modelling matrix multi-cracking evolution of ......modelling matrix multi-cracking evolution of...

Ceramics-Silikáty 63 (1), 21-31 (2019)www.ceramics-silikaty.cz doi: 10.13168/cs.2018.0042

Ceramics – Silikáty 63 (1) 21-31 (2019) 21

MODELLING MATRIX MULTI-CRACKING EVOLUTION OFFIBRE-REINFORCED CERAMIC-MATRIX COMPOSITES

CONSIDERING FIBRE FRACTURELI LONGBIAO

College of Civil Aviation, Nanjing University of Aeronautics and AstronauticsNo.29 Yudao St., Nanjing 210016, PR China

E-mail: [email protected]

Submitted May 1, 2018; accepted July 17, 2018

Keywords: Ceramic-matrix composites (CMCs), Matrix multi-cracking, Interface debonding, Fibre fracture

In this paper, the matrix multi-cracking evolution of fibre-reinforced ceramic-matrix composites (CMCs) considering fibre fracture have been investigated using the critical matrix strain energy criterion. The shear-lag model combined with the fibre fracture model and fibre/matrix interface debonding criterion is adopted to analyse the fibre and matrix axial stress distribution inside the damaged composite. The effects of the fibre volume fraction, the fibre/matrix interface shear stress, the fibre/matrix interface debonded energy, the fibre Weibull modulus and the fibre strength on the stress-dependent matrix multi-cracking development are discussed. The experimental matrix multi-cracking evolution of the unidirectional SiC/CAS, SiC/CAS-II, SiC/SiC, SiC/Borosilicate and mini-SiC/SiC composites are predicted.

INTRODUCTION

Ceramic materials possess high specific strengthand specific modulus at elevated temperatures. Buttheir use as structural components is severely limitedbecauseoftheirbrittleness.Continuousfibre-reinforcedceramic-matrix composites (CMCs), by incorporating fibres in ceramic matrices, not only exploit theirattractivehigh-temperaturestrength,butalsoreducethepropensity for catastrophic failure [1, 2]. These materials have already been implemented on some aero enginecomponents[3].Theenvironmentinsidethehotsectionof the components is harsh and a composite is typically subjected to complex thermomechanical loading, which can lead to matrix multi-cracking [4, 5]. These matrix cracks form paths for the ingress in the environmentoxidising the fibres and leading to premature failure [6-9]. The density and openings of these cracks depend on thefibrearchitecture,thefibre/matrixinterfacebondingintensity and the applied load [10]. It is important to developanunderstandingofthematrixmulti-crackingdamagemechanismstoanalysetheoxidationbehaviourinside of the CMCs. [11] Many researchers performed experimental and theoretical investigationson thematrixmulti-cracking evolution of fibre-reinforced CMCs. Pryce and Smith[12] investigated the quasi-static tensile behaviour of

unidirectionalandcross-plySiC/calciumaluminosilicate(CAS)glass-ceramiccomposites.Thefirstmatrixcrac-king stress is predicted using the Aveston-Cooper-Kelly (ACK) theory [13], and the relationship between the matrix cracking density and the stiffness reduc-tion is analysed with an increasing strain. Beyerle et al. [14] investigated the mechanical characteristic ofthe unidirectionalSiC/CAS-II composite, and thefirstmatrix cracking stress and the composite ultimate strength are predicted using the micromechanical models. However, the evolution of the matrix multi-cracking and modulus reduction show a differencebetween the experimental data and theoretical analysis without considering the fibres failure. Holmes andCho [15] investigated the effect of the matrix crackspacing on the surface temperature rising from the unidirectional SiC/CAS-II composite. It was foundthat the onset of frictional heating under cyclic loading coincideswiththefirstmatrixcrackingstress,andtheextent of frictional heating increases as the averagematrixcrackspacingdecreasesatagivenfatiguepeakstressandstressratio.Okabeetal.[16]investigatedthefailure process of the unidirectional SiC/Borosilicatecomposite under tensile loading. The relationship betweenthematrixmulti-crackingevolutionandstress/strain curve is analysed, and it is found that the firstmatrix cracking stress is close to the knee point of the

https://doi.org/10.13168/cs.2018.0042

Li L.

22 Ceramics – Silikáty 63 (1) 21-31 (2019)

nonlinearity in the tensilestress/straincurve.Smithetal. [17] investigated thedamageaccumulation ina2Dwoven SiC/SiC composite using electrical resistance.Itwas found that the resistancechange in theSiC/SiCcompositeissensitivetomatrixcracking[18].Gowayedet al. [19] investigated the feasibility of utilising theshear-lag theory to estimate the matrix crack density in a fabric reinforced2DSiC/SiCcomposite.Thematrixcracking density was highly sensitive to fibre volumefractionalongtheloadingdirectionandthefibre/matrixinterfaceshearstrengthbetweenthefibresandmatrix.Ogasawara et al. [20] investigated the experimentalmatrix multi-cracking of an orthogonal 3D woven Si–Ti–C–O fibre/Si–Ti–C–O matrix composite usingmicroscopic observation. The inelastic tensile stress/strainbehaviour isgovernedbymatrixmulti-crackingin the transverse fibre bundles at a low stress,matrixmulti-cracking in longitudinal fibre bundles at anintermediate stress, and fibre fragmentation at a highstress.Morscheretal.[21]investigatedtheoccurrenceof matrix cracks in a melt-infiltrated 3D orthogonalarchitectureSiC/SiCcompositeundertensionparalleltothe Y-direction which is perpendicular to the Z-bundle weave direction using acoustic emissions (AE). Thematrix cracking stress range depended upon the Z-di-rection bundle size and the local architecture. Solti et al. [22] developed an approach of a criticalmatrix strainenergy (CMSE) criterion to analyse the matrix multi-cracking evolution, in which the maximum fibre/matrix interface shear strength criterion was adopted to determine the interface debonded length during matrix multi-cracking. However, following the arguments ofGao et al. [23] and Stang and Shah [24], the fracture mechanics approach is preferred to the shear strength approach for thefibre/matrix interfacedebondingpro-blem. Rajan and Zok [25] investigate the mechanicsof a fully bridged steady-state matrix cracking in uni-directional CMCs under shear loading. The studies mentionedabove,however,donotconsidertheeffectoffibredebondingonthematrixmulti-crackingevolutioninfibre-reinforcedCMCs. In thispaper, thematrixmulti-crackingevolutionof fibre-reinforced CMCs considering fibre fractureis investigated using the critical matrix strain energycriterion.Theshear-lagmodelcombinedwiththefibrefracture model and fibre matrix interface debondingcriterionisadoptedtoanalysethefibreandmatrixaxialstress distribution inside the damaged composite. The effects of the fibre volume fraction, the fibre/matrixinterfaceshearstress,thefibre/matrixinterfacedebon-ded energy, the fibre Weibull modulus and the fibrestrength on the stress-dependent matrix multi-cracking evolutionarediscussed.Theexperimentalmatrixmulti-crackingevolutionoftheunidirectionalSiC/CAS,SiC/CAS-II, SiC/SiC, SiC/Borosilicate and mini-SiC/SiCcomposites are predicted.

