Journal 01 the Ceramic Societv 01 Japan 106 :10J 968-973 (1998)

Finite Element Method Simulation of Transverse Bridging InFiber Reinforced Composites

Paper

Mare-Oliver NANDY, Nabuyuki TOHYAMA, Byung-Nam KIM, Manabu ENOKI,Siegfried SCHMAUDER* and Terua KISHI

Research Cellter 101'Advanced Scie1lceand Tech1l010gy,The Unil'ersity 01 Tokyo, 4-6-1, Komaba, Meguro-ku, Tokyo 153-8904*Staatliche -'1aterialprÜlungsa1lstalt 0IPA), U1Iiversity01 Stultgart, PIaffenwaldring 32, 70569 Stul/gart, Germany

*~HI5sHt:~f~g.*Nf(=d3tt -Q ~ -5::'; AI\'- A 71) "):; ::';/)'(]) FEM y 2.::l L.-- Y:3 ::.;

Mare-Oliver Nandy . JiL1J~z . ~ ~Pi~' tl '# . Siegfried Sehmauder* . f=ft ~:Ji;:lE

*J?::*:"t:JtYlffiH"tJ5!iVi'Tlilf':!i:-t /';7 -, 153-8904 *J?:ffl) EI~g~t~ 4-6-1*Staatliche Materialprüfungsanstalt (MP Al, University of Stuttgart, Pfaffenwaldring 32, 70569 Stuttgart, Germany

Delamination cracks in ceramic composite materials may be bridged by misaligned or inclined fibers at ashallow angl,e. The in situ observation of delamination cracks in a Si-Ti-C-O fiber-bonded ceramic com­posite material reveals that the bridging fibers are subjected to increasing tensile stresses as the crack open­ing displacement becomes larger. These stresses cause a crack closure pressure that is considered to con­tribute to steady state transverse fracture toughness. To relate the crack closure pressure to the materialproperties of fibers, matrix and their interface, a two-dimensional FEM model of a misaligned fiber bridgingthe crack wake at a shallow angle was constructed. Crucial mechanisms such as fiber debonding and fric­tional sliding along the debonded interface as weH as matrix chipping were included. The crack closurepressure was simulated as a function of COD and the influences of these mechanisms were discussed. Thetoughening eifect of bridging fibers was estimated and the obtained results were compared to the experimen­tal data for a Si-Ti-C-O fiber-bonded ceramic composite material.

[Received April 2, 1998; Accepted July 13, 1998J

Key-words .' Delamination, Transuerse fiber bridging, Crack closure pressure

1. IntroductionContinuous fiber-reinforced ceramic matrix composites

are susceptible to delamination cracking due to the pro­nounced anisotropy in fracture resistance parallel and nor­mal to the fiber orientation. Although delamination crack­ing is not necessarily followed by catastrophic failure, it hasto be considered as an important damage mechanismbecause it reduces the structural stiffness. Moreover, thecompressive strength as weil as the shear strength of thematerial may decrease considerably. Thus, a betterunderstanding of the mechanisms that govern delaminationcracking is necessary to develop methods that contribute tosuppress it.

It has been reported that the infiuence of reinforcingfibers on the transverse fracture toughness of ceramicmatrix composites is an important factor for delaminationcrack growth.l),2) Some amount of misalignments which isintentionally introduced to uni-directional fiber-reinforcedcomposites leads to an interaction between delaminationcracks and fibers. Misaligned fibers or fiber bundles bridgethe delamination crack at shallow angles and contribute totransverse fracture toughness which has the desirableeffect of suppressing further growing of the crack. As thecrack opening displacement increases, the bridging fibersstart to debond from the matri.x and peel offthe crack faces,resulting in a limitation of the closure pressure and thusallowing greater displacement ranges than pin-jointedfibers would do.

The purpose of this paper is to construct a two-dimen­sional finite element method (FEM) model of a fiber bridg­ing the crack wake at a shallow angle in order to simulatethe correlation between closure pressure, matrix chippingand fiber debonding during delamination cracking. Themodel relates the crack closure press ure to basic consti­tuent, fiber geometry and interface properties. Simulationswere performed and discussed by using the experimental

968

reference data for a Si- Ti-C-O fiber-bonded ceramic com­posite material. The results were also compared to amechanical model wh ich had been developed by Kaute etaJ.3l The present study contributes to the development of aprocedure that allows the analysis of delamination crackingat elevated temperatures.

