MECHANI MATERIALS ELSEVIER Mechanics of Materials 29(1998)111-121 Effect of microstructural parameters on the fracture behavior of fiber-reinforced ceramics Yu-Fu Liu a*, Chitoshi Masuda Ryoji Yuuki b National Research Institute for Metals, 1-2-1 Sengen, Tsukuba 305, Japan nstitute of Industrial Science, University of Tokyo, Tokyo, Japan Received 30 May 1997 Abstract A bridging law which includes both interfacial debonding and sliding properties in fiber-rei ceramics is applied to fiber bridging analysis and crack growth problems by treating bridging fibers as a distribution of closure stress. A numerical method to solve distributed spring model of a penny-shaped crack is provided to determine the bridging stress, debond length, crack opening displacement and stress intensity factor. By introducing fracture criteria of the composite and fiber, crack growth behavior in R-curve for the penny-shaped crack are simulated and the effects of such microstructural parameters as interface debonding toughness, compressive residual stress, frictional sliding stress, and fiber volume fraction on the R-curve are quantified in an explicit manner. On the basis of R-curve results, the toughening mechanism of fiber-reinforced ceramics is discussed. @1998 Elsevier Science Ltd. All rights reserved Keywords: Microstructural parameters; Fracture behavior; Fiber-reinforced ceramics 1. Introduction approximation levels exist with regards to bridging laws(Marshall and Cox, 1985; Budiansky et al The contribution of fiber bridging in fiber-rein- 1986 Budiansky and Amazigo, 1989, Budiansky et forced ceramics to toughness enhancement is widely al., 1995; Meda and Steif, 1994), inter facial proper accepted and the toughness increment due to fiber ties( Gao et al., 1988; Hutchinson and Jensen, 1990), bridging is governed by the constitutive relation fiber and composite anisotropy(Hutchinson and between the fiber bridging stress and crack opening Jensen, 1990; Marshall, 1992; Luo and Ballarini, isplacement, ie, bridging law. In the bridging law, 1994), among arguments on bridging length scales the mechanical properties of the fiber-matrix inter-(Bao and Suo, 1992; Cox, 1993; Cox and Marshall face play an important role. Various treatment and 1994), initial flaw size and specimen geometrical effect( Cox and Marshall, 1991; Budiansky and Cui, 1994). Early studies (for example, Aveston et al 1971) used one parameter, i. e, constant shear fric- tional stress. to deal with the interfacial resistance (formerly Associate Professor) effect, which is presumably applicable to composite 0167-6636/98/S19.00@ 1998 Elsevier Science Ltd. All rights reserved PS0167-6636(98)00009X
Mechanics of Materials 29 1998 111–121 Ž . Effect of microstructural parameters on the fracture behavior of fiber-reinforced ceramics Yu-Fu Liu a,), Chitoshi Masuda a , Ryoji Yuuki b,1 a National Research Institute for Metals, 1-2-1 Sengen, Tsukuba 305, Japan b Institute of Industrial Science, UniÕersity of Tokyo, Tokyo, Japan Received 30 May 1997 Abstract A bridging law which includes both interfacial debonding and sliding properties in fiber-reinforced ceramics is applied to fiber bridging analysis and crack growth problems by treating bridging fibers as a distribution of closure stress. A numerical method to solve distributed spring model of a penny-shaped crack is provided to determine the bridging stress, debond length, crack opening displacement and stress intensity factor. By introducing fracture criteria of the composite and fiber, crack growth behavior in R-curve for the penny-shaped crack are simulated and the effects of such microstructural parameters as interface debonding toughness, compressive residual stress, frictional sliding stress, and fiber volume fraction on the R-curve are quantified in an explicit manner. On the basis of R-curve results, the toughening mechanism of fiber-reinforced ceramics is discussed. q 1998 Elsevier Science Ltd. All rights reserved. Keywords: Microstructural parameters; Fracture behavior; Fiber-reinforced ceramics 1. Introduction The contribution of fiber bridging in fiber-reinforced ceramics to toughness enhancement is widely accepted and the toughness increment due to fiber bridging is governed by the constitutive relation between the fiber bridging stress and crack opening displacement, i.e., bridging law. In the bridging law, the mechanical properties of the fiber–matrix interface play an important role. Various treatment and ) Corresponding author. 