composite properties

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AIAA J O U R N A LVol. 32, No. 8, August 1994

Interlaminar Stresses in Composite Laminates UnderOut-of-Plane Shear/BendingTaehyoun Kim* and Satya N. At lur ifGeorgia Institute of Technology, Atlanta, Georgia 30332

An approximate m ethod is developed to investigate the interlaminar stresses near the free edges of beam-typecomposite laminate structures subjected to out-of-plane shear/bending. Th e method is based upon admissiblefunction representations for in-plane stresses that contain a linear variation in the longitudinal direction.Closed-form solutions of all stress components are sought by minimizing the complementary energy with respectto the unknown functions. The resulting solutions satisfy the stress equilibrium, compatibility, and all of theboundary conditions. Numerical examples are given for both cross-ply and angle-ply laminates. It is found thatinterlaminar stresses under the shear/bending, particularly those for angle-ply laminates, may exhibit substan-tially different characteristics than under uniaxial loading or under pure bending.

Nomenclatureb = half-laminate width/i, /2 = u n k n o w n functions for the ply stresses c r 2 2an d o\2g (k) = unknown function for the ply stress an inth e fcth pl yh = laminate thicknessh{k\ h2k} = unknown funct ions for the strain en in the& th pl yL = laminate lengthA W 1 , 2 , 3 = eigenvalue so lutions to the characteristicequation (64)n = tota l number of pl iesP ( X I ) - arbitrary loading distributed in thelongitudinal direction x\ (force/unit area)S j j = ply compliances/ ( A r ) = thickness of the fcth pl yV - applied out-of-plane shear per unit width orlaminate volu me (force/unit width)X [ = longitudinal coordinatex2 = transverse coordinatex2 = normalized transverse coordinateA T 3 = o ut-of-plane coo rdinate with origin at thebo t t o m o f each pl yc i i 2 2 * 3 3 =normal strains723, To, 712 = shear strains^ 2 3 , v\3, ^ 1 2 = poisson rat iosH c = complementary energyH C = complementary energy defined by Eq. (58)a /y = far-field stressesat = companion stresses = l o ngitudinal lo cation o f shear load

IntroductionIN th e past, there ha s been a great deal o f research on thebehavior o f interlaminar stresses near free edges o f com-posi te laminates. The most frequently studied are the inter-laminar stresses at the straight edges under uniaxial loading o rpure bending. The metho ds employed vary from finite differ-

ences1"4 and finite elements5'8 to e igenfunction expansions.9*10Ano ther grou p o f m ethods makes use of assumed equi l ibratedstress representations and the use of the principle o f m inimu mcomplementary energy to satisfy compatibil ity.11"13 This tech-nique particularly al lows a simple an d efficient yet accuratetool fo r estimating th e interlaminar stresses. Most recently,th e method was generalized by accounting for the mismatchesin Poisson's ratios and the coefficients of mutua l influencesthat m ay further exist between different plies in the through-thickness direction.14 General ly, it is known that the stressesunder either type of loadings, uniaxial tension or pure bend-ing, are kno wn to share similar characteristics.Investigations of other types of loadings have been rela-tively rare. Tang 15 predicted interlaminar stresses o f unifo rml yloaded rectangular composite plates analytically bu t ignoredthe longitudinal variation in the boundary-layer part of thesolut ion. Mu rthy and Chamis8 examined interlaminar stressesunder various loadings such as in-plane and out-of-planeshear/bending an d torsion, using a three-dimensional finiteelement method. The characteristics that typify the stresses,particularly under o ut-of-plane shear/bending, however, wereno t clearly po inted ou t, an d so m e of the results need co mpari-sons with other analyses. Kassapoglou13 added the effects o fth e out-of-plane shear to his closed-form solut ions fo r com-bined extension/bending loading. The formulat ion, a l thoughaccurate fo r plates that are homogeneously anisotropic , doesnot adequately model the mismatches between the laminateand the individual laminae pro perties.In this paper, interlaminar stresses near straight free edgeso f beam-type compo site laminate structures under o ut-o f-planeshear/bending are investigated using an approximate methodbased on equilibrated stress representations and the use of theprinciple o f minimum complementary energy. The presentanalysis is different from th e previo us assumed stress methodsin that it includes th e longitudinal degrees o f freedom in thestress distributions, which adds much to the complexity of thef o rmu la t i on . As a result , the unknowns in the resulting stressexpressions are obtained by solving an eigenvalue problemwhose coefficients are not constants but may depend on the


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K I M A N D ATLURI: INTERLAMINAR STRESSES IN COMPOSITES 1701

f Ply k 1

~* 1 .....

