Wiley Encyclopedia of Composites. Part 3

497 Pages·191.481 MB·English

by  Nicolais L. et al.

FRACTURE: INTERLAMINAR Combining Equations (1) and (2) and using the con-
cept of compliance C=δ/P it leads to the well-known
MARCELOF.S.F.DEMOURA Irwin–Kiesrelation,
AND RUIM.GUEDES
EngenhariadaUniversidadedo G= P2 dC (3)
Porto, Porto,Portugal 2B da
whereaisthecracklength.Duringpropagation,thevalues
INTRODUCTION of G given by Equation (3) provide the energy necessary
to crack growth (G ), thus characterizing the material
c
Delamination is one of the most severe problems con- fracturebehavior.
cerning laminated composites. In fact, the mechanical Infractureanalysis,crackgrowthcantakeplaceunder
properties of these materials can be drastically reduced three different modes as shown in Fig. 2. The mode I
in presence of delamination that may develop between representsanopenmodeandtheothers(modesIIandIII)
adjacent plies. This kind of damage is based on the fail- aretheshearmodes.InmodeIItherelativedisplacement
ureofthethinresinlayerexistingbetweentwopliesand ofthecrackfacesisperpendiculartothecrackfront,andin
usuallyisaresultoflowvelocityimpactormanufacturing modeIIIthereferreddisplacementisparalleltothecrack
defects.Delaminationisconsideredaverydangerousdam- front.Inthemajorityofrealsituationstheappliedloading
agemodeincompositematerials.Itisaninternaldamage originatesacombinationofmodesatthecracktip,which
difficulttodetectandcanreducesignificantlytheresidual impliesthatamixed-modecriterionshouldbeconsidered
propertiesofthematerial.Consequently,theinterlaminar inordertobettersimulatethedamagepropagation.
fracturecharacterizationacquiresremarkableimportance
intheframeworkofcompositesstructuresdesign.
MODEI—DCBTEST
FRACTUREMECHANICS The double-cantilever beam (DCB) test (Fig. 3) is stan-
dardized[2–4]formodeIinterlaminarfracturecharacter-
Delamination can be viewed as a crack propagation phe- ization.Thetestconsistsofloadingaprecrackedspecimen
nomenon, thus constituting a typical application of frac- inordertopromotepuremodeIloadingatthecrackfront.
ture mechanics concepts. There are two types of fracture The load is applied with a cross-head speed between 1
mechanics criteria. The criteria can be based on stress and 2mm/min through piano hinges, such that the load
intensity factors or on the energetic concepts. Since com- maintainsverticalduringthetest.
positesareheterogeneousmaterial,theenergeticmethods Table 1 presents the typical dimensions to be used. It
of fracture are more adequate. In fact, the stress inten- shouldbenotedthatthethickerspecimensshouldbeused
sity factor is a local parameter which can be affected in tough composites in order to avoid nonlinear behavior
by the heterogeneity. On the other hand, the energetic inducedbylargedisplacements.Theprecracklength(a )
0
quantities constitute global fracture parameters giving a isusuallyobtainedbyinsertingathinfilmofTeflonatthe
more accurate idea of the fracture process in the mate- mid-plane of the plate during its fabrication. The values
rial and not being drastically affected by local material ofload,applieddisplacement,andcracklength(P,δ,and
variations. a)areregisteredinordertocalculatethefractureenergy
Thebasicprinciplesoflinearelasticfracturemechanics duringloadhistory.Thecracklengthduringpropagation
wereproposedbyGriffith[1].Thisauthorestablishedthat isfrequentlymeasuredbymarkingthespecimenedgeand
crackgrowthwilloccurwhentheenergyavailableatthe usingatravelingmicroscope.
crack tip (G—strain energy release rate) and due to the
appliedload,matchtheenergynecessarytocrackpropa- ClassicalDataReductionSchemes
gation(G —criticalstrainenergyreleaserate)whichisa
c
Therearetwoclassicaldatareductionschemesfrequently
materialproperty.Thestrainenergyreleaserateisgiven
usedtoobtainthefractureenergyinmodeI(G ).Thecom-
by: Ic
pliance calibration method (CCM) is based on Equation
dW dU (3),whichrequiresthedefinitionofthefunctionC=f(a).
G= − , (1)
dA dA Acubicpolynomialisgenerallyusedtofittheexperimental
C–acurve,thusallowingastraightforwarddetermination
whereW istheworkperformedbyexternalloading,U is of dC/da. Alternatively, the corrected beam theory (CBT)
the strain energy of the body, and dA is the variation of isalsocommonlyused[5].Inthiscase,itisassumedthat
cracked area. Considering a general body with constant G canbeobtainedfrom
Ic
thicknessB(Fig.1)loadedonthedirectionperpendicular
tocrackplane: 3Pδ
G = , (4)
Ic 2B(a+|(cid:3)|)
1
W=Pδ and U= Pδ (2)
2
WileyEncyclopediaofComposites,SecondEdition.EditedbyLuigiNicolaisandAssuntaBorzacchiello.
