Analysis of Reflective Cracks in Airfield Pavements Using a 3-D Generalized Finite Element Method

20 Pages·2010·2.14 MB·English

Analysis of Reflective Cracks in Airfield
Pavements Using a 3-D Generalized Finite
Element Method
J.Garzon, C.A.Duarte† andW. Buttlar
DepartmentofCivilandEnvironmentalEngr.,
UniversityofIllinoisatUrbana-Champaign,
NewmarkLaboratory,
205NorthMathewsAvenue,
Urbana,IL61801,USA
{jgarzon2,caduarte,buttlar}@illinois.edu
†CorrespondingAuthor
RÉSUMÉ.
ABSTRACT. Predictionandsimulationofload-relatedreflectivecrackinginairfieldpavements
require three-dimensionalmodels in order to accurately capture the effects of gear loads on
crack initiation and propagation. In this paper, we demonstrate that the Generalized Finite
ElementMethod(GFEM)enablestheanalysisofreflectivecrackinginathree-dimensionalset-
tingwhilerequiringsignificantlylessuserinterventioninmodelpreparationthanthestandard
FEM.Assuch,itprovidessupportforthedevelopmentofmechanistic-baseddesignprocedures
forairfieldoverlaysthatareresistanttoreflectivecracking.
TwogearloadingpositionsofaBoeing777aircraftareconsideredinthisstudy.Thenumerical
simulationsshowthatreflectivecracksinairfieldpavementsaresubjectedtomixedmodebe-
haviorwithallthreemodespresent.Theyalsodemonstratethatundersomeloadingconditions,
thecracksexhibitsignificantchanneling.
MOTS-CLÉS:
KEYWORDS:Fracture;Reflectivecracks;Finiteelementmethod;Generalizedfiniteelementmethod;
Extendedfiniteelementmethod
RoadMaterialsandPavementDesign.VolumeX-n X/2009,pages1à15
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1. Introduction
Veryoften,asphaltoverlaysareusedtorestoresmoothness,structure,andwater-
proofing benefits to existing airfield pavements. However, the controlling factor for
overlaylifespanisoftenreflectivecrackinginthenewoverlaycausedbystresscon-
centrationsinthevicinityofjointsandcracksintheunderlyingpavement.Reflective
crackingcanleadtoroughness,spalling,andmoistureinfiltrationatjointsandcracks,
which can greatly reduce the life expectancy of the overlay. Advanced spalling can
alsosignificantlyincreaseFODpotential.
Historically,reflectivecrackingisthoughttobedrivenbyboththermal(non-load
associated)andmechanicalgearloading(loadassociated)forces[BOZ02,KIM02].
Toolsarenowavailabletoaidthedesignerintheselectionofanappropriateasphalt
binder and mixture to resist thermally induced cracking. This includes both thermal
(top-down)cracking[AAS09b,AAS09a],andthermallyinducedreflectivecracking,
or bottom-up cracks, emanating from the vicinity of cracks and joints in the under-
lyingpavement[DAV10].However,recentresearchhasindicatedthatthesuccessful
predictionandabilitytodesignagainstreflectivecrackingliesintheabilitytounder-
standandaccountforload-relatedreflectivecrackingmechanisms[BAE08,DAV07].
Load-related reflective cracking mechanisms are more difficult to predict through
modelingandsimulation,asitisnecessarytostudythisphenomenoninthreedimen-
sions(3-D)inordertoaccuratelycapturetheeffectsofgearloadsoncrackinitiation
andpropagation.Previousresearchhasalsodemonstratedtheneedtodirectlyaccount
forthepresenceandpropagationofcracks,inordertoavoidmeshdependentnumeri-
calresults.Ina3-Dsetting,thismeansthatallmodesofcrackingmustbeconsidered,
i.e.,ModeI(opening),ModeII(shearing),andModeIII(tearing).Theneedfortrue
3-D modeling of reflective cracking complicates the development of computational
models using standard finite element methods. The Generalized or Extended Finite
ElementMethod[BAB94,BAB97,ODE98,DUA00,MO9˜9,SUK00,MO0˜2]adds
flexibility to the FEM while retaining its attractive features. A brief review of the
GeneralizedFEM(GFEM)ispresentedinSection3.
