Table Of ContentDAMAGE MODELING OF
COMPOSITE STRUCTURES
DAMAGE
MODELING OF
COMPOSITE
STRUCTURES
Strength, Fracture, and
Finite Element Analysis
P L
ENGFEI IU
AssociateProfessor,Oceancollege,ZhejiangUniversity,
Zhoushan,China
Elsevier
Radarweg29,POBox211,1000AEAmsterdam,Netherlands
TheBoulevard,LangfordLane,Kidlington,OxfordOX51GB,UnitedKingdom
50HampshireStreet,5thFloor,Cambridge,MA02139,UnitedStates
Copyright©2021ElsevierInc.Allrightsreserved.
Nopartofthispublicationmaybereproducedortransmittedinanyformorbyanymeans,electronic
ormechanical,includingphotocopying,recording,oranyinformationstorageandretrievalsystem,
withoutpermissioninwritingfromthepublisher.Detailsonhowtoseekpermission,further
informationaboutthePublisher’spermissionspoliciesandourarrangementswithorganizations
suchastheCopyrightClearanceCenterandtheCopyrightLicensingAgency,canbefoundatour
website:www.elsevier.com/permissions.
Thisbookandtheindividualcontributionscontainedinitareprotectedundercopyrightbythe
Publisher(otherthanasmaybenotedherein).
Notices
Knowledgeandbestpracticeinthisfieldareconstantlychanging.Asnewresearchandexperience
broadenourunderstanding,changesinresearchmethods,professionalpractices,ormedical
treatmentmaybecomenecessary.
Practitionersandresearchersmustalwaysrelyontheirownexperienceandknowledgein
evaluatingandusinganyinformation,methods,compounds,orexperimentsdescribedherein.In
usingsuchinformationormethodstheyshouldbemindfuloftheirownsafetyandthesafetyof
others,includingpartiesforwhomtheyhaveaprofessionalresponsibility.
Tothefullestextentofthelaw,neitherthePublishernortheauthors,contributors,oreditors,
assumeanyliabilityforanyinjuryand/ordamagetopersonsorpropertyasamatterofproducts
liability,negligenceorotherwise,orfromanyuseoroperationofanymethods,products,
instructions,orideascontainedinthematerialherein.
LibraryofCongressCataloging-in-PublicationData
AcatalogrecordforthisbookisavailablefromtheLibraryofCongress
BritishLibraryCataloguing-in-PublicationData
AcataloguerecordforthisbookisavailablefromtheBritishLibrary
ISBN:978-0-12-820963-9
ForinformationonallElsevierpublications
visitourwebsiteathttps://www.elsevier.com/books-and-journals
Publisher:MatthewDeans
AcquisitionsEditor:DennisMcGonagle
EditorialProjectManager:FernandaA.Oliveira
ProductionProjectManager:PrasannaKalyanaraman
CoverDesigner:GregHarris
TypesetbySPiGlobal,India
Preface
In2019,Imovedtoafamouseastislandcity,Zhoushan,Zhejiangprov-
ince in China, to pursue my new teaching and research career, where
OceanCollegeofZhejiangUniversitycreatesanexcellentworkplatform
withnewoffice,originallabs,aswellasfreshairandseafood.Everything
ineach-daylifeatZhoushancampusisvaluableandinspiring,althoughit
is also quite challenging. Again, I met many enthusiastic collaborative
partners in the new discipline. Particularly, my specialty changes from
mechanical engineering, solid mechanics, chemical machinery to ocean
and ship engineering, which creates a multidisciplinary opportunity for
me. Of course, one invariable topic is just about my research direction
in thearea of advanced composite materials and structures, which obvi-
ouslyshowsstrongadaptivecapacityforvariousengineeringapplication.
Because my graduate study was on fiber-reinforced metal matrix com-
posites at School of Aeronautics and Astronautics, Zhejiang University,
Istartedtolovemagiccompositematerialswhichhassomanyadvantages
because I always believe in the philosophy that one plus one is greater
thantwo,whichareboundtoinnovatefutureindustryandpeople’slife.
