cylilldricul bucklingIn this cxanlplc e consider an isotropic thin la}er delaminated from a thick plate in

its entire \\ idth. Due to symmetry. only one half of the length of the plate strip is modeledwith the cylindrical bending assumptions and a nonunifonn mesh of seven elements. The

cylindrical bending assumption is satisfied by restraining aJl degrees off r ~ e d o m

in the y-direction. The baselaminale is considered to be much more rigid than the thin delaminatedlayer so that it will not buckle or denect during the postbuckling of the delaminated layer.First, an eigenvalue (buckling) analysis is performed to obtain the buckling load andcorresponding mode shape. Then a nonlinear solution for the postbucJcling configurationis sought. Excellent agreement is found in thejump discontinuity displacements U and Wacross the delamination. The values of delamination opening Wand strain energy releaserate G are shown in Figs 3 and 4 as a function of the applied load, where t n is the criticalstrain at which buckling occurs for a delamination length of see Yin, )988).

4.3. A. isyIJu etric circular delanlinationAxisymmetric buckling of a circular, isotropic, thin-film delamination can be reduced

to a onc-dimensional boundary-value problem by means of the classical plate theory (CPT).One quadrant of a square plate of total width with a circular delamination of radius a

1.0 ::0 0.1

0.1.2U.. 0.4

> 0.2 c0 -.-

J. N. CI()sed-form Present study

Fig. 3. ~ f a,imum delamination r tninl U (or a thin film buckled delamination.

2.0

N.-----...,,...1.8

1 4

1.2 Closed-formo a P re se nt s tu dy r

, .00 .0 .0 2.0 4 5t ~ c i n e n e ~ c yre ;ecse re te

Fig. 4. Strain energy rekas


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382 E. J. BAkHlJlO ;and J. N. RLOUY

is modeled. The layer-wise clements are capable of representing discontinuities of thedisplacements at the interface bet\\ een layers. The symmetry boundary conditions used are:

u O, y) = 1/(0. y) = U O .I = 0 x, 0 = i x, 0 = J/ J x, 0 = 0

with i = I, Nand j = It , D; \vhere V is the number of layers and D is the numberof delaminations through the thickness; in this example. , e ha, c D = I. The boundary ofthe delamination is specified by setting the jump discontinuity variables UJ, yJ and JyJ tozero on the boundary of the delamination and ,\ hercver the plate is not delaminated. Theboundary of the plate is subjected to the follo\\ ing boundary conditions, which produce astate of axisymmetric stress on the circular delamination:

u b,) ) = UJ b,) ) = ; N ~ b , = I irf x,b) = Vi :c,b) = ; : ,b) = I i

where Ii is a uniformly distributed compressive force per unit length. The same materialproperties are used for the delaminated layer of thickness t and for the substrate of thickness h I . To simulate the thin-film assumptions, a ratio It = 100 is used. First an eigenvalueanalysis is performed to obtain the buckling load and corresponding mode shape. Then aNewton-Raphson solution for the post-buckling is sought. The Jacobian derivative method(see Barbero an d Reddy, 1990) is used at equilibrium solution to compute the distributioDof the strain energy release rate G s) along the boundary of the delamination. Fo r thisexample, G s) is a constant. Its value, shown in Fig. 5 as a function of the applied load, isin excellent agreement with the approximate analytical solution see Yin, 1985).

4.4. Circular delanlinalion under unidirectional loadIn this example we analyze a circular delamination, centrally located in a square plate

of lotal width 2c and subjected to a uniformly distributed in-plane load NIl. The exampleis considered because results or a three-dimensional analysis are available (Whitcomb, 1988)in the literature. Compared to the last example, this problemdoes not admit an axisymmetricsolution. Therefore, the distribution G s) varies along the boundary of the delamination.

A quasi-isotropic laminate [45/0/90] of total thickness = 4 mm and a circular delamination of diameter 2a located at : = 0.4 mm is considered. The material properties usedare those of AS4/PEEK: = 134 GPa, E J = 10.2 GPa, G I2 = 5.52 GPa, G 2 ) = 3.430Pa,Va2 = 3 As is well kno\l, n, the quasi-isotropic laminate exhibits equivalent isotropic

1.2 1 -0 model a 0 Present study

o

0.1

0.4

o

12.0 r

Fig. 5. Strain energy release: rate for a bUl llcd thin film. a i nlmetric d c l n ~ l n ~ l l o nas a functionor the inplane l o ~ d


