bu, J, SoWI Slrwtwa Vel ~. So, ). rP, )7J..}U, 1991tn•.- .. GfU& In&.lia.

ao~'61).~1 IJ 00+ .00C 199. Pcrp__ Praa pic

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

E. J. BARBERot and J. N. REDDvtDepanm~nl of Engio«ring~ and Mechanics. Virginia Polytechnic Institute and

Stale Uni\·crsity. Blackshurg. VA 2-«>61·0219. U.S.A.

(R~r~n·~d 10 Jun~ 1990: in r~f·;.f("d furm 30 NOl·<-n,~,. 1990)

Abstnd-The la)'er-\\;se laminate theory or Reddy is extended to account for multiple delami­nations between la)·en. and the associat~ computational model is developed. DeJaminatior.sbct"un lalers of composite plates are modeled byjump discontinuity conditions at the interra~,

Geometric nonlinearity is included to capture la)'cr buckling. The strain energy release rate dis­tribulion alonl the boundary ofdelaminations is computed by ~ novel aJlorithm. Thecomputationalmodel prcKnled is \-alicbled through sa'eral numerical examples. .

I. 1~'TRODUcnON

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~$tbe kept 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 ofan individual lamina dictates the aspect ratio ofan 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 et 01., 1989; Barbero et al., 1990a,b). Fur­thermore, it has the advantage of all plate theories in the sense that it is a two-dimensionaltheory, and does not sutTer from aspect ratio limitations associated with 3-D finite e~ement

models_The layer-\\·ise representation of the displacements through the thickness .has proven

to be successful. Yu (1959) and Durocher and Solecki (1975) considered the case of a three­layerplatc. ~1au (1973), Sriniv3s (1973). Sun and \Vhitncy (1972), and Seide (1980) derivedtheories 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 and piece-wise linear furctions\vcre used. Reddy·s theory is chosen in this \\'ork hecause of the generality it o~ers inmodeling delaminations.

Delaminations bet\\·een laminae are common defects in laminates, usuaJly de\"~Joped

t Present))· Assistant Professor. IXpartment of ~fechaniC31and Aerospace Engineering. West Virginia l:niver­sity. t.forgantown. \VV. U.S.A.

: Oiflon C. Garvin Professor.

373

either during manufacturing or 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<arrying cap~city in both the pre- and post-buck lingregimes. However, under certain circumstances. the gro\\·th of dclaminations can bearrested. An efficient use of laminated composite structures requires an understanding of thedelamination onset and growth. An analysis methodology is necessary to nlodel compositelaminates in the presence of delaminations.

Self-similar growth of the delamination along an interface bct\\'ccn layers is suggestedby the Janlinated nature of the panel. It \\"as noted by Obreimoff (1980) and Inoue andKobatake (1959) that axial compressive load applied in the direction of thc delaminationpromotes further growth in the same direction. One-din,cnsion~ll and two-dinlCn!\ionalmodels for the delamination problem \vere proposed by Chai (1982). Simitses el al. (1985).Kachanov (1976), Ashizawa (1981), and Sallam and Simitses (1985). According to thesemodels, the delamination can grow only after the debonded portion of the laminate buckles.However, the delamination can also grow due to shear modes II and III.

The spontaneous growth of a delamination \\'hile the applied load is constant is calledIfunstable growth". If the load has to be increased to promote further delamination, thegrowth is said to be "stable growth". The onset of delamination growth can be followedby stable growth, or unstable indefinite gro\\·th or even unstable growth followed by arrestand subsequent stable growth.

In most studies the buckling load of the debonded laminate is calculated using bifur­cation analysis (sec Chai, 1982; Simitses et a/., 1985; \Vebster. 1981 ; Bottega and Maewal,1983). Bifurcation analysis is not appropriate for debonded laminates that have bending­extension coupling, as noted by Simitses el al. (1985). Even laminates that arc originallysymmetric, once delaminated, experience bending-extension coupling. In general, delami­nations are unsymmetrically located with respect to the midplane and the resulting dclami­na-ted layers become unsymmetric. Therefore. in-plane compressive load produces lateraldeflection and the primary equilibrium path is not trivial (II- #: 0). Furthermore. bifurcationanalysis does not permit computation of the strain energy release rate.

Nonlinear plate theories have been used to anal)'zc the post-buckling behavior ofdebonded laminates. Bottega and Mac\val (1985), Yin (1985). and Fci and Yin (1985)analyzed the problem of a circular plate \\'ith concentric.. circular delamination. The vonKarman type of nonlinearity has been used in these studies.. Most of the analyses performed have been restricted to relatively simple nlodels. Thenlaterial was assumed to be isotropic in most cases and orthotropic in a fc\v. thus precludingthe possibility of analyzing the influence of the stacking sequence and bending-extensioncoupling.

