A Micromechanics-Based Damage Model For The Strength Prediction of Composite Laminates

0.61 MB·English

by  Camanho, Pedro P.

IIIEuropeanConferenceonComputationalMechanics
Solids,StructuresandCoupledProblemsinEngineering
C.A.MotaSoareset.al.(eds.)
Lisbon,Portugal,5–8June2006
A MICROMECHANICS-BASED DAMAGE MODEL FOR THE
STRENGTH PREDICTION OF COMPOSITE LAMINATES
PedroP.Camanho1,JoanA.Mayugo2,PereMaim´ı2 andCarlosG.Da´vila3
1DEMEGI,FaculdadedeEngenharia,UniversidadedoPorto
RuaDr. RobertoFrias,4200-465,Porto,Portugal
e-mail: [email protected]
2 AMADE,EscolaPolite`cnicaSuperior,UniversitatdeGirona
CampusMontilivi,17071Girona,Spain
e-mail: {pere.maimi,ja.mayugo}@udg.es
3NASALangleyResearchCenter
Hampton,VA,U.S.A.
e-mail: [email protected]
Keywords: ContinuumDamageMechanics,Fracture,CompositeMaterials.
Abstract. A new damage model based on a micromechanical analysis of cracked [±θ◦/90◦]
n s
laminatessubjectedtomultiaxialloadsisproposed. Themodelpredictstheonsetandaccumu-
lation of transverse matrix cracks in uniformly stressed laminates, the effect of matrix cracks
on the stiffness of the laminate, as well as the ultimate failure of the laminate. The model also
accounts for the effect of the ply thickness on the ply strength. Predictions relating the elastic
propertiesofseverallaminatesandmultiaxialloadsarepresented.
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PedroP.Camanho,JoanA.Mayugo,PereMaim´ıandCarlosG.Da´vila
1 INTRODUCTION
Theaerospace industry is committed to improvethe performance of aircraft whilst reducing
emissionsandweight. Suchagoalcanbeachievedbytheuseofadvancedpolymer-basedcom-
posite materials, that have excellent properties for aerospace applications, such as low density,
andfatigueandcorrosionresistance.
The design procedure used for advanced composite structures relies on a ’building-block’
approach[1],wherealargenumberofexperimentaltestsareperformedthroughouttheproduct
development process. The use of improved analytical or numerical models in the prediction
of the mechanical behavior of composite structures can significantly reduce the cost of such
structures. Such models should predict the onset of material degradation, the effect of the non-
criticaldamagemechanismsonthestiffnessofthelaminate,andultimatestructuralfailure.
Strength-based failure criteria are commonly used to predict failure in composite materi-
als [2]-[7]. Failure criteria predict the onset of the several damage mechanisms occurring in
compositesand,dependingonthelaminate,geometryandloadingconditions,mayalsopredict
structuralcollapse.
In multidirectional composite laminates, damage accumulates during the loading process.
Final failure occurs as a result of damage accumulation and stress re-distribution. The ulti-
mate failure load is higher than the initial damage predicted by strength-based failure criteria.
Furthermore, stress- or strain-based failure criteria cannot represent size effects that occur in
quasi-brittlematerials[8].
Simplified models, such as the ply discount method where some scalar components of the
stiffnesstensorarereducedtoapproximatelyzerowhendamageispredicted,areoftenusedby
theindustrytopredictultimatelaminatefailure. However,thesemethodscannotrepresentwith
satisfactory accuracy the progressive reduction of the stiffness of a laminate as a result of the
accumulationofmatrixcracks.
Constitutive laws based on Continuum Damage Mechanics (CDM) have been proposed to
predict the material response, from the onset of damage up to final collapse [9]-[14]. Although
the existing CDM models can predict accurately the evolution of damage, they usually rely
on fitting parameters that need to be measured at laminate level, such as the critical values of
thermodynamicforces[14].
Three-dimensionalmodelsbasedonCDMusematerialpropertiesmeasuredattheplylevel.
Damage mechanisms such as transverse matrix cracks are represented by strain-softening con-
stitutive models and bands of localized deformation. The implementation of strain-softening
models in finite element models causes convergence difficulties during the iterative solution
procedure. Furthermore, strain-softening models require regularization techniques to provide
mesh-independentsolutions[15,16].