In this paper, we demonstrate that the GFEM enables the analysis of reflective
crackingina3-Dsettingwhilerequiringsignificantlylessuserinterventioninmodel
preparation. As such, it provides support for the development of mechanistic based
designproceduresforairfieldoverlaysthatareresistanttoreflectivecracking.
2. FormulationofProblem
Inthispaper,weanalyzethreedimensionalreflectivecracksassuminglinearelas-
ticisotropicmaterialbehavior.Thissectionpresentsthestrongandweekformulations
for this class of problem. The solution methodology based on the Generalized FEM
(cf.Section3)is,however,applicabletoothermaterialmodels,suchasvisco-elastic
orvisco-plasticbehavior.
3-DAnalysisofReflectiveCracks 3
2.1. EquilibriumEquations
Considerathree-dimensionaldomainW withboundary¶ W decomposedas¶ W =
¶ W u ¶ W s with ¶ W u ¶ W s =0/. The domain geometry is arbitrary and may have
∪ ∩
cracks.Theequilibriumequations,constitutiveandkinematicrelationsaregivenby
sss +bbb = 000 inW (1)
▽·
sss = CCC:eee
eee = uuu
s
▽
wheresss istheCauchystresstensor,bbbarebodyforces,eee isthelinearstraintensor,
s
▽
isthesymmetricpartofthegradientoperatorandCCCistheHooke’stensor.Thematerial
propertiesareassumedconstantwithineachlayerofthepavement(cf.Figure7).
Theboundaryconditionsprescribedin¶ W areasfollows
uuu = uuu¯ on¶ W u
sss nnn = tt¯t on¶ W s
·
wherennnisanoutwardunitnormalvectoron¶ W anduuu¯ andtt¯t areprescribeddisplace-
mentsandtractions,respectively.
2.2. WeakForm
ThePrincipleofVirtualWorkfortheproblemdescribedaboveisgivenby
Finduuu H1(W )suchthat vvv H1(W )
∈ ∀ ∈
sss (uuu):eee (vvv)dxxx+h uuu vvvdsss= tt¯t vvvdsss+h uuu¯ vvvdsss (2)
ZW Z¶ W u · Z¶ W s · Z¶ W u ·
where H1(W ) is a Hilbert space defined on W and h is a penalty parameter. We use
thepenaltymethodtoimposedisplacementboundaryconditionsduetoitssimplicity
and generality. The computation of an approximation uuuh of the solution uuu using the
GeneralizedFEMisdescribedinthenextsection.
3. TheGeneralizedFiniteElementMethod
TheGeneralizedFEMcanberegardedasafiniteelementmethodwithshapefunc-
tionsbuiltusingtheconceptofapartitionofunity.Thismethodhasitsoriginsinthe
worksofBabuškaetal.[BAB94,BAB97,MEL96](underthenames“specialfinite
elementmethods”,“generalizedfinite elementmethod”and“finiteelementpartition
of unity method”) and Duarte and Oden [DUA95, DUA96b, DUA96c, DUA96a,
ODE98](underthenames“hpclouds"and"cloud-basedhpfiniteelementmethod”).
4 RoadMaterialsandPavementDesign.VolumeX-n X/2009
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Several meshfree methods proposed in recent years can also be viewed as special
casesofthepartitionofunitymethod.Detailsonthemathematicalformulationofthe
GFEM canbefoundin,e.g.,[MEL96,BAB97,DUA00,ODE98,STR01].Inthis
section,wesummarizethemainconceptsofthemethod.
The linear finite element shape functions j a , a =1,...,N, in a finite element
meshwithN nodesconstituteapartitionofunity,i.e.,
N
(cid:229) j a (xxx)=1
a =1
forallxxxinadomainW coveredbythefiniteelementmesh.Thisisakeypropertyused
inpartitionofunitymethodsand,inparticular,intheGFEM.