The Rayleigh-Ritz method has been used by Chai (1982). Chai tl al. (1981). andShivakumar and Whitcomb (1985) to obtain approximate solutions to simple problems.Orthotropic laminates were considered by Chai and Babcock (1985) and circular delami·nations by Webster (1981).

The finite ~lement method was used by \Vhitcomb (1981) to analyze through-widthdelaminated coupons. Plane-strain elements \"ere used to model sections of beams, or platesin cylindrical bending. The analysis of delaminations of arbitrary shape in panels requiresthe use of three-dimensional elements, \\'ith a considerable conlputational cost. A three­dimensional. fully nonlinear finite element analysis \\"as used by \Vhitcomb (1988). \\'hereit was noted that .'... plate analysis is potentially attractive because it is inherently n1uchless expensive than 3D analysis:· Plate elements and multi-point constraints (MPC) havebeen used by \Vhitcomb and Shivakumar (1987) to study delamination buckling and byWilt et al. (1988) to study free-edge delamina tions. This approach is inconvenient in manysituations. First, the MPC approach requires a large number of nodes to simuhlte llctualcontact bet\veen laminae. Second, a ne\\' plate element is added for each delamination. TheMPC approach becomes too complex for the practical situation of multiple delaminationsthrough the thickness. Third, all plate elements have their middle surface inrthe same plane,which is unrealistic for the case of delaminated laminae that h3VC their middle surfllce atdifferent locations through the thickness of the plate.

374 E. J. BARutRO and J. N. RrIID\"

\::::t-t;;.:-->~~

!. .I!

(kbmi~u~n in conlrositc bminatet 375

The laycr-\\'isc theory of Reddy (1987) is extended here to model the kinematics ofmultiple cJel~minationso The theory is applied to embedded delaminations that are entirelyseparated from th~ base laminate after buckling. Numerical results are presented for anumber of problems and the results are compared to existing solutions.

~. TilE lAYER·\VlSE LA~IISATE PLATE THEORY

Increased use of laminated composite plates has motivated the development of refinedplate theories to overcome certain shortcomings of the classical laminate theory_ The first­ordcran~ 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 (.~, )-) coordinat~ 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 (lI .. l\ Ir) are the displacements of a point (x, )',0) on the reference plane of thelaminate. and U and V are functions \\'hich vanish on the reference plane:

U(x,y,O) = V(x,y,O) = 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/JJ (=)

ja I

"Vex. y,:) = L I'J (x, y)tjJi (:).J. I

(3)

\\'h~re u' and fl arc undetermined coefficients and ¢' are any piece-\vise continuous functionsthat ~l[isfy the 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, 198~). through the thickness. In that case <pi denote thegloOJI interpolation functions. and IIi and IJ are the global nodal values of U and V (andpossihly their derivatives) at the nodes through the thickness of the laminate.

.. '" ... ::. _. ~.~ "-1.

376 E. J. BAItI£ItO ~nd J. S. REUUY

J. A ~tODEL FOR THE STl:DY OF D(l:\~l1NAT10SS

,r 3.1. IntroductionDelamination buckling in laminated plates subjected to in-plane compressive loads is

wellreco~ nized as a limiting factor on the performance of coniposite struct ures. While theaccuracy (",f the analysis is of paramount importance to the correct evaluation of damagein composItes, the cost of analysis precludes the use of three-dimensional models, Thissection de"ls with the fOrJlluhuion of a lanlin;ltcd plate theory that c~n handle nlultiplcdelaminations in composite plates.

3.2. Fornlul,:1 iOIl of the ,heoryModeling of delaminations in lanlinatcd composite plates requires an appropriatekinematical description to allow for separation and slipping. This can be incorporated intothe layer-wise theory by proper modification of the expansion of the displacements (I)through the thickness. The layer-wise theory can be extended to model the kinematics of alayered pla~et with provision for delaminations, by using the following expansion of thedisplacements through the thickness of the plate:

N Du.(x,)',=) = u(.t,)·)+ L t/li(:)UJ{.t.y)+ L H'(:)U/(x,y)J-. i- •

,\' DU2(X, Y. z) = vex, Y) + L ;i(:)LJ (x, y) + L Hi(:) Vi (x, y)

J. I j- ID

Ul(X,)1, z) = ,v(x.)') + L Hi(z) Jyi(.1:,)')J-a

(5)

where the step functions HI are computed in terms of the Heaviside step functions fI as:

(

',--

Hi(:) =H(:-:J) = 1 for Z ~ Zj

HI(:) = H(:-:/) = 0 for : <:1' (6)