Alternative methods based on the combination of elastic analysis of cracked plies and finite
FractureMechanicsprovidethebasisforanaccuraterepresentationoftheresponseofcompos-
ite materials [17]-[28]. Micromechanical models have been developed to predict the initiation
andevolutionoftransversematrixcracksundereitherin-planeshearortransversetension. Gen-
eralizationsofthesemodelsarerequiredforthemoreusualcaseofmultiaxialloading.
Inordertopredictultimatefailure,amicromechanicalmodelneedstobeusedincombination
with a fiber failure criterion. Furthermore, a general methodology to predict laminate strength
should be able to predict matrix cracking under high values of transverse compression. These
damagemechanismsusuallycorrespondtothefinalfailureoflaminatesunderuniformstatesof
stress[6].
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PedroP.Camanho,JoanA.Mayugo,PereMaim´ıandCarlosG.Da´vila
The objective of this work is to define a new damage model based on micromechanical
modelsoftransversematrixcrackstopredicttheonsetandevolutionoftransversematrixcracks
under multiaxial loading. A new constitutive model is derived based on the thermodynamics
of irreversible processes. The model proposed herein represents transverse matrix cracks as
distributeddamage. Themodelisabletopredicttheonsetandpropagationofmatrixtransverse
cracksundermultiaxialloadingaswellasthefinalfailureofuniformlystressedlaminateswhere
aperiodicdistributionoftransversematrixcrackscanbeassumed.
2 MICROMECHANICALMODEL
The proposed continuum damage model is based on two major components: a set of stress
basedfailurecriteriaandamicromechanicalmodeloftransversematrixcrackinginmultidirec-
tionallaminates.
The failure criteria define the onset of transverse matrix cracking, i.e. the activation of the
damage variables. A micromechanical model of transverse matrix cracks is required to define
theevolutionofthedamagevariables.
Several micromechanical models of transverse matrix cracks that have been proposed in the
literature [19, 28] can be used within the framework developed here. The micromechanical
model proposed by Tan and Nuismer [17, 18] accounts for the effects of the adjoining plies on
the homogenized elastic properties of a cracked ply. This micromechanical model is the basis
ofthedevelopmentspresentedinthiswork.
Using the assumption of generalized plane strain, Tan and Nuismer [17, 18] developed a
model able to relate the density of transverse matrix cracks in a central 90o ply to the homoge-
nized elastic properties of that ply. The model developed by Tan and Nuismer was used for the
predictionoftheevolutionoftransversematrixcracksundereitherin-planeshearortransverse
tensilestresses.
The laminates under investigation are symmetric and balanced with a [±θ/90o] layup con-
n s
taining a periodic distribution of transverse matrix cracks, as shown in Figure 1. The micro-
mechanical analysis of a balanced symmetrical laminate requires the division of the laminate
in two sub-laminates: the 90o layers in the middle layer (sublaminate 1), and the outer plies
(sublaminate2).
x
L n
0
L q,2 e 9 ,1 et
t a
a n
nim im
albus albus
z h(2) h(1)
h
Figure1: One-quarterofthelaminate.
Sublaminate 1 is taken as transversely isotropic. The outer layers, sublaminate 2, may con-
tainseverallayerswithdifferentfiberorientationsbutmustbeorthotropicinthelaminateglobal
system(x,yandzcoordinates).
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PedroP.Camanho,JoanA.Mayugo,PereMaim´ıandCarlosG.Da´vila
2.1 CONSTITUTIVETENSOR
ThestiffnesstensorofthebalancedandsymmetriclaminateshowninFigure1canbewritten
asafunctionofthedensityoftransversematrixcracks(1/L)inthesublaminate1:
 
¯ ¯
A(L) A(L) 0
11 12
A¯(L) = A¯(L) A¯(L) 0  (1)
21 22
¯
0 0 A(L)
66
¯
wheretheterms A (L),i,j = 1,2,6areobtainedfromtheTanandNuismermodel[18].
ij
The undamaged stiffness matrix Q¯ of the laminate is obtained from lamination theory using
the undamaged stiffness matrix of sublaminate 1, Q¯(1), and the stiffness matrix of sublaminate
2,Q¯(2):
(cid:179) (cid:180)
1
Q(1) = h¯Q¯ −h(2)Q(2) (2)
ij h(1) ij ij
withh¯ = h(1) +h(2).
