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A01-25337 42nd AIAA/ASME/ASCE/AHS/ASC Structures,Structural Dynamics,an d Materials C onferenceand E xhibitSeattle, WA 16-19 April 2001 AIAA-2001-1640

A Mi c r ome c ha n i c s B a s e d F i n i te E l e me n t Mod e l for the Prediction ofCompressive Fai lure in Notched Laminated Composi tesJunghyun Ahn1 and Anthony M Waas2Composite Structures LaboratoryDepartment of Aerospace Engineering,University o f Michigan, Ann Arbor, Ml 48109-:

Abst ractA micromechanics based analysis to predictdamage initiation in compressively loadedsymmetric notched multiply laminates at efevatedtemperature under remote biaxial loading isdescribed. The finite element method inconjunction with the commercial code ABAQUSis used to solve the governing system ofequations. The results obtained for the predictionsare compared against a set of experimentalresults. A unified model that captures damageinitiation and which describes failure mode bytracking the response of each lamina is presented.The prediction of the model is found to comparefavorably against the experimental data.

1. IntroductionLaminated fiber reinforced polymer matrix

composites (PMC's) are finding increased use in abroad range of industrial applications. Applicationsin the national infrastructure (such as all-composite bridges over interstate highways), theautomotive industry, medical, and the aerospaceindustry are some examples of where thesematerials are earmarked for several uses.Composite materials are superior toconventional monolithic materials for a number ofreasons, noteworthy amongst these being theirmechanical properties. Ashby and Jones [(1994)]1have formulated performance indices that can beused to select a material for a given

application based on satisfying certain objectivemeasures. PMG's appear to be. the material ofchoice for a number of such applications. Issuessuch as damage tolerance and durability,performance degradation duejo ageing, fatigueunder multiaxial loads aact response at elevatedtemperature are currently being researched with aview to expanding the confidence levelsassociated with PMC applications.

Experience with previous applications ofPMC's for rotor blades in the helicopter industry,pressure vessels, and other similar situations thatcall for superior tensile stiffness and tensile fatiguelife have shown the superior performance ofPMC's in a tensile environment. In contrast^compressive strength of PMC's is known to beless attractive [Mathews and Rawlings, (1994)]2,[Waas and Schultheisz, (1995)f.The response of composite laminates whensubjected to mechanical loads is influenced by thematerial type (fiber and matrix) and configuration(stacking sequence, lay-up). In addition to thesefactors, geometrical parameters (cut-out, notch,thickness change, etc.) and loading characteristics(multiaxial, thermal, cyclic loading, etc) also affectthe overall performance of composite laminates.One of the challenging tasks in the analysis anddesign with PMC based structural laminates is toimprove the predictive capability of the mechanicalresponse of a composite structure that contains a

discontinuity in the form of a cutout [Starnes et al.,1 Research Fellow, Aerospace Engineering, University o f Michigan, Ann Arbor, member, AIAA.Copyright 2001 by Junghyun Ahn, published by the Am erican Institute of Aeronautics and Astronautics,Inc; with permission2 Professor of Aerospace Engineering, Associate Fellow, AIAA.

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(1979)]4 or a sudden thickness discontinuity suchas the termination of reinforcement, under variousdesign environments (loading, temperature,moisture, etc.).Of concern in the present paper is thedevelopment of a micromechanics based model topredict the onset of failure in laminated compositeplates containing a stress raiser in the form of acircular hole under multiaxial loading at room andelevated temperature. Both material degradation(due to temperature) and stress concentrationeffects were included in the analysis. Thedevelopment of the model relies on previousexperimental work reported by the authors [Ahn

and Waas, (2001 )]5. The predictions of the modelare found to be in good agreement with the resultsreported in [Ahn andWaas, (2001 )]5.2. Modeling of C o m p r e s s i v e Response of aMultiply Multidirectional Laminate

As shown in a companion paper [Ahn andWaas, (2001 )]5, when a multidirectional laminatecontaining a circular hole is subjected to remotecompressive loads, failure in the form of amicrostructural instability is initiated in the vicinityof the hole, which is the area of highest stressconcentration. For cross-ply laminates, the failureis triggered by 0-degree ply instability,characterized by fiber kinking. For the case ofquasi-isotropic laminates, which consist of zeroand 45-degree plies, two types of failuremechanisms compete against each other. Theseare failure initiation due to fiber instability in thezero plies and failure initiated by matrix shearresponse, responsible for interface (fiber/matrix)cracking and/or shear failure in the 45 degreeplies.