AGFEM shapefunctionf a i isbuiltfromtheproductofaFEshapefunctionj a
andanenrichmentfunctionLa i
f a i(xxx)=j a (xxx)La i(xxx) (nosummationona ), (3)
wherea isanodeinthefiniteelementmesh.Figure1illustratestheconstructionof
GFEMshapefunctions.
(cid:0)
L
(cid:1)(cid:2)
i
(cid:1)
(cid:3)
(cid:1)
Figure1.ComputationofaGFEMshapefunction(bottomsurface).Thefiniteelement
shapefunctionj a isshownatthetop,whiletheenrichmentfunctionLa i isshownas
the middle surface. The resulting function at the bottom, f a i is the generalized FE
shapefunction.
Thisprocedureallowstheconstructionofshapefunctionswhichcanapproximate
wellthesolutionofaproblem.Featuressuchasdiscontinuitiesormaterialinterfaces
arerepresentedbyajudiciouschoiceofenrichmentfunctionsLa i(xxx),insteadoffinite
elementmesheswithelementfacesfittingthem.
3-DAnalysisofReflectiveCracks 5
Severalenrichmentfunctionscanbehierarchicallyaddedtoanynodea inafinite
elementmesh.Thus,ifD isthenumberofenrichmentfunctionsatnodea ,theGFEM
L
approximation,uuuh,ofafunctionuuucanbewrittenas
uuuh(xxx) = (cid:229)N (cid:229)DL uuua if a i(xxx)= (cid:229)N (cid:229)DL uuua ij a (xxx)La i(xxx)
a =1i=1 a =1i=1
= (cid:229)N j a (xxx)(cid:229)DL uuua iLa i(xxx)= (cid:229)N j a (xxx)uuuah(xxx)
a =1 i=1 a =1
where uuua i, a =1,...,N, i=1,...,DL, are nodal degrees of freedom and uuuha (xxx) is
a local approximation of uuu defined on w a = xxx W :j a (xxx)=0 , the support of
thepartitionofunityfunctionj a .Inthecaseof{afi∈niteelement6part}itionofunity,the
supportw a (oftencalledacloudorapatch)isgivenbytheunionofthefiniteelements
sharingavertexnode xxxa [DUA00]. The GFEM approximation uuuh isthe solutionof
thefollowingproblem
Finduuuh XXXh(W ) H1(W )suchthat, vvvh XXXh(W )
∈ ⊂ ∀ ∈
sss (uuuh):eee (vvvh)dxxx+h uuuh vvvhdsss= hhh¯ vvvhdsss+h uuu¯ vvvhdsss (4)
ZW Z¶ W u · Z¶ W s · Z¶ W u ·
where,XXXh(W )isasubspaceofH1(W )generatedbytheGFEMshapefunctionsdefined
in(3).
The local approximations uuuah, a = 1,...,N, belong to local spaces c a (w a ) =
smpeannt{oLriab}aDis=Li1sdfuenficnteiodnosnfothreaspuaprptiocrutlsawrlao,caal=spa1c,e..c.,aN(w.Ta )hedespeleencdtisoonnotfhteheloecnarlicbhe--
havior of the function uuu over the cloud w a . The case of 3-D fractures is discussed
below.
3.1. Enrichmentfunctionsfor3-Dcracks
Inthissection,enrichmentfunctionsfor3-Dcracksarebrieflydescribed.Further
details can be found in [PER09a]. Three types of enrichment functions are consid-
ered:
(i)Enrichmentforcloudsw a thatdonotintersectthecracksurface.
In this case, the elasticity solution uuu is continuous and can be approximated by
polynomial functions. Our implementation follows [ODE98, DUA00] and the en-
richments functions {La i}Di=L1 for a cloud associated with node xxxa =(xa ,ya ,za ) is
givenby
{La i}Di=L1=(cid:26)1,(x−haxa ),(y−haya ),(z−haza ),(x−ha2xa )2,(y−h2aya )2,...(cid:27) (5)
withha beingascalingfactor[ODE98,DUA00].
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(ii)Enrichmentforcloudscompletelycutbythecracksurface.
