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AIAA JOURNALVol. 40, No. 2, February 2002

Prediction of Compressive Failure in Laminated Compositesat Room and Elevated Temperature

Junghyun Ahn and Anthony M. Waas

University of Michigan, Ann Arbor, Michigan 48109-2140

An experimental and analytical investigation of the notched strength of ber reinforced polymermatrix compos-

ite laminatessubjected to remote uniaxialand biaxial loads is presented. Experimentalresults pertaining to failure

mechanisms, critical loads, and critical strains for cross-ply and quasi-isotropic laminates at room and elevated

temperature (200C) are presented. A micromechanics based nite element analysis that uses readily available

constituent properties for predicting notched strength of general multilayered composite laminates is developed

and used to predict theexperimentallymeasured failureloads.Good agreementbetween experimentand analysisis

reported. The present methodology is an attempt to remove the empirical nature of composite strength prediction.

Introduction

LAMINATED ber-reinforced polymer matrix composites(PMCs) are nding increased use in a broad range of indus-

trial applications. Applications in the national infrastructure (suchas all-composite bridges over interstate highways), the automotiveindustry, the medical sector, and the aerospace industry are someexamples of where these materials are earmarked for several uses.

Compositematerials are superior to conventionalmonolithicma-terials for a number of reasons, noteworthy among these being theirmechanical properties. Ashby and Jones1 have formulated perfor-mance indices that can be used to select a material for a givenapplication based on satisfying certain objective measures. PMCsappearto be thematerialofchoicefora numberof such applications.Issuessuch as damage toleranceand durability,performance degra-dation due to aging, fatigue under multiaxialloads, and response at

elevated temperatureare currently being researched.Experiencewith previousapplicationsofPMCs forrotor bladesinthe helicopterindustry,pressurevessels,and othersimilarsituationsthat call for superior tensile stiffness and tensile fatigue life haveshown PMCs superior performance in a tensile load environment.In contrast, the compressive strength of PMCs is known to be lessattractive.2;3

The response of compositelaminateswhen subjected to mechan-ical loads is inuenced by the material type (ber and matrix) andconguration (stacking sequence and layup). In addition to thesefactors, geometrical parameters (notches and thickness discontinu-ities) and loading characteristics (multiaxial, thermal, cyclic load-ing, tonamea few) also affect the overall performanceof compositelaminates.One of thechallengingtasks in theanalysisand designof

PMC-based structural laminates is to improve strength prediction,particularlyfor situationsinvolvingnonhomogeneousstress states4

under different design environments.Previouswork related to the presentinvestigationwas undertaken

by Sandhu,5 Daniels et al.,6 Soutis et al.,7 Martin et al.,8 and Whamand Palazotto.9 Martin et al.8 investigated failure in thermoplasticlaminates and provided comparisons between theory and experi-ment. The effect of a central notch on failure in thermoplastic lam-inates was examined by Daniels et al.6 and Wham and Palazotto.9

Received 3 November 2000; revision received 10 July 2001; acceptedfor publication 11 July 2001. Copyright c 2001 by Junghyun Ahn andAnthony M. Waas. Published by the American Institute of Aeronautics andAstronautics, Inc., with permission. Copies of this paper may be made for

personalor internaluse, on conditionthat thecopierpay the $10.00per-copyfee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers,MA 01923 ; include the code 0001-1452/02 $10.00 in correspondence withthe CCC.

Research Fellow, Composite Structures Laboratory, Department ofAerospace Engineering. Member AIAA.

Professor, Composite Structures Laboratory, Department of AerospaceEngineering. Associate Fellow AIAA.

In addition to these investigations, Waas and Schultheisz3 providea comprehensive review of previous work related to compressivefailure in composite laminates.

The organization of the present paper is as follows. First, detailsof the uniaxialand biaxialexperimentsconductedon notchedplanarcompositelaminateswill be presented.Results obtainedfrom exam-ining different types of laminates will be described, with importantobservationsdeducedfrom the tests to ascertain the laminate resid-ual stiffnessproperties.This is followedby a discussionof the effectof the environment on the observed failure mode. Next, the devel-opment of a micromechanics-basednite element model to predictthe onset of failure in notched laminated composites at room andelevatedtemperature,when subjected to remote planaruniaxialandbiaxial loads, is presented. Both material degradation (due to tem-perature) and structural (stress concentration) effects are accountedfor in the analysis. The predictions of the model are found to be ingood agreement with the measured experimentalresults.

Details of the Experiments

Khamseh and Waas10 have pointed out difculties associatedwith stress concentrations that are encountered in the design ofa cruciform-shaped planar specimen for in-plane biaxial loadingof composite laminates. To select the size of the specimen in thepresent experiments, a model of the notched laminated plate andgrips (steel) was generatedusing the ABAQUS niteelementanal-ysis (FEA) softwarepackage.This stress analysisprogram was usedto examine the effect of several specimen geometries on the stressconcentrations due to the applied loads. The loads were applied atthe boundaries of the loading arms by use of displacement con-straints. Eight noded parabolic plane stress elements were used to

model the plate material and steel grips. Ply properties were en-tered from which ABAQUS calculates the equivalent plane stressconstants using classical lamination theory. From the results of theFEA, it was concluded that a cruciform conguration matching thedimensions given in Fig. 1 was the desirable specimen shape. Theword desirable is used to identify with a stress state for whichan in-plane region in the interior of the specimen (shown as thecrosshatched area in Fig. 1) is not inuenced by the effects of thefar-eld edges (i.e., edge curvatures identied as Rc in Fig. 1) dueto the loading.