In eqn (5) 41(z) arc linear Lagrange interpolation functions, N is the number of layers usedto model the laminate and D is the number ofdelaminations. The jUlnps in the displacementsat thejth delaminated interface are given by VI, Vi and IVi . Using the step functions HJ(z),wecan model any nunlber ofdelaminations through the thickness; the nUlllber ofadditionalvariables is equal to the number of dclaminations considered. At delaminated interfaces,the displacements on adjacent layers remain independent, allo\\'ing for separation andslipping.Although nonlinear effects are important, rotations and displacements are not expectedto be so large as to require a full nonlinear analysis. Only the von Karman nonlinearityin the kinematic equations needs to be considered.The linear strains of the theory are

I (ell ev) J;\' '(ClI i CIJ) I D (CUi i: Vi)t = - -+ - +-"f/J' -+- +-~H' - +-"1 2 OJ' ex 27 cy ex 2 '7 cy c.'(

cHJ£: = 0 because -~-:::& 0

co:

(7)

Delamination in composite laminates

I [Cl~. ,t; c~ D aWl]£,_ = - - + ~-v'+'HJ-. · 2 C)· 7' c: 7' 0)'

377

(8)

where the underlined terms are due to the introduction of the delamination variables VI,VI, WI.The nonlinear portion of the strains are:

I (OlV)J t ~ ~(H''HJaW' aWl) aw~ ,aW'".1 = - - +-,--~ --- +-L,H-2 ax 2 i I ax ax ax I ax

'Ix: = '1,: = 'I: =o.

The virtual strain energy is now given by

(lOa)

where

dUcL is the contribution of the classical linear termsdUC.¥L is the contribution of the von Kar-n:tan classical nonlinear termslJUDL is the contribution of the new linear terms [underlined in eqn (8)]

dUD,YL is the contribution of the new nonlinear terms [underlined in eqn (9)]. (lOb)

The contribution of the conventional displacements to linear terms is:

I { cou N I cbui olJu N J edt! (c~u o~v)~UCL = N.r -;- + L N~ --;:- + N.r -;-;- + L N.r-;- + N~J -;-;- + -;-n u.t j_ I U.. u) J- I u:c t/) uX

N (cui at!) cc5U' N CO)V N }+ L N~,. ~+~ +Qz-a + L Q~ul+QJ"~+ L Qf.vl-qO)~' dA. (11)J- I (jJ u X .T j _ I ()Y j _ I

The contribution of the von Karman nonlinear terms is

(12)

318 E. J. BA'UJERO and J. N. Rwuy

The contribution of the new linear terms is

, D f. {. CcSW; ... cc5'Vi elJU' cdVJ (06U1 C6J1')}~UDL = L Q~-~- + Q~-~- +M~-~- +M~~ +M'q -a- + -~- dxdy,_ I' v..t (ly (J.t &I) Y (IX

(13a)

where

'f·",.-

The contribution of the new nonlinear terms is

i{D [ ; (au, O~Wi ccSu- aWl) ;(0.". a~w' 06K' aWl)~UDNL = L M x -;- -~- +~-~- +M,. ~ -~- + -;-~Q i vX vX u:C (J.t: uy uY lJ)' V)'

; (OlV Ob Wi a~u' aWi eu- ecS Wi C~K' aWI)] D D. I •+Mz, -a -a-+~~+-;--~-+-a -;- +LLiMlx 'Y (IX uY U)' (I."C 'Y uX I J

X (aw; a~wJ + a~W' OWJ) + iMii(OWi 006Wi + C~W' aWJ)~ ~ h ~ , ~ ~ ~ ~

"(OW; a~wJ aWi CcSW)} ~n+M'J ---+---- \U£~1 0)' O.."C O.."C 0)'

where

The boundary conditions of the theory are given below:

(13b)

(14a)

(14b)

1 .

Geometric Forceu Nxnx+NZTn,v Nz,nx+N¥n,w Q.,nx+Q,.n,.II N~nx+N~.rnxI"J N~ ...n.~+ !t'~n,.

V jA-f~n.~ + I\f~.,.n.r

Vi ~f~J.nx + 4\f~.IJ,.

Wi Q~\":I1.1'+ Q.~:n.r. (IS)The lanlinate constitutive equations can be obtained in the usual manner (see Reddy,1988; Barbero, 1989).

3.3. Fracture 111eclzanics anal)'sisThe deformation field obtained from the laycr-\1r'ise theory is used to compute the strainenergy release rates along the boundary of the delamination. Delminations usually exhibitplanar gro\\'th, i.e., the crack grows in its original plane. However, the shape of the crack

Dclamin~tion in composite laminates 379rnay vary ,,"ith time. For example. a crack shape initially elliptic usually grows with variablea'l"=ct ralio. Thcrcror~. the ~rack gro\\-th is not self-similar. Ho\\·e\~r, a self-similar virtualcra~k cXh:nsion \\-ill ~ assumed in order to compute the distribution of the strain energyrc:J~as~ ratc G (s) along the boundary of the delamination.