1 Deceased formerly Associate Professor . Ž . approximation levels exist with regards to bridging laws Marshall and Cox, 1985; Budiansky et al., Ž 1986; Budiansky and Amazigo, 1989; Budiansky et al., 1995; Meda and Steif, 1994 , interfacial proper- . ties Gao et al., 1988; Hutchinson and Jensen, 1990 , Ž . fiber and composite anisotropy Hutchinson and Ž Jensen, 1990; Marshall, 1992; Luo and Ballarini, 1994 , among arguments on bridging length scales . ŽBao and Suo, 1992; Cox, 1993; Cox and Marshall, 1994 , initial flaw size and specimen geometrical . effect Cox and Marshall, 1991; Budiansky and Cui, Ž 1994 . Early studies for example, Aveston et al., . Ž 1971 used one parameter, i.e., constant shear fric- . tional stress, to deal with the interfacial resistance effect, which is presumably applicable to composite 0167-6636r98r$19.00 q 1998 Elsevier Science Ltd. All rights reserved. PII S0167-6636 98 00009-X Ž
Y-F. Liu et al./ Mechanics of Materials 29(1998)111-121 systems with weak or unbonded sliding interfaces. A Effects of key microstructural parameters ensuing typical bridging law, P v8, was obtained(Marshall from an axisymmetrical unit-cell model are quanti and Cox, 1985), with p and 8 representing the fied and discussed with an emphasis on the resis- edging traction and crack opening displacement, tance curve respectively. The Coulomb friction law was also used to describe frictional sliding resistance at debond interface( Budiansky et al., 1986; Gao et al., 2. Analysis of the bridging effect by the dis 1988; Hutchinson and Jensen, 1990). Besides friction tributed spring model at the interface, the importance of interface bonding and debonding phenomena encourages the use of 2. 1. Analytical model interface bond strength or toughness, based on the debonding stress and energy release rate concepts of Fig. I shows the analytical model used in this the debonding crack(for example, Hutchinson and present study. A penny-shaped crack with an initial Jensen, 1990). Various forms of bridging laws have length of co in a unidirectionally aligned fiber rein- een developed to incorporate effects of scattered forced ceramic is assumed to extend to c under fiber strength(Thouless and Evans, 1988; Cox and monotonically increasing loading, o with intact Marshall, 1991), stick and slip conditions(Meda and fibers left behind to bridge the crack wake. due to Steif, 1994; Liu, 1995; Liu and Kagawa, 1996), fiber bridging, stress intensity at the crack tip is time-dependence(Marshall, 1992, Cox and Rose, reduced and the crack opening is restrained and 994), large scale sliding(Xia et al., 1994)and further propagation of the crack is retarded. Fibers surface roughness(Parthasarathy et al., 1994). Many with a volume fraction of f are assumed to have a researchers have successfully characterized the me- single-valued strength, implying that fiber failure chanical behavior of fiber-reinforced ceramics with occurs only at the main crack wake. The problem of various levels of approximation, providing fruitful interest is to evaluate the stress intensity factor in the results and better understanding of fiber-reinforced presence of bridging fibers, and then propagating behavior of the main crack. one main feature of the However, effects of interfacial properties in the present study is the application of a bridging law resence of both interface debonding and sliding on which incorporates both friction and debonding at the fracture process of fiber-reinforced ceramics re- the fiber-matrix interface. The obtained results will main less explored. In Budiansky et al. (1995), a be expressed in terms of fracture resistance curve new bridging law was derived that included the influe nce of debonding toughness as well as friction based on modified shear-lag analysis. However, only A the toughness and strength problem under a steady- Frictional sliding and debonding state, i.e., long crack limit, were considered there Besides, most of the analyses were expressed in complicated formulas and the many parametric de pendencies occurring during the fracture process were not explicit enough. The aim of the present study to address the toughness enhancement problem in the fracture process of fiber-reinforced ceramics when debonding toughness as well as friction is present. Using the energy release rate derived previously and a method to determine the debond length(Yuuki and Liu, 1994)as a bridging law, we will look at a partial bridging configuration and toughening mech anisms, with particular attention given to the fracture Fig. 1. Schematic diagram of a penny-shaped crack in fiber-rein- esistance curve accompanying an initial flaw growth. forced ceramics