The dominant failure mode for this type oflaminates has been reported as zero ply kinkingfailure [Ahn and Waas (1999)]6. However, it is notclear whether post mortem investigation or surfaceimaging of a particular laminate system (fiber,matrix, lay-up) can pinpoint the general failureinitiation mechanism. Based on these facts, it wasdecided to investigate both possibilities of fiberkinking and matrix shear failure. Furthermore, it ishypothesized that the in-plane failure mode is thedominant mode of failure for the symmetric quasi-isotropic laminates. These assumptions arediscussed in more detail in [Ahn and Waas,2000)]5.

The cross-ply laminates (Figure 1) werestudied first, followed by the quasi-isotropiclaminates (Figure 2). Unlike the cross-ply

laminates (0 and 90 plies), the quasi-isotropiclaminates have three kinds of lamina (0 45, and90 plies). Therefore, it is necessary to separatethe two families of lamina in the region of study(the 090 lamina and the 45 lamina).The failure initiation analysis based onmicromechanics is carried out in a manner similarto that described in [Ahn and Waas, (1999)]6. Theregion of failure near the cutout and correspondingto the zero plies are isolated and analyzed usingthe finite element method. The size of this regionis left as an unknown of the problem. A convergedsolution is obtained when the salient features ofthe micromechanics based analysis shows nodifference with respect to the size of the areameshed.For the FEA analysis, a rectangular mesh

containing 22,650 elements (4 node quadrilateralplane strain) and 23,028 nodes, with two degreesof freedom (u and v) per node was chosen. TheABAQUS commercial solver was used. Theinitial mesh size (termed "baseline model")corresponds to a rectangular region that is 300fiber diameters (d f) in length and 142 fiberdiameters in width. The elements were of length 2df, and width 0.5 df, such that two layers ofelements constitute a single fiber. The matrix wasalso modeled with two adjacent elements, albeitthe two matrix elements differed in width from thetwo fiber elements, taking into account thedifference in thickness of the matrix such that thefiber volume fraction (V f), corresponds to thespecimen (55 %). Convergence and mesh sizedependency was checked by incrementallyincreasing the mesh size until no substantialchange in the salient features (and values)associated with the response resulted. The resultsof the mesh sensitivity study are shown in Figure 4and Table 4 for a configuration of the currentanalysis (hole diameter 0.5 in.). As can be seen inTable 4, the maximum limit load (our analysis usesthis value as the failure initiation point) isinsensitive to mesh size, and the plateau load alsoshows insignificant change as mesh size varies.Based on this mesh sensitivity study, the analysismodel was deemed to have satisfactoryconvergence for the present study.

Each microregion is situated as shown shadedin Figure 1 and Figure 2, where the symmetryplane (y = 0) is marked as indicated. TheLekhnitskii's orthotropic, elasticity [Lekhnitskii,(1968)]7 solution is used to compute thedisplacement fields along the edgescorresponding to the microregion. These

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displacement fields are computed correspondingto a unit far-field load in the case of uniaxialloading and unit far-field proportional loads in thecase of biaxial loading. Along the bottom edge ofthe microregion (marked as AD), equilibrium nodalequivalent loads are enforced. That is, the edgeAD is under load control, and the remaining edgesare under displacement control, so that fibers candeform according to the requirement of far-fieldequilibrium. It turns out that, if an analysis isconducted with the edge AD free of tractions, theresults obtained for the salient features associatedwith the deformations within this microregion donot change. This is not surprising since thedominant loading is compression along the fibers,whereas the tractions along the bottom edge(edge AD) of the microregion are negligiblecompared to these dominant compressivestresses. Nevertheless, the objective is to obtainthe limit load that the microregion can sustain andthe corresponding far field stresses.