THEORETICAL ANALYSIS

Stress analysis

To analyse the stress distributions in the sand matrix of the damaged composite, a unit cell is extracted from the CMCs, as shown in Figure 1. The unit cell containsasinglefibresurroundedbyahollowcylinderof the matrix. The fibre radius is rf, and the matrix radius is R (R = rf/Vf1/2). The length of the unit cell is lc/2,whichishalfofthematrixcrackspacing.Thefibre/matrix interface debonded length is ld. At the matrix cracking plane, fibres carry all the stress (σ/Vf, where σdenotesthefar-fieldappliedstressandVf denotes the fibrevolumefraction).Theshear-lagmodeladoptedbyBudiansky, Hutchinson and Evans [26] is obtained toperform the stress and strain calculations in thefibre/matrix interface debonded region (x ∈ [0, ld]) and inter-face bonded region (x ∈ [ld, lc/2]).Thefibreaxialstressσf (x), the matrix axial stress σm (x)andthefibre/matrixinterface shear stress τi (x) are determined using the followingequations:

(1)

(2)

(3)where Tdenotes thestresscarriedbythe intactfibres;Vm denotes thematrix volume fraction; τi denotes the fibre/matrixinterfaceshearstress;ρ denotes the shear-lag model parameter; and σfo and σmo denote the fibreand matrix axial stress in the interface bonded region, respectively.

( )i df

fd d c

fo fo i df f

2 , 0,

( )2 exp ,

2

T x x lr

xl x l lT x lr r

τ

σσ σ τ ρ

− ∈= − + − − − ∈

,

( )f i df m

md d cf

mo fo i dm f f

2 , 0,

( )2 exp

2

V x x lrV

xl x l lV T x l

V r r

τ

σσ σ τ ρ

∈= − − − − − ∈

, ,

( )i d

i d d cfo i d

f f

,( )

2 exp ,2 2

x lx l x l lT x l

r r

τ

τ ρσ τ ρ

∈

= − − − − ∈

0,

,

Interface friction sliding Debonding tip

Matrix

X

L/2Ld

σm (z)

σ/Vf σ

τiFiber

Figure1.ThematerialpropertiesoftheSiC/CAS,SiC/CAS-II,SiC/SiC,SiC/Borosilicateandthemini-SiC/SiCcomposites.

Modelling matrix multi-cracking evolution of fibre-reinforced ceramic-matrix composites considering fibre fracture


(4)

(5)

where Ef, Em and Ecdenotethefibre,matrixandcom-positeelasticmodulus,respectively;αf, αmandαc denote thefibre,matrixandcompositethermalexpansioncoef-ficient, respectively; and ∆T denotes the temperaturedifference between the fabricated temperature T0 and the testing temperature T1(∆T=T1 −T0). The possibility of fibre failure within the matrixduetothestatisticalnatureofthefibrestrengthcanbeaccountedforbyusing theWeibullanalysis.The two-parameter Weibull model is adopted to describe thefibrestrengthdistribution,andtheGlobalLoadSharing(GLS) assumption is used to determine the stress carried bytheintactandfracturefibres.[27]

(6)

where 〈Tb〉 denotes the stress carried by brokenfibres;and P(T)denotesthefibrefailureprobability.

(7)

where mdenotesthefibreWeibullmodulus,whichde-scribesthevariationinthefibrestrength;andσc denotes thefibrecharacteristicstrengthofalengthδcofthefibre.[27]

(8)

where σ0denotesthefibrestrengthofalengthofl0. Whenafibrebreaks,thestresscarriedbythefibredrops to zero at the position of the break. Similar to the case ofmatrix cracking, the fibre/matrix interfacedebondsandthestressbuildsupinthefibrethroughtheinterface shear stress. During the process of loading, the stressinabrokenfibreTb as a function of the distance x fromthebreakcanbewrittenbythefollowingequation:

(9)

Inorder tocalculate theaveragestresscarriedbybrokenfibres 〈Tb〉, it is necessary to construct the pro-bability distribution F(x) of the distance xofafibrebreakfromthereferencematrixcrackplane,provided thatabreak occurs within a distance ±lf. For this conditional probability distribution, Phoenix and Raj [28] deduced thefollowingequationbasedonWeibullstatistics.

(10)

where(11)

Theaveraging stresscarriedbybrokenfibres 〈Tb〉 duringtheprocessofloadingusingEquations9and10leadstothefollowingequation:

(12)

SubstitutingEquations7and12intoEquation6,itleadsintothefollowingequation:

(13)

UsingEquation13,thestressT carried by the intact fibresat thematrixcrackingplanecanbedetermined.SubstitutingtheintactfibrestressTintoEquation7,therelationshipbetweenthefibrefailureprobabilityandtheapplied stress can be determined.