Fig. 1 Ply coordinate system.

Stress Equ ilibrium EquationsConsider, for example, a composite laminate that iscantilevered at one end and subjected t o an ou t -o f -p lane shearl oad V (force/uni t width) concentrated at x\ = % (see Fig. 1) .The laminate consis ts of n plies and can have general unsym-metr ic layups . It is a s sumed that th e width (2b) of the beam issma l l compared with i ts length, and the shear load is dis-tr ibuted un i f o r mly over the width. It is also assumed that localconstrain ing effects in the vicinity o f the root of the canti levercan be ignored . For convenience, the fo l l owi ng t r ans f o r ma-t ions are in troduced for the width coordinate x2:b x2x2 = * for x2 > 0 (1)

x2 = fo rBoth the free edges at x2 = b are then defined by x2 = 0. Thelocal vertical coordinate x3 is defined as being zero at thebottom of each p ly . For a geometr ica l ly l inear theory , thestress equi l ibr ium equations (a/y> 7 = 0), in the absence o f bodyforces, have th e f o r m

1 dai2 dg 13 _h dx 2 dx3 ~1 3022 d(723 _ Qh dx2 dxi

_h dx2

#033

(2)

(3 )

(4)where m i n u s and plus signs are used for the positive andnegative x2, respectively. Assuming that the laminate has ef-fective length o f , th e traction b o u n d a r y cond i t i ons that m u s tbe enforced in the complementary energy principle are

(7 = 1, 2, 3)(7 = 1,2,3)

(5)(6)

It is seen that since al l of the three equilibrium equat ions arecoup led , at least three stresses need to be k n o w n a priori toobtain th e remaining components.

the contrary, bo t h components are expected to exist near th efree edges to meet th e traction free boundary conditions (6).Acco r d ing to the classical laminate plate theo r y (CLPT) orth e first-order shear deformation theory , the in-plane far-fieldstress c o m p o n e n t s in the /r th ply can be expressed as16- V[ (Xl - 0 - (*i - V[ (Xl - {) - (*! -V[ (Xl - f ) - (*!

(8)(9)

(10)where th e coefficients A$k\ B$k\ an d C\k) ar e obtained for au n i t m o m e n t M x = 1 and with al l other m o m e n t s equal tozer o , an d

0 if 0X i - if ( 1 1 )

It is emphasized that an y higher order shear def o r mat i ontheor ies can also be invoked, but on ly th e zer o th - an d first-or-de r theories are considered in the present analysis .The rem aining far-field stress c o m p o n e n t s , i.e., the ou t -o f -plane stress c o m p o n e n t s due to the in-plane gradient field ar eobtained by integrating the stress equilibrium equations (2-4)with the previous express ions for a ff , d^p, and d$. Treatingthe applied shear as an impulse l oad at x\ = , one can sum-marize those components as

) = - ( E l f *

[ k)xl ) V(xl - ) < - !>

- V[\ - (x, - S ) < \ (12)V[\ - (x, - f )(] (13)

(14)where

0 if 0 < xi (jc3 = f W ) = V( Xl - ) < - ! > ( top surface)W (*3 = 0) = 0 (bottom surface) (j = 1, 2, 3)

= l ,2 ,3 , A : = 1,2, . . . , /z - 1)

(20)

(21)