Several specimen congurations corresponding to different val-ues of Rc were studied

10 using the FEA. The results were used toarrive at an optimum value of Rc based on the condition that thespecimen with a hole does not contain effects associated with the

stress concentration at the hole interfering with nonuniformities inthe stress eld generated on account of the edge curvatures. Forthe composite plate we studied, this working area translated intoa 6.35-cm (2.5-in.) square region in the middle of the cruciformconguration.

The test specimens were made of graphite/toughened epoxy ma-terial containing Hercules IM7 (intermediate modulus) ber and

346

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Table 1 Zero-ply material properties of the 48-ply graphite/977-3 epoxy composites

Material E11 , Msi/GPa E22, Msi/GPa G12, Msi/GPa 12 Thickness, in. (mm)

IM7/977-3 epoxy 23.5/162.0 1.21/8.3 0.72/5.0 0.30 0.0052 (0.132)IM7 ber 42/289.6 0.25 2.756E04 (0.007)

977-3 epoxy 0.7/4.8 0.34 2.444E04 (0.006)

Table 2 Laminate material properties as determined from CLT using the propertiesfrom Table 1

Laminate type Stacking sequence Ex x , Msi/GPa Eyy , Msi/GPa Gx y , Msi/GPa xy

Quasi isotropic [C45=0=45=90]6s 8.828/60.867 8.828/60.867 3.372/23.249 0.309

Cross ply [0=90]12s 12.401/85.502 12.404/85.502 0.7119/4.908 0.029

Fig. 1 Specimen conguration and dimensions. 5.5 in. (139.7 mm) 5:5 in. (139.7mm) cruiciform specimens, Rc = 0.25 in. (6.35mm), HER-CULES IM-7/977-3 (toughened epoxy resin), 48-ply plate, 0.5-in. (12.7-

mm) hole diameter, 0.25-in. (6.35-mm) thickness, cross-ply: (0/90)12s ,and quasi-isotropic: (+45/0/ 45/90)6s .

977-3 toughened epoxy matrix, designed for operation at elevatedtemperatures. In Table 1, the lamina properties are given, basedon data provided by the manufacturer. Two types of stacking se-quenceswere investigated:crossply [0=C90]12s and quasiisotropic[C45=0=45=90]6s. In Table2, Laminatematerialproperties,calcu-lated using generalizedclassical laminationtheory (CLT) are listed.The linearelasticplanestresssolutionthatcorrespondsto thepresentconguration, as given by Lekhnitskii,11 was used to calculate thestress eld corresponding to the uniaxial and equibiaxial loadingused in the experiments. This solution were veried with the strain

data (away from and near the hole edge) that were recorded in theroom temperature test for each laminate type. Such a check yieldedgood agreement between test data and the two-dimensionalelastic-ity solution. The details of the elasticity solution, the biaxial loadframe used for the experiments,and the test procedure are given byAhn.12

The in-plane dimensionsof thetest specimensare givenin Fig. 1.The thicknessof the laminateswas 0.635-cm(0.25-in.) with a 0.013-cm (0.005-in.) tolerance.In the centerof the steel grips, a 0.724-cm(0.285-in.) channel, 3.048 cm (1.200 in.) in depth, was machinedalong the length of the grip. The specimen was bonded to the wallsof this channel with the use of Devcon brand plastic steel puttyadhesive,treated with release agentto allow for ease of separationofthe two materialsat theend of an experiment.TheDevconputtyacts

as an interfacebetweenthe specimenedge andthe surfaceon whichthe specimensits, effectivelysmoothening out the interfacesurfaceirregularitiesbetween the two materials and ensuring a smooth loadtransfer.

To perform a compression test at elevated temperature, afeedback-controlled radiation-heating element with an insulationtunnel was designed. Special strain gauges for high temperature

Fig. 2 Strain gauge locations on the specimen.

were used for the test. To ensure a steady temperature in the en-closed test chamber, four locations in the test chamber were usedto monitor temperature. The heating rate was limited to be veryslow (2C=min). In addition, thermocouple gauges placed insidethe specimen cutout edge and on the specimen surface were usedto verify that the specimen temperature was fairly even (less than1C change from location to location). Circulating water through acopper pipe wrapped aroundboth parts cooledthe load cell and theloading frame. The maximum temperature of the tests was limitedto 200C (392F).

A 50,000-lbf load cell was used to obtain a load time history ofthe far eld applied load. Surface strain measurements at variouslocations on the specimen were obtained via strain gauges. Straingauges were mounted along the edges of the hole, as well as in the

correspondingfar-eld region. A set of two additionalstrain gaugeswere mounted inside the hole, along the wall thickness, to recordchanges in strain in the out-of-plane(through-the-t hickness) direc-tion, revealing failure initiation. Strain gauge locations are shownin Fig. 2. Strain gauge readings, along with load cell readings, weremonitored and recorded using a commercial data analysis softwarepackage. The data sampling rate was ve samples per second at aloading rate of 0.0001 in./s.