Interface debonding

Whenthematrixcrackingpropagatestothefibre/matrixinterface,itdeflectsalongtheinterface.Afrac-ture mechanics approach is adopted in the present ana-lysis.Thefibre/matrix interfacedebondingcriterionisdeterminedusingthefollowingequation:[23]

(14)

where ζd denotes the fibre/matrix interface debondedenergy; F = πrf2σ/Vf) denotes the fibre load at thematrix cracking plane; wf(0)denotesthefibreaxialdis- placement on the matrix cracking plane; and v(x) deno-testherelativedisplacementbetweenthefibreandthematrix. Theaxialdisplacementsofthefibreandthematrix,i.e., wf (x) and wm(x), are determined by the following equations:

(15)

(16)

The relative displacement between the fibre andthe matrix, i.e., v(x), is determined by the followingequation:

fVσ = T [1 – P(T)] + 〈Tb〉 P(T)

11c

f c

1 expmm TT

V Tσσ

σ

++ = − −

( ) ( )dfd i0

f d d

0 14 2

lw v xF dxr l l

ζ τπ

∂ ∂= − −

∂ ∂∫

( )( )

( )

c /2 mm

m

2 2 mo c df i f fd d fo i

f m m m m f

22

l

x

m

xw x dx

E

l lV rVl x l TrV E E V E r

σ

στσ τ

ρ

=

= − + − − − −

∫

( )( )

( )

c /2 mm

m


f m m m m f

22

l

x

m

xw x dx

E


σ

στσ τ

ρ

=

= − + − − − −

∫

( )( )

( )

c /2 mm

m


f m m m m f

22

l

x

m

xw x dx

E


σ

στσ τ

ρ

=

= − + − − − −

∫

( )( )

( ) ( )

c /2 ff

f

2 2 fo c di fd d d fo i

f f f f f f

22

l

x

xw x dx

E

l lrT l x l x l TE r E E E r

σ

στσ τ

ρ

=

= − − − + − + − −

∫

( )( )

( ) ( )

c /2 ff

f


f f f f f f

22

l

x

xw x dx

E


σ

στσ τ

ρ

=

= − − − + − + − −

∫

( )( )

( ) ( )

c /2 ff

f


f f f f f f

22

l

x

xw x dx

E


σ

στσ τ

ρ

=

= − − − + − + − −

∫

11/1 1

0 0 i 0 f 0c c

f i

,

mm mm ml r lr

σ τ σσ δ

τ

+ + = =

i

f

2rτ

Tb (x) = x

( )( )

[ ]1 1

ff c f c

1 exp , 0,m m

T x TF x x lP T l lσ σ

+ + = − ∈

( )( )

1c

b

1m P TT T

T P Tσ + − = −

ff

i2r T

lτ

=

( )1

c1 exp

mTP Tσ

+ = − −

= + − ∆Τ( )ffo f c fc

E EE

σ σ α α

( )mmo m c mc

E EE

σ σ α α= + − ∆Τ

Li L.


(17)

Substituting wf (x = 0) and v(x) intoEquation 14,leadstothefollowingequation:

(18)

Solving Equation 18, the fibre/matrix interfacedebonded length ld is determined by the following equation:

(19)

Matrix multi-cracking

Soltietal.[22]developedthecriticalmatrixstrainenergy (CMSE) criterion to predict the matrix multi-crackingevolutioninfibre-reinforcedCMCs.Thecon- cept of a critical matrix strain energy presupposes the existence of an ultimate or critical strain energy. Beyond the critical value of thematrix strain energy, asmoreenergy is entered into the composite with increasing applied stress, the matrix cannot support the extra load and continues to fail. The failure is assumed to consist of the formation of new cracks and the fibre/matrixinterface debonding, to make the total energy within the matrixremainconstantandequaltoitscriticalvalue. The matrix strain energy is determined using the followingequation:

(20)

where Am is the cross-section area of the matrix in the unit cell. Substituting the matrix axial stresses in Equation2 intoEquation20, thematrix strainenergyconsideringthematrixmulti-crackingandfibre/matrixinterface partially debonding, is described using the followingequation:

(21)

(21)

When the fibre/matrix interface completely de-bonds, the matrix strain energy is described using the followingequation:

(22)

Byevaluatingthematrixstrainenergyatacriticalstress of σcr, the critical matrix strain energy of Ucrm can be obtained. The critical matrix strain energy is describedusingthefollowingequation:

(23)

where k (k ∈ [0,1]) is the critical matrix strain energy parameter; and l0 is the initial matrix crack spacing and σmocrisdeterminedusingthefollowingequation:

(24)

where σcr is the critical stress corresponding to the com-posite’s proportional limit stress, i.e., the stress at which the stress-strain curve starts to deviate from linearitydue to damage accumulation of the matrix cracks [29]. The critical stress is defined to be the Aveston-Cooper-Kelly matrix cracking stress [13], which was determined using the energy balance criterion, invol-ving the calculation of the energy balance relationship before and after the formation of a single dominant crack.TheAveston-Cooper-Kellymodelcanbeusedtodescribe the long-steady-state matrix cracking stress, corresponding to the proportional limit stress of the tensile stress-strain curve. The Aveston-Cooper-Kelly matrix cracking stress is determined using the following equation:[13]

(25)

where ζmdenotesthematrixfractureenergy.However,as microcracks exist in the matrix when CMCs were cooled down from the high fabrication temperature to room temperature, due to a thermal expansion coefficient misfit between the fibre and the matrix,these microcracks are short-matrix-cracking, and the cracking stresses of these microcracks lie in the linear regionoftensilestress-straincurve[30,31].Withanin- creasing of the applied stress, the matrix microcracks can propagate into long-matrix-cracking. The matrix crackingstressoftheAveston-Cooper-Kellymodelwasused to determine the critical matrix strain energy.

2 2 22c i c i i f f f i

d d df m m f m m f f f c f

04 4 2

E E T r T r T r Tl lrV E E V E E E E E E

τ τ τ σ τζ

ρ ρ

+ − + − − − =

2 2 22c i c i i f f f i

d d df m m f m m f f f c f

04 4 2

E E T r T r T r Tl lrV E E V E E E E E E

τ τ τ σ τζ

ρ ρ

+ − + − − − =

( ) ( ) ( )

( ) ( )

f m

2 2c i f c dd d fo i

f f m m f m m f f

2

v x w x w x

E r E lT l x l x TE rV E E V E E r

τσ τ

ρ

= −

= − − − + − −

( ) ( ) ( )