©2012JohnWiley&Sons,Inc.Published2012byJohnWiley&Sons,Inc.
1
2 FRACTURE:INTERLAMINAR
P Table1. TypicalDimensionsoftheDCB
δ Specimen
L(mm) B(mm) h(mm) a0(mm)
125–150 20–25 2–5 45–50
a B
C1/3
(mm/N)1/3
Δ a1 a2 a3 a (mm)
Figure1. Crackedbodyunderuniaxialload. Figure4. Schematic representation of the crack length correc-
tionduetorootrotationintheDCBtest.
where various toughening mechanisms take place, such
as microcracking, plastification namely in tough compos-
ites and fiber bridging, the crack tip is not undoubtedly
defined during its propagation. As a result, the classical
data reduction schemes based on compliance calibration
or beam theories could not provide accurate estimations
offractureenergysincetheydependoncracklengthmea-
Mode I Mode II Mode III
surements. In order to overcome the referred difficulties,
Figure2. Failuremodes. analternativedatareductionschemebasedoncrackequiv-
alent concept is presented. The method does not require
crack length measurements and accounts for the energy
dissipated at the FPZ. This nonnegligible FPZ affects
the measured toughness since a nonnegligible amount
of energy is dissipated on it. Consequently, its influence
shouldbetakenintoaccount,whichdoesnotoccurwhen
therealcracklengthisusedinaclassicaldatareduction
scheme(Fig.5).
Figure3. TheDCBtest.
where(cid:3)isacorrectionforcracktiprotationanddeflection. a + |Δ|
(cid:3)isdeterminedfromalinearregressionanalysisof(C)1/3
versusadata(Fig.4). ae
Itshouldbeemphasizedthatbothmethodsdependon
crack length measurements during its propagation. This a + |Δ| + FPZ
is not easy to perform and remarkable errors can occur,
affecting the measured G . In fact, owing to the devel- Figure5. SchematicrepresentationoftheFPZandcrackequiv-
Ic
opment of a nonnegligible fracture process zone (FPZ), alent(ae)concept.
FRACTURE:INTERLAMINAR 3
Compliance-BasedBeamMethod(CBBM) whichcanbeusedinsteadof(cid:3)inEquation(6).Aniterative
procedure involving Equations (6) and (7) must be used
Considering the Timoshenko beam theory the C=f(a)
todetermineE .Duringpropagationtheequivalentcrack
relationshipcanbewrittenas[6,7]: f
length is obtained from Equation (5) as a function of
8a3 12a the current specimen compliance registered during the
C= + (5) test and considering a instead of a. The solution of this
E Bh3 5BhG e
1 13 equation can be found using the mathematical software
®
This equation constitutes an approach based on beam Matlab and is presented in Appendix A. The energy
theoryanddoesnotaccountforalleffectsinfluencingthe releaserateinmodeIcanbeobtainedusingEquations(3)
specimen behavior. For example, it is known that stress and(5)
concentrations arise around the crack tip, which is not (cid:2) (cid:3)
6P2 2a2 1
takenintoconsiderationbythisapproach.Toovercomethe G = e + (8)
referreddiscrepancies,anequivalentflexuralmodulus(Ef) I B2h h2Ef 5G13
canbeobtainedfromEquation(5)consideringtwoinitial
Thepresentedmethodologyallowsobtainingtheenergy
conditions taken from the experimental tests: the initial
release rate G only as a function of the P–δ data. For
cracklengtha andinitialcomplianceC .Moreover,this I
0 0
this reason, it is designated by compliance-based beam
approachtakesintoaccountthevariationofthematerial
method (CBBM). Using this method it is not necessary
propertiesbetweendifferentspecimenssinceitisbasedon
to measure the crack length during propagation since
theexperimentallymeasuredC .Thus,E canbeobtained
0 f the calculated equivalent crack length is used instead of
fromEquation(5),
the real one. Another advantage relies on the fact that
(cid:2) (cid:3)
12(a +|(cid:3)|) −1 8(a +|(cid:3)|)3 ae includes the effect of the FPZ, which is not taken
Ef = C0− 5B0hG 0Bh3 (6) into account when the real crack length is considered.
13 Following this procedure, an entire R-curve [G =f(a )]
I e
is obtained allowing to get fracture energy G from its
where (cid:3) is the root rotation correction for the initial Ic
plateau. The problem of spurious high initial toughness
cracklength.Inthebeamtheory,itisassumedthateach
inducedbyanimproperstarterdelamination(e.g.,aresin
arm of the DCB specimen is an encastred beam whose
pocketformedattheendoftheinsert)isovercome,since
lengthisequaltothecracklengtha.Thisparameter((cid:3))
canbeachievedbyalinearregressionofC1/3=f(a).The the initial pop-in in the R-curve can be ignored and the
real toughness can be associated to the plateau value.