Inthiscase,thecracksurfacedividesw a intotwosub-domains,w a+ andw a− such
thatw a =w a+ w a−.Theenrichmentsfunctionsaretakenas
∪
{La i}Di=L1=(cid:26)H,H (x−haxa ),H (y−haya ),H (z−haza ),H (x−ha2xa )2,H (y−h2aya )2,...(cid:27)
(6)
wherethefunctionH (xxx)denotesadiscontinuousfunctiondefinedby
H(xxx)= 1 ifxxx∈w a+ (7)
(cid:26) 0 otherwise
These discontinuous functions can approximate discontinuities of the elasticity
solutionuuuinsideafiniteelement.
(iii)Enrichmentforcloudsthatintersectthecrackfront.
Terms from the asymptotic expansion of the elasticity solution near crack fronts
are used as enrichment functions in clouds that intersect the crack front. Their de-
scriptionismoreelaboratedthanthepreviousonesandwerefertheinterestedreader
toreferences[PER09a,DUA00,DUA01].
With the above enrichments, the finite element mesh is not required to fit crack
surfacessincethediscontinuitiesandsingularitiesareapproximatedbytheenrichment
functions,nottheFEMmesh.ThisisincontrastwiththeFEMandgreatlyfacilitate
the solution of fracture mechanics problems in complex 3-D domains such as those
foundinairfieldpavements.ThisfeatureoftheGFEMisillustratedinFigures2and
11.
3-DAnalysisofReflectiveCracks 7
Figure2.Three-dimensionalcrackinapavement.Thecrackcanbeinsertedintoan
existingfiniteelementmeshwithoutrequiringmodificationsontheFEmeshoriginally
designedtorepresentanuncrackedpavement.Thecrackisapproximatedbydiscon-
tinuousenrichmentfunctions,nottheFEMmesh.
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4. Inclinedellipticalcrack
ThisexampleisaimedatverifyingtheGFEMmethodologyinamixedmodefrac-
ture case. The problem consists of an inclined elliptical crack of dimensions a/c=
9/15embeddedinacubeofedgesizeb,asillustratedinFigure3.Inordertoreduce
the finite domain effect on the solution, the relation c/b=3/40 is set. The Young’s
modulususedinthisexampleisE =1.0whilethePoisson’sratioissetton =0.25.
Thecracksurfacehasaninclinationa =30 withrespecttothehorizontalaxis.The
◦
domainissubjectedtoauniformtensiletractions =1intheverticaldirection.Figure
3showsthemodelandtheinitialcoarsemeshuse◦dinthisexample.Inthediscretiza-
tion of the solution, localized mesh refinement on elements that intersect the crack
frontisappliedalongwithnodalenrichments.Polynomialenrichmentoforder p=3
(cf.Eq.(5))isappliedovertheentiredomain,discontinuousstepfunctionenrichment
(cf. Eq. (6)) is applied at the nodes of the elements that are cut by the crack surface
andtheasymptoticsolutionenrichmentisusedatelementscutbythecrackfront.The
ratio of the element size, L, to crack length, a, for elements along the crack front is
0.054<L/a<0.088afterthemeshrefinementisapplied.