A flow chart of the analysis procedure isindicated in Figure 3(a). First, an elasticeigenvalue analysis of the microregion is carriedout in order to obtain the eigenmodes of themicroregion. Next, using the eigenmodeassociated with the smallest non-zero eigenvalueas a perturbation to an otherwise perfect mesh, aresponse analysis is carried out using the Riksmethod option provided in ABAQUS This is donefor a series of imperfection magnitudes. Theeigenmodes provide the perturbation shape butnot the absolute magnitude of perturbation. Thus,the imperfection magnitude must be specified bythe user. In the present work, this is achieved asfollows; as shown in Figure 3(b), the maximumamplitude of the lowest eigenmode (which occursalong the edge AD) is chosen such that the fibermisalignment angle < |> is approximately in the range0.05-1. Since A, is known, 8 is chosen such that(| ) assumes the intended value. After several runscorresponding to different values of arecompleted, the load maxima associated with theRF1 vs. LPF curves are plotted as a function ofimperfection magnitude. Here, RF1 stands for theresultant compressive load of the microregion andLPF stands for load proportionality factor, whichindicates the amount of generalized load appliedto the microregion expressed as a fraction of thetotal load. Then, by extrapolation of the maximumload, the value corresponding to the perfect case(no imperfection) is obtained. The perfect casecorresponds to perfectly straight fibers and thiscase yields an upper bound for the attainable loadmaximum.

The fiber (IM7) is assumed to be linearlyelastic (see Table 1), and the matrix (3270toughened epoxy) property within the laminate isevaluated from the +457-45 coupon test, which willbe described in the next section. Thus, the in-situelastic plastic matrix behavior is incorporated inthe present analysis. The matrix is modeled as aJ2 incremental flow theory solid with isotropichardening.

The cases studied in this paper correspond tothose from the experiments as reported in [Ahnand Waas (2001)] . Thus, for each laminate, twodifferent runs are performed, one at roomtemperature and one at elevated temperature. Foreach temperature, uniaxial and biaxial loadingcases are investigated. In this manner, we havestudied four different cases for each laminate type.Of course, in the quasi-isotropic case, there arefour additional cases, since both the zero ply andthe 45-degree ply are investigated. Tables ofresults corresponding to the above cases are asshown in Table 2 and Table 3. The resultsobtained for different imperfection magnitudes arealso shown in these tables.3. In-Situ Matrix Characterization

To evaluate the material properties of thematrix within the laminate (in-situ matrixproperties), a (45)M coupon test (Figure 5) atroom and high temperature was performedfollowing the procedure of ASTM-D3518-76[ASTM Standard, (1982)]8. The elastic propertiesof the lamina and the complete shear stress-shearstrain behavior of an IM7/3270 in the principalmaterial coordinate system were obtained fromthis test. The procedure used to generate the dataconsists of subjecting a (45)m angle ply laminateto uniaxial compression and measuring thelaminate strains ^and e^ and the applied remotestress on the laminate. Note that, for this test,Y x y = 6XX ~ syy - The data shown in Figure 6 from thistest can be used to extract the complete nonlinearshear stress - shear strain response of the in-situmatrix (3270) as discussed below. We begin byassuming that the 3270 matrix material can bemodeled as an elastic-plastic solid obeying thesmall strain J2 flow theory of plasticity [Lubliner,(1998)]9. Then, from the elastic (linear) portion ofthe curve in Figure 6, we first obtain the inplanelamina shear modulus G12. In the lamina principalcoordinates, we know that,


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(1)Y\2 ~

And

(2)

. 2Thus, using (1) and (2) above and thedefinitions of equivalent stress, a and equivalent

plastic strain increment, dep [Lubliner, (1998)]9,the data in Figure 7 can be used to construct aplot of a against sp.