Now, my research pays more attention to thermoset and thermoplastic
fiber-reinforcedresinmatrixcompositeandstructures,andIambuilding
myresearchgroupwithpostdoctoralstudents,doctoralstudents,gradu-
ate students, and undergraduates, who concentrate on some practical
composite engineering structures including composite wind turbine
blade, and composite pressure vessel. Of course, lightweight design
andmanufacturingapplicationofcompositematerialsinshipandocean
structuresdeservesfavorableexpectation,althoughserveoceanenviron-
mentsarecomplicated.Afterall,solongeastcoastlineinChinaprovides
a potential to develop lightweight composite ocean structures and tech-
niques. We believe composites will make an important breakthrough in
thefield of ship and oceanengineering.
In recent years, my group addressed theoretical modeling, numerical
analysis, and nondestructive test of laminated and adhesive composite
structures. Now, an important task is to apply these methods and tech-
niques to lightweight design, manufacturing, and in-serve performance
analysis of practical fiber-reinforced resin matrix composite structures.
In 2019, I received the book invitation from Dr. Dennis Mcgonagle on
MechanicsofMaterials,ElsevierPress.Ijustattempttomakeasummary
on our recent fundamental work although it is so tiny. Currently,
xi
xii
Preface
laminatedandadhesiveprocessingforcompositematerialsshowsadvan-
tagesthatbringsabouthighstrengthandstiffnessaswellaslowdensityin
theareasofmanufacturing,assembly,andrepairofcompositestructures.
However, failure mechanisms of composites are complicated, which
includes intralaminar damage and interlaminar delamination as well as
the failure of adhesive joints. Various damage models are introduced to
provide theoretical and techniquesupports for damagetolerance design
and lifetime prediction of composite structures, with the purpose of
reducing theweightandcost.Ofcourse,thesemodelscanbedeveloped
orarebeingimprovedinordertoreachhigh-efficiencyandhigh-fidelity
simulation as well as optimal design of composite structures and their
components. Before this work, there have been many publications on
damage models of composite laminates. However, little work integrates
both the intralaminar damage and interlaminar delamination as well as
adhesive failure of composite structures systematically under different
loads and environments, including intralaminar damage constitutive
models,failurecriteria,anddamageevolutionlawsaswellasvirtualcrack
closuretechnique,cohesivemodels,andextendedfiniteelementmethod
for predicting delamination and adhesivefailure. In addition, numerical
techniques for implementing progressive failure analysis using implicit
andexplicit finite elementanalysisaswellas some important numerical
issuesincludingnumericalconvergenceandmeshsensitivityarealsosig-
nificantlyrequiredtobedeeplydiscussed.Furthermore,multiscaledam-
agebehaviorsofcompositestructuresalsorequiremorepowerfultheories
and numerical technique. It is therefore necessary and timely to publish
thisbookthatoutlinesthelinksbetweenvariousdamagemodels,numer-
ical algorithms, and failure mechanisms of composites. Although it is
absolutely not comprehensive, the book would be partly a collection of
publications of some world-leading experts and authors in relevant
researchareas.Therefore,itwillcoveradvancedknowledgeoncomposite
mechanics, making it informative reading for the experts, and, concur-
rently, it can serve as a fruitful reference source for peers intending to
work in this area.
Iwouldappreciateallmygraduatestudents,undergraduatestudents,
andcollaborativecolleagueswhoare/weresozealousinspendingbeau-
tifultimeonacademicdiscussionandexchange aswellascollaboration.
Thisstimulatesmyresearchmotivationandconfidence,andenhancesmy
thinkingabilityprogressively.Iwouldalsoappreciatethestrongsupport
from my family—my parents, my wife, and two daughters. Finally, I
thanktheinvitationbyElsevierPress.Noretreatbutforwardformywork
andlife.