Assumingthatthedegradationduetothetransversematrixcracksonlyoccurinsublaminate
1,thedamagedstiffnesstensoroflaminate1isgivenas:
(cid:179) (cid:180)
1
A¯(1)(L) = h¯A¯ (L)−h(2)Q(2) (3)
ij h(1) ij ij
whereA¯(1)(L)(i,j = 1,2,6)arethescalarcomponentsofthestiffnesstensorofsublaminate1.
ij
These components are a function of the distance between transverse matrix cracks (L, defined
inFigure1).
Having defined A¯(1)(L), it is possible to calculate the effective transverse modulus, Poisson
ij
ratio,andshearmodulusofthe90◦ ply:
(cid:104) (cid:105)
2
A¯(1)(L)
12
E¯(1)(L) = E¯(1)(L) = A¯(1)(L)− (4)
2 x 11 A¯(1)(L)
22
A¯(1)(L)
υ¯(1)(L) = υ¯(1)(L) = 12 (5)
21 xy A¯(1)(L)
22
G¯(1)(L) = G¯(1)(L) = A¯(1)(L) (6)
12 xy 66
The quotient υ21 is not a function of damage. This observation is in agreement with several
E2
othermodels,suchastheonesproposedbyLawsetal. [23,24],andNguyen[25].
The plane stress compliance tensor of the damaged sublaminate 1, H(1), only contains two
componentsthatdependonthedensityoftransversematrixcracks: H(1)(L)andH(1)(L). The
22 66
tensorH(1),isestablishedasafunctionofthedensityoftransversematrixcracks, L,as:
 
1 −υ21 0
 E1 E2 
H(1)= −υ12 H(1)(L) 0  (7)
E1 22
0 0 H(1)(L)
66
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PedroP.Camanho,JoanA.Mayugo,PereMaim´ıandCarlosG.Da´vila
with:
1 A¯(1)(L)
H(1)(L) = = 22 (cid:104) (cid:105) (8)
22 E¯2(1)(L) A¯(1)(L)A¯(1)(L)− A¯(1)(L) 2
11 22 12
1 1
H(1)(L) = = (9)
66 G¯(1)(L) A¯(1)(L)
12 66
The evolution of the homogenized elastic properties of the cracked ply (sublaminate 1) as
a function of β = 1/(2L) is shown in Figure 2 for a [±25◦/90◦] laminate with the material
3 s
propertiesshowninTable1.
E (GPa) E (GPa) G (GPa) υ
1 2 12 12
163.4 11.9 6.2 0.30
Table1: Elasticpropertiesoftypicalcarbon/epoxycomposite.
Figure2: Effectiveelasticpropertiesofthe90◦plyasafunctionofthedensityofmatrixcracks.
Figure2showsthattheYoungmodulusofsublaminate1,E ,isnotaffectedbythepresence
1
of transverse matrix cracks. Furthermore, the curves E (β)/E and υ (β)/υ are coincident
2 2 21 21
(Figure2).
2.2 ONSETOFMATRIXTRANSVERSECRACKS
Theonsetoftransversematrixcrackinginaplyunderthecombinedeffectofin-planeshear
stresses and transverse tensile stresses needs to be predicted using an appropriate failure crite-
rion. Thefailurecriterionshouldbeestablishedintermsoftheactualstrengthsofaplywhenit
isembeddedinamultidirectionallaminate(in-situstrengths).
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PedroP.Camanho,JoanA.Mayugo,PereMaim´ıandCarlosG.Da´vila
2.2.1 In-situstrengths
A methodology to predict the onset of matrix transverse cracking must be able to predict
the in-situ strengths of a ply. Both the transverse tensile and in-plane shear strengths of a ply
embedded in a multidirectional laminate are higher than the corresponding values measured in
unidirectionallaminates[29].