The virtual crack extension method postulates that the strain energy release rate canbe computed from th~ strain energy t~ and the representative crack length Q as:

G(s) =dU/da. (16)

In the actual implementation or the virtual crack extension method, however, eqn (16) isapproximated by a quotient 6UIt:.a that in the limit approximates the vaNe of the strainenergy release rate G. In the Jacobian derivative method (Barbero and Reddy, 1990), theG(s) is computed \\"ithout approximating the derivative, so it does not require that wechoose the magnitude of the virtual crack extension !la.

The boundary of a delamination is modeled as boundary conditions on the delami­nation variables Ui, Vj, and Iyi, by setting them to zero. A self-similar virtual crackextension of the crack (delamination) is specified for the nodes on the boundary of thedelamination. i.e.• the t\\'o components of the normal to the delamination boundary arespecified for each node on the boundary. The JOM is then used at each configuration (orload step) to compute G(s) from the displacement field.

3.4. Finite-ele,,,eIJ' nlolJelThe generalized displacements (u, D, lV, II, 01, Ul, VI, WI) are expressed over eachelement as a linear combination of the I\vo-dimensional interpolation functions .pi and thenodal v~llucs (II,. L·,. u·,.II:.l'!. Vi, VI, n'l) as ronows:

•(lit l" " .. ul

, r J• Vi, Vi, U'i) = L (Ui , Vi, ""i' uf, vi, U1, Vi, Wnt/li (17)

;. I

'W'here III is the number of nodes per element. Using eqn (17) in eqn (lOa), we obtain thefoJlo\ving finite element equations:

k " k::! k l2 k: 3 kkJ {A} {q},v

ki' kii k~;. k:!) k2J {AI} {ql}I' 10

k~' k_~'1 k,~.;" k:) k') {AN} = {("} (18a).,.

,\" ,"D

k:' k)~ k J2 k)) kJ ) {3 1} {qJ}II IN II ID

kA' k)~ kJ,~ k)) k)) {!D} {qD}01 DI DD

\\'here

{6 }T= {li It t:., ". It ••. , u., t'"., K·.}{11/} T - { j .j j J}- II,. L I, ... t II"" 11ft

'6,}T - 'VI J/j J~/i VJ Vi WI} (18b)\ - \ I· I' I,···, III' "" '"

and th~ subnlatrices [k"]. [k):], [k}'). (k;:!], [k;)l, [k~']t [k!/l 'With i.j= 1, ... ,N andr,s= I, ... ,D are given in Barbero (1989). The load vectors {q}, {ql} ... {~}, and{qll .... , {(jn} are analogous to {6}, {~I}, ... , {~N} and {AI}, ... , {~D}, respectively [seeeqn (18b)]. The nonlinear algebraic system is solved by the Newton-Raphsori algorithm.The components of the Jacobian matrix are also given in Barbero (1989).The nonlinear equations are linearized to formulate the eigenvalue problem associated\\"jth hifurcation (buckling) analysis:

. ..,,,:.

380 E. J. 8AIt8£ItO and J. S. R1.UUY

([KDJ - i.(Kc;)· cJ) =0 (19)

.-- where (KDl is the linear part of the direcl stiffness matrix and (KG) is the geometric stiffnessmatrix.

4. ~L'~tERICAL RESULTS

Several exa::-.ples ,; re presented in order to validate the proposed formulalion. Ana­lytical solutions ~an be developed for simple cases and lhey are used for conlparison withthe more general Jppro~,ch presented here.

4.1. Square thin filnl de/(JjJlinationA thin layer delami'lated from an isotropic square plate (flexural rigidity, D) by aconcentrated load P at its center is considered. The delamination is also assumed to besquare with side 2a. Tht: base laminate is assumed to be rigid \\ith respect to the thindelaminated layer. An ar. alytical solution for the linear denection and strain energy releaserate can be derived assuming that the delaminated layer is clamped to the rigid baselaminate. Due to the biaxual symmetry, a quadrant of the square delamination is analyzedusing 2 x 2 and Sx 5 mesh~s of nine-node elements. Either the clamped boundary conditionis imposed on the boundar y ofthe delamination or an additional band of elements with aclosed delamination is placed around the delaminated area to simulate the nondelaminaledregion. Both moeels produce consistent results for transverse deflections and average strainenergy release ra:~s. A fine mesh is necessary to obtain a smooth distribution of the strainenergy release rat: G along the boundary of a square delamination. The linear solution forP = 10 compares well with the analytical solution (see Fig. I). The linear and nonlinearmaximum delamination opening Wand average strain energy release rate Gilt' as a functionof the applied load P are shown in Fig. 2. It is evident that the membrane stresses caused

10.0,......--------....__ Uneor P.10__ Nonlinear p.SO

_ _ _ _ Nonlineor P.,0000000 Closed-form P.10

1.0

2.0

"aa 1.0

"-od 4.0-0

0.0 +-------------,,..=11I-..0.0 0. , 0.2 O.J 0... o.~

x/aFig. I. Distrib~:lon or the strain energ), rele35e rate along the boundary or a squ3re delamination

due to concentrated J03d.

t.-f

7.0 ----------....