( ) ( )

f m

2 2c i f c dd d fo i

f f m m f m m f f

2

v x w x w x

E r E lT l x l x TE rV E E V E E r

τσ τ

ρ

= −

= − − − + − −

2 2f m m f f f m f m f m m f

d d2 2 2c i c i f c i

12 2 4r V E r r V V E E T rV E El T T

E E V Eσ

ζτ ρ ρ τ τ

= − − − − +

2 2f m m f f f m f m f m m f

d d2 2 2c i c i f c i

12 2 4r V E r r V V E E T rV E El T T

E E V Eσ

ζτ ρ ρ τ τ

= − − − − +

( )cm

2m m m0

m

12

l

AU x dxdA

Eσ= ∫ ∫

( )

( )

22 cm f i

m d d mo dm m f

c df f imo fo d

m f m f

2

c df f ifo d

m f m f

43 2

/ 22 2 exp 1

/ 22 exp 22

f

f

lA VU l l lE V r

r l lV VT lV rV r

r l lV VT lV rV r

τσ

τσ σ ρ

ρ

τσ ρ

ρ

= + −

+

–

− − − − − − −

− + − − − − −

1

( )

( )

22 cm f i

m d d mo dm m f

c df f imo fo d

m f m f

2

c df f ifo d

m f m f

43 2

/ 22 2 exp 1

/ 22 exp 22

f

f

lA VU l l lE V r

r l lV VT lV rV r

r l lV VT lV rV r

τσ

τσ σ ρ

ρ

τσ ρ

ρ

= + −

+

–

− − − − − − −

− + − − − − −

1

( )23

m c i fm c d c

m f m

, , / 26A l VU l l lE rV

τσ

= =

2mocr

crm m 0m

12

U kA lE

σ=

( )mmocr cr m c mc

E EE

σ σ α α= + − ∆Τ

( )

12 2 3f f c i m

cr c c m2f m m

6V E E ErV E

τ ζσ α α

= − − ∆Τ



The energy balance relationship to evaluate thematrixmulti-crackingevolutionisdeterminedusingthefollowingequation:

(26)

The matrix multi-cracking evolution versus theapplied stress canbe solvedbyEquation26when thecritical matrix cracking stress of σcrandthefibre/matrixinterface debonded length of ld are determined by Equations19and25.

DISCUSSION

TheceramiccompositesystemofSiC/CASisused forthecasestudyanditsmaterialpropertiesaregivenby[14]: Vf = 30 %, Ef = 200 GPa, Em = 97 GPa, rf=7.5μm,ζm = 6 J·m-2, ζd = 0.8 J·m-2, τi = 20 MPa, αf = 4 × 10-5/°C,αm = 5 × 10-5/°C,∆Τ=–1000°C,m = 4, and σc = 2.0 GPa.

Effectofthefibrevolumefraction

Thematrix cracking density, the fibre/matrix in-terface debonded length (2ld/lc) and the broken fibresfraction for the different fibre volume fractions (i.e.,Vf = 30 % and 35 %) are shown in Figure 2. When thefibrevolumefraction isVf = 30 %, the matrixcrackingdensityincreasesfrom0.15/mmatthefirstmatrixcrackingstressof201MPato3.9/mmatthesaturationmatrixcrackingstressof310MPa;thefibre/matrix interface debonded length (2ld/lc) increases from 0.8%to75.7%;andthebrokenfibresfractionincreasesfrom 0.4 % to 9.4 %. When thefibrevolume fraction isVf = 35 %, the matrixcrackingdensityincreasesfrom0.19/mmatthefirstmatrixcrackingstressof235MPato4.5/mmatthesaturationmatrixcrackingstressof360MPa;thefibre/matrix interface debonded length (2ld/lc) increases from 0.8%to52.2%;andthebrokenfibresfractionincreasesfrom 0.4 % to 3.9 %.

Withanincreasingfibrevolumefraction, thefirstmatrix cracking stress, the matrix saturation cracking stress and the cracking density increase, and the matrix cracking evolveswith a higher applied stress; and thefibre/matrix interfacedebonded length and thebrokenfibresfractiondecrease.

Effectofthefibre/matrixinterface shear stress

Thematrixcrackingdensity,thefibre/matrixinterface debonded length (2ld/lc) and the broken fibresfraction for the different fibre/matrix interface shearstress (i.e., τi = 10 and 15 MPa) are shown in Figure 3. When the fibre/matrix interface shear stress is τi = 10 MPa, the matrix cracking density increases from 0.2/mmat thefirstmatrix cracking stressof147MPato 3.3/mm at the saturation matrix cracking stress of 217 MPa; the fibre/matrix interface debonded length(2ld/lc) increases from 0.7 % to 100 %; and the broken fibresfractionincreasesfrom0.1%to9.4%. When the fibre/matrix interface shear stress is τi = 15 MPa, the matrix cracking density increases from 0.16/mmatthefirstmatrixcrackingstressof177MPato 3.6/mm at the saturation matrix cracking stress of 274 MPa; the fibre/matrix interface debonded length(2ld/lc) increases from 0.8 % to 92.6 %; and the broken fibresfractionincreasesfrom0.2%to9.4%. With an increasing fibre/matrix interface shearstress,thefirstmatrixcrackingstress,thematrixsatu-ration cracking stress and the cracking density increase, the matrix cracking evolves with a higher appliedstress; and the fibre/matrix interface debonded lengthdecreases.

Effectofthefibre/matrixinterface debonded energy

Thematrix cracking density, the fibre/matrix in-terface debonded length (2ld/lc) and the broken fibres

0

1

2

3

4

5

Mat

rix c

rack

ing

dens

ity (m

m-1

)

Vf = 30 %Vf = 35 %

200 240 280 320 3600

0.2

0.4

0.6

0.8

1.0

2ld/l c

Stress (MPa)200 240 280 320 360

Stress (MPa)200 240 280 320 360

Stress (MPa)

0

2

4

6

8

10

Bro

ken

fibre

s fra

ctio

n (%

)

Figure2.Theeffectofthefibrevolumefractionon:a)thematrixcrackingdensityversustheappliedstresscurves;b)thefibre/matrix interface debonding length (2ld/lc)versus theappliedstresscurves;andc) thebrokenfibresfractionversus theappliedcyclescurves.

a) b) c)

( ) ( )m cr c d crm cr 0, , ,U l l U lσ σ σ> =

Li L.