2. MaterialThe material used in this study was a unidirectional Si­

Ti-C-O fiber-bonded ceramic composite (Tyrannohex;Ube Industries Co., Ud., Japan) synthesized from pre-ox­idized Si- Ti-C-O fibers by hot-pressing at 1750°C under apressure of 40 MPa. The interstices between the fiberswere packed with an oxide material which existed on thesurface of the pre-oxidized Si- Ti-C-O fibers. The volumefraction of the oxide matrix is about 10 vol% and thus muchlower than that of common ceramic-matrix fiber reinforcedmaterials (30-70 vol%). Aligned turbostratic carbon wasformed in si/u as the result of a chemical reaction at thefiber-matrix interface during hot-pressing.4) This carbonlayer plays an important role as regards the toughness pro­perties of the material since it controls the slidingresistance of debonded fiber-matrix interfaces.

The material properties at room temperature reportedfor the Si- Ti-C-O fiber-bonded composite material are sum­marized in Table 1. The transverse fracture toughness wasdetermined by performing Double-Torsion (DT) -tests withpre-notched specimens.51 The orientation of fibers wasparallel to the direction of crack propagation. The fracturetoughness can be determined by making use of the strain­energy release rate, GJc. The corresponding equation yields

1 Pe2 dCGI =--- (1)

e 2 t da

where Pe is the criticalload for crack propagation in a DT­specimen, t is the thickness of the specimen and dC/da is

Mare-Oliver NANDY et a1. JO/lrnal of Ihe Ceramie Soeiely of Japan 106 [10J 1998 969

Table 1. Constituent, Interface and Composite Properties of Si­Ti-C-O Fiber-Bonded Composite

180 BunseIl et al.lI)

10-20 Nandy et al.IO).

Evans et al. 9)

Sakuma12)

0.2

0.9

95

1.55

9.45

Interfacia! shear resistanee, t' (MPa)

Fiber volume fraetion,f,

Interface debonding energy, F; (11m')

Fiber modulus, EI (GPa)

Matrix modulus, Em (GPa)

Fiber diameter, d (/LfTl)

Fiber strength, a,., (GPa)

3. Modelling of fiber bridgingA two-dimensional FEM model of a fiber bridging the

the relation between the compliance of the specimen, e,and the length of the initial notch, a. dei da as weil as Pe hadto be determined experimentally by using the compliancecalibration method as described in Ref. 5. The averagevalue after 7 tests was Grc = 12.2 J 1m2. Further ex­periments with double-cantilever beam (DCB) specimens6)revealed the values of approximately 12.5 J 1m2. The frac­ture surfaces of the specimens were observed by a scanningelectron microscope (SEM). Although the crack pro­pagated mainly along the fiber-matrix interface, fiberswere observed that bridged the crack in a shallow angle.The number of these fibers was estimated to be about 61mm2. The strength and the Weibull parameters for the Si­Ti-C-O fibers were determined by performing tensile testswith single fibers at a gauge length of 25 mm. Fiberdiameters were measured by SEM for every single fiber.

In order to obselTe delamination cracking, in situ SEMfour-point bending tests with pre-notched beam specimenswere performed. After the load-displacement curve reach­ed a maximum value, delamination cracking occurred.Figure 1 (a) shows a SEM micrograph of the zone near thenotch with severe damage due to delamination cracks.Fibers that bridge adelamination crack at a shallow anglecan be seen from Fig. 1 (b). The profile of these fibersoriginally follows the curved outline of a beam in bending,straightening out as the crack opening displacement in­creases. The cause for fiber bridging is fiber waviness.2)

Composite transverse fraelure toughness, (MpaF m) 1 - 1.5

crack wake at a shallow angle during transverse fracturewas constructed, as shown in Fig. 2. The mesh was refinedin regions where high stress gradients are expected. Inorder to adjust the model to the real conditions, thegeometry of in situ observed bridging fibers was taken intoaccount. These observations suggest that the fibers areslightly curved where they are connected to the matrix.This curvature was taken into consideration as weil as thegeometrical dimensions of the Si- Ti-C-O fibers. The initialbridging angle of the modeled fiber has an angle of rp = 12°,the initial fiber length 10 is 30 j.lm and the fiber diameter 10j.lm. It should be noted that initial angles other than 12° mayoccur, too. But from in situ observations it can be seen thatfibers with angles much larger than 12° hardly occur due toa sufficient alignment during processing, whereas misalign­ed fibers with very shallow angles do not bridge the crack.Thus, the chosen initial angle can be considered as the mostfrequent bridging angle.