Experimental Procedure

The 48-ply composites were tested under uniaxial and 1:1 (infar-eld displacement) biaxial loading at both room temperature(21C) and 200C to understand the failure mechanisms and howthey are affected by temperature. The catastrophic nature of the

failure in composites and the additional limitation placed on thevisual investigation of failure initiation and progression suggesteda d isplacement-controlled mode for the t ests (as opposed to theload control). This facilitates examination of specimens loaded toinitial (local) failure but recovered prior to global failure. The in-uence of hole size on the failure mode had been established andveried before.10;13 Therefore, in the present study, one hole size

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Table 3 Cross-ply laminate experimental data

Specimen Temperature, Loading Failure initiation Maximum holetype C type load, ksi/MPa strain, " Failure mode

Cross ply 21 Uniaxial 44/303 7500 Kink at 0-deg edgeCross ply 200 Uniaxial 19/131 2700 Kink at 0-deg edgeCross ply 21 Biaxial 37/255 6600 Kink in 0, 90-deg edgeCross ply 200 Biaxial 18/124 2080 Kink in 0, 90-deg edge

Table 4 Quasi-isotropic laminate experimental data

Temperature, Loading Failure initiation Maximum holeSpecimen type C type load, ksi/MPa strain, " Failure mode

Quasi isotropic 21 Uniaxial 21/145 6600 Shear failure (C45=45)Quasi isotropic 200 Uniaxial 19/131 3900 Shear failure (C45=45)

Quasi isotropic 21 Biaxial 26/179 4443 Shear failure (C45=45)Quasi isotropic 200 Biaxial 24/165 2872 Shear failure (C45=45)

Fig. 3 Experimental strain results for equibiaxial displacement load-ing of a cross-ply laminate plotted against remote strain (room tem per-ature).

(0.5 in. diam) was chosen for all specimens. Each test consists ofloading a particular type of laminate until global failure, to esti-mate the ultimate strength of the specimen. This information isused subsequently to aid in unloading similar specimens that havefailure initiated but that have not reached their ultimate load car-rying capacity. A sudden nonlinear (with respect to a far-eld loadcomponent) increase in the through-the-thickness strain inside thehole (strain gauge placed through the thickness on the hole edge)indicates that there is failure initiation at the hole edge of the spec-

imen. At this point, there may not be a sign of global failure of thespecimen.

Note that in the elevated temperature experiments, during theheating cycle, the loading piston was separated from the speci-men grips initially, to ensure free thermal heating, with freedomforunconstrained thermal expansion (or contraction). Thermal strainswere measured and recorded.These values were used for establish-ing the laminate orthotropic thermal expansion coefcients.12

A typical strain history is shown in Fig. 3 for a cross-ply lam-inate loaded in equibiaxial compression at room temperature. Asexpected,the response is symmetric becausethe cross-plylaminatehas equal stiffness along the loading directions. Detecting local-ized failure was done by followingthe through-the-thickness strain,as shown in Fig. 4, for a uniaxially loaded cross-ply specimen at

elevated temperature. The test was stopped as soon as any nonlin-ear strain increase was observed in the through-the-thickness strainreading. The unloaded specimens were sectioned and observed un-der an optical microscope to examine the failure initiating mecha-nisms. Both the in-plane and out-of-planeviews of the failed regionwere examinedand digital photomicrographswere acquired.A sim-ilar procedure was used to study the remainder of the specimens.

Fig. 4 Experimental strain results for a uniaxially loaded cross-plylaminate at elevated temperature.

Room Temperature Results

Themechanismoffailurein cross-plylaminateswas kinkbanding(also identied before by a number of researchers; see Refs. 13

23). The values of far-eld stress at failure for different loadingconditionsareindicatedin Table3. Failureinitiatesin thezero plies,at the hole edge, and propagatesin a direction that is perpendicularto the direction of applied far-eld load. Both, inplane and out-of-plane kink banding occurs in the zero plies (Fig. 5). For uniaxialloading at room temperature, the surface strain at the hole edge in

the direction of applied load at failure initiation is 7500 " andthe corresponding far-eld strain is approximately 3000 ". Whenthe damage propagates in the form of kink bands, the ber/matrixinterfaceis completelysevered in a numberof bers within thekinkbands. In addition, the kink banding also introduces interlaminarseparationbetweenthe kinkedply and the neighboring90-degplies.

Quasi-isotropic laminate specimens fail by both inplane shearfailure and kink banding simultaneously (Fig. 6). Whether kinkbands form in the 0-deg plies or ber/matrix interfacefailureoccursin the 45-deg plies is determined by 1) the specimen characteris-tics (stacking sequence, ber/matrix interfacetoughness,and initialber misalignment) and 2) the in situ shear response of the ma-trix. In the present case, it is possible that shear failure along theber/matrix interface led to catastrophic failure of the specimens.

Although there was evidence of kink band formation observed af-ter the test, the ber/matrix interfacial failure in the 45-deg plieswas seen to be extensive, starting at the hole edge and propagatingoutward. A summary of the critical quantities at failure of quasi-isotropic laminates is given in Table 4.