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bc:havior hen lo ad ed in it s p la ne. The bending be ha v io r , h o w e v e r , d e p e n d s o n th e orientat io n. To avo id co n lp li c: lt i o ns in the in te rp re ta tio n or th e resu lts in tro d u c e d b y the n on -iso tropic bending b e h a v io r . an equ ivalen t iso tr opic m a te r ia l is used by W h itc o m b (1988),where the equ iv alen t stif fn ess co m p on en ts o f the 3 0 elastic ity a re fo u n d fro m th e re la ti o n :

.. f k l a m in JI .o n in co m f O silt h lm in ~t es 313

Due to the tran sverse in co m pressib ili ty used in th is \\:o rk , I t IS m o re c o n v e n ie n t andcusto

m ary to w ork \v ith th e reduced stiffness. E quivale n t m a te ria l p ro p e r t ie s c an fo u n ddirectly ,from t h e A - m a t r i x o f th e quasi-iso trop ic la m in a te as fo llo w s. F ir s t . c o m p u t e theequivalen t reduced stiffness coeffic ien ts

w here h is the to ta l th ic kn e ss or th e pla te a n d A i } a re th e ex ten s io n a l stiffnesses. N e x t, th eeq uivalen t m aterial p ro p e r t ie s c a n found as:

= Qu t G I2 = Q 3 l

G = Q E E ; . 1 2 =

QI I ~

An eigenvalue a n a ly s i s reveals th a t the de la m ina ted p o r t io n o f t h e p la te b uck les a tN x = 286 816 N m I fo r = S mm and a t N . = 73,666 N m I fo r = 30 m m . T h em axim um tr an sve rse o p e n i n g o f the delam ina tio n as a fu n c t io n o f th e ap p lied in -p lan estra in t. t is show n in Fig . 6. T h e differences observed \\ ith th e resu lt s o f W h it c o m b (1988)a re due to th e fact th a t in th e la tte r an artific ia ny zero transv e rse de flection is im p osed onth e base la m inate to red u ce th e com pu ta ti o na l cost o f the th ree -d im en s io n a l finite elem entso lu tio n . T he differences a re m o re im p o rtan t for = 30 m m , a s in d ic a te d by th e dashedline in Fig. 6 w hich re p re se n ts the tr ansverse deflection o f the m id p la n e o f th e p la te . T h esquare sym bols d e n o te th e to ta l openin g (o r gap) o f the d e la m in a t io n , w h ile the solid linere presents the o pen ing re p o rt e d by \V hitc om b (1 988) w ith u = o. T h e d ifferences i n th e

L I , . . - - - - - - - - - - -

.2

3 D e e ~ e~ ts00 00 0 ~ e n j n r C ef e c t. o n \U

E 1. 0

E- 0 .8 0

0 . 4

0 .2I

0. 0 - - - - - - _ _ _ _ _ c 30

0 . 2 - - - . . , - . . - _ _ 1 2 .0 3.0 4 .0 5 .0

s t ra in t .

Fig_ 6. a ~ i m um tran sv e rse o pe n ing l o f a cir cu lar d e am in at io n o f d iam e ter 2a in a sq u a re pl a tesu bjecte d to in -p la n e load ~\ as a fu nctio n o f th e in p Jan e u n if o rm st ra in .


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,=0.003

.=0.002

.=0.005

-100.00. 0

0.0 - - - - - - = : ; ; ; ; - - - - - - : = ~ _Present study____ 3 elements

10.0 20.0 JO.O 'O.O 50.0

s :I I 0 [ m m ] ( O < ~ < 1 I / 2 )Fil. 7. Distribution or the strain ener,y r e l ~rate along the bound31 ) of a circular delamination

O i ~various values of tbe applied in-plane axial strain.

384 E. J. B.\RlIlItO and J. N. R.uuy

700.0

800.0 8 \ Nt500.0400.0

C(s) 300.0 ;..Nw200.0

-----OO.O

total opening Wand trar. iverse deflection 11- have an influence on the distribution of thestrain energy release rate, as can be seen in Fig. 7. Both solutions (i.e., solutions of 3-Delements and the present 2D elements) coincide for the sma)) delamination of radius Q = mm, as shown in Fig. 8. For the larger radius Q = 30 n1m, the assumption ,r = 0 is no