According to the J2 flow theory of plasticitywith a Mises-Henky yield condition, the ratio of theincrement of each plastic strain component to itscorresponding deviatoric stress componentremains constant, which is [Lubliner, (1998)]?,(3)

Using (3), and the relation between

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For the case of quasi-isotropic laminates(Figure 2), the responses look different for eachlamina (zero ply and 45 ply). Again, the zero plyresponse is similar to that of the cross-ply modelbut the far field load at which kink banding initiatesis different The 45-degree ply is modeled in amanner that is similar to the zero ply microregion,but the microregion boundaries are taken to bealong the material principal directions (In this casethe 45 degree and 135 degree directions-seeFigure 2). Again, displacement boundaryconditions are enforced on the edges AB, BC andCD, while edge AD is subject to equivalentequilibrium nodal loads. As loading proceeds, theRF1 vs. LPF relation is very nearly linear (Figure13 (a)) and shows no sign of leading to a loadmaximum preceding a sudden load drop. At thesame time, the corresponding plots for theequivalent plastic strain sp, vs. LPF (Figure 13 (b),sampled at the location with coordinates (0.025in., 0.276 in.)) shows that the matrix elementscontained between two fibers along the 45-degreedirection and close to the cutout undergoincreasing amounts of shearing. As loadingproceeds, more and more of the matrix elementsundergo large amounts of shearing.Simultaneously, the RF1 vs. LPF plot for the zeroply microregion approaches a maximum load.Thus, the scenario is as follows; the zero plyreaches a maximum load, at which stage kinkbanding is about to initiate in this ply. At the sametime, the integrity of the 45-degree ply iscompromised due to the large amounts of plasticstraining. Thus, the laminate has a "choice" inselecting the failure path that corresponds to thelargest release of energy. As observedexperimentally, a fiber/matrix interfacial crack isseen to dominate the failure within the 45 degreeply, but not without some incident kink banding inthe zero plies. Thus, in these cases, the maximumload corresponds to the initiation of kink bandingwhich then triggers the mode of fiber/matrixinterfacial failure that is observed experimentally inthe 45-degree plies. Clearly, we need to have anaccurate knowledge and measurement of thecritical fiber/matrix interfacial fracture toughness(or the in-situ matrix fracture toughness, since thecrack meanders between the fiber matrix interfaceand along the matrix) to carry out a fracturemechanics based energy release rate analysis ofthe 45 degree ply to ensure that it is indeed themode of failure for which the largest amount ofenergy is released.

An analysis that includes strain energy releaserate computation is relegated to the future, but

details of such an analysis in the context of doublecantilever beam specimens is reported in Songand Waas [Song and Waas (1995)]11. For now, weobserve the good agreement between the far-fieldload corresponding to the maximum loadprediction and the experimentally measured failureloads as indicated in Table 2, especially for thecase of small imperfection ( c j ) = 0.8). This lendsconfidence to our contention that zero ply kinkingis the dominant and hence governing mode offailure initiation in notched quasi-isotropiclaminates. Such a conclusion, based only onexperimental results, has been reached earlier byseveral previous investigators-see for example,Soutis, Fleck and Smith [(1991)]12 and Khamsehand Waas [(1997)]13.