112 Y.-F. Liu et al.rMechanics of Materials 29 1998 111–121 ( ) systems with weak or unbonded sliding interfaces. A typical bridging law, p;'d , was obtained Marshall Ž and Cox, 1985 , with . p and d representing the bridging traction and crack opening displacement, respectively. The Coulomb friction law was also used to describe frictional sliding resistance at a debond interface Budiansky et al., 1986; Gao et al., Ž 1988; Hutchinson and Jensen, 1990 . Besides friction . at the interface, the importance of interface bonding and debonding phenomena encourages the use of interface bond strength or toughness, based on the debonding stress and energy release rate concepts of the debonding crack for example, Hutchinson and Ž Jensen, 1990 . Various forms of bridging laws have . been developed to incorporate effects of scattered fiber strength Thouless and Evans, 1988; Cox and Ž Marshall, 1991 , stick and slip conditions Meda and . Ž Steif, 1994; Liu, 1995; Liu and Kagawa, 1996 ,. time-dependence Marshall, 1992; Cox and Rose, Ž 1994 , large scale sliding Xia et al., 1994 and . Ž. surface roughness Parthasarathy et al., 1994 . Many Ž . researchers have successfully characterized the mechanical behavior of fiber-reinforced ceramics with various levels of approximation, providing fruitful results and better understanding of fiber-reinforced ceramics. However, effects of interfacial properties in the presence of both interface debonding and sliding on the fracture process of fiber-reinforced ceramics remain less explored. In Budiansky et al. 1995 , a Ž . new bridging law was derived that included the influence of debonding toughness as well as friction, based on modified shear–lag analysis. However, only the toughness and strength problem under a steadystate, i.e., long crack limit, were considered there. Besides, most of the analyses were expressed in complicated formulas and the many parametric dependencies occurring during the fracture process were not explicit enough. The aim of the present study is to address the toughness enhancement problem in the fracture process of fiber-reinforced ceramics when debonding toughness as well as friction is present. Using the energy release rate derived previously and a method to determine the debond length Yuuki and Ž Liu, 1994 as a bridging law, we will look at a . partial bridging configuration and toughening mechanisms, with particular attention given to the fracture resistance curve accompanying an initial flaw growth. Effects of key microstructural parameters ensuing from an axisymmetrical unit-cell model are quantified and discussed with an emphasis on the resistance curve. 2. Analysis of the bridging effect by the distributed spring model 2.1. Analytical model Fig. 1 shows the analytical model used in this present study. A penny-shaped crack with an initial length of c in a unidirectionally aligned fiber rein- 0 forced ceramic is assumed to extend to c under monotonically increasing loading, s with intact fibers left behind to bridge the crack wake. Due to fiber bridging, stress intensity at the crack tip is reduced and the crack opening is restrained and further propagation of the crack is retarded. Fibers with a volume fraction of f are assumed to have a single-valued strength, implying that fiber failure occurs only at the main crack wake. The problem of interest is to evaluate the stress intensity factor in the presence of bridging fibers, and then propagating behavior of the main crack. One main feature of the present study is the application of a bridging law which incorporates both friction and debonding at the fiber–matrix interface. The obtained results will be expressed in terms of fracture resistance curve. Fig. 1. Schematic diagram of a penny-shaped crack in fiber-reinforced ceramics
Y-F. Liu et al./ Mechanics of Materials 29(1998)111-121 2. 2. Bridging law in consideration of interfacial verified that the energy release rate showed qualita- friction and debonding toughness (Yuuki and Liu, tive agreement with finite element analysis results in 1994) determining debond length and predicted results agreeing well with previous experimental data, al An axisymmetrical fiber-matrix model having a though singular stress and strain fields at the debond debond length of I and a constant sliding stress, T crack tip were simplified in deriving the energy between 0<:<I is used(Fig. 2). R is the fiber release rate for the debond crack(Yuuki and Liu, radius and Rm is the matrix radius, and fiber volume 1994). This energy release rate expression is a sec- fraction,f, is defined as f=(R/Rm). The outer ond-order function of the debond length and will be boundary conditions of the cylindrical model are set outlined below to stress-free. A uniform tensile stress, o, is applied Extruding bridging fiber and surrounding matrix to the upper end of the model. With the traditional as an axisymmetrical cylinder, the crack opening shear-lag method and energy balance arguments, we displacement, u, is expressed as(Yuuki and Li have derived an all-inclusive energy release rate, G 1994) by using a Lame solution from Hutchinson and Jensen(1990)after considering an infinitesimal ad vance of the debond crack( Gao et al., 1988 Sigl and b,F 2(1) Evans. 