determination of (cid:3) can be performed by slightly loading
Moreover, the specimen modulus (E ) is not a measured
the specimen with three different initial crack lengths f
(Fig.6)inordertodefinetheC1/3=f(a )linearregression property but rather a computed one (Eq. 6) as a function
0
oftheinitialcompliance,thusaccountingforthematerial
(Fig.4).Alternatively,Hashemietal.[8]proposedanother
variabilitybetweendifferentspecimens.Theonlymaterial
form of determining the root rotation effects by altering
property required in this approach is the shear modulus
thecracklengthusingtheparameter(cid:3)
I
G . However, from Equation (8) it is straightforwardly
(cid:4) 13
(cid:5) (cid:7) (cid:2) (cid:3) (cid:8) √ concludedthatthetermcontainingG issmallrelatively
(cid:3)I=h(cid:5)(cid:6)11EGf13 3−2 1+(cid:4)(cid:4) 2 and(cid:4)=1.18 GE1f3E3 btoetuhseeodnaenodftEhfa.tTihtiissmneoatnnsecthesastaarytytpo1i3mcaelavsaulureeiotffGor13ecaacnh
(7) specimen.
Themethodwasappliedtointerlaminarfracturechar-
acterizationofcarbon–epoxylaminatesinordertoassess
δ , P theeffectoffiberbridgingonthemeasuredfractureenergy
1 1 δ2, P2 δ , P [9].Fiberbridgingoccursduetodebondedfibersbridging
3 3
the two surfaces and increases the fracture toughness
owing to the energy consumed in the debonding process.
Inordertoassesstheinfluenceoffiberbridgingoninter-
laminarfracturebehavior,bridgingfiberswerecutduring
crack propagation using a sharp razor blade. This proce-
dureinducedasawtoothload–displacementcurve(Fig.7).
FromtheRcurvesobtainedbymeansoftheCBBM,itwas
verified that the bridging phenomenon is responsible for
60% of energy being dissipated during crack propagation
(Fig.8).Itshouldbenotedthatfiberbridgingisfrequently
viewed as a feature of testing unidirectional laminates
a
03 anddoesnotoccurobligatorilyinstructuralapplications.
a02 Consequently, the values provided by cutting fibers are
a more realistic since fiber bridging lead to nonconserva-
01
tive toughness values. This analysis also emphasizes the
Figure6. Schematic representation of the compliance calibra- advantageoftheCBBM,asthismethodprovidesentireR
tionasafunctionofthecracklengthintheDCBtest. curvesthusfacilitatingthistypeofanalysis.
4 FRACTURE:INTERLAMINAR
60
50
40
P (N) 30
20
Normal test
10 Cutting fibers
0
0 5 10 15 20 25
δ (mm)
Figure7. TypicalP–δcurvesofinterlaminarfractureDCBtests
withandwithoutcuttingfiles.
1.4
Normal test
1.2
Cutting fibers
1
m)
m 0.8
N/
G (I 0.6
0.4
0.2
Figure9. SchematicrepresentationsofthemodeIItests.
0
40 50 60 70 80 90 100
a (mm)
e
not requiring special devices to be performed. It consists
Figure8. R curves obtained from interlaminar fracture char- of a three-point bending test on a precracked specimen.
acterization with (three specimens) and without cutting fibers Consequently,theENFtestisthemostpromisingcandi-
(threespecimens).
datetostandardizationforinterlaminarmodeIIfracture
characterization.
MODEII
ENFTest
Delaminations are prone to propagate under mode II TheENFtestconsistsofloadingonathree-pointbending
loading. Effectively, the bending phenomenon of multidi- fixture a precracked specimen, which produces a shear
rectional laminates leads to shear loading at interfaces loading at the crack tip. The load is applied with a
between different oriented plies induced by mismatch cross-head speed between 0.5 and 1.0mm/min through
bending of these plies. As a result, interlaminar fracture acylindricalpinofabout5mmdiameter.Theprecrackis
characterization under mode II loading is fundamental usually executed by means of a thin film of Teflon at the
to accurately predict the susceptibility of the material to mid-planeoftheplateduringitsfabrication.
delamination. Contrarily to what happens with mode I, One of the drawbacks of the ENF is related to unsta-
there is no ISO standard for mode II interlaminar frac- ble crack propagation. Carlsson etal. [13] demonstrated
ture,whichisexplainedbyseveraldifficultiesinherentto that a0≥0.7L to get crack growth stability by analyzing
this loading mode. In reality, phenomena such as unsta- the sign of dGII/da. This means that longer specimens
blecrackgrowthandremarkabledifficultiesonthecrack (2L=200 mm) are preferable (Table 2). Effectively, in
lengthmonitoringduringpropagationimpededtheselec- shorter specimens the referred relationship implies that
tion of an adequate testing method. The most common theprecracktipisclosetothecentralloadingpoint,which
usedtests(Fig.9)aretheend-notchedflexure(ENF),the canaffectthenaturaldevelopmentoftheFPZandinduce
end-loaded split (ELS), and the four-point ENF (4ENF) a spurious increase of GIIc. The values of load, applied
test. In the 4ENF the crack length measurement during displacement,andcracklength(P,δ,anda)areregistered
propagation is not necessary, but a complex set-up and duringthetest.Thecracklengthismonitoredbymarking
problems related to remarkable friction effects [10] ren- thespecimenedgeandusingatravelingmicroscope.