At both ends, the fiber is embedded in the matrix(thickness: 5 j.lm), which consists of Si02 glass. Residualstresses due to the mismatch in the coefficient of thermal ex-

Fig. 1. SEM micrograph of delamination cracks in unidirectionalSi- Ti-C-O fiber-bonded composite (a) and fibers bridging adelamination crack in a shallow angle (b).

lshikawa et al. 4)

2.95

25

400 Ishikawa.et al. 4)

17

Gauge length, l,q (mm)

Weibull modulus, m

Composite tensile strength, 0 (MPa)

Composite fraelure loughness, Klo (MPaF m)

I'~-

x

Fig. 2. Schematic illustration of the FEM model of a curved fiber bridging a crack in a shallow angle. The bridging fiber is subjected tohigh tensile stresses. Fiber-matrix debonding and matrix chipping are indicated as weil as the boundary conditions of the model. The fiberis loaded by increasing .du in y direction.

970 Finite Element Method Simulation of Transverse Bridging in Fiber Reinforced Composites

pansion and the fabrication procedure were neglected inthis study. As indicated in Fig. 2, two mechanisms wereassumed to occur when the crack opening displacement isincreased: Debonding of the fiber from the interfacetogether with sliding along the debonded interface and chip­ping of the matrix. Applying basic fracture mechanics, thedebond length, Xd, is calculated by the fo11owing equation.2)

d ( 212 .)Xd=- u- !:i K1Ic (2)4r "y d

where U is the tensile stress acting where the fiber entersthe matrix, dis the fiber diameter, r is the interfacial shearresistance, and Kbc is the mode II interface fracturetoughness. Note that Eq. (2) which is used for a relativelythin matrix layer in the present study was origina11y derivedfor a single fiber embedded in an infinite matrix. In the pre­sent case of a thin matrix layer, the thickness as we11as themechanical boundary conditions of the thin matrix layeraffect Khc. The right term of Eq. (2) and thus Khc deter­mines the onset of interfacial debonding. Therefore, the useof Eq. (2) in the present form may cause some uncertain­ties.

For simplicity, we assurne that r is constant and thatthere is no difference between the debond shear stress andthe frictional shear stress.7) Thus, by applying the basicequation of frictional fiber sliding,8) the stress at the end ofthe debonded part of the fiber, u (Xd), can then be calculatedby

(3)

Because u depends on the crack opening displacement,the debond length remains zero at the beginning of thebridging process. When the tensile stress in the fiber ex­ceeds a certain threshold value (from Eq. (3) for Xd=O),

debonding occurs and the debond length, Xd, increases.The in situ SEM observation of delamination cracks has

shown that the matrix below the fiber starts to chip offwhen the bridging angle of the fiber attains a certain value(about 12°). Since the glass matrix of the Si- Ti-C-O fiber­bonded material is very brittle, its fracture was assumed tooccur when the mean tensile stress exceeds 100 MPa, avalue reported für the tensile strength of Si02 glass.

Within the FEM model, these two mechanisms wererealized as fo11ows:When debonding at the fiber-matrix in­terface occurred according to Eq. (2), the corresponding de­bond length was calculated and a crack with an openingmuch sma11er than the fiber diameter and with a lengthequal to the calculated value of Xd was generated. The stressreduction of the fiber due to slippage relative to the fiber­matrix interface was then calculated by Eq. (3), with theaverage value for u computed by FEM. At the location ofthe fiber-matrix interface where matrix chipping was ex­pected to occur, the two surfaces of the crack were modeledin terms of a contact problem. When the crack openingdisplacement was increased, the fiber was distorted,loading the matrix similarly to a bar in bending. As soon asthe mean tensile stress exceeded the tensile strength of theglass matrix at a certain node, a11the out er elements of thisnode were removed, resulting in a longer bridging fiberlength and areduction of the bridging angle.

The model was loaded step by step by increasing thecrack opening displacement. For each step, debonding andchipping were calculated and the geometry of the modelwas modified in the fo11owing step. At the beginning of thesimulation, a sma11step width was chosen, but after severalsteps with increased crack opening displacement, largerstep widths were set to reduce the time of calculation.