1989: Hutchinson and Jensen. 1990: Yuuki and Liu, 1994). Influences of various important pa- where ef=50(ar-am)dr. Here, AT is the tem- rameters on the energy release rate were demon- perature change from bonding, a's and b, are mate strated and its physical significance clarified. It wa rial- and geometry-relevant parameters given in Hutchinson and Jensen(1990). Other parameters are explained in Fig. 2 Eq.(1) is a nonlinear function of the debond Matrix length, 1, which needs to be determined. Assuming that the applied stress, o, is fixed, the debond length Material Properties upon the applied stress is obtained from the follow- Fiber: Er, Vr,af, af ing condition Matrix: Em, Vm. am G.≥G where Gic is the debonding toughness obtained ex- perimentally and G is normalized as Z▲ G G E R e a,BE Roof 向1a2Ene 2f ic cylindrical model: (a)longitudinal section, (b)transverse section. Also shown are material and geometric where a's and b's are again material- and definitions. Material properties are indicated in this figure, where geometry-relevant parameters given in Hutchinson E and v are Youngs moduli and Poissons ratios, respectively, and subscripts, f and m, refer to fiber and matrix, respectively; ar and Jensen(1990). Within the two roots of Eq (2) is thermal expansion coefficient and subscripts, r and z, for a only the small one is physically meaningful and dicate radial and axial directions, respectively should be taken as the critical debond length when
Y.-F. Liu et al.rMechanics of Materials 29 1998 111–121 ( ) 113 2.2. Bridging law in consideration of interfacial friction and debonding toughness Yuuki and Liu, ( 1994) An axisymmetrical fiber–matrix model having a debond length of l and a constant sliding stress, t , between 0FzFl is used Fig. 2 . Ž . R is the fiber f radius and Rm is the matrix radius, and fiber volume Ž .2 fraction, f, is defined as fs R rR . The outer f m boundary conditions of the cylindrical model are set to stress-free. A uniform tensile stress, s , is applied to the upper end of the model. With the traditional shear–lag method and energy balance arguments, we have derived an all-inclusive energy release rate, G ,i by using a Lame solution from Hutchinson and ´ Jensen 1990 after considering an infinitesimal ad- Ž . vance of the debond crack Gao et al., 1988; Sigl and Ž Evans, 1989; Hutchinson and Jensen, 1990; Yuuki and Liu, 1994 . Influences of various important pa- . rameters on the energy release rate were demonstrated and its physical significance clarified. It was Fig. 2. Axisymmetric cylindrical model: a longitudinal section, Ž . Ž . b transverse section. Also shown are material and geometric definitions. Material properties are indicated in this figure, where E and n are Young’s moduli and Poisson’s ratios, respectively, and subscripts, f and m, refer to fiber and matrix, respectively; a is thermal expansion coefficient and subscripts, r and z, for a indicate radial and axial directions, respectively. verified that the energy release rate showed qualitative agreement with finite element analysis results in determining debond length and predicted results agreeing well with previous experimental data, although singular stress and strain fields at the debond crack tip were simplified in deriving the energy release rate for the debond crack Yuuki and Liu, Ž 1994 . This energy release rate expression is a sec- . ond-order function of the debond length and will be outlined below. Extruding bridging fiber and surrounding matrix as an axisymmetrical cylinder, the crack opening displacement, u, is expressed as Yuuki and Liu, Ž 1994 :. 2 1 1 s t l T us ya lq qa ´ l , 1Ž . 1 2z ž / b F E ER 2 m mf T DT Ž f m where ´ sH a ya .d . Here, DT is the tem- z0z T perature change from bonding, a’s and b are mate- 2 rial- and geometry-relevant parameters given in Hutchinson and Jensen 1990 . Other parameters are Ž . explained in Fig. 2. Eq. 1 is a nonlinear function of the debond Ž . length, l, which needs to be determined. Assuming that the applied stress, s , is fixed, the debond length upon the applied stress is obtained from the following condition: G GG , 2Ž . i ic where G is the debonding toughness obtained ex- ic perimentally and G is normalized as: i 2 G E 2 t l i m G˜i 23 ' sŽ . b qb Rfs 2 ½ ž / s ž / Rf T lt 1 a a 1 2z m ´ E y yq Rfs 2 f 2 2s 2 T 1yfa a E 1 2 mz ´ q q , 2-1 Ž . ž / 2 f 2s 5 where a’s and b’s are again material- and geometry-relevant parameters given in Hutchinson and Jensen 1990 . Within the two roots of Eq. 2 , Ž . Ž. only the small one is physically meaningful and should be taken as the critical debond length when