PengfeiLiu
Ocean College, Zhejiang University,Zhoushan, China
C H A P T E R
1
Cohesive models and implicit
finite element analysis for
delamination analysis of
composite laminates with
different fracture modes
1.1 Introduction
Currently, carbon/glass fiber reinforced polymer composites have
beenincreasinglyusedinareassuchasaerocraft,ocean,andnewenergy
vehiclesduetoaseries offavorableadvantagesincludinghighstiffness-
to-weight ratios and designability. Two important applications in ocean
engineering are composite wind turbine blade and composite pressure
vessel, in which the composite skin in the blade is laminated structure
and the composite pressure vessel belongs to laminated winding struc-
ture. They are generally designed and manufactured by deploying fiber
layups in different orientations for different layers. These stacked angle-
plylayersareexpectedtoresistloadsthatcomefromdifferentdirections.
However, laminated properties of composites bring about complicated
failuremechanisms.Inmostcases,interlaminardelaminationofcompos-
itescausedbyweakshearstrengthofpolymermatrixleadstoseveredeg-
radationofphysicalandmechanicalperformancesofcomposites.Besides,
delamination is usually coupled with intralaminar transverse matrix
cracking[1].
Thedelaminationissueofcompositesisessentiallyrepresentedbythe
crackinitiationandpropagation.Theenergydissipationnearthecracktip
duetocrackpropagationisthemainreasonforcatastrophicmaterialfail-
ure, and the mechanical response of composites is largely affected by
1
DamageModelingofCompositeStructures Copyright©2021ElsevierInc.Allrightsreserved.
https://doi.org/10.1016/B978-0-12-820963-9.00014-6
2
1. Cohesivemodelsandimplicitfiniteelementanalysis
interlaminarfracturebehaviors.Toanalyzeinterlaminarfractureproper-
ties,someadvancedfracturetheorieshavebeendeveloped.Forexample,
thevirtualcrackclosuretechnique(VCCT)[2–4]andthecohesivetheory,
which are increasingly applied to delamination analysis of composites.
While VCCT is efficient in calculating the energy release rate for crack
propagationwithinthelinearelasticfracturemechanics,itcannotpredict
the crack initiation. By comparison, Dugdale and Barenblatt [5, 6] origi-
nally proposed the cohesive process zone conception near the crack tip
that describes discrete fracture as the material separation across the sur-
face.Inthecohesivemodel,thenonlinearfracturebehaviorsarecontrolled
by an assumed traction-displacement jump (or called separation) law,
which can be essentially considered as a projection of continuous bulk
constitutivemodelintothediscontinuousinterface,accordingtothevar-
iationalstressboundary/initialvalueproblemwithaninterfacedisconti-
nuity. One of the major advantages for the cohesive model is that it can
predict both the crack initiation and propagation, effectively. Besides, it
can avoid the consideration for the complicated stress singularity at the
crack tip. If the width of the fracture zone in this discrete representation
isapproximatedtobezero,thelengthscaleassociatedwithfailureisinfi-
nitely smaller than that in thestructure [7].
Now, many popular cohesive models were developed to predict the
delamination crack initiation and propagation of composite laminates
fromthephenomenologicalperspective.Particularly,numericalapplica-
tion of cohesive models in delamination analysis of composites was
reviewed [8–12], which were expected to represent true governed
traction-displacementjumpconstitutiverelationshipsandenergydissipa-
tionmechanismsduetocrackpropagationtolargeextentsthataregener-
allyidentified by experimentalapproaches. On one hand, many triangle
cohesive models were developed for mixed-mode fracture delamination
of composites by many researchers [13–22]. Specially, a discrete triangle
cohesive modelfor delamination analysis wasdeveloped[23].Although
the triangle cohesive models have achieved success in predicting the
delaminationgrowthofcomposites,theycannotrepresentaccuratelytrue
nonlinearshapeofcohesivezonesuchasthefiberbridgingmechanismsin
composites.Ontheotherhand,thepotential-basedexponentialcohesive
models for mixed-mode fracture were developed[24–26].