In order to calculate the crack c10sure pressure exerted

by bridging fibers, two terms have to be considered: Theforce per fiber, f(u), and the number of bridging fibers perunit area, n (u). The force per fiber can be calculated fromthe average tensile fiber stress obtained by the FEM model.Considering that the fiber in the two-dimensional FEMmodel has a rectangular cross section, whereas the crosssection of the actual fiber is circular, the force per fiber isdetermined as

nd2f (u) =- u(u) sin rp (4)

4

where u(u) is the average tensile fiber stress that evolvesas the crack opening displacement u increases, and rp is thebridging angle as indicated in Fig. 1. The initial nu mb er ofbridging fibers per unit area has to be estimated by SEM. Itdepends on the misalignment and volume fraction of fibersand diminishes by fiber failure. However, n (u) was con­sidered to be constant in this study (n (u ) = no) for thefo11owing reason: The strength distribution of ceramicfibers is usually described by Weibull statistics. Thestrength quoted for the Si- Ti-C-O fibers in Table 1 is validfor the gauge length of 25 mm and the failure probability of0.632. For a constant survival probability and fiberdiameter, the strength is a function of fiber length:

u (I )llm-= ~ (5)Urei I

where u and Urei are defined as those stresses at which thefailure probability of the fibers of length land referencelength Irei is 0.632, respectively; m is the Weibull modulus.Inserting the values for Urei, lrei and m given in Table 1, abridging Si- Ti-C-O fiber with the length of 0.5 mm wouldbe able to sustain tensile stresses of about 6 GPa. Becausethe transverse fracture toughness is expected to be verylow, it can be assumed that the fiber peels off the crackfaces rather than failing by tensile stresses. SEM observa­tions indicate that the length of bridging fibers ranges from0.01 to 0.3mm. With bothf(u) and n(u)=no, the crackc10sure pressure yields the simple expression

p(u) =f(u)no (6)

The c10sure press ure is used to determine the contribu­tion of the bridging fibers to the steady state transverse frac­ture toughness G1c=To+LlGIc by using the fo11owing rela­tion:

LlGIc=[P(U)du (7)

where Ui is the crack opening displacement at the end of thefully developed bridging zone remote from the crack tip andTo is the mode I interface fracture resistance. Thus, Eq. (7)allows to relate the transverse fracture toughness proper­ties to the crack c10sure pressure and thus to the materialproperties of fibers, matrix and their interface.

4. Results and discussionSimulations were performed for interfacial shear

strengths of 20 and 75 MPa, respectively. These valuesrepresent the lower half of the common range of r in fiberreinforced ceramic composites,91 whereas experimentalresults101 indicate that in the case of the Si- Ti-C-O fiber­bonded material used in this study r is lower than 20 MPa.Figure 3 shows the maximum tensile fiber stress, Umax,

which occurs at the root of the fiber, and the average tensilefiber stress, u, which is measured at the center of the bridg­ing length of the fiber, as a function of the crack openingdisplacement. At the beginning, when the crack openingdisplacement (COD) is very small, both the maximum ten­sile fiber stress as well as the average tensile fiber stress, in­crease more or less linearly with increasing COD. For com-

Mare-Oliver NANDY et a1. Journal 0/ tile Ceramic Society 0/ Japan 106 [10J 1998 971

Fig. 3. Maximum tensile fiber stresses and average tensile fiberstresses as functions of crack opening displacement for interfacialshear strengths of r=20 and 75 MPa. Initial debonding occurs atU = Ud' Us indicates the steady state. and at U = Uj the fiber stressdecreases due to pronounced fiber-matrix debonding at Uo (seeFig.4).

parison, the corresponding stress-COD curves of a curvedbar derived fram Beam-Column Theory are plotted in thesame figure. The reason for the considerable difference bet­ween the FEM model and the calculated curves might bethe fact that the matrix below the fiber was not consideredin the latter. However, it can be seen that the slopes of thestress-COD curves reach similar values for both amax and a.