The main differencein response of the laminates between the re-mote uniaxial and the equibiaxialdisplacementloading is not in the

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Fig. 5 Cross-ply laminate failure mechanism (biaxial, room temperature).

Fig. 6 Quasi-isotropic laminate failure mechanism (biaxial, room temperature).

mechanism of failure,but the magnitude of far-eld loadingneces-sary to achieve failure initiation. Remote equibiaxial compressionloading tends to relieve the compressive stress gradient away fromthe hole, as compared with uniaxialloading. For the cross-plycase,this combined effect due to biaxial loading does not show much ofan effect on the magnitudeof the failureinitiationstressbecause thefailure mechanism is initiated by a ber instability. However, thequasi-isotropic specimens show different far-eld failure initiationloads depending on whether the remote loading is uniaxial or equi-biaxial. This is because the failure mechanism is governed by thestate of shear in the matrix, which is a strong function of ply angle

and overall stacking.In summary, in cross-ply laminates, because there are only

0-/90-degplies,failure initiationoccurs by kinkbandingin the0-degplies, starting from the hole edge at a position that is perpendicu-lar to the direction of loading. Ideally, both kink bands (on eitherside of the hole) should form simultaneously, but due to materialnon-homogeneityor a slight perturbationof the symmetry of load-

ing, one of the two kink bands forms rst, followed by the other.The kink band formation leads to interlaminar cracking betweenthe kinked ply and the adjacent 90-deg ply. Quasi-isotropic lam-inates show a competition between shear failure in the C45=45lamina and the 0=90 lamina. The experimental results show thatquasi-isotropic laminates fail by interfacial failure along the 45-deg plies; however, it is postulatedthat this mechanism is triggeredby kink banding in the zero plies, as modeled later.

Elevated Temperature Results

At elevated temperature, a new parameter emerges to inuence

the mechanism of failure. Because the matrix properties undergodegradation as temperature increases, the matrix-sensitivemechan-ical properties of laminates such as the instantaneous compositetransversemodulusand the instantaneouscompositeshear modulusare affected at elevated temperature. An examination of the shearresponseof thein situ matrixas a function of temperatureprovidesaconvenientmeans to establish the manner in which matrix-sensitive

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Table 5 Results of the mesh convergence with hole diameter 0.5 in. (12.7 mm)

Maximum Maximum resultantDimensions imperfection stress on Plateau stress

Mesh used X length Y length size microregion, kpsi/MPa on microregion

Baseline 300df 142df 1.81df 37/255 29Mesh A 375df 142df 1.81df 37/255 28Mesh B 300df 177df 1.81df 37/255 27Mesh C 375df 177df 1.81df 37/255 26Mesh D 450df 213df 1.81df 37/255 29

properties are degraded with increase in temperature. Furthermore,due to the mismatch in the coefcients of thermal expansion, thenature of the ber/matrix interface is also changed. In addition,the matrix-rich region between plies that control the interlaminarstrength is also affected.

As in the room temperature tests, failure initiation is sensed bya suddenincrease in the through-the-thickness (out-of-planemode )strains,which allows the unloadingof a specimenfor postfailuremi-croscopicexaminationofregionsneartheholeedge.Cross-plylami-natesshow extensivekinkformation justas in the room-temperaturecase. Compared to the room-temperature case, failure occurs at alower load (see Table 3). Again, the kink banding initiates at the

hole edge and propagates in a direction perpendicularto the direc-tion of loading (Fig. 5). Noticeably, the out-of-plane failure is veryextensive. This observation may point to the reduced interlaminarstrength at elevated temperature,for the reasons discussed earlier.

Quasi-isotropic laminates (Table 5) show a combination of dif-ferent failure modes; however, ber/matrix shear failure is still thedominant failure mechanism for damage propagation. A band ofdamage consistently propagates along the C45=45 line from the0-deg hole edge, in the form of interfacial ber/matrix cracking,indicating that this is the form of energy release at failure.

Modeling of Compressive Response of MultiplyMultidirectional Laminates

As observedin the experiments,when a multidirectionallaminatecontaininga circular hole is subjected to remote compressiveloads,failure in the form of a microstructuralins tability is initiated in thevicinityof thehole,which is thearea of higheststressconcentration.For cross-ply laminates,the failure is triggered by a 0-degply insta-bility, characterizedby ber kinking. For the case of quasi-isotropiclaminates,which consist of 0- and 45-degplies, two typesof failuremechanisms compete against each other. These are failure initia-tion due to ber instability in the zero plies, and failure initiated bymatrixshearresponse,responsibleforinterface(ber/matrix) crack-ing and/or matrix shear failure. The dominant failure mode for thistype of laminates has been reported as zero-ply kinkingfailure.10;23

However,itisnotclearwhetherpostmorteminvestigationorsurfaceimaging of a particular laminate system (ber, matrix, and layup)

can pinpointthe general failure initiationmechanism.Based on this,it was decided to investigate both possibilities of ber kinking andmatrix shear failure. Furthermore, it is hypothesized that the in-plane kinking failure mode is the dominant mode of failure for thesymmetric quasi-isotropiclaminates.