The thick models described in [30] are used. The tensile transverse in-situ strengths of
sublaminate(1)is[30]:
(cid:115)
8G
Forathinembeddedply: Y = Ic (10)
T πtΛo
22
√
Forathickply: Y = 1.12 2Yud (11)
T T
where Yud is the tensile transverse strength measured in a unidirectional test specimen, t is the
T
thicknessofsublaminate1,G isthemodeIintralaminarfracturetoughnessandΛ◦ isdefined
Ic 22
as:
(cid:181) (cid:182)
1 υ2
Λ◦ = 2 − 21 (12)
22 E E
2 1
Thein-situshearstrengthsareobtainedas[29]:
(cid:115)
(1+χφG2 )1/2 −1
S = 12 (13)
L
3χG
12
whereχistheshearresponsefactordefinedin[29],andtheparameterφiscalculatedaccording
totheconfigurationoftheply:
(cid:161) (cid:162)
12 Sud 2 (cid:161) (cid:162)
Forathickply: φ = L +18χ Sud 4
G L
12
48G
Forathinply: φ = IIc (14)
πt
where Sud is the shear strength measured using an unidirectional test specimen and G is the
L IIc
modeIIfracturetoughness.
2.2.2 Failurecriterionforthepredictionoftransversecrackingundermultiaxialloading
In general, a ply represented by sublaminate 1 in Figure 1 is subjected to transverse tensile
stresses and in-plane shear stresses. Under pure in-plane shear or pure transverse tension, the
onset of transverse matrix cracking is predicted by comparing the components of the stress
tensorwiththerespectivein-situstrengths(definedintheprevioussection).
Under multiaxial loading, it is necessary to use a failure criterion to predict the onset of
matrix cracking. The LaRC04 [4, 5] failure criteria are a function of the components of the
stresstensorandin-situstrengths. Fortransversetension,thecriterionusedis:
σ σ σ
(1−g) 22 +g( 22)2 +( 12)2 −1 ≤ 0withσ ≥ 0 (15)
22
Y Y S
T T L
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PedroP.Camanho,JoanA.Mayugo,PereMaim´ıandCarlosG.Da´vila
where g = GIc. G is the mode II component of the fracture toughness associated with matrix
GIIc IIc
transversecracking.
Under moderate values of transverse compression, the cracks propagate along the laminate
thicknessdirection,andtheLaRC04failurecriterionis:
1 (cid:173) (cid:174)
|σ |+ηLσ −1 ≤ 0withσ < 0 (16)
12 22 22
S
L
whereηL isthecoefficientoflongitudinalinfluencedefinedin[4,5].
2.3 Evolutionofmatrixtransversecracksundermultiaxialloading
2.3.1 Transversetension
Tan and Nuismer [18] obtained a closed-form expression that defines the evolution of trans-
versematrixcracksunderuniaxialstressstates(eithertransversetensionorin-planeshear).
To predict failure under multiaxial loading, i.e. when the lamina is simultaneously under
tensile and in-plane shear strains, ε and γ respectively, it is necessary to derive a relation
xx xy
between the density of transverse matrix cracks and the applied multiaxial strain state. It is
assumedthattherelationbetweenthetensileandshearstrainsisconstantthroughouttheloading
history:
γ
xy
κ = withε > 0 (17)
xx
ε
xx
whereκisthemultiaxialstrainratio.
Consideraplywithcrackdensity L,asshowninFigure 3a).
2L L L
a) Crack density 2L b) Crack density L
Figure3: Progressionoftransversematrixcracks.
Thestrainenergyinalaminatecelloflength2Lsubjectedtotransversetensionandin-plane
shear just prior to fracture, U , can be established as a function of the strain tensor and of the
2L
crackdensityas:
(cid:163) (cid:164)
U = 2hLb E (L)ε2 +A (L)γ2 (18)
2L x xx 66 xy
where E (L) is the axial stiffness of the laminate with a crack density of 1/(2L) and b is the
x
specimenwidth;A (L)isthelaminateeffectiveshearstiffnesscorrespondingtoacrackdensity
66
of 1/(2L), 2h and 2L are the laminate thickness and the distance between two consecutive
transversematrixcracks,respectively.