1.0 -O/o"P

·r

__ NonJir.eor- - wnecr

0.0 +----r--"I..---.--a-"'..--,.....OOO

1.0

'.0

~.o

wO/c' p4.0

GAvO/p2 J.O

p

Fig. 2. ~1a~imL.~ delamination opening U· anu 3\Crag\.' slrJin energy rc1~aS4: rate according tononlinear analysis of a square c:lJrninJlion.

·., DcI.1mln~lion in comrosite LlminaltS 381

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

4.2. Thill fib" 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. Thecylindrical bending assumption is satisfied by restraining aJl degrees of fr~edom in the y­direction. 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 2a (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 2b with a circular delamination of radius a

"" 1.0...::-,..

0• 0.1.......,

~~

c: 0.1.2U..! 0.4~

"-> 0.2.,c0"-.-

N~ ~ J.-N.__ CI()sed-form• • • •• Present study

Fig. 3. ~fa ,imum delamination 0rtninl U' (or a thin film buckled delamination.

2.0 ~---------~

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

1.4

1.2 __ Closed-formo a ~:: Present study ·r

, .00 .0 ~ .0 2.0 30 4.0 5.0St~cin ene~c;y re;ecse rete

Fig. 4. Strain energy rekas<: rate for buckled thin film delamination.

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) = Lei (x, 0) = J/ J (x, 0) =0

with i = I, ...• Nand j = It ..• , D; \vhere IV 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,)') =0; N~(b,)·) = -Iirf(x,b) = Vi(:c,b) = 0; N...(:(,b) = -Ii

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 and Reddy, 1990) is used at equilibrium solution to compute the distributioDof the strain energy release rate G(s) along the boundary of the delamination. For 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 ora three-dimensional analysis are available (Whitcomb, 1988)in the literature. Compared to the last example, this problem does 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, GI2 =5.52 GPa, G2) = 3.430Pa,Va2 =0.3. As is well kno\l,'n, the quasi-isotropic laminate exhibits equivalent isotropic

1.2 -- 1-0 model000 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 dclan~ln~llonas a functionor the in·plane lo~d.

bc:havior ,,-hen loaded in its plane. The bending behavior, however, depends on the orien­tation. To avoid conlplic:ltions in the interpretation or the results introduced by the non­isotropic bending behavior. an equivalent isotropic material is used by Whitcomb (1988),where the equivalent stiffness components of the 3-0 elasticity are found from the relation:

.., fklaminJI.on in comf'Osilt hlmin~tes 313

Due to the transverse incompressibility used in this \\:ork, It IS more convenient andcustomary to work \vith the reduced stiffness. Equivalent material properties can be founddirectly,from theA-matrix of the quasi-isotropic laminate as follows. First. compute theequivalent reduced stiffness coefficients

where h is the total thickness or the plate and Ai} are the extensional stiffnesses. Next, theequivalent material properties can be found as:

£" = Qu(t- ~~~)GI2 = Q3l

Gll =Qss

Q"E.. -E;."12=-----QIIE1~

An eigenvalue analysis reveals that the delaminated portion of the plate buckles atNx = 286,816 N m- I for Q = IS mm and at N. = 73,666 N m- I for a = 30 mm. Themaximum transverse opening of the delamination as a function of the applied in-planestrain t.t is shown in Fig. 6. The differences observed \\'ith the results of Whitcomb (1988)are due to the fact that in the latter an artificiany zero transverse deflection is imposed onthe base laminate to reduce the computational cost of the three-dimensional finite elementsolution. The differences are more important for a = 30 mm, as indicated by the dashedline in Fig. 6, which represents the transverse deflection ,,,. of the midplane of the plate. Thesquare symbols denote the total opening (or gap) of the delamination, while the solid linerepresents the opening reported by \Vhitcomb (1988) with u' = o. The differences i.n the

LI,..----------"-'

'.2

__ 3-D e!e~e~ts

00000 C~enjn9 r• Cef!ect.on \U •~

E 1.0E

- 0.8

.. 0.'

~ 0.4

0.2 I..0.0 - - - - - - _ _ _ _ _ c-30...............

-0.2 +---..,-..- -_--P-_......0.0 1.0 2.0 3.0 4.0 5.0

• strain t.