The cross-ply laminates (Fig. 7) were studied rst, followed bythe quasi-isotropiclaminates(Fig.8). Unlike the cross-plylaminates(0- and 90-deg plies), the quasi-isotropiclaminates have three kindsof lamina (0-, 450- and 90-deg plies). Therefore, it is necessaryto separate the two families of lamina in the region of study (the0-=90-deg lamina and the 45-deg lamina).

The failure initiationanalysisbased on micromechanicsis carriedout in a manner similar to that described by Ahn and Waas.23 Theregion of failure near the cutout and correspondingto the zero plies

are isolated and analyzedusing the nite element method. The sizeof this region is left as an unknown of the problem. A convergedsolutionis obtainedwhen the salient featuresof the micromechanicsbased analysis shows no difference with respect to the size of themeshed area.

For the FEA analysis, a rectangular mesh containing 22,650 el-ements (four-node quadrilateral plane strain) and 23,028 nodes,

Fig. 7 Cross-ply laminate microregion for FEA.

with two degrees of freedom (u and v) per node was chosen.The ABAQUS commercial solver was used. The initial mesh size(termed baseline model) correspondsto a rectangular region that is300 ber diameters df in length and 142 ber diameters in width.The elements were of length 2df , and width 0.5df , such that twolayers of elements constitute a single ber. The matrix was also

modeled with two adjacent elements,albeit the two matrix elementsdiffered in width from the two ber elements, taking into accountthe difference in thickness of the matrix such that the matrix vol-ume fraction Vm corresponds to the specimen (55%). Convergenceand mesh size dependency were checked by incrementallyincreas-ing the mesh size until no substantialchange in the salient features(and values) associated with the response resulted. The results ofthe mesh sensitivity study are shown in Fig. 9 and Table 5, for aconguration of the current analysis [hole diam 0.5 in. (12.7 mm)].As can be seen in Table 5, the maximum limit load (our analysisuses this value as the failure initiation point) is insensitive to meshsize, and the plateau load also shows insignicant change as meshsize varies. Based on this mesh sensitivitystudy, the analysismodelwas deemed to have satisfactoryconvergencefor the present study.

Each microregionis situatedas shown in Figs. 7 and 8, where thesymmetry plane (y D 0) is marked as indicated. The Lekhnitskii11

solution corresponding to a notched laminate of innite extent isused to computethe displacementelds alongthe edgescorrespond-ing to the microregion. These displacement elds are computedcorrespondingto a unit far-eld load in the case of uniaxial loadingand unit far-eld proportional loads in the case of biaxial loading.

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Fig. 8 Quasi-isotropic laminate microregion for FEA.

Fig. 9 Effect of mesh size on load-shortening behaviorof microregion.

Along the bottom edge of the microregion (marked as AD), equi-librium nodal equivalent loads are enforced. That is, the edge ADis under load control, and the remaining edges are under displace-mentcontrol,so that bers candeformaccordingto the requirementof far-eld equilibrium. It turns out that, if an analysis is conductedwith theedgeAD freeof tractions,the results obtainedfor the salientfeatures associated with the deformations within this microregiondo not change. This is not surprising because the dominant loadingis compressionalong the bers,whereas the tractionsalongthe bot-tom edge (edge AD) of the microregion are negligiblecompared tothese dominant compressive stresses. Nevertheless,the objective isto obtain the limit load that the microregion can sustain, as well asthe correspondingfar-eld stresses.

A owchart of the analysis procedure is given in Fig. 10a. First,an elastic eigenvalue analysis of the microregion is carried out toobtain the eigenmodes of the microregion. Next, by the use of theeigenmodeassociated with the smallestnonzeroeigenvalueas a per-turbationto an otherwiseperfectmesh,a responseanalysisis carriedout usingthe Riksmethodoptionprovidedin ABAQUS. Thisis donefor a series of imperfection magnitudes. The eigenmodes provide

Table 6 Comparison of experiment and analysisfor cross-ply laminate

Un iaxial , Un iaxial, Biax ial, Bi ax ial,ksi/MPa ksi/MPa ksi/MPa ksi/MPa

Case (21C) (200C) (21C) (200C)

Analysis 62/427 30/207 67/462 35/241 0:8 deg 40/276 21/145 46/317 25/172Experiment 44/303 19/131 37/255 18/124

Table 7 Comparison of experiment and analysis

for quasi-isotropic laminate

Un iaxial , Un iaxial, Biax ial, Bi ax ial,ksi/MPa ksi/MPa ksi/MPa ksi/MPa

Case (21C) (200C) (21C) (200C)