Y-F. Liu et al./ Mechanics of Materials 29(1998)111-121 the applied fiber stress, o/f, is equal to the fiber tigliano's theorem, the crack opening displacement is tensile strength, of(Yuuki and Liu, 1994). Eq (2) obtained as(Sneddon and Lowengrug, 1968) is an additional equation relating 1, o and other microstructural parameters, and combination of Eqs 4(1-2)c u( r) (1)and (2) yields a new bridging law 丌BE Plod 2.3. Solution of the distributed spring model where EC=Er+(I-f)Em and v=fvr +(1 As in previous studies raised above, the bridging f)vm. Here, E and v are the Young's moduli and fibers in Fig. 1 are replaced by an equivalent trac- Poissons ratios, respectively, and subscripts, f and tion-displacement law defined in Eqs. (1) and(2), m, express quantities for fiber and matrix,respec- with the new traction and displacement boundary tively, B is an orthotropic factor of the composite, conditions along the crack surface remaining to be which is near "1.0 for most engineering combina- satisfied. Assuming that a bridging region occurs tions(Budiansky and Amazigo, 1989; Budiansky and between Co <x< c as shown in Fig. 3, the stres 994) intensity factor of the crack tip may be expressed as A bridging law represented by Eqs. (1) and(2), (Law,1975;Sih,1985) associated with Eqs. (3)and(4), will be used below to grasp comprehensively effects of various mi- p(x)xdx crostructural parameters including the interface fric- (3) tion and debonding toughness. Egs.(1)and(2)may nience as where p(x)is the bridging traction shown in Fig. 3 and is smeared out as a continuous function of u(x)=f(o(x), I (x) (co <x<c) position, P(x)=fo (x)=o(x)(Figs. 2 and 3) a(x)=g(1(x)) Taking the composite as an equivalent trans- versely isotropic body and making use of Cas- In order to obtain a full profile of the bridging problem, it is necessary to solve Eqs. ( 1)-(4)simul taneously, with four unknowns, u, o, I and KI,to be determined. Since complicated non-linearity is involved in these equations, a numerical method is usually needed. A solution method through discretiz- Transversely Isotropic Material tion, x, is used while many other solution methods may be found in the literature(Marshall and Cox 1987; McCartney, 1987, Budiansky and Amazigo 1989: Cox and Marshall, 1991; Budiansky and Cui 1994) Fig. 4 shows the flow chart for solving the present bridging problem numerically. An unknown of bridg- ing stress, o(x), is discretized over normalized co- ordinates, X=x/c, with N equal divisions; within one division, o(X) is linearized as Fig. 3. Distributed spring model that treats bridging fibers as a distribution of stresses to close the crack face X≤X≤X1+1
114 Y.-F. Liu et al.rMechanics of Materials 29 1998 111–121 ( ) the applied fiber stress, srf, is equal to the fiber tensile strength, sfu Ž . Ž. Yuuki and Liu, 1994 . Eq. 2 is an additional equation relating l, s and other microstructural parameters, and combination of Eqs. Ž. Ž. 1 and 2 yields a new bridging law. 2.3. Solution of the distributed spring model As in previous studies raised above, the bridging fibers in Fig. 1 are replaced by an equivalent traction–displacement law defined in Eqs. 1 and 2 , Ž. Ž. with the new traction and displacement boundary conditions along the crack surface remaining to be satisfied. Assuming that a bridging region occurs between c Fx-c as shown in Fig. 3, the stress 0 intensity factor of the crack tip may be expressed as Ž . Lawn, 1975; Sih, 1985 : c c pxx 1 Ž . d x K s2( ( sy2 ,3 H Ž . I p p ' 2 0 1yx where p xŽ . is the bridging traction shown in Fig. 3 and is smeared out as a continuous function of position, p xŽ . Ž . Ž .Ž . sfs x ss x Figs. 2 and 3 . f Taking the composite as an equivalent transversely isotropic body and making use of CasFig. 3. Distributed spring model that treats bridging fibers as a distribution of stresses to close the crack face. tigliano’s theorem, the crack opening displacement is obtained as Sneddon and Lowengrug, 1968 : Ž . 