When debonding occurs, the stress-COD curve of theFEM model deviates from linearity. The critical COD for in­itial debonding, marked as Ud in Fig. 3, was the same forboth interfacial shear strengths. This can be explained bysolving Eq. (2) for Xd =0 which yields the critical value of aat the fiber root where initial debonding occurs. Thisthreshold value that causes debonding is only dependent onK\rc and the fiber diameter, d, but not dependent on r.Therefore, the onset of debonding is the same for bothr=20 and 75 MPa, respectively.

K\Ic is calculated from the mode Ir interface debondenergy, TI> which has been reported to exhibit very lowvalues (0.1 to 0.2 ]lm2 according to Ref. 9) in the case ofceramic matrix composites. In the present simulation, a con­stant value of Khc=0.19 MPa rm was assumed whichgives T,=0.2 ]lm2. It should be mentioned that KIIc wasestimated and that by varying Klrc different values of Bd

would be obtained.After initial debonding occurred, the fiber stress con­

tinues to rise with increasing COD, more pronounced forr=75 MPa than for r=20 MPa. At a certain COD, as in­dicated approximately by Us in Fig. 3, the stress reaches asteady state, though oscillating around a mean value whichis again higher for r= 75 MPa. Simulations were also per­formed up to larger COD values, however, a considerablechange of the fiber stress was not observed. Figure 4 showsthe total debond length as a function of the COD. It can beseen from this figure that the lower interfacial shearstrength causes greater debond lengths. This is in agree­ment with the fact that the stress reduction by frictionalsliding is lower for a lower interfacial shear strength. Ascan be expected, Fig. 4 indicates that the total debondlength is given by a linear function of the COD. Progressivedebonding causes the bridging length of the fiber to in­crease, resulting in stress relaxation as can be seen fromFig. 3 and Fig. 4: At the COD marked by Uo in Fig. 4 thefiber-matrix debonding exhibits a pranounced leap, caus­ing both ama.x and a to decrease considerably, as marked byUl in Fig. 3. However, it should be noticed that debonding isusuallya continuous process and thus the sizable variationsof the debond length shown in Fig. 4 are a result of the step-

605020 30 40Total Debonding Length [~m]

10

........

........0....0

= 20 MPa ""

't ~ ,P"" '0 •.••..•~o 9/ ", ....•........o 9-,;"''''' _•••.....•

....t ~~.\......•.....-0 ,"'

o P t "t = 75 MPa,../ •.....•

......etb.Q- •...... Uo

.•.....0.0005 0.001 0.0015 0.002 0.0025 0.003

Crack Openlng Displacement, u [mml

't = 20 MPa

---""""0 p.....

.....• ...-:: •.•...

•......J'~ .--:.::.

~~ß" ...;.-eg

;1"/.:;

/,,!,,/ •,/. ~~ ••• "(=75MPa

o~ 0o

60

70 ~------------------------,

o

o

10

E2; 50.<::

So

ai 40..J"Cc:

~ 30o(; 20Ö>-

50E2;

"ö>-

10

..s 40C>c:~..J

C> 30c:'e.a.

t5 20

60 ~------------------------~

wise calculation applied in this simulation.On the other hand, the debond length affects the chipping

length. The matrix starts spalling off because it can onlywithstand a certain amount of downward contact force ex­erted by the fiber. In this approach, the mean tensile stresswas taken as the criterion for matrix fracture which mightbe an underestimation of the actual susceptibility of thematrix to this downward force. The evolution of matrixchipping is illustrated in Fig. 5 where the total chippinglength is plotted as a function of the total debonding length.It can be seen from this figure that the evolution of matrixchipping follows a linear law with a similar slope for boththe interfacial shear strengths of 20 and 75 MPa. However,the maximum of the total length of matrix that chippedaway was larger for r=20 MPa. The reason for thisbehavior is the longer debond length (Fig. 4): when thefiber debonds from the matrix, the downward force on thematrix increases due to a longer lever arm, since the part ofthe matrix which already debonded from the fiber is loadedsimilar to a bar in bending. Chipping is a discontinuous pro­cess which substantially contributes to the oscillation of thestress-COD curve (see Fig. 3). When apart of the matrixspalls off, the free length of the fiber increases, causingstress relaxation. Thus, debonding, interfacial sliding andmatrix chipping are interacting with the stress-CODcurves, as shown in Fig. 6, where the average debondstress, ad (measured over the fiber diameter at the fiberroot), and the average tensile fiber stress, a (measured atthe center of the bridging length of the fiber) are plotted asa function of the COD for r=75 MPa. Without taking fric­tional sliding into account, the ad exhibits considerablyhigher values than a. However, deduced by the frictionalslipping component according to Eq. (3), ad tends tooscillate around a.