Analysis 38/262 22/152 44/303 24/165 0:8 deg 27/186 15/103 34/234 18/124Experiment 21/145 19/131 26/179 24/165

the perturbationshape, but not the absolute magnitude of perturba-tion. Thus, the user must specify the imperfectionmagnitude.In thepresentwork, this is achieved as follows: As shown in Fig. 10b, the

maximum amplitude of the lowest eigenmode (which occurs alongthe edge AD) is chosen such that the ber misalignment angle isapproximately in the range 0.05 1 deg. Because is known, ischosen such that assumes the intended value. After several runscorrespondingto differentvalues of are completed,the load max-ima associated with the resultant compressive load on edge AB ofthe microregionor reaction force (RF1) vs the load proportionalityfactor (LPF) (fraction of the total applied load) curves are plottedas a function of imperfection magnitude. Then, by extrapolation,the value correspondingto the perfectcase (no imperfection) is ob-tained. The perfectcase correspondsto perfectly straight bers,andthis case yields an upper bound for the attainable load maximum.The perfect case was also analyzedseparately (with straightbers)as a nonlinear response problem, and it yielded the same result for

the maximum load and plateau load as that obtained by extrapo-lation. As shown by Ahn and Waas,23 the present problem is onewhere the loading on the microregionboundariesis nonunifrom.Asa result,this loadnonuniformityessentiallyproducesthe sameeffectas that of an initial geometrical imperfection of the bers (such asber misalignment). Therefore,evenin theabsenceof an initial bermisalignment, the perfect microregion response analysis exhibits alimit load type behavior.

The ber (IM7) is assumedto be linearlyelastic (seeTable1),andthe matrix (3270 toughened epoxy) property within the laminate isevaluatedfrom the C45=45 coupon test, which will be describedin the next section. The matrix is modeled as a J2 incremental owtheorysolidwith isotropichardening.Thus,the in situelastic-plasticmatrix behavioris incorporatedin the presentanalysis.

The cases studied herein correspond to those from the experi-ments reported earlier. Thus, for each laminate, two different runsare performed,one at room temperature andone at elevatedtemper-ature. For each temperature, uniaxial and biaxial loading cases areinvestigated. In this manner, we have studied four different casesfor each laminate type. Of course, in the quasi-isotropic case, thereare four additional cases because both the zero ply and the 45-degply are investigated. Tables 6 and 7 show results corresponding tothe preceding cases. The results obtained for different imperfectionmagnitudes are also shown in Tables 6 and 7.

In Situ Matrix Characterization

To evaluate the material properties of the matrix within the lam-inate (in situ matrix properties), a .45/ns coupon test (Fig. 11)

at room and high temperature was performed following the pro-cedure of American Society for Testing and Materials ASTM-D3518-76 (Ref. 24). The elastic properties of the lamina and thecomplete shear-stress/shear-strain behavior of an IM7/3270 in theprincipal material coordinate system were obtained from this test.The procedure used to generate the data consists of subjecting a.45/ns angle ply laminate to uniaxial compression and measuring

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Fig. 10a Imperfection sensitivity analysis procedure.

Fig. 10b Imperfection magnitude denition based on (maximumdeection angle).

Fig. 11 The +45/ 45 coupon test of a specimen taken from the cross-ply laminate.

the laminate strains "x x and "yy and the applied remote stress on thelaminate. Note that, for this test, x y D "x x "yy . The data shown inFig. 12 from this test can be used to extract the complete nonlinearshear-stress/shear-strain response of the in situ matrix (3270) asdiscussed hereafter. We begin by assuming that the 3270 matrixmaterialcan bemodeledas an elasticplasticsolidobeyingthesmallstrain J2 ow theory of plasticity.25 Then, from the elastic (linear)portion of the curve in Fig. 12, we rst obtain the in-plane lamina

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Fig. 12 The +45/ 45 compression experiment result.

Fig. 13 Shear stress 12 vs shear strain 12 curve for IM7/3270 (roomtemperature) basedonJ2 ow theoryof plasticity andthedatainFig.12.

Fig. 14 In situ shear stress vs shear strain response of the matrix andthe deduced uniaxial stress vs uniaxial strain response of the matrix(room temperature).

shear modulus G12. In the lamina principal coordinates, we knowthat

"11 D "22 D ."x x C "yy /=2; 12 D 2"12 D "x x "yy (1)

11 D x x =2; 22 D x x =2; 12 D xx =2 (2)

Thus,usingEqs. (1) and (2) andthedenitionsofequivalentstressN and equivalentplastic strain increment dN"p (in the Appendix), thedata in Fig. 12 can be used to construct a plot of N against N"p .

Accordingto the J2 ow theoryof plasticitywith a MisesHenkyyield condition, the ratio of the increment of each plastic straincomponentto its correspondingdeviatoricstress componentremainsconstant, which is (see Ref. 25)

d"p

i j

Oi j D d (3)

Using Eq. (3), and the relation between N and dN"p , the nonlin-ear portion of the shear-stress 12/shear-st rain 12 response curvefor a single lamina can be extracted. The curve thus obtained forIM7/977-3 system is shown in Fig. 13. The instantaneous slope ofthe curve in Fig. 13 is the tangent shear modulus of the laminaG12.12/. Therefore, we next use the approximateHalpinTsai rela-tions (see Ref. 26) to extract the variation of the in situ matrix shearmodulus Gm .12/ by using Gm .12/,

Gm D G 12

1 C 2 2 f

1 2 f

1; 2 D

G 12 f Gm

G 12 f C 2 Gm(4)

and 2 D 1 forrandom packing of bers. With Gm .12/ so obtained,the in situ matrix shear-stress/shear-strain curve is as shown inFig. 14. In a similar manner, the in situ uniaxialstress/strain curvein

Fig. 15 In situ shear stress vs shear strain response of the matrix and

the deduced uniaxial stress vs uniaxial strain response of the matrix(high temperature).