2 2 4 1Ž. Ž. yn c 4 1yn c ' 2 u x Ž . s s 1yx y pbEc c pbE = 1 d s pt s Ž .dt H H , 4Ž . ' ' 2 2 22 x s yx s 0 yt where Ecf m f sfE qŽ. Ž 1yf E and nsfn q 1y f .nm. Here, E and n are the Young’s moduli and Poisson’s ratios, respectively, and subscripts, f and m, express quantities for fiber and matrix, respectively; b is an orthotropic factor of the composite, which is near ;1.0 for most engineering combinations Budiansky and Amazigo, 1989; Budiansky and Ž Cui, 1994 .. A bridging law represented by Eqs. 1 and 2 , Ž. Ž. associated with Eqs. 3 and 4 , will be used below Ž. Ž. to grasp comprehensively effects of various microstructural parameters including the interface friction and debonding toughness. Eqs. 1 and 2 may Ž. Ž. be rewritten, respectively, for descriptive convenience as: u x Ž . Ž . Ž . Ž . Ž. sfŽ . s x ,lx c0Fx-c , 5 s Ž . Ž . Ž. x sglx Ž . . 6 In order to obtain a full profile of the bridging problem, it is necessary to solve Eqs. 1 – 4 simul- Ž. Ž. taneously, with four unknowns, u, s , l and KI, to be determined. Since complicated non-linearity is involved in these equations, a numerical method is usually needed. A solution method through discretizing the equivalent bridging stress, s Ž . x , over position, x, is used while many other solution methods may be found in the literature Marshall and Cox, Ž 1987; McCartney, 1987; Budiansky and Amazigo, 1989; Cox and Marshall, 1991; Budiansky and Cui, 1994 .. Fig. 4 shows the flow chart for solving the present bridging problem numerically. An unknown of bridging stress, s Ž . x , is discretized over normalized coordinates, Xsxrc, with N equal divisions; within one division, s Ž . X is linearized as: XyXi s Ž. Ž . X s s ys qs , iq1 i i D X FXFX , 7Ž . i iq1
Y-F. Liu et al./ Mechanics of Materials 29(1998)111-121 where A is an incremental range and o is the discretized stress at a grid point X-I(Fig. 5). At the crack tip, on+I=o(1), whose value is dictated by qs.(1)and (2) to satisfy the crack tip condition 0.0L 1) with the following auxiliary integral variables T=ssin 8, S=x/cos (8) Fig. 5. Discretizing bridging stress at the bridging zone as a linear function of position Eq(4) may be transformed to 4( 丌E tion of /(X) into Eq. (5)results in an expression of 4(1-v2)cx arccos do Id( X )as a function of o. The obtained expression is again substituted into Eq. (9)to yield the following E 0}mnd(9)a(x)-+-“2 TE。6V1-x2=B(a)(10) The debond length, I(X, ) at X, is calculated as a function of o according to Eq(6). Then, substitu- The remaining terms in Eq. (9)is also expressed as a function of o, leading to the following equa tIon 4(1-v2)cr rarccos x do E Discretize o(x):01,02,,ON X sin6sin6d6≡A as a function ofσ Integration with regards to variables, 0 and , ar carried out by the trapezoidal rule For each discretized grid between co/csX<I it follows from Eqs. (10)and(11)that A=B() Substitute COD eqution Eq. (12)is a series of nonlinear equations with N unknowns, which is solved to obtain o by Newtons Calculate unknows a with Newton's iteration method iterative method. In actual calculation, many initial values of o were tested and unique convergent solutions were achieved. Once o is obtained other nd /(x) unknowns are easily determined(Fig. 4) 2. 4. Comparison with Marshall and Cox's results Calculation of KI To verify the above procedure for solving bridg ng problems, results of distributed bridging str for the same problem were compared to that of Fig. 4. Flow chart to solve the distributed spring model that Marshall and Cox (1987). Results shown in Fig. 6 considers debonding toughness as well as frictio were for a bridging law of p v8 with a partial
Y.-F. Liu et al.rMechanics of Materials 29 1998 111–121 ( ) 115 where D is an incremental range and s is the i discretized stress at a grid point X Ž . Fig. 5 . At the iy1 crack tip, sNq1ss Ž . 1 , whose value is dictated by Eqs. 1 and 2 to satisfy the crack tip condition Ž. Ž. uŽ . 1 . With the following auxiliary integral variables, tss sin u , ssxrcos f Ž . 8 Eq. 4 may be transformed to: Ž . 2 4 1Ž . yn c ' 2 u x Ž . s s 1yx p Ec 2 4 1Ž . yn cx arccos x df y H 2 p Ec 0 cos f = pr2 x H p sin u sin udu . 9Ž . ž / 0 cos f The debond length, l XŽ ., at X is calculated as a i i function of s according to Eq. 6 . Then, substitu- Ž . i Fig. 4. Flow chart to solve the distributed spring model that considers debonding toughness as well as friction. Fig. 5. Discretizing bridging stress at the bridging zone as a linear function of position. tion of l XŽ . Ž. into Eq. 5 results in an expression of i u XŽ . as a function of s . The obtained expression is i i again substituted into Eq. 9 to yield the following Ž . relation in normalized coordinates: 2 4 1Ž . yn c 2 u XŽ . Ž. Ž. i i ii y s(1yX 'B s . 10 p Ec The remaining terms in Eq. 9 is also expressed Ž . as a function of s , leading to the following equa- i tion: 4 1yn2 Ž . cX arccos X df H 2 p Ec 0 cos f pr2 X =H s sin u sin udu'A s . 