Fig. 4. Total debond lengths as functions of crack openmgdisplacemtnt for r=20 and 75 MPa.

Fig. 5. Total chipping length as a function of the total debondinglength for r=20 and 75 MPa.

.!s.'

-<>- Max. fiber stress, 20 MPa

-+- Av. fiber stress, 20 MPa

. 1::".. Max. fiber stress. 75 MPa

....•... All. fiber stress, 75 MPa

- •. Max. stress, column lheory

-Av. stress, column theory

'~"'~"'* ····~····~···~····~···l

0.001 0.0015 0.002 0.0025 0.003

Crack Openlng Displacement, u [mm]

0.0005

400

200

1600

1400

1200

"i 1000:E

;; 800'"eu; 600

972 Finite Element Method Simulation of Transverse Bridging in Fiber Reinforced Composites

450

---.-Av. fiber stress~Av. debond stress

..•.. Av. debond sttess ind. frictional slipping

0.10.02 0.04 0.06 0.08

Crack Opening Displacement, u [mm]

Kaute er aJ., K.cm=O.7 MpaFm

~.,/'?

-05 MPaFm .. "":

Kaute et a/~' K,,,,,- , " .' i.- _." "... .

" /'

~".. FEM Simulation

'---- ..... "... .,.'/

-:,. ....

o

o

1. 2.5::?~~ 2viIn

~ 1.5.cCl:Iof- 1"51l~ 0.5

u.

0.003

:e...e.

...

0.0005 0.001 0.0015 0.002 0.0025

Crack Openlng Displacement, u [mm]

100

50

o

o

400

350

300'ii"Q.

~250In

~200u;

150

Fig, 6. Average debond stress with and without frictional slipp­ing as functions of crack opening displacement for r= 75 MPa. Theaverage fiber stress is also plotted for comparison.

Fig. 8. Predicted contribution of fiber bridging to the transversefracture toughness of the Si- Ti-C-O fiber-bonded compositematerial.

Fig. 7. Crack closure pressures plotted as functions of crack open­ing displacement. The result obtained by the FEM simulation forr=20 MPa are compared to the analytical model of Kaute et aJ.2),3)The arro\\'s mark the onset of debonding.

The results obtained by the FEM simulation have beencompared to the bridging model developed by Kaute et a1.in Ref. 2 (see Appendix). The model employs both simplemechanics and Weibull statistics to express the crackc10sure pressure as a function of the material properties offibers, matrix and their interface. Figure 7 shows thec10sure pressure as a function of COD for both the FEMca1culation performed in this study and the model by Kauteet al. for r=20 MPa. The c10sure pressure was ca1culatedby using Eqs. (4) and (6), and the initial number of fibersbridging the crack wake was set no=6 mm-2 in Eqs. (Al)and (A2). For a mode I fracture toughness of K1cm=O.7MPa I!TI of the matrix which is normally reported forglass, the difference between the FEM ca1culation and themodel is considerable. However, taking into account thatpores are quite frequent between fibers, resulting in a reduc­ed fracture toughness of the matrix, the difference is lesscritical. Assuming Klcm = 0.5 MPa I!TI, the agreement bet­ween the models is improved. Especially, the COD-valuesat which the curve starts deviating from linearity are quitesimilar. For much lower values of Klcm' that means muchhigher porosity, the agreement between the models wouldbe even better, but it has to be noted that the criterion formatrix chipping used for the FEM model in this study isvery simple and should be refined prior to this argument.

The contribution of bridging fibers with no = 6 mm -2 totransverse fracture toughness is ca1culated using Eq. (7).This contribution, Ll Glc, is the area under the p (u) vs. ucurve, which is obtained employing Eq. (6). The result ofthis ca1culation is presented in Fig. 8. The FEM results andthe model of Kaute et al. with Klcm = 0.5 MPa rm exhibit a

0.035

0.030

'ii"Q.

~ 0.0250.