longer valid, and differences can be observed in Fig. 8, although the maximum values of Gcoincide and the shapes of the distributions of G are quite sinliJar. Mesh refinement. withat least two elements close to the delamination boundary, must be used to account for thesudden changes in deflections and slope in a narro\y region close to the delaminationboundary, similar to the phenomena described by Bodner (1954). Distributions of the strainenergy_releaseralesG (s}along the boundary of the delamination are sho-\\ n in Figs 7 and8 for the different delamination radii, a = 15 mm an d a = 30 m m, respectively. The values = 0 corresponds to x = a, y = 0) and s = an/2 to .\ = 0,) = a . r\egalivc values of G sindicate that energy should be provided to advance the delamination in that direction.Negative values are obtained as a result of delaminated surfaces that come in contact. thuseliminating the contribution of mode [ of fracture but not of modes II and III. The presentanalysis does not include contact constraints 1nd therefore layers may overlap. As notedby Whitcomb 1988), the strain energy release rate G s in the region \vithout overlap isnot significantly affected by imposing contact constraints on the small overlap area.

4 5 Unidirectional delal lillQted graphite ep o C r plateA circular delamination of radius a = 5 in. in a square plate of side c = )2 in.. made

of unidirectional g r a p h i t e ~ p o x y _is considered next. The thickness of the plate is = 0.5

t O O C ~

c.=O.CC3

=0 002

S nI

~ \ : ~ .IN.t ~ ) - i / f I

; I~ I

0. 0 L . - _ ~ ~ : : : ~ ~ - -_ _ Present s ~ u c y____ 3 elements

400.0 ~

300.0 j

0 = 1 5-100.0 0 .0

- --5 0 -,-, ......or- .o...--........-S-.O....-.....2-0 1.0 -..... -, ~ 5 1

S = of l ~ m 0


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D d .J n l i c J t l o n in c o n l o s i l c lanlinatC S 3851 1 .0

1 0 .0

9 .0 O = C O N,Ncr

8 '\. 0

7 .0

6 .0

N r5 .0

1 . 0 0 5 0 . 0 0.5 1 . 0lo o d ra tio : r = N . - N , / N . + N ,

F i I . 9 . Buckling load \ ersus the ratio o rin -p lan e loads and N for a circuJar deJamination.

in ., a n d th e d e la m in a tio n is locate d a t a d is ta n c e I = 0.005 in . from the surfa ce. A nig e n v a lu e an a ly sis is used to o b ta in the m a g n itu d e o f the in -p la n e lo ad un d e r which theh in d e la m in a te d lay er buckles. D ue to the o r th o t r o p ic n a tu r e o f th e m ateria l used in thisx a m p le , it is in te re s ti n g to s tu d y the effect o f d if fe ren t c o m b in a t io n s o f lo ad s z an d I l e t us d e n o t e th e lo a d ra tio a s

T h e m a g n i tu d e o f the buckJin g lo ad as a fu n c ti o n o f th e lo a d ra tio w ith < r < 1,s s h o w n in F ig . 9. T h e d is tr ib u tio n o f the s tr a in e n e rg y rele ase r a t e G s a lon g the b ou nd ary ttc J 0 < t J < n /2 is p lo tted in Figs 1 0 -1 2 fo r th e lo a d ra tio s , = - I, 0 , and 1, ando r se v e ra l v a lu e s o f the app li ed in -p lane lo a d , in m u lt ip le s i o f th e buck ling load cr F o r= I (Le ., N .: = N a n d r = 0) it is c le a r th a t (see F ig . 10) t h e d e la m in a tio n is likely tor o p a g a te in a the d ire c tio n a p p ro x im a te ly p e rp e n d ic u la rto th e lo a d d irec tio n . In trod u c tionf lo a d in th e d ire c tio n p erp en d ic u la r to fibers c a u se s the m a x im u m value o f G to alignlo se r to th e .x-axis . N o te th a t b o th the m a g n itu d e a n d th e sh a p e o f the d is trib u tio n o f G s h a n g e a s th e lo a d ra tio chan ges see F ig . 13). T h e p lo ts su g g e s t th a t the pro pa ga tio n o rr r e s t o f d e J a m in a t io n s is grea tly in fluenced by th e a n is o t r o p y o f the m aterial and theo m i n a n t lo a d .

1 0 .0 o y

5 0 .0 ~ _ = ~ ~

r = 1 : = N I N e r5 .0 - -o o r -. .. .. .. . r- - - r - . . . . . - . . - - .- - .- . . .- -

0 . 0 1 .0 2 .0 3 . 0 4 .0 5 . 0 6 . 0 7 . 0 8 . 0 = o lp m m O< ~ < . / 2 )

F ig . 10 . D i s t r i b u t i o n o f th e s t ra in e n e r g ~ r l ~ s ra t e a lo n g th e b o u n d a r y o f a c ir c u la r delam inatio no r \ a r io u s v alu es o f a p p lie d Joad V

.