11 Ž . ij j 0 ž / cos f Integration with regards to variables, u and f, are carried out by the trapezoidal rule. For each discretized grid between c rcFX-1, 0 it follows from Eqs. 10 and 11 that Ž. Ž. Aij j i i s sB Ž. Ž. s . 12 Eq. 12 is a series of nonlinear equations with Ž . N unknowns, which is solved to obtain s by Newton’s i iterative method. In actual calculation, many initial values of s were tested and unique convergent i solutions were achieved. Once s is obtained, other i unknowns are easily determined Fig. 4 . Ž . 2.4. Comparison with Marshall and Cox’s results To verify the above procedure for solving bridging problems, results of distributed bridging stress for the same problem were compared to that of Marshall and Cox 1987 . Results shown in Fig. 6 Ž . were for a bridging law of p;'d with a partial
116 Y-F. Liu et al./ Mechanics of Materials 29(1998)111-121 Fig. 7 shows a distribution of the crack opening displacement(COD), with the debonding toughness, Gie, altered. Larger Gic produces smaller COD, with L.5 COD increasing significantly at the unbridged zone Figs. 8-10 indicate effects of Gi on distributions of the bridging stress, interface debond length and stress intensity factor, respectively. These results demon 0.5 · Present study strate that Gis has a large impact on the bridging chall and Cox effect and large Gic yields small debond length large bridging stress, as such a small stress intensity factor. These tendencies seem to agree with practical Normalized position, x/c situations. The influences of interfacial sliding stress Fig. 6. Comparison of bridging stress distributions to results by at the debond interface, on the bridging stress distri Marshall and Cox(1987) when the solution method presented this study is applied to a bridging law of bution and stress intensity factor are shown in Figs I 1 and 12, respectively. It is obvious that the effect of interfacial sliding stress possesses the same ten- bridging zone, co/c=0.156 and a normalized ap- dency as that of Gic. Effects of other microstructural plied stress of 2=0.90. The obtained distributed parameters, such as fiber volume fraction and resid- edging stresses were in reasonable agreement, al ual stresses, were reported in Liu(1995), with larger though discretizing and solution procedure were dif- residual stresses and smaller fiber volume fraction ferent between the present study and that of Marshall yielding smaller bridging stress and larger stress and Cox(1987). Some degree of error appeared in Intensity factor plotting the results of Marshall and C read directly from one figure in the published work Taking into account such a factor. it is concluded 4. Crack growth analysis and discussion that confidence in the current analytical procedure is gained through the comparison Since aligned fiber-reinforced ceramics behave like ductile materials, R-curve is utilized to identify its fracture process, despite of its initial crack and 3. Analytical results of bridging effect geometrical dependence. In this section, fracture cri teria for composite and bridging fiber are considered Bridging effects are analyzed based on the mate- to analyze the growth of an initial penny-shaped rial properties listed in Tables 1 and 2(Ceramic crack on the basis of the above bridging effect Source, 1989). The material parameters essentially computation. The obtained results are expressed in orrespond to a combination of Al,O, matrix rein- the form of R-curve, with influences of various forced with continuous SiC fiber. Other altered pa- microstructural parameters on R-curve identified and rameters are indicated in the corresponding figure Table I Material properties for numerical study Materials Fiber(sic) Matrix(Al,O3) Youngs modulus, E(GPa) 4 Coefficient of thermal expansion,a(×10°/K) Matrix toughness, Kc(MPay m) Fiber strength, af(GPa Initial matrix crack length, co(mm) 3.0
116 Y.-F. Liu et al.rMechanics of Materials 29 1998 111–121 ( ) Fig. 6. Comparison of bridging stress distributions to results by Marshall and Cox 1987 when the solution method presented in Ž . this study is applied to a bridging law of p;'d . bridging zone, c rcs0.156 and a normalized ap- 0 plied stress of Ss0.90. The obtained distributed bridging stresses were in reasonable agreement, although discretizing and solution procedure were different between the present study and that of Marshall and Cox 1987 . Some degree of error appeared in Ž . plotting the results of Marshall and Cox, which were read directly from one figure in the published work. Taking into account such a factor, it is concluded that confidence in the current analytical procedure is gained through the comparison. 