fractionforthedifferentfibre/matrixinterfacedebondedenergy (i.e., ζd = 0.5 and 1.0 J·m-2) are shown in Figure 4. When thefibre/matrix interfacedebonded energyis ζd = 0.5 J·m-2, the matrix cracking density increases from 0.13/mm at the first matrix cracking stress of 201MPa to 3.6/mm at the saturationmatrix crackingstressof320MPa;thefibre/matrixinterfacedebondedlength (2ld/lc) increases from 0.9 % to 79.7 %; and the brokenfibresfractionincreasesfrom0.4%to9.4%. When thefibre/matrix interfacedebonded energyis ζd = 1.0 J·m-2, the matrix cracking density increases from 0.18/mm at the first matrix cracking stress of 201MPa to 4.1/mm at the saturationmatrix crackingstressof304MPa;thefibre/matrixinterfacedebondedlength (2ld/lc) increases from 0.8 % to 74.7 %; and the brokenfibresfractionincreasesfrom0.4%to9.4%. With increasing fibre/matrix interface debondedenergy, the first matrix cracking stress remains thesame, the matrix cracking saturation stress decreases, and the saturation matrix cracking density increase, and therateofmatrixcrackingdevelopment increasesduetothedecreaseofthefibre/matrixinterfacedebondingratio.

EffectofthefibreWeibullmodulus

Thematrix cracking density, the fibre/matrix in-terface debonded length (2ld/lc) and the broken fibresfraction for the different fibre Weibull modulus (i.e., m = 3 and 5) are shown in Figure 5. When the fibre Weibull modulus is m = 3, the matrixcrackingdensityincreasesfrom0.18/mmatthefirstmatrixcrackingstressof201MPato3.9/mmatthesaturationmatrixcrackingstressof298MPa;thefibre/matrix interface debonded length (2ld/lc) increases from 0.8%to85.7%;andthebrokenfibresfractionincreasesfrom 1.2 % to 17 %. When the fibre Weibull modulus is m = 5, the matrixcrackingdensityincreasesfrom0.18/mmatthefirstmatrixcrackingstressof201MPato4.2/mmatthesaturationmatrixcrackingstressof310MPa;thefibre/matrix interface debonded length (2ld/lc) increases from 0.8%to70.4%;andthebrokenfibresfractionincreasesfrom 0.1 % to 5.3 %. WithanincreasingfibreWeibullmodulus,thefirstmatrix cracking stress remains the same, the saturation matrix cracking stress and the cracking density increase; andthefibre/matrixinterfacedebondedlengthandthebrokenfibresfractiondecrease.

0

1

2

3

4

Mat

rix c

rack

ing

dens

ity (m

m-1

)

200 240 280 320 3600

0.2

0.4

0.6

0.8

1.0

2ld/l c

Stress (MPa)200 240 280 320 360

Stress (MPa)200 240 280 320 360

Stress (MPa)

0

2

4

6

8

10

Bro

ken

fibre

s fra

ctio

n (%

)

ζf = 0.5 J m-2

ζf = 1.0 J m-2

Figure4.Theeffectofthefibre/matrixinterfacedebondedenergyon:a)thematrixcrackingdensityversustheappliedstresscurves;b)thefibre/matrixinterfacedebondinglength(2ld/lc)versustheappliedstresscurves;andc)thebrokenfibresfractionversustheappliedcyclescurves.

a) b) c)

100 150 200 250 300 350 4000

0.5

1.0

1.5

2.0

3.0

3.5

4.0

2.5

Mat

rix c

rack

ing

dens

ity (m

m-1

)

Stress (MPa)100 150 200 250 300 350 400

0

0.2

0.4

0.6

0.8

1.0

Stress (MPa)100 150 200 250 300 350 400

0

2

4

6

8

10

Bro

ken

fibre

s fra

ctio

n (%

)

Stress (MPa)

τi = 10 MPaτi = 15 MPa

2ld/l c

Figure3.Theeffectofthefibre/matrixinterfaceshearstresson:a)thematrixcrackingdensityversustheappliedstresscurves;b)thefibre/matrixinterfacedebondinglength(2ld/lc)versustheappliedstresscurves;andc)thebrokenfibresfractionversustheappliedcyclescurves.

a) b) c)



Effectofthefibrestrength Thematrix cracking density, the fibre/matrix in-terface debonded length (2ld/lc) and the broken fibresfraction for different fibre strengths (i.e.,σc = 2.0 and 2.5 GPa) are shown in Figure 6. Whenthefibrestrengthisσc = 2.0 GPa, the matrix cracking density increases from 0.18/mm at the firstmatrix cracking stress of 201 MPa to 4.1/mm at thesaturationmatrixcrackingstressof304MPa;thefibre/matrix interface debonded length (2ld/lc) increases from 0.8%to74.7%;andthebrokenfibresfractionincreasesfrom 0.4 % to 9.4 %. Whenthefibrestrengthisσc = 2.5 GPa, the matrix cracking density increases from 0.18/mm at the firstmatrix cracking stress of 201 MPa to 4.3/mm at thesaturationmatrixcrackingstressof317MPa;thefibre/matrix interface debonded length (2ld/lc) increases from 0.8%to67.6%;andthebrokenfibresfractionincreasesfrom 0.1 % to 2.7 %. Withan increasingfibre strength, thefirstmatrixcracking stress remains the same, the saturation matrix cracking stress and the cracking density increase; and the fibre/matrix interface debonded length and thebrokenfibresfractiondecrease.

DISCUSSION

The experimental and theoretical matrix cracking density,thefibre/matrixinterfacedebondedlength(2ld/lc) andthebrokenfibresfractionversustheappliedstressforthedifferentCMCs,i.e.,unidirectionalSiC/CAS[12], SiC/CAS-II [14], SiC/SiC [14], SiC/Borosilicate [16]andmini-SiC/SiC [32] composites are predicted usingthe present analysis, as shown in Figures 7 ~ 11. The material properties of the CMCs are listed in Table 1. For the SiC/CAS composite, the matrix crackingevolutionstartsfromtheappliedstressof160MPaandapproaches saturation at the applied stress of 278 MPa; thematrix crackingdensity increases from0.3/mm tothesaturationvalueof7.1/mm;thefibre/matrixinterfacedebonded length increases from 0.7 % to 82.6 %; and the brokenfibresfractionincreasesfrom0.01%to2.3%,asshown in Figure 7. FortheSiC/CAS-IIcomposite,thematrixcrackingevolutionstartsfromtheappliedstressof260MPaandapproaches saturation at the applied stress of 354 MPa; thematrix cracking density increases from0.7/mm tothesaturationvalueof9.3/mm;thefibre/matrixinterfacedebonded length increases from 0.3 % to 40.5 %; and the