ders it unsuitable for this purpose. The ELS test also
requiresaspecialset-upincludingalinearguidancesys-
Table2. TypicalDimensionsoftheENF
tem allowing horizontal movement of translation of the
Specimen
clampinggripduringloading[11].Inaddition,theclamp
is a source of variability [11] and the specimen is prone 2L(mm) B(mm) 2h(mm) a0(mm)
to undergo large displacements inducing nonlinearities 150–200 10–25 3–5 50–70
[12].TheENFtestisundoubtedlythesimplesttoexecute,
FRACTURE:INTERLAMINAR 5
ClassicalDataReductionSchemes. TheCCMisthemost where
useddatareductionscheme.Acubicrelationshipbetween
3L 3L
the compliance (C) and the measured crack length (a) is C =C− and C =C − . (15)
c 0c 0
usuallyassumed[14] 10BhG13 10BhG13
Combining Equations (3) and (12), G =f(a ) can be
C=D+ma3, (9) II e
obtainedas:
where D and m are constants. GIIc is then obtained from 9P2a2
Equations(3)and(9) GII= 16B2h3eE . (16)
f
G = 3P2ma2. (10) Using this methodology crack measurements are
IIc 2B unnecessary. Experimentally, it is only necessary to
registerthevaluesofappliedloadanddisplacement.The
Alternatively,thebeamtheorycanbeused.Wangand
method accounts for FPZ effects through a and for the
Williams[15]proposedaCBTbasedon, e
influenceofspecimenvariabilityontheresultsbymeans
9(a+0.42(cid:3) )2P2 of the Ef. A typical value for the shear modulus G13
GIIc= 16B2h3EI (11) can be used owing to its minor influence on the results
1 relatively to E . Following this procedure a complete
f
where(cid:3)I isgivenbyEquation(7),being0.42(cid:3)I thecrack R-curve[GII=f(ae)]isobtained.ThevalueofGIIc canbe
length correction to account for shear deformation. The estimatedfromitsplateau.Inaddition,sincethismethod
applicationofbothmethods(CCMorCBT)requirescrack provides the R-curve, the problems related to unstable
lengthmeasurementduringpropagation.However,aprob- crack growth and the spurious increase of GIIc at crack
lemintrinsictoallfracturecharacterizationtestsinmode starting advance due to nonnatural character of crack
II is the remarkable difficulty to rigorously monitor the createdbyinsertfilmsaremitigated.
crack since it tends to close during propagation. In addi- The method was applied to experimental determi-
tion, under mode II loading, a pronounced FPZ usually nation of GIIc of carbon/epoxy prepreg (Texipreg HS
®
develops.Consequently,theenergydissipatedonitshould 160 RM from SEAL ). The CBBM was applied to the
beaccountedforwhichreinforcesthenecessityofacrack experimentalload–displacementcurveandtherespective
equivalentmethod. R-curve[GII=f(ae)]wasobtained.Anumericalvalidation
using cohesive zone modeling (CZM) was also performed
Compliance-Based Beam Method (CBBM). The CBBM (Fig. 10). The CZM allows the simulation of damage
wasalsoappliedtotheENFtest.Themethodisbasedon initiationandgrowth[12].ThevalueofGIIc(≈1.0N/mm)
specimen compliance, beam theory, and crack equivalent was taken from the plateau of the experimental R-curve
concept. Considering the Timoshenko beam theory the and inputted in the numerical model. Good agreement
C=f(a)relationshipbecomes[16] betweenthenumericalandexperimentalP–δ curveswas
obtained (Fig. 11). The application of the CBBM to the
3a3+2L3 3L
C= + (12)
8E Bh3 10G Bh
1 13
Theinitialcompliance(C )andcracklength(a )canbe
0 0
usedtoestimatetheflexuralmodulusofeachspecimento
beusedinsteadofE
1
(cid:2) (cid:3)
3a3+2L3 3L −1 Figure10. MeshoftheENFspecimen.
E = 0 C − (13)
f 8Bh3 0 10G Bh
13
400
This procedure takes into account the variableness of
Numerical
the material properties between different specimens and
several effects that are not included in beam theory, for 300
example,stressconcentrationnearthecracktipandcon-
ttahcetsbpeetwcimeeennthbeehtwavoiaorrmasn.dIncofancste,qtuheensetlpyhtehneomPe–nδacaufrfevcet, P (N) 200 Experimental
even in the elastic regime. Using this methodology their
influenceisaccountedfor,throughthecalculatedflexural 100
modulus.DuringpropagationaFPZdevelopsaffectingthe
material compliance. In order to account for it the cur-
rentcomplianceCisusedtoestimateanequivalentcrack 0
lengtha throughEquations(12–13), 0 1 2 3 4 5
e
δ (mm)
(cid:9) (cid:2) (cid:3) (cid:10)
C 2 C 1/3
a = c a3+ c −1 L3 (14) Figure11. NumericalandexperimentalP–δ urvesofoneENF
e C0c 0 3 C0c test.