ThestressintensityfactorsformodesI,II,andIII ofaninclinedellipticalcrack
embedded in an infinite domain are used as references. These SIFs are given by
[TAD00]
s sin2g √p a a 2 14
KIinf. = ◦ E(k) (cid:20)sin2q +(cid:16)c(cid:17) cos2q (cid:21)
s sing cosg √p ak2 k 1
KIiInf. = − ◦ 1 (cid:20)B′cosw cosq +Csinw sinq (cid:21)
a 2 4
sin2q + cos2q
(cid:20) (cid:16)c(cid:17) (cid:21)
s sing cosg √p a(1 n )k2 1 k
KIiInIf. = ◦ − 1 (cid:20)Bcosw sinq −C′sinw cosq (cid:21)
a 2 4
sin2q + cos2q
(cid:20) (cid:16)c(cid:17) (cid:21)
whereB,C,K(k)andE(k)aredefinedas
B=(k2 n )E(k)+n kK(k), C=(k2 n k2)E(k) n k2K(k),
′ ′ ′
− − −
K(k)= p2 dj , E(k)= p2 1 k2sin2j dj ,
Z0 1 k2sin2j R0 q −
q −
andk2=1 k2, k =a/c andq is a parametric angle representing a point A onthe
′ ′
crackfront−(cf.Figure3).Fortheexamplesolvedinthissection,g =p /2 a =p /3
andw =0. −
The extracted stress intensity factors for mode I, II, and III using the GFEM
methodologyarepresentedinFigure4.Theseresultsarecomparedwiththeanalytical
3-DAnalysisofReflectiveCracks 9
(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:1)
(cid:2)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:3)(cid:9)(cid:11)(cid:12)(cid:13)(cid:10)(cid:14)(cid:3)(cid:9)(cid:15)(cid:10)(cid:16)(cid:17)(cid:3)(cid:17)(cid:18)(cid:16)(cid:19)(cid:20)
(cid:22) (cid:2)(cid:2) (cid:3)(cid:3)
(cid:21)
(cid:22) (cid:9)
(cid:26)(cid:16)(cid:27)(cid:24)(cid:18)(cid:15)(cid:25)
(cid:22)
(cid:22)
(cid:22)
(cid:23)(cid:10)(cid:16)(cid:19)(cid:17)(cid:24)(cid:18)(cid:15)(cid:25)
Figure3. Cubesubjectedtotopandbottomuniformtensiletractionsandrepresenta-
tionofinclinedellipticalcrack.
0.8
0.6
0.4
F
SI
d
e
zaliz 0.2
m
or
N
0
KI inf.
-00.22 KKIIIIIIIiinnff..
KI GFEM
KII GFEM
KIII GFEM
--00.44
0 50 100 150 200 250 300 350
Parametric Angle (deg.)
(cid:28)
Figure4.StressintensityfactorsformodesI,II,andIII.
10 RoadMaterialsandPavementDesign.VolumeX-n X/2009
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solutionforaninfinitedomainandexcellentagreementcanbeobservedbetweenthe
twosolutionsforallthreemodes.
5. AnalysisofReflectiveCracksinAirfieldPavements
Sincethepurposeofthisstudyistoanalyzeratherthandesignatypicalpavement
sectionofanairport thatservesthe Boeing777aircraftwas selectedtobe modeled.
ThemodelgeometryandpavementcrosssectionsfortheunderlyingPCCarearbitrar-
ilyselectedasthestandardsectionofrunway4L-22RatO’HareAirport.
Athree-dimensionalfiniteelementmeshwascreatedtorepresenttheairportpave-
mentanditslayers.Also,atwo-dimensionalfiniteelementmeshwascreatedtorep-
resentthegeometryofthecracksurface.Bymodelingthe3-Dpavementmeshandin-
sertingthe2-Dcracksurfacemeshjustabovetheconcretejoint,areflectivecracking
studyinasphaltoverlaywasperformedwithTheGeneralizedFiniteElementMethod
describedintheprevioussection.ModesI,IIandIIIstressintensityfactorsarecom-
putedalongthecrackfrontinordertoinvestigatethethree-dimensional,mixedmode
crackdrivingmechanismsinatypicalairfieldpavement.
5.1. ModelGeometry
ThepavementmodelencompasseshalfoftwoadjacentslabsandonePCCjointas
illustratedbytheshadedregioninFigure5.Thesimulatedairfieldpavementiscom-
posedofan20cmasphaltoverlay;23cmconcreteslabs,eachwithplandimensions
of 6 m 5.7 m (PCC), and;an 28 cmcement treatedbase (CTB). A saw cutof 13
×
mmwidthismodeledinthePCClayer.
Thedimensionsofthecracksurfaceare2.54cmofhightand5cmofdepth.The
curvedportionofthecrackhasaradiusof2.54cmandstartsandthemidpointofits
depth.Figure6showsthegeometryofthecracksurface.
5.7 m
66mm
Figure 5.Airfield pavement analyzed. The shaded square represents the portion of
pavementmodeledinthisstudy.