5. Concluding RemarksThe results of the present micromechanicsbased global-local finite element analysis showsthat the fundamental failure mechanisms of asymmetric multidirectional laminate is kink bandformation due to a structural instability of thefibers, especially for laminates containing pliesaligned along the remote load directions (cross-plyand quasi-isotropic laminates). The cross-plylaminates have only 0 and 90-degree lamina, andthe majority of load is carried by simplecompression of these laminae. These laminae failby fiber microbuckling, (on account of a graduallyweakening matrix in shear), which in turnprogresses to kink band formation due to fiberrotation, followed by band broadening and thetriggering of failure in other plies.For the quasi-isotropic laminate, the majorityof the load is still carried by the zero ply lamina.However, the difference between this laminate andthe cross ply laminate is the angled ply (45 ply inthis case). This 45 ply acts as a "buffer zone"(lower stiffness, but better shear response). Basedon the analysis result, it is possible to postulatethat although the matrix layers between any two

fibers and situated near the zone of high stressand strain gradient are undergoing excessiveplastic deformation (implies the possibility ofactivating small cracks at the fiber/matrix interfaceor within the matrix) the integrity of the laminate isstill controlled by the zero-plies. As soon as thezero ply reaches an instability limit, twopossibilities are encountered. Either energy isreleased by the spreading of the kink band withinthe zero plies or through interfacial fiber/matrixfailure within the now compromised 45-degreeplies. The "winner" of this competition is governed


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by the fiber/matrix interfacial fracture toughnessand/or the in-situ matrix fracture toughnesscompared against the zero ply kink bandtoughness (defined as the energy released perunit advancement of the kinked band). Clearly, inthis case, the kink banding triggers the failuresince until the zero ply undergoes a load drop, theco-existing 45-degree ply has no possibility ofinitiating the fiber/matrix interfacial failure. Thus,the failure load prediction based on the zero plymaximum loads provides a good agreement withthe corresponding experimental data.6. Acknowledgments

We are grateful for the partial support from theAFOSR and the University of Michigan for thiswork. This paper is based, in part on the Ph.D.thesis of the first author submitted and defended inthe Aerospace Engineering Department,University of Michigan, in December 1999.7. Reference s1. Ashby M.F. and Jones, D., "EngineeringMaterials 2", Pergamon, 19942. Mathews, F. L. and Rawlings, R., "CompositeMaterials: Engineering and Science",Chapman and Hall, London, 19943. Waas, A. M., and Schultheisz, C. R.,"Compressive failure in Composites, Part II",Progress in Aerospace Science, Vol.32, pp. 43

-78, 1995.4. Starnes, J., Rhodes, M.D., and Williams, J.G.,"Effect of Impact Damage and Holes on theCompressive Strength of a Graphite/EpoxyLaminate", Nondestruct ive Evaluation an dFlaw Criticality for Composite Materials, editedby R. B. Pipes, ASTM STP 696, pp. 145-171,1979.5. Junghyun, Ahn and A.M. Waas, "FailureMechanisms of a multidirectional laminates atroom and elevated temperature", to appear inAIAA Journal, 2001.6. Junghyun Ahn and A. M. Waas Finite ElementModel for Compressive Failure of NotchedUniply Composite Laminates under RemoteBiaxial Loads: ASME Journal of EngineeringMaterials and Technology: Vol.121 July, 1999.7. Lekhnitskii S., "Theory of Elasticity of anAnisotropic body", Text, Godden-Day, 1968.8. "D 3410-75 Standard Test Method forCompressive Properties of Unidirectional orCross-ply Fiber-Resin Composites", 1982,Annual Book of ASTM Standards, Part 36 ,

American Society for Testing and Materials,Philadelphia, PA, 1982, 872-880.9. Jacob Lubliner, "Plasticity Theory", Text,Prentice Hall, 1998.10. Isaac M. Daniel and Ori Ishai, "Engineering

Mechanics of Composi te Materials", OxfordUniversity Press, New York, 1994.11. S. J. Song and A. M. Waas, "An EnergyMethod based Model for Mixed Model Failureof Laminated Composites", AIAA Journal, vol.