6 FRACTURE:INTERLAMINAR
2.0 P c
1.5 Numerical
m) Loading lever
m
N/ 1.0
G (II
0.5 Experimental Specimen
0.0
45 50 55 60 65
a (mm) Base
e
Figure12. Numerical and experimental R curves of one ENF
test.
Figure13. Schematicrepresentationoftheloadingconditionsin
theMMBtest.
numerical P–δ curve also provided similar values of G
IIc
fromtheplateauoftheRcurves(Fig.12),whichvalidates
theorycanbeusedtogetsimpledatareductionschemes.
the CBBM as an adequate data reduction scheme to be
FromFig.15,theloadcomponentsare:
appliedtotheENFspecimen.
(cid:2) (cid:3) (cid:2) (cid:3)
3c−L c+L
P = PandP = P. (17)
MIXEDMODE(I+II) I 4L II L
ThemodecomponentsareestimatedusingtheCBTto
In the majority of real applications composite structures
accountforthebeamrootrotationatthecracktip,which
behave under mixed-mode loading. These mixed-mode
ispronouncedinmodeI,andsheardeformationinthetwo
conditionsarisefromexternalloadingandalsofrommate-
modes[20]
rial anisotropy. Consequently, it is fundamental to study
delamination growth under mixed-mode (I + II) load-
12(a+h(cid:3) )2P2 9(a+0.42(cid:3) h)2P2
ing conditions in order to establish adequate criterion. G = I I; G = I II (18)
I B2h3E II 16B2h3E
Several different tests have been proposed to this goal 1 1
[17]: the cracked lap shear (CLS), the edge delamina-
whichleadtothefollowingmoderatio
tiontension(EDT),theArcan test,andasymmetricDCB
(cid:2) (cid:3) (cid:2) (cid:3)
test. Sze´krenyes and Uj [18] also proposed the single-leg G 4 3c−L 2 a+h(cid:3) 2
bending (SLB) and the end-loaded split for mixed mode I = I (19)
G 3 c+L a+0.42(cid:3) h
(ELS-MM). All of them intend to induce simultaneously II I
normal and shear stresses at the delamination plane.
being(cid:3) givenbyEquation(7).Equation(19)allowsdefin-
I
However, several limitations are common in these tests:
ingthevalueofcasafunctionofapreselectedmoderatio
mode mixity varying with the crack length, small range
(G /G ) for a given precrack length a . The specimen
I II 0
ofmodemixities,complexloadingsystems,andabsenceof
dimensionsaresimilartotheonesusedfortheENFtest
closeformsolutionsfordatareduction.
(Table2).Theloadisappliedwithacross-headspeedinthe
range of 0.5–2.0mm/min being transmitted to the speci-
MMBTest
menviabearing-mountedrolleratthespecimenmid-span,
Themixed-modebending(MMB)testproposedbyReeder andabondedpianohingetoinducecrackopening(Fig.14).
and Crews [17,19] revealed to be able to overcome the Asinthepreviouscases(DCBandENF),thestartercrack
majorityofthereferreddrawbacksanditwasselectedfor isgeneratedbyathinfilmofTeflon.Thepropagationcan
standardization [20]. The MMB test can be viewed as a bestableorunstabledependingonthemodemixitybeing
combination of the DCB and ENF tests. In fact, the test usedinthetest.Generally,theinstabilityincreasesasthe
consists of adding an opening mode load to a mid-span presenceofmodeII.
loaded ENF specimen (Fig. 13). Using one applied load Thefractureenergycomponentsandmodemixturecan
at the end of the lever (Fig. 13), a downward load is becalculatedfromEquations(17–19),whichrequirecrack
appliedtothespecimenmid-spancreatingmodeII,while lengthmonitoringduringitspropagationor,inthecaseof
anupwardforceisappliedtothesplitendofthelaminate unstablepropagation,theconsiderationoftheinitialvalue
creating mode I. The test is relatively easy to execute correspondingtocrackstartingadvance.Alternatively,the
andprovidesaneasyalterationofmixed-moderatioonly equivalentcrackconceptdescribedforDCBandENFtests
changingtheleverlengthoftheapparatus(parametercin canbeappliedtogetthefractureenergycomponentsfrom
Fig.14),sincethisreflectsonthealterationoftherelative therespectiveRcurves[21].Inthiscase,thedisplacement
magnitude of the two resulting loads on the specimen at the specimen extremity (δ ) is equal to the mode I
E
(Fig. 15). Moreover, theload appliedto thespecimen can component (δ ) and should be measured during the test
I
be separated into modes I and II components, and beam inordertodefineC =δ /P .Similarly,thecomplianceof
I I I
FRACTURE:INTERLAMINAR 7
Yoke
Loading lever
Yoke
Saddle
Loading
lever
c
Specimen
Figure14. SchematicrepresentationoftheMMBtestset-up.