Afterthepropagationoftransversematrixcracks,Figure3b),thestrainenergyintheoriginal
unitcelloflength 2Lis:
(cid:183) (cid:181) (cid:182) (cid:181) (cid:182) (cid:184)
L L
U = 2hLb E ε2 +A γ2 (19)
L x 2 xx 66 2 xy
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PedroP.Camanho,JoanA.Mayugo,PereMaim´ıandCarlosG.Da´vila
where E (L/2) and A (L/2) are respectively the axial stiffness and the laminate effective
x 66
shearstiffnesscorrespondingtoacrackdensitydefinedby L.
The energy required to generate a new matrix crack in a ply equals the loss of strain energy
ofthelaminate[18]. Therefore,thedifferencebetweenequation(18)andequation(19)isequal
totheenergyreleasedbythesublaminate1:
∆U = U −U =
2L (cid:189)(cid:183)L (cid:181) (cid:182)(cid:184) (cid:183) (cid:181) (cid:182)(cid:184) (cid:190)
L L
= 2hLb E (L)−E ε2 + A (L)−A γ2 (20)
x x 2 xx 66 66 2 xy
∆U = 2h(1)bG = 2h(1)bG +2h(1)bG (21)
c I II
where G is the mixed-mode fracture toughness of sublaminate 1 under tensile (mode I) and
c
shear(modeII)loading. From(20)and(21):
(cid:183) (cid:181) (cid:182)(cid:184) (cid:183) (cid:181) (cid:182)(cid:184)
L L h(1)G
E (L)−E ε2 + A (L)−A γ2 = c (22)
x x 2 xx 66 66 2 xy hL
Using the definition of κ given in (17), equation (22) can be re-written as a function of the
strains:
(cid:189)(cid:181) (cid:181) (cid:182)(cid:182) (cid:183) (cid:181) (cid:182)(cid:184)(cid:190)
L L h(1)G
E (L)−E +κ2 A (L)−A ε2 = c (23)
x x 2 66 66 2 xx hL
or:
(cid:189) (cid:183) (cid:181) (cid:182)(cid:184) (cid:183) (cid:181) (cid:182)(cid:184)(cid:190)
1 L L h(1)G
E (L)−E + A (L)−A γ2 = c (24)
κ2 x x 2 66 66 2 xy hL
Therelationsbetweenthenormalandshearstrainsandthecrackdensityareobtainedas:
(cid:115)
h(1)G 1
ε = c(cid:163) (cid:161) (cid:162)(cid:164) (cid:163) (cid:161) (cid:162)(cid:164) (25)
xx hL E (L)−E L +κ2 A (L)−A L
x x 2 66 66 2
(cid:115)
h(1)G κ2
γ = c(cid:163) (cid:161) (cid:162)(cid:164) (cid:163) (cid:161) (cid:162)(cid:164) = κε (26)
xy hL E (L)−E L +κ2 A (L)−A L xx
x x 2 66 66 2
Equations(25)and(26),areestablishedasafunctionofthemixed-modefracturetoughness
G thatneedstobedefined.
c
The mixed-mode fracture toughness is defined in terms of the mode I and mode II compo-
nentsas:
G = G +G (27)
c I II
From(27)and(22):
(cid:183) (cid:181) (cid:182)(cid:184)
h(1)G L
I = E (L)−E ε2 (28)
hL x x 2 xx
(cid:183) (cid:181) (cid:182)(cid:184)
h(1)G L
II = A (L)−A γ2 (29)
hL 66 66 2 xy
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PedroP.Camanho,JoanA.Mayugo,PereMaim´ıandCarlosG.Da´vila
Dividingequations(28)and(29)by(23)and(24),respectively:
(cid:163) (cid:161) (cid:162)(cid:164)
G E (L)−E L
A = I = (cid:163) (cid:161) x(cid:162)(cid:164) (cid:163)x 2 (cid:161) (cid:162)(cid:164) (30)
I G E (L)−E L +κ2 A (L)−A L
c x x 2 66 66 2
(cid:163) (cid:161) (cid:162)(cid:164)
G κ2 A (L)−A L
A = II = (cid:163) (cid:161) 6(cid:162)6(cid:164) (cid:163) 66 2 (cid:161) (cid:162)(cid:164) (31)
II G E (L)−E L +κ2 A (L)−A L
c x x 2 66 66 2
A andA aretheratiosbetweenthemodeIandmodeIIcomponentsoftheenergyrelease
I II
rateandthemixed-modefracturetoughness.