~. 0.020:IInIn"c: 0.015

~:I~ 0.010U

0.005

0.000

Kaute etj, ~cm= 0.7 MPa.rm

:f - - - - - - - - - - - - - - - - - - - - - -i' Kaute er al., K.c;m",O.5 MPa.rm'~- ..L" _

!t/It/

//

0.0005 0.001 0.0015 0.002 0.0025 0.003

Crack Opening Displacement, u [mmJ

similar characteristic. At a crack opening dis placement of0.05 mm, for example, the contribution of the bridgingfibers to the transverse fracture toughness of the Si- Ti-C­O fiber-bonded composite material is predicted to be Ll GIc

=1 J/m2• Comparing this value to the experimentally ob­tained transverse fracture toughness of the material (GIc=

To+LlG1c), exhibiting values between 12.2 and 12.5 J/m2, itappears that the predictions of these models are quitereasonable. Since the number of bridging fibers is sm all inthe Si- Ti-C-O fiber-bonded composite material, the con­tribution of the bridging fibers to the transverse fracturetoughness remains on a relatively low level. Fiber failuredoes not seem to be significant as long as the crack openingdisplacement remains small. However, for largerdisplacements, fiber failure may occur, reducing the closurepressure and leading to an upper limit of the rising R-curve.Since such large COD-values are definitely unacceptable forany kinds of technical applications, it seems to be sufficientto consider only the initial rising part of the R-curve wberefiber failure can be neglected. In this range, the FEM modelallows reasonable predictions of the transverse fracturetoughness of fiber reinforced composites.

5. Conclusions(1) Delamination cracks in fiber reinforced ceramic

composites are found to be bridged by fibers or fiberbundles in a shallow angle. These bridging fibers or fiberbundles have been shown to contribute to the transversefracture toughness by exerting a closure pressure on thecrack faces.

(2) A model for a single bridging fiber was applied tothe Finite Element Method taking into account thegeometrical properties of a curved fiber. The material pro­perties of a Si- Ti-C-O fiber-bonded ceramic compositematerial were used for the model calculations.

(3) The crack closure press ure as a function of thecrack opening displacement was simulated. It was foundthat the debonding and matrix chipping processes lead tostress-COD curves that differ significantly from thosecalculated by the beam-column theory. The crack closurepressure, debond length and the amount of matrix chippingare depending on the interfacial shear strength of the com­posite. The predicted contribution to the transverse frac­ture toughness of the Si- Ti-C-O fiber-bonded ceramic com­posite material in the FEM model yielded satisfactoryresults.

AppendixAccording to Ref. 2, the c1osure' pressure induccd by bridging

fibers along the crack faces is described by the following two equa­tions; the first one is valid for the early stage of the bridging pro-

Mare-Oliver NANDY et al. Journal 0/ the Ceramic Society 0/ Japan 106 [JO] 1998:;;:c

973

(Al)

cess before the crack c10sure pressure, p (u), reaches its maximum:

[3TiEfd (U) 71d2 /2TEflo (U)2]p(u) =no -- - +- ,1-- -64,1/ 10 4 I d 10

Due to matrix chipping, the maximal fiber force is limited to a cer­tain value. As the matrix peel-off process exposes the fiber moreand more to the maximum fiber stress, the number of fibersdecreases via fiber failure. At this state, the c10sure pressure is ex­pressed by

p(u) = hnoKIcmCiß3/2 exp [

x (u-u*) ] (A2)

where Cißis a mean crack length parallel to the fiber over which thesteady state force/max acts, ab stands for the mean maximum ben­ding stress at the ro ot of the fiber, Cb is a dimensionless correctionfactor smaller than unity to allow the comparison between themean bending stress and the reference tensile stress, Ipmax is thesteady state bridging angle and u* is the reference crack openingup to which the bridging fibers remain intact. In the present studyDIawas 0.002 mm, Cb=0.7, and ab=5.42 GPa. In situ observationsindicate Ipmax is about 12° and u* about 0.0135 mm. All otherparameters are given in Table 1.

Acknowledgments The authors would like to thank NEDO tor

their financial contribution and Ube Industries Co., Ltd. tor supply­ing the material. M.-O. Nandy wishes to acknowJedge the tinancial

support through the German National SchoJarship Foundation andDaimler-Benz AG.

References

1) S. M. Spearing and A. G. Evans, Acta Metall. Mater., 40,2191-99 (1992).

2) D. A. W. Kaute, H. R. Shercliff and M. F. Ashby, ActaMetall. Mater., 41, 1959-70 (1993).

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