P c L
Loading lever
c +L LP LcP 3c4 −L LP c +L LP c 4+L LP
Specimen = +
L a
c 2+L LP c 2−L LP 3c4 −L LP c 2+L LP c 4+L LP sFiisgoufrtehe15M.MSBupseprepcoismiteinon. loadinganaly-
MODEIII
Linear criterion
0.3
G G
0.25 I + II =1 ShearmodesIIandIIIplayanimportantrolewhenlam-
G G
Ic IIc inated composites are submitted to bending loads, as is
m) 0.2 the case of low velocity impact. In fact, the mismatch
m
N/ 0.15 of elastic properties between differently oriented layers
G (I 0.1 [23] leads to important shear stresses at the interfaces,
thusoriginatingdelamination.Furthermore,Geetal.[24]
0.05
demonstrated that mode III has a considerable influence
0 on delamination growth under compression loads. The
0 0.2 0.4 0.6 0.8 1 authorsusedadigitalspecklecorrelationmethodtoobtain
GII (N/mm) thein-planedisplacementsofcompressedlaminatedcom-
posite panels containing a preset elliptical delamination.
Figure16. TypicalrepresentationintheG versusG spacefor
I II
severalmixed-modecombinations[21]. They concluded that the mode III strain energy release
rate G had almost the same contribution to the total
III
energy release rate as G . However, the research dedi-
II
catedtomodeIIIinterlaminarfracturecharacterizationis
stillratherincipientmainlyduetotestingdifficulties.Sev-
themodeIIcomponent(C )willbedefinedfromδ =δ + eraldifferenttestshavebeenproposed[25–27],butmost
II II C
δ /4 [22], which requires the monitoring of the mid-span of them have some significant problems that hinder con-
I
displacement δ . From C and C , the CBBM can be sistent measurements. For instance, the split-cantilever
C I II
easily applied considering Equations (5–8) and (12–16) beam (SCB) [24] and the 4ENF [26] tests present non-
formodesIandIIstrainenergyreleaseratecomponents, negligible spurious mode II at the specimen edges which
respectively. The values of energy components kept from affectsGIIIcmeasurements.ThemodifiedSCB[27]ischar-
the plateaus of both R curves [G =f(a ) and G =f(a )] acterizedbyacombinationofloads(Fig.17),whichcancels
I e II e
are then used in a representation of G versus G space, thebendingmomentatthecracktip,thuseliminatingthe
I II
in order to get a failure criterion to be used in damage spuriousmodeIIcomponent.However,thequitecomplex
toleranceanalysesofcompositestructures(Fig.16). loadingsystemandthehighspecimenstiffnesshinderan
8 FRACTURE:INTERLAMINAR
pins (length l in Fig. 18). The authors also verified that
the magnitude of the mode II strain energy release rate
is small when compared to the mode III component and
L
concludedthatECTtestisadequateforinterlaminarfrac-
ture characterization in mode III. Ratcliffe [30] proposed
analternativespecimengeometryandlay-uptominimize
thepresenceofmodeII.
The data reduction schemes based on CCM require
2P
the establishment of the relation C=f(a). However, in
the ECT test the crack propagates in the central region
B P
which precludes its monitoring at the specimen edges,
2P as commonly done in modes I and II tests. In addition,
P the crack propagates nonuniformly in the central region
delimitated by the pins [29,31], which complicates the
a/2
2h a definitionofanundoubtedlycracklength.Thealternative
is testing specimens with different precrack lengths in
order to establish the function C=f(a) [29], which is
Figure17. Schematicrepresentationoftheloadingconditionsin
expensiveandlaborious.
themodifiedSCBspecimen.
Toovercomethereferreddrawbacksanalternativedata
reductionschemebasedonspecimencompliancehasbeen
easyandaccurateevaluationofthecriticalstrainenergy recentlyproposed[32].Themethodrequiresthetestingof
releaserateinmodeIII,G . two specimens (one with a precrack and another without
IIIc
aprecrack)andisbasedontheequationpreviouslydevel-
ECTTest opedbyLee[28],usingclassicallaminationtheory(CLT),
The edge crack torsion (ECT) test initially proposed by
Lee[28]constitutesthemostpromisingoptiontomeasure 1 4B(D ) (cid:11) (cid:12) (D )
= 66 I 1−(1−2s)(a/B) ands= 66 II (20)
GIIIc. This test consists of a plate with a precrack along C b2l (D66)I
its mid-plane under torsion by the action of four pins
(Fig. 18). The author considered [90/(±45)n/(∓45)n/90]s where (D66)I and (D66)II are the torsional stiffness terms
laminates with n ranging between 1 and 4. In order to for the uncracked laminate and cracked half laminate,
avoid highly nonlinear behavior of the P–δ curve related respectively. The author assumed that the entire speci-
tolargedeflection,theauthorarguedthatn=3orn=4 men undergoes a uniform rotation. The total torque was
shouldbeused.The(±45)pliesprovidetorsionalstiffness then computed from the sum of the torques acting on
andstrengthtothespecimens.Eachhalfofthespecimen the cracked and uncracked parts of the specimen. The
contains a lay-up symmetric to the mid-plane to avoid torsional stiffness of each part was obtained from CLT.
spuriousthermalmismatcheffects.Itwasassumedthata This assumption violates continuity conditions between
pure mode III loading exists. However, using the virtual the cracked and uncracked parts of the specimens. On
crack closure technique, Li etal. [29] demonstrated that the other hand, there are other issues not accounted for
there is some spurious mode II near the specimen edges, in Equation (20), for example, stress concentration at
althoughapuremodeIIIexistsintheregionbetweenthe the crack tip and spurious mode II, that influence the
Z
y
P
x L
l
t a0
Precrack
b
B
Figure18. TheECTspecimen.