Fig_ 6. Ma~imum transverse opening lI' of a circular de!amination of diameter 2a in a square platesubjected to in-plane load ~\', as a function of the in·pJane uniform strain.

£,=0.003

£.=0.002

£.=0.005

-100.00.0

0.0 "'------=:;;;;------:=~_Present study____ 3-0 elements

10.0 20.0 JO.O 'O.O 50.0

s :II 0" [mm] (O<~<1I'/2)

Fil. 7. Distribution or the strain ener,y rel~ rate along the bound31')" of a circular delamination(Oi~ various values of tbe applied in-plane axial strain.

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

700.0

800.0 8'\ Nt500.0

400.0

C(s) 300.0 ~ ~ ;..Nw

200.0

-----iOO.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 2·D elements) coincide for the sma)) delamination of radius Q = ISmm, as shown in Fig. 8. For the larger radius Q = 30 n1m, the assumption ,r = 0 is nolonger 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_release ralesG(s}along the boundary of the delamination are sho-\\'n in Figs 7 and8 for the different delamination radii, a = 15 mm and a = 30 mm, respectively. The values = 0 corresponds to (x = a, y = 0) and s = an/2 to (.\" = 0,)' =a). r\egalivc values of G (s)indicate 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 delal1lillQted graphite-epo."C.r plateA circular delamination of radius a = 5 in. in a square plate of side 2c = )2 in .. made

of unidirectional graphite~poxy_ is considered next. The thickness of the plate is " = 0.5

"

t.=O.OC~

c.=O.CC3

£,=0.002

S !n~ I ••

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

; I

~------' I

0.0 L.-_~~"":::'~~--­__ Present s~ucy

____ 3-0 elements

400.0 ~

300.0 j

0=15-100.00""'.0"""""'-"--5""0"'"-,-,,,......or-".o...--.....,...-S-·.O....., -......, 2-0"""1.

0"'-'....., -,2~51.0

S = ofl (~m) (0<,,<rr/2)

200.0

G(s) N~

100.0

Fig. ~. DislnbJtion of slrain energy release r~te 3!ung the bvundary of a (ircuLJr ddanlinatiun (orvarious \'aluC'S of jr.·rlane 3'\1:]) strain.

Dd.JnlicJtlon in conl"osilc lanlinatC'S 38511.0

10.0

9.0 O=CON,

Ncr

8'\8.0

7.0

6.0

~ ~ ...N.r5.0

-1.0 -0.5 -0.0 0.5 1.0lood ratio: r = (N.-N,)/(N.+N,)

FiI.9. Buckling load \'ersus the ratio orin-plane loads N. and N.. for a circuJar deJamination.

in., and the delamination is located at a distance I = 0.005 in. from the surface. Aneigenvalue analysis is used to obtain the magnitude of the in-plane load under which thethin delaminated layer buckles. Due to the orthotropic nature of the material used in thisexample, it is interesting to study the effect of different combinations of loads Hz and HI.·let us denote the load ratio as

The magnitude of the buckJing load as a function of the load ratio " with -I < r < 1,is shown in Fig. 9. The distribution of the strain energy release rate G(s) along the boundarys ~ ttc/J, 0 < t/J < n/2 is plotted in Figs 10-12 for the load ra tios or , = - I, 0, and 1, andfor several values of the applied in-plane load, in multiples i. of the buckling load Ncr. Forr = I (Le., N..: = Nand N.r = 0) it is clear that (see Fig. 10) the delamination is likely topropagate in a the direction approximately perpendicular to the load direction. Introductionof load in the direction perpendicular to fibers causes the maximum value of G to aligncloser to the .x-axis. Note that both the magnitude and the shape of the distribution ofG(s)change as the load ratio changes (see Fig. 13). The plots suggest that the propagation orarrest of deJaminations is greatly influenced by the anisotropy of the material and thedominant load.

10.0 oy------------

~

"(5 0.0 ~_-:=:::::::-~~

r = 1" := NINer- 5.0 +--oor-........,r---r-.....-..--.--,.-...-- ~0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

$ = olp [mm] (O<~<"./2)Fig. 10. Distribution of the strain energ~ rcl~Jsc rate along the boundary of a circular delaminationfor \'arious values of applied Joad IV.

------- -,,-,.....~-----

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

20.0 ..,.-------------.

10.0

0.0

~

'"'-'<.,)

-10.0

r • 0A • N/N.,-20.00.0 1.0 2.0 3.0 4.0 5.0 1.0 7.0 8.0

s • ctp [mm) (O<tp<n/2)

Fig. II. Distribution of the strain energy release rate along the boundar)' o( a circular delaminationfor abc ratio r - O.

r • -1A • N/Ner

- 20'°0.0 1.0 2.C 3.0 •.0 5.0 6.0 7.0 8.0

s • o~ (mm) (0<,<,,/2)

-10.0

-15.0

..... ".0c

, 10.0.DW S.O

~ 0.0'-'U -5.0

30.0 ,.-------------.