0

1

2

3

4

5

Mat

rix c

rack

ing

dens

ity (m

m-1

)

m = 3m = 3

200 240 280 320 3600

0.2

0.4

0.6

0.8

1.0

2ld/l c

Stress (MPa)200 240 280 320 360

Stress (MPa)200 240 280 320 360

Stress (MPa)

0

5

10

15

20

Bro

ken

fibre

s fra

ctio

n (%

)

0

1

2

3

4

5

Mat

rix c

rack

ing

dens

ity (m

m-1

)

200 240 280 320 3600

0.2

0.4

0.6

0.8

1.0

2ld/l c

Stress (MPa)200 240 280 320 360

Stress (MPa)200 240 280 320 360

Stress (MPa)

0

2

4

6

8

10

Bro

ken

fibre

s fra

ctio

n (%

)

σc = 2.0 GPaσc = 2.5 GPa

Figure5.TheeffectofthefibreWeibullmoduluson:a)thematrixcrackingdensityversustheappliedstresscurves;b)thefibre/matrix interface debonding length (2ld/lc)versus theappliedstresscurves;andc) thebrokenfibresfractionversus theappliedcyclescurves.

Figure6.Theeffectofthefibrestrengthon:a)thematrixcrackingdensityversustheappliedstresscurves;b)thefibre/matrixinterface debonding length (2ld/lc)versus theappliedstresscurves;andc) thebrokenfibres fractionversus theappliedcyclescurves.

a) b) c)

a) b) c)

Li L.


brokenfibresfractionincreasesfrom0.11%to1.2%,asshown in Figure 8. For the SiC/SiC composite, the matrix crackingevolutionstartsfromtheappliedstressof240MPaandapproaches saturation at the applied stress of 290 MPa; the matrix cracking density increases from 1.4/mmto the saturation value of 15.8/mm; the fibre/matrixinterface debonded length increases from 0 to 27.8 %; andthebrokenfibresfractionincreasesfrom0.07%to0.41 %, as shown in Figure 9.

For the SiC/Borosilicate composite, the matrixcracking evolution starts from the applied stress of220 MPa and approaches saturation at the applied stress of 340 MPa; the matrix cracking density increases from 0.2/mm to the saturation value of 6.4/mm; the fibre/matrix interface debonded length increases from 0.8 % to100%;andthebrokenfibresfractionincreasesfrom0.2 % to 14 %, as shown in Figure 10. For themini-SiC/SiCcomposite, thematrix crac-kingevolutionstartsfromtheappliedstressof135MPa

Table1.ThematerialpropertiesoftheSiC/CAS,SiC/CAS-II,SiC/SiC,SiC/Borosilicateandthemini-SiC/SiCcomposites.

Items SiC/CAS[12] SiC/CAS-II[14] SiC/SiC[14] SiC/Borosilicate[16] mini-SiC/SiC[32]

Ef (GPa) 190 200 200 230 160Em (GPa) 90 97 300 60 190Vf 0.34 0.4 0.4 0.31 0.25rf(μm) 7.5 7.5 7.5 8 6.5αf (10-6/°C) 3.3 4 4 3.1 3.1αm(10-6/°C) 4.6 5 5 3.25 4.6τi (MPa) 10 25 50 7.6 15ζd (J·m-2) 0.4 1.8 2.8 0.2 0.4m 5 5 5 5 5σc (GPa) 2 2 2 2 2

140 180 220 260 300 340 3800

1

2

3

4

6

7

8

5

Mat

rix c

rack

ing

dens

ity (m

m-1

)

Stress (MPa)140 180 220 260 300 340 380

Stress (MPa)140 180 220 260 300 340 380

Stress (MPa)

0

0.2

0.4

0.6

0.8

1.0

0

0.5

1.0

1.5

2.0

2.5

Bro

ken

fibre

s fra

ctio

n (%

)

Experimental dataPresent analysis

2ld/l c

240 260 280 300 320 340 360 3800

2

4

6

8

12

10

Mat

rix c

rack

ing

dens

ity (m

m-1

)

Stress (MPa)

0

0.1

0.2

0.3

0.4

0.5

0

0.2

0.4

0.6

0.8

1.4

1.0

1.2

Bro

ken

fibre

s fra

ctio

n (%

)


2ld/l c

240 260 280 300 320 340 360 380Stress (MPa)

240 260 280 300 320 340 360 380Stress (MPa)

Figure7.a)theexperimentalandtheoreticalmatrixcrackingdensityversustheappliedstresscurves;b)thefibre/matrixinterfacedebonded length (2ld/lc)versustheappliedstresscurves;andc)thebrokenfibresfractionversustheappliedstresscurveoftheunidirectionalSiC/CAScomposite.

Figure8.a)theexperimentalandtheoreticalmatrixcrackingdensityversustheappliedstresscurves;b)thefibre/matrixinterfacedebonded length (2ld/lc)versustheappliedstresscurves;andc)thebrokenfibresfractionversustheappliedstresscurveoftheunidirectionalSiC/CAS-IIcomposite.

a) b) c)

a) b) c)



and approaches saturation at the applied stress of 240 MPa; the matrix cracking density increases from 0.1/mm to the saturation value of 2.4/mm; the fibre/matrix interface debonded length increases from 1 % to89%;andthebrokenfibresfraction increasesfrom0.03 % to 3.35 %, as shown in Figure 11.