Companion Stress ShapesThe in-plane parts of the companion stresses (72C2 and af 2, inth e boundar y region, are assumed as fo l lows:(22)(23)2(*Z>

where f\(x 2) an d /2(x2) are unknown functions of the nondi-mensional coordinate x2 an d will be determined from th eprincip le of minimum complementary energy. {Note thatf\(x2) an d f2(x2) are left as undetermined functions in theassumed equilibrated solutions a/J [see Eqs. (51-56)]). Withat and a * defined as Eqs. (9) and (10), the previous expres-sions for the companion in-p lane stresses guarantee globalequilibrium of total in-plane stresses. Although the in-planeshear stress (23) satisfies the t ip bo undary condit ion (5), thereexists a restraint in Eq. (22) that the transverse normal s tress< 7 2 2 shou ld also vanish at the tip of the laminate. According toRose and Herakovich,14 this is a valid expression not account-ing for the local mismatches in Poisson's ratios. The boundaryconditions (6) require

/2(0)=-1l im/ i ( fe) = 0Iim/2(x2) = 0

(24)(25)

The second conditions in Eqs. (24) and (25) insure that th ecompanion parts ar e zero fa r away from th e free edges. Theprevious in-plane components and all other out-of-plane com-ponents that f o l l ow will be made to satisfy th e g lobal equi-l ibrium automat ica l ly. No functional form is assumed for thein-plane stress af l9 but as will be seen later, it can be invokedf rom a compatibility equation.The first of the out-of-plane companion stresses, a23, isobtained directly by integrating th e second equilibrium equa-t i on (3) as

'

) V[\ - < x i -V[(Xl - {) - to -

and bo t tom surfaces and the stress continuity at all ply inter-faces. Also, vanishing of a2c3 at the free edges requires(28)

Next, to obtain a3c3 one has to k n o w th e first term in thethird equilibrium equation (4). This can be done by taking a 3from th e first equilibrium equat ion (2) as(29)

where the two integrals include appropriate constants to sat-isfy stress continuity at each ply interface. Since no impu lse isexpected in aft, one can assume

(30)where g has yet to be determined. Hence, upon substitutingthe expression (30) an d differentiating Eq. (29) with respect to*i, o ne obtains for the fcth ply

dx {V[\ - (Xl - (31)

It is no ted that since the derivatives of any o ut-of-p lane s tresswith respect to J C i are also con t inuous at each pl y interface, th eintegral in Eq. (31) m u s t contain appropriate constants tosatisfy this continui ty . Final ly , upo n substi tu ting Eqs. (26) and(31) into the last equilibrium equation (4) and integrating withrespect to *3, one obtainsV(Xl -

22

*hfi*V[\-

1 ,where

V{(Xl - 0 - ( A T , -?)


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K IM AND ATL URI: INTERL AMINAR STRESSES IN COMPOSITES 1703

8x 28x 3

'8x18x3

(36)

(37)The various stress an d strain c om pone nt s a re re lated throug hth e f o l l owi n g stress-strain relations:

en 2 2 3 3723713712

(*) S\\ S\2


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1704 K IM A N D A T L U R I: I N T E R L A M I N A R STRESSES IN COMPOSITES

Total Stress Field SummarySo far, all of the stress component s have been found. Thefar-field components are given in Eqs. (8-10) and Eqs. (12-14).The companion parts are given in Eqs. (22), (23), and (26) an dEqs. (46), (47), and (50). The total stress field including thefar- field an d companion parts can be summarized as f o l l ows :

(k ) _

V[( Xl - & - (Xl - (51)(52)

r> = I _R(* ) , SI L tP(k ) . $f0M. , ^iL fLL 3 sfr ' 3 +s


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KI M AND ATLURI: INTERLAMINAR STRESSES IN COMPOSITES 1705

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1X 2 / hFig. 5 ( 7 3 3 at first interface; [ 45]55, entire transverse region.