Fig. 16 Typical microregion response plot of cross-ply laminate.

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354 AHN AND WAAS

Fig. 17 Deformed microregion shape (equivalent plastic strain contour plot).

compressionfor the matrix is also obtained. This uniaxialresponsecurve was used in the input data le for the micromechanics basedFEA. This curve is also shown in Fig. 14. The same procedure is

applied to the high temperature case, and the results obtained forthe matrix stress/strain curves are shown in Fig. 1 5.

FEA Results and Interpretation

A typical load response behaviorof a microregion within a cross-ply laminate model is shown in Fig. 16, and a series of deformedplots of the microregion showing the initiation and propagation of

damageintheformofkinkbandingisshowninFig.17.Thenumbersindicated in Fig. 16 correspond to the series of deformation plotsshown in Fig. 17. The microregion response follows a linearpath up

to point. Althoughlocal matrix yielding (in areas of the microre-gion near the cutout) is indicated before the attainment of point,the total integrityof this regionis not affected much fromthe matrixyielding because 1) the area of yielding is small compared to theoverall size of the microregion and 2) the ber rotation is small upto the point of maximum load (point). Thus, the resultantRF1 vsLPF relation is linear up to the maximum load point, although this

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AHN AND WAAS 355

Fig. 17 Deformed microregion shape (equivalent plastic strain contour plot) (continued).

is notnecessarilywhat could be expected;thatis, in generalthe RF1vs LPF behavior is problem dependent. As loading increases, thebers in the areas where the matrixhas becomesofter start to rotate,resulting in a drop of the resultant RF1 computed at the symmetry

plane (indicated by point in Fig. 11). As deformation progresses,the region of matrix yielding (and corresponding ber rotation) isexpanding,propagatingupward away from the areas near the cutout(until it reaches point ; see Fig. 11 for the deformed congura-tion). Once the region expands away from the hole to an area wherethe effect of the stress gradient has diminished,there is no tendencyfor further propagation,and,thus, the load assumes a fairly constant

value (plateau load). After this point, deformation becomes stabi-lized (ber rotation stops), and the band of kinked bers starts tobroaden in the x direction without any further sudden drop in thereactionforce(). Thescenariojust describedis quite typicalof the

responseof the microregionsfor all of the casescorrespondingto thecross-ply laminate as well as to the zero-ply microregion within aquasi-isotropiclaminate. The far-eld load (or loads correspondingto biaxial tests) correspondingto point is read off and tabulatedin Table 1 for all of the cross-ply laminate cases studied herein.

For the case of quasi-isotropic laminates (Fig. 2), the responseslook differentfor eachlamina (0plyand45ply). Again,the zero-ply

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356 AHN AND WAAS

Fig. 17 Deformed microregion shape (equivalent plastic strain contour plot) (continued).

a)

b)

Fig. 18 Microregion response result for a quasi-isotropic laminate,

45-deg ply. f The equivalent plastic strain is sampled at the point withcoordinate [0.021 in. (0.533 mm), 0.271 in. (6.88 mm)].g

response is similar to that of the cross-ply model, but the far-eldload at which kink banding initiates is different. The 45-deg ply ismodeled in a manner that is similar to the zero-ply microregion,

but the microregion boundaries are taken to be along the materialprincipaldirections (in this case the 45- and 135-deg directions,seeFig. 8). Again, displacement boundary conditions are enforced onthe edges AB, BC, and CD, whereas edge AD is subject to equiva-lent equilibriumnodal loads. As loading proceeds, the RF1 vs LPFrelation is very nearly linear (Fig. 18a) and shows no sign of lead-ing to a load maximum preceding a sudden load drop. At the sametime, the correspondingplots for the equivalent plastic strain "p vsLPF [Fig. 18b, sampled at the location with coordinates (0.025 in.,0.276 in.)] shows that the matrix elements contained between anytwo bers along the 45-deg direction and close to the cutout un-dergo increasing amounts of shearing. As loading proceeds, moreandmore of thematrix elementsundergo large amountsof shearing.Simultaneously, the RF1 vs LPF plot for the zero-ply microregionapproaches a maximum load. Thus, the scenario is as follows: thezero-ply reaches a maximum load, at which stage kink bandingis about to initiate in this ply. At the same time, the integrity ofthe 45-deg ply is compromised due to the large amounts of plasticstraining. Thus, the laminate has a choice in selecting the failurepath that corresponds to the largest release of energy. As observedexperimentally, a ber/matrix interfacial crack is seen to dominatethe failure within the 45-deg ply but not without some incident kinkbanding in the zero plies. Therefore, for quasi-isotropic laminates,the maximum loadcorrespondsto theinitiationof kink bandingthatthen triggers the mode of ber/matrix interfacial failure that is ob-served experimentally in the 45-deg plies. Clearly, we need to havean accurate knowledgeand measurementof the critical ber/matrixinterfacial fracture toughness (or the in situ matrix fracture tough-ness becausethe crack meanders between the ber matrix interfaceand along the matrix) to carry out a fracture mechanics based en-ergy release rate analysis of the 45-deg ply. Such an analysis willdetermine the mode of failure for which the largest amount of en-ergy is released. A fracture mechanics based analysis is relegatedto the future, but details of such an analysis in the context of double

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AHN AND WAAS 357

cantilever beam specimens is reported by Song and Waas.27 Fornow, we observethe goodagreementbetween the far-eld load cor-respondingto the maximum load predictionand the experimentallymeasured failure loads as indicated in Table 3, especially for thecase of small imperfection ( D 0:8 deg). This lends condence toour contention that zero-ply kinking is the dominant and, hence,governing mode of failure initiation in notched quasi-isotropiclaminates.