CONCLUSIONS

In this paper, the effect of fibre fracture on thematrix multi-cracking development of the CMCs hasbeeninvestigated.Theshear-lagmodelcombinedwiththefibre fracturemodel and thefibre/matrix interfacedebondingcriterionhasbeenadoptedtoanalysethefibreand matrix axial stress distribution inside the damaged

200 240 280 320 360 400 440 200 240 280 320 360 400 440 200 240 280 320 360 400 4400

1

2

3

4

6

7

8

5

Mat

rix c

rack

ing

dens

ity (m

m-1

)

Stress (MPa) Stress (MPa) Stress (MPa)

0

2

4

6

8

12

14

16

10

0

0.2

0.4

0.6

0.8

1.2

1.0

Bro

ken

fibre

s fra

ctio

n (%

)


2ld/l c

240 260 280 300 3200

4

8

12

16

20

Mat

rix c

rack

ing

dens

ity (m

m-1

)

Stress (MPa)

0

0.1

0.2

0.3

0

0.1

0.3

0.2

0.4

0.5

Bro

ken

fibre

s fra

ctio

n (%

)


2ld/l c

240 260 280 300 320Stress (MPa)

240 260 280 300 320Stress (MPa)

100 140 180 220 260 3000

0.5

1.0

1.5

2.0

3.0

2.5

Mat

rix c

rack

ing

dens

ity (m

m-1

)

Stress (MPa)100 140 180 220 260 300

Stress (MPa)100 140 180 220 260 300

Stress (MPa)

0

1

2

3

4

0

0.2

0.4

0.6

0.8

1.0

Bro

ken

fibre

s fra

ctio

n (%

)


2ld/l c

Figure10.a)theexperimentalandtheoreticalmatrixcrackingdensityversustheappliedstresscurves;b)thefibre/matrixinterfacedebonded length (2ld/lc)versustheappliedstresscurves;andc)thebrokenfibresfractionversustheappliedstresscurveoftheunidirectionalSiC/Borosilicatecomposite.

Figure9.a)theexperimentalandtheoreticalmatrixcrackingdensityversustheappliedstresscurves;b)thefibre/matrixinterfacedebonded length (2ld/lc)versustheappliedstresscurves;andc)thebrokenfibresfractionversustheappliedstresscurveoftheunidirectionalSiC/SiCcomposite.

Figure11.a)theexperimentalandtheoreticalmatrixcrackingdensityversustheappliedstresscurves;b)thefibre/matrixinterfacedebonded length (2ld/lc)versustheappliedstresscurves;andc)thebrokenfibresfractionversustheappliedstresscurveofthemini-SiC/SiCcomposite.

a) b) c)

a) b) c)

a) b) c)

Li L.


composite. The effects of the fibre volume fraction,thefibre/matrix interface shear stress, thefibre/matrixinterface debonded energy, the fibreWeibullmodulusand the fibre strength on the stress-dependent matrixmulti-cracking development have been discussed.Theexperimental matrix multi-cracking development oftheunidirectionalSiC/CAS,SiC/CAS-II,SiC/SiC,SiC/Borosilicateandthemini-SiC/SiCcompositeshavebeenpredicted.● With an increasing fibre volume fraction, the first

matrix cracking stress, the matrix saturation cracking stress and the cracking density increase, and the matrixcrackingevolveswithahigherappliedstress;and the fibre/matrix interface debonded length andthebrokenfibresfractiondecrease.

● Withanincreasingfibre/matrixinterfaceshearstress,thefirstmatrixcrackingstress, thematrixcrackingsaturation stress and the saturation matrix cracking densityincrease,thematrixcrackingevolveswithahigher applied stress; and the fibre/matrix interfacedebonded length decreases.

● With an increasing fibre/matrix interface debondedenergy, the firstmatrix cracking stress remains thesame, the matrix saturation cracking stress decreases, and the saturation matrix cracking density increase, andtherateofmatrixcrackingdevelopmentincreasesduetoadecreaseinthefibre/matrixinterfacedebon-ding ratio.

● With an increasingfibreWeibullmodulus andfibrestrength,thefirstmatrixcrackingstressremainsthesame, the saturation matrix cracking stress and the crackingdensityincrease;andthefibre/matrixinter-facedebonded lengthand thebrokenfibres fractiondecrease.

Acknowledgements

The work reported here is supported by the Funda-mental Research Funds for the Central Universities (Grant No. NS2016070).

REFERENCES

1. Christin F. (2002): Design, fabrication, and application of thermostructuralcomposites (TSC) likeC/C,C/SiC,andSiC/SiC composites. Advanced Engineering Materials,4(12),903-912.doi:10.1002/adem.200290001

2. Naslain R. (2004): Design, preparation and properties of non-oxide CMCs for application in engines and nuclear reactors:anoverview.CompositesScienceandTechnology,64(2),155-170.doi:10.1016/S0266-3538(03)00230-6

3. Schmid, S., Beyer S., Knabe H., Immich H., Meistring R., Gessler A. (2004): Advanced ceramicmatrix compositematerials for current and future propulsion technology applications. Acta Astronautica, 55(3-9), 409-420. doi: 10.1016/j.actaastro.2004.05.052

4. Cox B. N., Marshall D. B. (1996): Crack Initiation in Fi-ber-Reinforced Brittle Laminates. Journal of the American Ceramic Society, 79(5), 1181-1188. doi: 10.1111/j.1151-2916.1996.tb08570.x

5. SevenerK.M.,TracyJ.M.,ChenZ.,KiserJ.D.,DalyS.(2017):Crack opening behavior in ceramicmatrix com-posites. Journal of the American Ceramic Society, 100(10), 4734-4747.doi:10.1111/jace.14976

6. Filipuzzi L., Camus G., Naslain R., Thebault J. (1994): Oxidationmechanismsandkineticsof1D-SiC/c/SiCcom-posite materials: I, an experimental approach. Journal of theAmericanCeramicSociety,77(2),459-466.doi:10.1111/ j.1151-2916.1994.tb07015.x

7. Lamouroux F., Naslain R., Jouin J. M. (1994): Kinetics and mechanismsofoxidationof2DwovenC/SiCcomposites:II, theoretical approach. Journal of the American Ceramic Society, 77(8), 2058-2068. doi: 10.1111/j.1151-2916.1994.tb07097.x

8. Verrilli M. J., Opila E. J., Calomino A., Kiser J. D. (2004): EffectofEnvironmentontheStress-RuptureBehaviorof a Carbon-Fiber-Reinforced Silicon Carbide Ceramic Matrix Composite. Journal of the American Ceramic Society, 87(8),1536-1542.doi:10.1111/j.1551-2916.2004.01536.x

9. HalbigM.C.,McGuffin-CawleyJ.D.,EckelA.J.,BrewerD.N.(2008):Oxidationkineticsandstresseffectsfortheoxidation of continuous carbon fibers within a micro-crackedC/SiCceramicmatrix composite. Journal of theAmerican ceramic society, 91(2), 519-526. doi: 10.1111/j.1551-2916.2007.02170.x