Currently,theexponentialcohesivemodels[24–26]andthepolynomial
cohesive models [27] have been used to represent nonlinear cohesive
behaviors. The advantage of the exponential cohesive models is that the
exponential function is continuous for the traction-displacement jump
cohesive relationship and their derivatives. It could be the reason for
theexponentialcohesivemodeltobecomethemostpopularforthemath-
ematical derivation and finite element implementation. In order to con-
sider further the damage effect, we develop a damage-type exponential
3
1.2 Stressboundaryvalueproblemwithaninterfacediscontinuity
cohesive model to predict mixed-mode delamination of composite lami-
nates, which introduces damage variables to describe the irreversible
damage evolution behaviors during the delamination of composites.
Numerical convergence using finite element analysis (FEA) is solved by
introducingtheviscouseffectandthemeshsizeeffectissolvedbyfollow-
ing the crack band model [28]. The four typical delamination cases:
(a) double cantilever beam (DCB), (b) end notch flexure (ENF),
(c)mixed-modebendingfracture(MMB),and(d)fixedratiomixedmode
(FRMM)areusedtovalidatethecohesivemodel.FortheMMBandFRMM
cases,thedamageevolutionlocusbetweenindividualfailuremodeisfur-
ther discussed.
1.2 Stress boundary value problem with an interface
discontinuity
LetusconsideraboundeddomainΩ(cid:1)ℝ3occupiedbyaninitiallylin-
earelasticbody,asshowninFig.1.1.Theboundary∂ΩofthedomainΩ
withtheoutwardnormalnΩissubdividedintoeitherNeumannorDirich-
let boundary conditions defined on ∂Ω and ∂Ω , respectively, such that
t u
∂Ω¼∂Ω [∂Ω and∂Ω \∂Ω ¼∅. Further, Ω is divided into Ω+ and Ω(cid:3)
u t u t
byaninternaldiscontinuityΓ suchthatΩ¼Ω+[Ω(cid:3)inthereferencecon-
d
figuration.ThedisplacementfielduacrossΓ isdiscontinuousbutcontin-
d
uous on Ω+and Ω(cid:3).
Neglecting theinertia and body force, thestrong form is given by the
balance of linear momentum and theboundary conditions
8
>><—(cid:4)σ¼0, in Ω+[Ω(cid:3)
u¼u , on ∂Ω
b u (1.1)
>>:σ(cid:4)nΩ¼Tb, on ∂Ωt
σ(cid:4)N¼tð½½u(cid:5)(cid:5)Þ, on Γ
d
Tb ∂Ω
t
Γ–
d
Γ–
d
Ω– Point X–
Γd+ Ω+ ∂Ωu
nΩ
N Point X+
FIG.1.1
Boundaryvalueproblemwithaninterfacediscontinuity.
4
1. Cohesivemodelsandimplicitfiniteelementanalysis
whereu :∂Ω !ℝ3isthefieldofprescribeddisplacementontheDirichlet
b u
boundaryandT :∂Ω !ℝ3isthefieldofprescribedtractionontheNeu-
b t
mann boundary. The Cauchy stress tensor is denoted by σ. The Cauchy
cohesive traction t which acts on the internal boundary Γ depends on
d
thedisplacementjump½½u(cid:5)(cid:5)attheinterfaceΓ .Thereferencenormalvector
d
Npoints from Ω(cid:3)to Ω+.
ThevirtualworkequationforthebodywithvolumeVcontainingthe
cohesive interface Γ isgiven by the following weak form
ð d ð ð
— δu:σdV+ t(cid:4)δ½½u(cid:5)(cid:5)dA¼ δu(cid:4)T dA (1.2)
X b
Ω+[Ω(cid:3) Γ ∂Ω
d t
1.3 Cohesive modeling for mixed-mode delamination of
composite laminates
The exponential cohesive traction-displacement jump relationship for
interfacefailureisused,asshowninFig.1.2.Theexpressionforeachfail-
ure mode is given by.
t ¼etc=½½u(cid:5)(cid:5)c½½u(cid:5)(cid:5)expð(cid:3)½½u(cid:5)(cid:5)=½½u(cid:5)(cid:5)cÞ, ði¼1,2,3Þ (1.3)
i i i i i i
where i¼1, 2, and 3 represent the normal and two tangential directions,
respectively. σ ¼tc and τ ¼tc¼tc are the maximum tractions in
max 3 max 1 2
thenormal and tangential directions, respectively.