FRACTURE:INTERLAMINAR 9
C=f(a)relationship.Totakeintoaccountthesefeatures,
thefunctionC=f(a)wasassumedtobegivenby:
1
=A(1−ma), (21)
C
where the parameters A and m include the effect of
specimen geometry and its torsional stiffness. These
parameters must be obtained from data fitting using the
initial compliances of an uncracked specimen (a=0 and
C=C ) and of the specimen being tested (a=a and
0(0) 0
C=C ),whichgives
0
(cid:2) (cid:3)
1 1 C
A= andm= 1− 0(0) . (22)
C a C
0(0) 0 0
Combining Equations (3), (21), and (22) the fracture
toughnessinmodeIIIisgivenby: Figure19. TheECTtestset-up.
(cid:13) (cid:14)
δ2 C −C
G = c 0 0(0) . (23)
IIIc 2la C C
0 0 0(0)
2500
The determination of G relies on the initial compli-
IIIc
ancesoftheuncrackedandcrackedECTspecimens,onthe 2000
initialcracklengthandonthecriticaldisplacementcorre-
1500
spondingtotheinitialcrackstartingadvance(δc).Itdoes N)
not involve compliance calibration as a function of crack P ( 1000
length, which requires several specimens with different
crack lengths. In this case, it is only necessary to have 500
two types of specimens: without a precrack and with the
0
chosen initial crack length. The detailed numerical anal-
0 1 2 3 4 5 6
ysis with CZM performed in Ref. 32, puts into evidence
δ (mm)
crack length dependency of G . This was explained by
IIIc
nonuniform damage propagation as well as the increase Figure20. Experimentalload–displacementcurvesoftheECT
of the width of the propagated crack and the FPZ area test.
with the initial crack length. It was also verified that an
initialcracklengthequalto41%(i.e.,18mm)ofspecimen
width(BinFig.18)leadstoexcellentagreementwiththe
MixedMode(I+III)
inputtedvalueinthecohesivedamagemodel.
Experimental tests (Fig. 19) were performed consider- Szekre´nyes [33] proposed the prestressed split-cantilever
inga[90/(±45)4/(∓45)4/90]s laminate.Theprecrackwas beam (PSCB) fracture specimen to induce a mixed-mode
createdbyinsertingTeflonfilmswith12.5μmofthickness (I + III) loading. The selected specimen is a combina-
during lamination. These Teflon films also contribute to tion of the modified SCB (Fig. 17) and the well-known
reducefrictionbetweencrackfaces,whichcouldaffectthe DCBspecimens.ThemodeIloadingisobtainedbyinsert-
measured GIIIc. The dimensions of the six tested speci- ing a steel roller between the specimen arms inducing a
menswere(Fig.18):L=108mm,B=44mm,l=76mm, fixed crack opening displacement. The mode III part of
b=38 mm, t=5.8 mm, and a0=18 mm. Three speci- the energy release rate is provided by the external load
mens withoutcrack were also tested inorder to evaluate usingaspecialrig.ThePSCBisabletoprovideanycom-
theinitialcomplianceofuncrackedspecimens[C0(0)]. bination of modes at crack initiation and a simple data
Figure 20 shows the load–displacement curves. The reductionschemebasedonbeamtheorycanbeused.How-
GIIIcwasdeterminedfromthecriticaldisplacementcorre- ever, several drawbacks are pointed to this specimen. It
sponding to maximum load (δc) and using Equation (23). presentsverylowcomplianceandthemoderatiochanges
Anaveragevalueof1.36N/mmwasobtainedforG with
IIIc with the crack length, the applied load and along the
acoefficientofvariationof7.6%.
crackfront.Finally,themoderatiocannotbefixedbefore
theexperimentaltestsareperformedsinceitdependson
MIXEDMODES(I+IIIANDII+III) the definition of the crack initiation and the respective
load.Actually,theload–displacementcurveispractically
InadditiontotheI+IImixedmodeitisofinteresttostudy insensitive to delamination growth, which makes initia-
themixedmodescontainingthemodeIII,thatis,theI+ tion very difficult to detect. As a consequence the author
IIIandII+IIIcombinations,toprovideacompleteenvelop recommended the use of transparent specimens to allow
of fracture criterion. There has been few works on such visual detection of initiation. The author recognized that
mixed-modecombinationsbutnostandardsarecurrently more research is needed to reduce the drawbacks of the
envisagedforinterlaminarfractureofcomposites. test.