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386 E. J. O. JlMIIU :.and S. Rruln

20.0 . . . .

10.0

0.0


15/16


16/16

8 8E. J. B A R . t A O and J. N. R ouy

R e d d y . J N. (1987). A ~ e n e r a l i z a t j o n o f tw o -d im ensiona l theories o f lam ina ted plates. C o m n ,u n . Ap p l wnwr.t lh . J 113-180.R ed dy . 1. N. (1988). Me< )anics o f com posite structu res. In F in i t J ~ n , ~ n l . n a l ys i s f o , n g i n ~ r r ; n 9 D t fig n E d ;tedy J. N . Reddy. C . S. K rishna M o o rlh y an d K . N . S cc tharam u). C h ap . 14. pp. ) ) 8 - 3 5 9 . S r rin g e r . Berlin.ed d y . J. N. (1989). On refined co m p u tat io n a l m od els

o f com posite lam in ates . Int. J N lln t r ~ f t t h . EngtJq 11 ,61-382 .R e d d y . 1. S . (1990). A re\ ew of refined theories o r lam ina ted com posite plates. S h o r k Vibr . Digrsl 21 7). 3 -11 .eddy. J . N B arbero . E. and Teply. J L. 1989). A plate bending clem ent based on a genera lized l a ~ r a t ela te theory. Int. J Nun l t r . A ftl lt 9 g 28. 221S-2392 .S.Jllam . S. and Sim itses. ;. 1. (1985). D elam ina tion buckling and grow th o r n t cross ply lam in ates. CI J ~i t l f .tr u t t . ~ 361-381.

Se ide , P. 1980). An imprc,ved approxim ate theory ror the bending o f l a m i n l te d plates. M tch . r u d o ) ' 5.451- 66.h iv akum ar, K. N. and W hitcom b. J. D. (1985). B uc ld in , o r a sub lam ina le in a quasi-iso trop ic co m ro sitea m in a te . J ompoJ MQI ~ . I 2 -1 8 .Sim itses , Q . J., Sallam. S .a t ,d Yin. W. L. 1985). Effect o r delam ination o ra lia U y lo:aded hom oacneous l a m i r u lc dla t e s . I J n 14 37-/4 40 .S r in iv as , S. (1973). Refined : nalysis o f com posite lam in a tes . J S o u n d Yib r . JO, 4 9 5 -5 0 7 .u n , C . T . and W hitney. J. ld . (1972). O n theories for the dynam ic response o f lam inated plates. A IA A P apero. 72 398.T o led a n o , A. and M ura kam i. H. (1987). A com posite p l a t e theory or arb itra ry la m in a lc configurations. J Ap p l .l ~ c h . S 4 t 181-189.W e b s

t e r . J. D. (1981). F la w critica lly o f c ir c u la r d is b o n d defects in com pressive lam inates. C C M S R eport 81..QJ.h i t c o m b J. D . 1981). f in ite elem ent analysis o r instab ility re la te d d elam in a tion grow th . J . Con pos f QI ~ r5, 113-180.\ \ h i tc o m b . J. D. (1988). Instability related d e la m in a t io n gro . . t h o em bedded and edge de lam ina tions. l A S AM 100655.W h itc o m b J. D. and Shivakum ar. K. N. 1987). S tra in -cnergy release-rate analysis o r a lam inate with a postuck led de lam ina tion . NASA T ~ 89091.Wi l t ~ T E M urth ytP . L . N . and C ham is. C . C . 1988). F rac tu re toughness co m p uta tio na l sim ulation o f generale la m in a tio n s in fiber com posites. A IA A P ap er 8 8 2 2 6 1 , pp. 391-401.i n W . - L 1985). A xisym m etric buckling and g ro w th o r a circular de lam ina tio n in a compreSSC d lam inate. Int. Solids Sl r u c l u r ~j 21, S03-S 14.

Y i n , W .-L . 1988). T he effects o r lam inated stru ctu re o n delam ination buck ling and grow th, J Con pos. AID ftr 1 .5 0 2 -51 1 .Y u , Y. Y . 1959). A new theory o f elastic sandw ich p l a t e s - o n e dim ensiona l case. J Ap p l . A/ te l l . 2 6,41 s-42 1 .

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