0.0200.015

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1X 2 / hFig. 6 0 2 3 at first interface; [ 45]ss, entire transverse region.

where iW po J o (59)N o t e that th e energy formu la t i on defined by Eq. (58) includesonly a half of the region of the laminate, either Q

To determine S/ an d 5/, first Eq. (66) is substituted into Eqs.(62) and (63) to yield

^=-Rsm*+Rm2++RRlom+* (/ = l* 2 > 3) (6?)Next, su bsti tu ting Eq. (66) into the boundary conditions (6 )yields

-1= Si + S 2 + S 3- 1= Sl +52 +S 3 (68)0 = Siml

Th e previous six equations in Eqs. (67) and (68) can be solvedfo r the corresponding S/ and 5, for a given A W / .The two u n k n o w n s / i and/2 are in general functions o f .According to the order of magnitude analysis, however , th eactual so lut ions should not change significantly with the shearload location. In fact , for most of the composi te laminatesinvestigated, th e roots of the characteristic equation (64) weref ound to be a lmos t identical to those of the pure bendingproblem.

Special CaseCross-Ply LaminateThe so lu t ion simplifies significantly for a cross-ply laminatesince there is no in-plane shear stress ai2 in this case. A s aresul t , /2 does no t exist an d


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1706 KIM AND ATLURI: INTERLAMINAR STRESSES IN COMPOSITES

with

an d"*1,2=

m ^m2 - m2-

(72)

(73)It is noted that , for a cross-ply l aminate, the exact symmetryand antisymmetry of the out-of-plane stresses in the transversedirection will be recovered.

Application to Arbitrary LoadingFor an arbitrary loading P ( X I ) that is distributed over0


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K IM AND ATLURI: INTERLAMINAR STRESSES IN COMPOSITES 1707

-0.1 0.1513 l ie

Fig. 10 < r i 3 at J c 2 = 0.05; [ 4S]ss, entire through-thickness region.

*2 = 0 whereas a23 has a complete ant isymmetry. Therefore,near th e negative free edge fa ^0), which i s no t shown here,a33 and an will have the same magnitudes and signs as the onesrepresented in the figures, and < 7 23 will have the same magni-tude with opposite sign.The next group o f figures, Figs. 5-7, represents no rmalizedinterlaminar stresses for [ 45] 5S lay-up at the bot tom of thetop p ly over th e entire transverse regio n, b


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1708 K IM AND ATLURI: INTERLAMINAR STRESSES IN COMPOSITES

Applied Mechanics, V o l . 49, Sept. 1982, pp. 541-548.10Wang, S. S., and Choi , I., "Boundary-Layer Effects in Compos-ite Laminates, Part-2: Free Edge Stress Solu t ions an d Basic Charac-teristics,"ASME Journal of Applied Mechanics, V o l . 49, Sept. 1982,pp. 549-560.HNishio ka , T . , and A t lur i , S . N . , "Stress Analysis of Holes inAngle-Ply Laminates: A n Efficient Assumed Stress 'Special-Hole-Ele-m e nt ' Approach and a Simple Estimation Method," Computers &Structures, V o l . 15, No. 2, 1982, pp. 135-147.12 Kassapog lou , C., and Lagace, P. A., "An Efficient Method forth e Calculat ion o f Inter laminar Stresses in C om p os i t e Materials,"ASME Journal of Applied Mechanics, Vol. 53 , Dec. 1986, pp .

744-750.13 Kassapog lou , C., "Determination of Inter laminar Stresses inComposite Laminates Under Combined Loads," Journal o f Re in-forced Plastics and Composites , V o l . 9, Jan. 1990, pp. 33-58.14Rose, C. A., and Herakovich, C. T., "An Approxim ate Solu t ionfo r Interlaminar Stresses in C om p os i t e Laminates," Composi tes E n-gineering, V o l . 3 , N o . 3 , 1993, pp . 271-285.15Tang, S., "Interlaminar Stresses o f Uniformly Loaded Rectangu-la r C om p os i t e Plates," Journal o f Composite Materials, V o l . 10 , Jan.1976, pp. 69-78.16 Jones , R. M ., Mechanics of Composite Materials, McGraw-Hil l ,N ew York, 1975.

This timely tu-torial is theculmination o fextensive par-allel research and a year o f col laborat iveef fo r t by three dozen e xcel lent research-ers. It s e r v e s as an important de parturepoint fo r near - term applications as wellas fur ther research. The text contains 25chapters in three parts: Structural Mod-

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