33, No. 4, 739-745, 1995.12. Soutis, C., Fleck, N. and Smith, F., "FailurePrediction Technique for Compression LoadedCarbon Fiber-Epoxy Laminate with an OpenHole", J. Composite Materials, 25, pp1476-

1498, 199113. A. Khamseh and A. Waas, "FailureMechanisms of Composite Plates with aCircular Hole under remote Biaxial PlanarCompressive Loads", ASME J. Materials andTechnology, vol. 119, pp56-64, 1997


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S.Tables an d Figures

Material

IM7/977-3EpoxyIM7 Fiber

977-3Epoxy

En(Msi

23.5

420.7

E22(Msi)

1.21

(MsO V12

0.72 V0.250.34

Thickness (in)0.00522.756E-042.444E-04

Table 1Zero ply material propert ies of the 48p ly graphite / 977- 3 e p o x y composites

Analysisd > 0.8

Experiment

Uniaxial(ksi)(25C)624044

Uniaxiai(ksi)(200 C)302119

Biaxial(ksi)(25 C)

674637

Biaxial(ksi)(200C)352518

Table 2 Cross-ply an alysis results

Table 3 Quasi-lsotropic ply analysis results

Analysis4 ) *0.8Experim

ent

UniaxiaI (ksi)(25 C)

382721

Uniaxial(ksi)(200C)221519

Biaxial(ksi)(25C)443426

Biaxial(ksi)(200C)

L 241824

MeshUsed

Basel!neMeshAMeshBMeshCMeshD

DimensionsX length xY length

300 d, x142 df375 dfx142 df300 df x177 df375 df x177d,450 d, x213d,

Maximum

Imperfection Size1.81 df1.81 df1.81 d,1.81 d,1.81 d,

MaximumResultantstress onMicroregion (Ksi)

3737373737

PlateauStressonMicroregion2928272629

Table 4 Results of the m e s h c on ve r ge n c e withhole diameter 0.5 in.

Figure 1 Cross-Ply Laminate Model


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Figure 2 Quasi-lsotropic / Angle Ply LaminatesModel

\Maximum deflection 6

/ \

Response of MicroregionImpufcetton

Extrapolation to Perfect Case

Figure 3(a) Imperfection Sensitivity AnalysisProcedure

Figure 3(b) Im perfection definition based on 4 >(maximum deflection angle)

300

250 -

160 -

50 -

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9LP F

Figure 4 Ef fect o f mesh size on load-shorteningbehavior of microregion


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Strain Gages

Figure 5. +45/-4S c oupon test o f a specimen takenfrom th e cross ply laminates

30000

25000

2QOQO

J? 15000-

10000

5000

Room Temperature (25 C)

High Temperature ( 2 0 0 C)

0 .00 001002003OB4 005ODEFigure 6. +451-45 compression experiment result

30000

25000

20000

15000

10000

5000

0.000 OU05 QD10 OJ015 0.020 OJ025 0.030 O X J 3 5

Figure 7. Equivalent stress vs. equivalent plasticstrain (room temperature)

18000160001400012000

10000ft 8000-

6000 4000

2000

O U 2 OJ 0 8 0.10Strain

Figure 8Shear Stress, r12vs. Shear Strain, yucurve for IM 7 / 3270 (room temperature) based o n J2Flow Theory of Plasticity and the data in Figure 7.

9000

8000 -7000 -

6000

5000 4000

30002000 1 0 0 0 -

situ Shear

0 . 0 0 001 002 003 0.04O.OS006007Strain

Figure 9. The in-situShear Stress vs. ShearStrain response of the matrix and thededuced Uniaxial Stress vs. Uniaxial StrainResponse of the Matrix (Room Temperature).


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In-situ Compression

0.04Stra in

Figure 10. The in-situ Shear Stress vs. S he a rSt ra in response of the matr ix and the deducedUniaxial Stress vs . Uniaxial Strain Response ofthe M atrix (High Temperature).

cr-uni3530 -

25

20 -

15 -

10

5 -

0.00 005010015020025LPF

Figure 11. Respon se Plot of TypicalCross-Ply Analysis


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Figure 13.45ply response result for a quasi-isotropfclaminate (The equivalent plastic strain is sampled atthe point with coordinate (0.021 in., 0.271 in.))

Figure 12 Deformed Microregion S h a p e


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