10.KuoW.S.,ChouT.W.(1995):MultipleCrackingofUni-directional and Cross-PlyCeramic Matrix Composites. Journal of the American Ceramic Society, 78(3), 745-755. doi:10.1111/j.1151-2916.1995.tb08242.x

11. Parthasarathy T. A., Cox B., Sudre O., Przybyla C., Cini-bulk M. K. (2018): Modeling environmentally inducedpropertydegradationofSiC/BN/SiCceramicmatrixcom-posites. Journal of the American Ceramic Society, 101(3), 973-997.doi:10.1111/jace.15325

12.PryceA.W.,SmithP.A.(1992):Behaviourofunidirectio-nalandcrossplyceramicmatrixcompositesunderquasi-static tensile loading. Journal of materials science, 27(10), 2695-2704.doi:10.1007/BF00540692

13.Aveston J., Cooper G.A., Kelly A. (1971). Properties ofFiber Composites, in: Conference Proceedings of National Physical Laboratory. IPC Science and Technology Press, pp.15−26.

14.Beyerle D. S., Spearing S.M., Zok F.W., Evans A. G.(1992): Damage and failure in unidirectional ceramic-matrix composites. Journal of the American Ceramic Society, 75(10), 2719-2725. doi: 10.1111/j.1151-2916.1992.tb05495.x

15.Holmes J. W., Cho C. (1992): Experimental Observa-tions of Frictional Heating in Fiber-Reinforced Ceramics. Journal of the American Ceramic Society, 75(4), 929-938. doi:10.1111/j.1151-2916.1992.tb04162.x

16. Okabe T., Komotori J., Shimizu M., Takeda N. (1999): MechanicalbehaviorofSiCfiberreinforcedbrittle-matrixcomposites. Journal of materials science, 34(14), 3405-3412. doi:10.1023/A:1004637300310

17. Smith C. E., Morscher G. N., Xia Z. H. (2008): Monitoring damage accumulation in ceramic matrix composites using electrical resistivity. Scripta Materialia, 59(4), 463-466.doi:10.1016/j.scriptamat.2008.04.033



18. Simon C., Rebillat F., Herb V., Camus G. (2017): Moni-toring damage evolution of SiCf/[SiBC] m composites using electrical resistivity: Crack density-based electro-mechanical modeling. Acta Materialia, 124, 579-587. doi: 10.1016/j.actamat.2016.11.036

19.GowayedY.,OjardG.,SanthoshU., JeffersonG. (2015):Modeling of crack density in ceramic matrix composites. Journal of Composite Materials, 49(18), 2285-2294. doi: 10.1177/0021998314545188

20.OgasawaraT., IshikawaT., ItoH.,WatanabeN.,DaviesI. J. (2001):Multiple Cracking and Tensile Behavior foranOrthogonal3-DWovenSi–Ti–C–OFiber/Si–Ti–C–OMatrix Composite. Journal of the American Ceramic Society, 84(7), 1565-1574. doi: 10.1111/j.1151-2916.2001.tb00878.x

21. Morscher G. N., Yun H. M., DiCarlo J. A. (2005): Matrix Cracking in 3D Orthogonal Melt-Infiltrated SiC/SiCComposites with Various Z-Fiber Types. Journal of the AmericanCeramicSociety,88(1),146-153.doi:10.1111/j.1551-2916.2004.00029.x

22. Solti J. P., Mall S., Robertson D. D. (1995): Modeling dama-ge in unidirectional ceramic-matrix composites. Com- positesscienceandtechnology,54(1),55-66.doi:10.1016/0266-3538(95)00041-0

23.GaoY.C.,MaiY.W.,CotterellB.(1988):Fractureoffiber-reinforced materials. Zeitschrift für angewandte Mathe-matik und Physik ZAMP, 39(4), 550-572. doi: 10.1007/BF00948962

24.Stang H., Shah S. P. (1986): Failure of fibre-reinforcedcomposites by pull-out fracture. Journal of materials science,21(3),953-957.doi:10.1007/BF01117378

25.RajanV. P., Zok F.W. (2014):Matrix cracking of fiber-reinforced ceramic composites in shear. Journal of the MechanicsandPhysicsofSolids,73,3-21.doi:10.1016/j.jmps.2014.08.007

26.Budiansky B., Hutchinson J. W., Evans A. G. (1986):Matrix fracture in fiber-reinforced ceramics. Journal ofthe Mechanics and Physics of Solids, 34(2), 167-189. doi: 10.1016/0022-5096(86)90035-9

27.Curtin W. A. (1991): Theory of mechanical propertiesof ceramic-matrix composites. Journal of the American Ceramic Society, 74(11), 2837-2845. doi: 10.1111/j.1151-2916.1991.tb06852.x

28.PhoenixS.L.,RajR.(1992):Overviewno.100Scalingsinfractureprobabilitiesforabrittlematrixfibercomposite.Acta Metallurgica et Materialia, 40(11), 2813-2828. doi: 10.1016/0956-7151(92)90447-M

29.LongbiaoL.(2017):Synergisticeffectsoffiberdebondingandfractureonmatrixcrackinginfiber-reinforcedceramic-matrix composites. Materials Science and Engineering: A, 682,482-490.doi:10.1016/j.msea.2016.11.077

30. Li L. B., Song Y. D., Sun Y. C. (2014): Modeling the tensile behaviorofunidirectionalC/SiCceramic-matrixcompo-sites. Mechanics of Composite Materials, 49(6), 659-672. doi:10.1007/s11029-013-9382-y

31. Li L. B., Song Y. D., Sun Y. C. (2015): Modeling the tensile behavior of cross-ply C/SiC ceramic-matrix composites.Mechanics of Composite Materials, 51(3), 359-376. doi: 10.1007/s11029-015-9507-6

32. Zhang S., Gao X., Chen J., Dong H., Song Y. (2016): Strength model of the matrix element in SiC/SiC com-posites. Materials & Design, 101, 66-71. doi: 10.1016/j.matdes.2016.03.166

modelling matrix multi-cracking evolution of ......modelling matrix multi-cracking evolution of...

Documents