Conclusions

Experimental results of a series of uniaxial and equibiaxialin-plane compressive loading experiments on notched compositelaminated plates under room temperature and elevated tempera-ture conditions were presented in this paper. Two types of 48-plygraphite/epoxy composites were studied to understand their com-pressive failure mechanisms. The laminate stacking sequence wasselectively designed to isolate the different failure mechanisms thatoperatein angle plies and plies oriented in the loading direction. Inthe specimens containingplies alongthe direction of the load (crossply), uniaxial compression leads to ber kink banding. This samescenario persists at both room and elevated temperatures.

The failure of the quasi-isotropic laminates involves a combina-tion of kink banding and ber/matrix interfacialfailure. The magni-

tude of remote failure initiation loads between the loading cases

N D

q12

.11 22/

2 C .22 33/2 C .33 11/

2C 3

212 C

223 C

231

dN"p D

rn29

hd"

p

11 d"

p

22

2C

d"p

22 d"

p

33

2C

d"p

33 d"

p

11

2iC 4

3

hd"

p2

12C d"

p2

23C d"

p2

31

io

(uniaxial or biaxial) at a given temperature is unlike that of cross-ply laminates. The interfacial failure persists either at C or

angle or both. Fibers crossing the line of failure were broken due tothe large amount of energy released as the interfacial crack propa-gates rapidly. At an e levated temperature, the same failure mecha-nismspersist,but at differentmagnitudesof far-eld loading.Underequibiaxialcompression,the remote load to initiatefailureat a giventemperature is higher than the uniaxial case, whereas for cross-plylaminatesthere is no appreciableincreasein loadbetweenthe uniax-ialand biaxialcases at a giventemperature.Furthermore,for a givenloading (uniaxial or biaxial), the failure initiationload decreases forcross-ply laminates, whereas there is not a noticeable decrease forthe quasi-isotropiclaminates.

From the observed results, and the loads reported in Tables 3 5,the following is concluded.In cross-ply laminates there is a notice-able decrease in the failure initiation loads between room and high

temperature, whereas such a reduction is hardly noticeable in thequasi-isotropic laminates. This observation shows that the mech-anism of failure initiation in cross-ply laminates (ber instabilityfollowed by kink banding) is strongly inuenced by the matrix be-cause the mechanical properties of the matrix have been degradedat high temperature.

On the other hand, the mechanism of failure initiation in quasi-isotropiclaminates is not affectedas much as cross-plylaminates bytemperature. It is seen that at a given temperature, biaxial loadingreduces the failure initiation load for cross-ply laminates, whereasfor quasi-isotropiclaminates this trend is reversed.

Results from a micromechanics-based global-local FEA wereused to understand the experimental results. Cross-ply laminateshave only 0- and 90-deg lamina, and the majority of load is car-

ried by simple compressionof these laminae. These laminas fail byber microbuckling(on accountof a graduallyweakening matrixinshear), which in turn progressesto kink band formation due to berrotation, followed by band broadening and the triggeringof failurein other plies.

For the quasi-isotropic laminate, the majority of the load is stillcarried by the zero-plylamina.However, the differencebetween this

laminate and the cross-plylaminate is the angled ply (45-deg ply inthiscase). This 45-deg ply acts as a buffer zone (lower stiffness,butbetter shear response). Based on the micromechanics-based FEAresults, it is possible to postulate that although the matrix layers be-tween any bers andsituatednear thezone of large stress andstraingradient are undergoingexcessiveplastic deformation, the integrity(catastrophic failure) of the laminate is still controlled by the stateof stress and deformation of the zero plies. As soon as the zero plyreachesan instability limit, two possibilitiesare encountered.Eitherenergy is released by the spreading of the kink band within the zeroplies or throughinterfacialber/matrix failure within the now com-promised 45-deg plies. The winner of this competition is governedby the ber/matrix interfacial fracture toughness and/or the in situmatrix fracture toughness compared against the zero-ply kink bandtoughness (dened as the energy released per unit advancement ofthe kinkedband). It is probablethat kink bandingtriggersinterfacialfailure becauseuntil the zero ply undergoesa loaddrop,the coexist-ing 45-deg ply has no possibilityof initiating the ber/matrix inter-facialfailure.Thus, the failure loadpredictionbased on the zero-plymaximum loads provides a good agreementwith the correspondingexperimental data, in both cross-ply and quasi-isotropic laminates.

Appendix: Effective Stress and Effective

Plastic Strain Increment

Acknowledgments

The authors are grateful for nancial supportfrom the Air ForceOfce of Scientic Research for the initial phases of this project.The supply of the laminates at reduced cost from Fiberite (courtesyof Anthony Bosch) is acknowledged with thanks.

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A. N. PalazottoAssociate Editor