10 FRACTURE:INTERLAMINAR
Pereira and de Morais [34] proposed a new test for loading. The problems related to unstable crack growth
measuringthemixed-mode(I+III)interlaminarfracture and crack monitoringduring its growth can be mitigated
toughness. The specimen adopted an eight-point bending by a judicious selection of its geometry and the use of a
plate(8PBP)withacross-plylay-upandamid-thickness proposeddatareductionschemesimilartotheonereported
edge predelamination at the standard 0/0 interface. The fortheDCBtest.
mode mixity can vary in a wide range by altering the The MMB test is the most suitable for interlaminar
load-point displacements imposed. However, a numeri- fracturecharacterizationundermixed-mode(I+II)load-
cal model based on finite element analysis is required ing.ThistestisacombinationoftheDCBandENFtests
for experimental data reduction, a somewhat ambiguous andprovidesaneasyalterationofthemodemixinawide
definitionofcrackinitiationregionandarelativelycompli- range,thusallowingtheestablishmentoffractureenvelop
catedfixtureisneededwhichcanbeviewedasimportant intheG versusG space.Inaddition,theproposeddata
I II
drawbacksofdevelopedtest. reductionschemesfortheDCBandtheENFtestscanbe
easilyappliedinthistest.
MixedMode(II+III) The mode III fracture characterization is under
Szekre´nyes [35] also proposed the mixed-mode (II + III) research and no standard test exists. The ECT test
appearstobethemostpromisingsolution,althoughsome
version of the prestressed end-notched flexure (PENF)
disadvantages are pointed and discussed. A new data
fracture specimen, which combines the well-known ENF
reductionschemebasedoninitialcomplianceandcritical
and the modified SCB (Fig. 17) specimens. The mode III
displacement is proposed instead of the classical and
part of the strain energy release rate is fixed by using a
cumbersomeCCM.
special rig, which loads the specimen in the plane of the
Finally, only few works were dedicated to the
delamination as shown in Fig. 17. The mode II compo-
mixed-mode cases involving the mode III. The proposed
nent is provided by the external load using a three-point
solutions are able to provide a wide range of mode mix
bending fixture. This combination of loads on the new
ratios. However, several drawbacks related to complex
beam-likespecimenisabletoprovideanycombinationof
the mode mix (II + III) at crack initiation. Nevertheless, experimental set-ups, variation of mode ratio as a
function of crack length, nonuniform distributions of
this test has some disadvantages. The mixed-mode ratio
variesasafunctionofcracklengthandtheappliedloadat strain energy components, and nonlinear effects require
crackinitiationwhichprecludesthepredefinitionofmode furtherresearchbeforeanystandardtestareenvisaged.
mixity.
De Morais and Pereira [36] presented a new test
methodologyforthemixed-mode(II+III)fractureofcar- APPENDIXA
bon/epoxy laminates. The specimens used were six-point
Equation(5)canbeexpressedas,
bending plates with cross-ply lay-up and the standard
0/0 interface. Finite element analyses were performed to
selectspecimengeometriessuitableformeasuringtheini- αa3e+βae+γ =0, (A1)
tiation critical strain energy release rate G over a wide
c
range of mode mix ratios. The main difficulties were the wherethecoefficientsα,β,andγ are,respectively
nonuniform distributions of G and G , data reduction
II III
requiresfiniteelementanalysisandtheconsiderablegeo- 8 12
α= , β = , and γ =−C. (A2)
metricnonlinearity.Asapositiveaspect,theauthorspoint Bh3E 5BhG
f 13
thatthetestreplicatesconditionsthatareclosertothose
ofactualapplications. UsingtheMatlab®softwareandonlykeepingthereal
solutionwehave,
SUMMARY
1 2β
a = A− (A3)
e
6α A
Interlaminar fracture characterization is fundamental in
thecontextofdelaminationinitiationandgrowthincom-
being
posites.Underthisperspective,themostprominentfrac-
ture tests for pure and mixed-mode loading are reviewed ⎛⎛ (cid:17) ⎞ ⎞
(cid:2) (cid:3) 1/3
inthissection. A=⎝⎝−108γ +12 3 4β3+27γ2α ⎠α2⎠ . (A4)
For mode I, the DCB test is standardized and is uni- α
versallyacceptedasbeingthemostadequate.Beyondthe
classicaldatareductionschemes,anewmethodbasedon
beam theory, specimen compliance, and crack equivalent
conceptispresented.Thisapproachissimpletoapplyand
REFERENCES
has several advantages namely when one of the assump-
tionsoflinearelasticfracturemechanics(negligibleFPZs) 1. GriffithAA.Thephenomenonofruptureandflowinsolids.
isnotcompletelysatisfied. PhilosTransRSoc1920;221A:163–198.
TheENFtestappearstobethemostappealingsolution 2. JISK7086:1993.Testingmethodsforinterlaminarfracture
to interlaminar fracture characterization under mode II toughnessofcarbonfibrereinforcedplastics.