The criterion proposed by Hahn [31] for the prediction of transverse matrix cracking under
transversetensileandin-planeshearloadsisused:
(cid:114)
G G G
(1−g) IIc +g I + II = 1 (32)
G G G
Ic Ic IIc
Substitutingequations(30)and(31)into(32)gives:
(cid:114)
A G A G (1−A )G
I c I c I c
(1−g) +g + = 1 (33)
G G G
Ic Ic IIc
Thepositiverealsolutionof(33)is:
(cid:195) (cid:115) (cid:33)
A (G −G )2 4 G G
G = G + I Ic IIc 1− 1+ Ic IIc (34)
c IIc 2 G A (G −G )2
Ic I Ic IIc
where A depends on the density of transverse matrix cracks, β, and on the multiaxial strain
I
ratio,κ(equation(30)).
Figure 4 shows the relation between the crack density β and ε for different multiaxial
xx
strainratios. A[±25◦/90◦] laminatewiththeelasticpropertiesshowninTable 1isusedinthe
3 s
predictions.
Figure4: Relationbetweenappliedstrain(ε andκ)anddensityofmatrixcracks.
xx
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PedroP.Camanho,JoanA.Mayugo,PereMaim´ıandCarlosG.Da´vila
The effects of multiaxial strain states on the crack density are clearly shown in Figure 4.
It can be observed in Figure 4 that the density of transverse matrix cracks increases with the
multiaxialstrainratioforafixedvalueof ε .
xx
2.3.2 Transversecompression
Transverse matrix cracks created by a combination of transverse compression and in-plane
shearcloseunder theeffectofthe compressivetransversestress. Whena crackcloses, itsfaces
can transmit normal tractions but shear tractions may cause slippage between the crack faces.
Therefore, it can be assumed that transverse matrix cracks only affect the shear stiffness of a
laminate.
Following the procedure described in the previous section, the energy released by the sub-
laminate1is:
(cid:183) (cid:181) (cid:182)(cid:184)
L
∆U = 2hLbγ2 A (L)−A = 2h(1)bG (35)
xy 66 66 2 IIc
Therelationbetweentheshearstrainandthecrackdensityisobtainedas:
(cid:115)
h(1)G
γ = (cid:163) IIc (cid:161) (cid:162)(cid:164) (36)
xy hL A (L)−A L
66 66 2
3 MICROMECHANICS-BASEDDAMAGEMODEL
3.1 Constitutivemodel
A damage model able to represent the onset and accumulation of a periodic distribution of
transverse matrix cracks should yield a compliance tensor similar to the one obtained from
the micromechanical model, equation (7). An appropriate damage model can be developed by
definingtheGibbsfreeenergyperunitvolume, Ψ ,as:
G
(cid:183) (cid:181) (cid:182) (cid:184)
1 σ2 σ2 σ2 υ υ
Ψ = 11 + 22 + 12 − 21 + 12 σ σ +
G 11 22
2 E (1−d )E (1−d )G E E
1 2 2 6 12 2 1
+(α σ +α σ )∆T +(β σ +β σ )∆M (37)
11 11 22 22 11 11 22 22
whereE ,E ,υ ,υ andG arethein-planeelasticorthotropicpropertiesofaunidirectional
1 2 12 21 12
lamina. The subscript 1 denotes the longitudinal (fiber) direction and 2 denotes the transverse
(matrix) direction. d and d are damage variables associated with transverse matrix crack-
2 6
ing. α and α are the coefficients of thermal expansion in the longitudinal and transverse
11 22
directions, respectively. β and β are the coefficients of hygroscopic expansion in the lon-
11 22
gitudinal and transverse directions, respectively. ∆T and ∆M are respectively the changes in
temperatureandmoisturecontentfromthestress-freestate.
Theproposedmodeldistinguishesbetweenactive(d )andpassivedamage(d )variables,
2+ 2−
correspondingtotheopeningorclosureoftransversematrixcracksunderloadreversalrespec-
tively. Toaccountfortheactivedamageundereitheropeningorclosure,thefollowingequation
isproposed:
(cid:104)σ (cid:105) (cid:104)−σ (cid:105)
22 22
d = d +d (38)
2 2+ 2−
|σ | |σ |
22 22
10