25.0 A-S.Q

20.0

(

Fig. 12. Distribution of the strain energy reJ~se rate aJ"ng the boundary or a circular delamination(or the ratio r = - I.

ITIJ]N,

C(s)

30.0

20.0

10.0

0.0

-10.0

- 20.0 -+-.....-,r--T--..,-...,.....,..-r--'I'--.,.--,-~..,..-,--,r--T--.

0.0 1.0 2.0 .3.0 4.0 5.0 6.0 7.0 8.0

s = c, [mm] (O<cp<1T/2),

Fig. 13. Distribution or L'1e strain energy release rate al"ng the boundary of a circular delaminationof radius a = 5 in. for \"3';OUS \·3l:.;e-s of the toad ratio, r.

387

(

Dclaminal101l in composite laminalns. SU~t~I.-'RY AND COl'CLUSIONS

A layer-wise theory and associated finite-clement model for the study ordclaminationsin laminated conlposite plates is developed. The same displacement distribution in theindividual layers is capable of representing displacement discontinuity conditions at inter­faces bet\\·een la)crs. The finite element model predicts accurate distributions of strainenergy release rates along the boundary of dclaminations of arbitrary shape. The modelcan be used to study multiple delaminations through the thickness or the plate.The layer-"'ise laminate plate theory provides an adequate framework for the analysisof laminated composite plates. Particularly, the layer-\vise linear approximation of thedisplacements through the thickness and the use of Heaviside step functions to modeldclaminations prove to be an effective approach for an accurate analysis of Jocal effects inlaminated composite plates. It must be noted, however, that the. computational cost of theproposed analysis makes it unattractive for prediction of global behavior when comparedwith conventional theories. For the prediction of local effects (Le., delaminations, inter­laminar stresses, etc.), the theory and formulation presented in this study shows its potentialas an alternative to three-dinlensional analysis. The model can also be used in a global­local analysis scheme wherein the local regions are modeled using the layer-wise theory andglobal regions are modeled using less refined theories, say the first-order laminate theory.Transition elements must be developed to join regions modeled by the layer-wise theory toregions modeled by less expensive theories in global-local analysis procedures.It is expected that accurate stress distributions obtained with this type of analysis,along with meaningful failure theories, will enable realistic prediction of failure initiationand propagation in composite laminates. Also, the layer-wise plate theory can be used asa post-processor to enrich the stress prediction of the first-order shear defonnation theory.Ackn()"'/~Jg('n,~nlS-The support of this work by NASA Langley Research Center and the U.S. Army throughGrant NAG·I·I030 and by NSF International Programs (U.S.-India Cooperatn-e Science Program) throughCirant INT·8908307 is gratefully ackno,,·ledged.

REFERENCESAshil;lW3. M. (1981). Fast interlaminar (racture or a compressh·eJy loaded composite containing a defect. FifthDOD-NASA Conf. on Fibrow Con,posi/~Sin SITlIClurD/ D"sign, Nev.' OrJ~ns. LA.&rbcro. E. J. (1989). On a generalized laminate theory wilh application to bending. vibration, and delaminationbuckling in composite laminates. Ph.D. Dissertation. Virginia PoJ)'technic Institute and State University,Blacksburg. VA.Barbero. E. J. and Reddy. J. N. (1990). The Jacobian derivath'e method for three-dimensional fracture mechanics.Cun,n,un. Appl. Nun,er. ,\(~/h. 6(7). S07-S18.Barbero, E. J., Reddy, J. N. and Teply. J. L. (19903). A general two-dimensional theory of laminated cylindricalshells. AIAA J. 28(3), 544-552.Bar~ro. E. J.• Reddy, J. N. and Teply, J. L. (l990b). An accurate determination of stresses in thick laminatesusing a generalized plate theory. Int. J. "'~,. M"It. Engng 29, 1-14.Bodner. S. R. (1954). The post-buckling bcha\;or of a clamped circular plate. Q. Appl. Afolh. 12, 397-401.BOlle~a. W. J. and ~tacwal. A. (1983). [)(Iamination buckling and growth in laminates. J. Appl. }'I~ch. SO, 184­IK9.