Thefracturetoughnessforeachfracturemodeisequaltotheareaunder
thetraction-displacement jump curve
ð
∞
t d½½u (cid:5)(cid:5)¼eσ ½½u (cid:5)(cid:5)c¼Gc (1.4)
1 1 max 1 1
0
ð ð
∞ ∞
t d½½u (cid:5)(cid:5)¼ t d½½u (cid:5)(cid:5)¼eτ ½½u (cid:5)(cid:5)c¼eτ ½½u (cid:5)(cid:5)c¼Gc¼Gc (1.5)
2 2 3 3 max 2 max 3 2 3
0 0
whereGc,Gc,andGcarethefracturetoughnessforfracturemode-I,II,and
1 2 3
III,respectively.
FIG.1.2
Exponentialtraction-displacementjumplaw.
5
1.3 Cohesivemodelingformixed-modedelaminationofcompositelaminates
1.3.1 Single-mode delamination model based on continuum
damage mechanics
Based on Eq. (1.3), the damaged cohesive traction-displacement jump
relationship isdeveloped
(cid:2) (cid:3)
½½u(cid:5)(cid:5) 1
t ¼etc=½½u(cid:5)(cid:5)c½½u(cid:5)(cid:5)exp (cid:3) i , ði¼1,2,3Þ (1.6)
i i i i ½½u(cid:5)(cid:5)c1(cid:3)d
i i
where d are damage variables.
i
The Helmholtz free energy ψ iswrittenas
1
ψ¼ψeðt,½½u(cid:5)(cid:5),d,TÞ+ψdðκ,TÞ, ψdðκÞ¼ aκ(cid:4)κ (1.7)
2
whereψeandψdarethefreeenergyrepresentingtheelasticdeformation
and damage evolvement of interface, respectively. T is the temperature
andκ isa vector of internal variable. aisa materialparameter.
Accordingtothethermodynamictheory,therelationshipsbetweenthe
thermodynamic conjugate forces (Y,B) and internal variables (d,κ) are
given by
∂ψ ∂ψ ∂ψ
t¼ , Y¼(cid:3) , B¼ ¼aκ (1.8)
∂½½u(cid:5)(cid:5) ∂d ∂κ
TheClausius-Duhemdissipationinequalityimposedbythecontinuum
thermodynamics on theevolution laws is[29].
γ¼t(cid:4)½½u_(cid:5)(cid:5)+Y(cid:4)d_ (cid:3)B(cid:4)κ_(cid:3)1—T(cid:4)q(cid:6)0 (1.9)
T
where γ isthe power of dissipationdue to damage. q isthe heat flux.
Ifthedissipationγreachesthemaximum,thedamageevolutionlawis
given by
∂∂Yγ ¼0)d_ ¼λ_d∂∂FYd, ∂∂Bγ ¼0)κ_ ¼(cid:3)λ_d∂∂FBd (1.10)
where Fd(Y,d,B) is the damage potential function and λd is the damage
consistencyparameter.TheyareassumedtoobeytheKuhn-Tuckerload-
ing/uploading conditions [29].
λ_d(cid:6)0, Fd(cid:7)0, λ_dFd ¼0 (1.11)
Here,adamagepotentialfunctionFd(Y,d,B)isproposedforeachfail-
ure mode ifthe temperature effect isneglected
FdðY,d,BÞ¼Y(cid:3)Y (cid:3)ðB(cid:3)B Þ (1.12)
c c
FromEqs.(1.11),(1.12),thedamagecriterionisnotsatisfiedandλ_d50
holdsatFd<0whilethefurtherdamageappearsandλ_d>0holdsatFd¼0.