Chai. H. (1982). The growth of impact damage in compressively loaded laminates. Thesis, California Institute orTechnology.Chai. H. and Babcock. C. D. (1985). Two-dimensional moocHing ofcompressive failure in delaminated laminates.J. Con'po.t..\fa/~r. 19, 67-98.Chait II .• Babcock. C. D. and Knauss, \V. G. (1981). One dimensional modelling of failure in bminated plateshy ddamination buckling. Int. J. Solitu StrUC1Uf"S 17, 1069-1083.Durocher, L. L. and Solecki, R. (1975). Steady·state \'ibrations and bending of transvcrsely isolropic three·layerplates. DC'(·c/op",tll/J in .\ftC"hanics. P,oc. 1411r .\fidM·l*J1C'rn .\ftclr. Con!, Vol. 8. pp. 103-124.rei. Z. and "in. W.-L. (1985). Axis)mmctric bucklir.g and growth or a circular delamination in a compressedlaminate. /tu. J. Solids Structures 21, 503-514.Inoue. I. and Kobatake. Y. (1959). ~fechanjcs of adhesive joints, part IV: Peeling t~t. Appl. Scient. Rts. SA,321.

Kach;Jnov, L. ~1. (1976). Separation failure ofcompos:te materials. PolynJtr .\{ech. J2, 812-815.~faut S. T. (1973). A refined laminated plate theory. J. Appl. ,\Itch. 40. 606-607."furakami. H. (1986). Laminated composite plale theor}' with impro'icd in·plane responses. J. App/. Mtch, 53,661-666.ObrC'imoff. J. W. (1980). The splitting strength of Mica. Proc. R. Soc. A127, 290.Reddy. J. N. (1984). Energy and VarWt;on.J1 .\ft-Ihods in Applied .\fechanics. John \Viley, New York.$~S 2.: )·1

.,

388 E. J. BAR.tAO and J. N. R£ouyReddy. J. N. (1987). A ~eneralizatjon of two-dimensional theories of laminated plates. Comn,un. Appl. ,'·wnwr.Mtlh. J. 113-180.Reddy. 1. N. (1988). Me< ')anics of composite structures. In Fini!t £J~n,~nl."nalysisfo, £ngin~rr;n9 Dt.fign (Ed;tedby J. N. Reddy. C. S. Krishna Moorlhy and K. N. Scctharamu). Chap. 14. pp. ))8-359. Srringer. Berlin.Reddy. J. N. (1989). On refined computational models of composite laminates. Int. J. Nlln'tr. ~ftth. EngtJq 11,361-382.Reddy. 1. S. (1990). A re\ ew of refined theories or laminated composite plates. Shork Vibr. Digrsl 21(7). 3-11.Reddy. J. N.• Barbero. E. 1. and Teply. J. L. (1989). A plate bending clement based on a generalized la~:r.ate

plate theory. Int. J. Nun ltr. Aftllt. £"9"g 28. 221S-2392.S.Jllam. S. and Simitses. (;. 1. (1985). Delamination buckling and growth or nat. cross..ply laminates. CIJ"'~i'tlf.Strutt. ~. 361-381.

Seide, P. (1980). An imprc,ved approximate theory ror the bending oflaminlted plates. Mtch. rudo)' 5.451-&66.Shivakumar, K. N. and Whitcomb. J. D. (1985). Bucldin, or a sublaminale in a quasi-isotropic comrositelaminate. J. CompoJ. MQI~'. I'. 2-18.Simitses, Q. J., Sallam. S.at,d Yin. W. L. (1985). Effect ordelamination oraliaUy lo:aded homoacneous lamirulcdplates. AIAA J. n 1437-/440.Srinivas, S. (1973). Refined .:, nalysis of composite laminates. J. Sound Yibr. JO, 495-507.Sun, C. T. and Whitney. J. ld. (1972). On theories for the dynamic response of laminated plates. AIAA PaperNo. 72..398.Toledano, A. and Murakami. H. (1987). A composite plate theory (or arbitrarylaminalc configurations. J. Appl.Al~ch.S4t 181-189.Webster. J. D. (1981). Flaw critically of circular disbond defects in compressive laminates. CCMS Report 81..QJ.Whitcomb, J. D. (1981). finite element analysis or instability related delamination growth. J. Con,pos.•'fQI~r.15, 113-180.\\'hitcomb. J. D. (1988). Instability..related delamination gro..·tho( embedded and edge delaminations. l'ASATM 100655.Whitcomb t J. D. and Shivakumar. K. N. (1987). Strain-cnergy release-rate analysis or a laminate with a post­buckled delamination. NASA T~1 89091.Wilt~T.E .• MurthytP. L.N. and Chamis. C. C. (1988). Fracture toughness computational simulation of generaldelaminations in fiber composites. AIAA Paper 88·2261, pp. 391-401.Yin,W.-L (1985). Axisymmetric buckling and growth or a circular delamination in a compreSSC'd laminate. Int.J. Solids Slruclur~j 21, S03-S 14.Yin, W.-L. (1988). The effects or laminated structure on delamination buckling and growth, J. Con,pos. AIDftr.11.502-511.Yu, Y. Y. (1959). A new theory of elastic sandwich plates-one dimensional case. J. Appl. A/tell. 26,41s-421.