3. Analytical results of bridging effect Bridging effects are analyzed based on the material properties listed in Tables 1 and 2 Ceramic Ž Source, 1989 . The material parameters essentially . correspond to a combination of Al O matrix rein- 2 3 forced with continuous SiC fiber. Other altered parameters are indicated in the corresponding figure. Fig. 7 shows a distribution of the crack opening displacement COD , with the debonding toughness, Ž . Gic ic , altered. Larger G produces smaller COD, with COD increasing significantly at the unbridged zone. Figs. 8–10 indicate effects of G on distributions of ic the bridging stress, interface debond length and stress intensity factor, respectively. These results demonstrate that Gic has a large impact on the bridging effect and large G yields small debond length, ic large bridging stress, as such a small stress intensity factor. These tendencies seem to agree with practical situations. The influences of interfacial sliding stress at the debond interface, on the bridging stress distribution and stress intensity factor are shown in Figs. 11 and 12, respectively. It is obvious that the effect of interfacial sliding stress possesses the same tendency as that of G . Effects of other microstructural ic parameters, such as fiber volume fraction and residual stresses, were reported in Liu 1995 , with larger Ž . residual stresses and smaller fiber volume fraction yielding smaller bridging stress and larger stress intensity factor. 4. Crack growth analysis and discussion Since aligned fiber-reinforced ceramics behave like ductile materials, R-curve is utilized to identify its fracture process, despite of its initial crack and geometrical dependence. In this section, fracture criteria for composite and bridging fiber are considered to analyze the growth of an initial penny-shaped crack on the basis of the above bridging effect computation. The obtained results are expressed in the form of R-curve, with influences of various microstructural parameters on R-curve identified and discussed. Table 1 Material properties for numerical study Materials Fiber SiC Matrix Al O Ž. Ž . 2 3 Young’s modulus, E Ž . GPa 420 400 Poisson’s ratio, n 0.19 0.23 -6 Coefficient of thermal expansion, aŽ . =10 rK 4.5 7.5 Matrix toughness, Kmc Ž . MPa6m y 2.5 Fiber strength, sfu Ž . GPa 1.0 y Initial matrix crack length, c0 Ž . mm 3.0
Y-F. Liu et al./ Mechanics of Materials 29(1998)111-121 Table 2 oG;=20.0N/m Basic microstructural parameter set for numerical study G;=12.0N Debonding toughness, Gic (N/m) C/c=0.5 =1000MPa △G;=6.0N/m Frictional stress, T(MPa) Temperature change, AT(K) Fiber volume fraction, f bE Altered parameters are indicated in the corresponding figure 4.1. Composite fracture criterion 0.0 Considering the plane-strain condition for a Normalized position, x/c Mode-I crack lying in the plane of transverse Fig. 8. Influences of G on the distribution of bridging stress. sotropy, Budiansky and Cui (1994)obtained the following expression of critical Mode-I stress inten- sity factor, Kic, on the basis of energy balance KIc is obtained from Eq (13). Then, a crack growth length of Ac is extended, leaving intact fibers be- E(1-m(1-f (13) bridging, the value of K, is reduced (d K/dc <0)so that gradually increasing loading is required to sat isfy KI=Kic. The newly obtained loading stress, where Kme is the critical Mode-I stress intensity t is used to describe fracture process of fiber-rein- factor for monolithic matrix. A criterion K,=Kic is used to simulate crack growth, where KI is the stress forced ceramics in the form of R-curve or g-c intensity factor calculated in Section 2 curve, where c=Co Ac. The R-curve approach is utilized in the current study and fracture resistance 4.2. Matrix crack growth analysis Kg, is defined as KR=20VC/T Fig. 13 shows the flow chart for crack growth simulation carried out in the present study. Firstly, Cracking extension becomes unstable under the cracking initiation stress, oo, for a given initial flaw following two situations while the relation KI-Mic size, co, is calculated as do=Kc/(2/c/T), where is satisfied. One is due to initiation of fiber fracture at the largest fiber stress obtained for each loading increment and this leads to next fiber fracture. and hen catastrophic cracking follows. Fiber fracture is x上 G:=200N/m o=100. 0MPa Fig. 7. Influences of G on the distribution of coD Fig. 9. Influences of G on the distribution of debond length
Y.-F. Liu et al.rMechanics of Materials 29 1998 111–121 ( ) 117 Table 2 Basic microstructural parameter set for numerical study Debonding toughness, Gic Ž . Nrm 6.0 Frictional stress, t Ž . MPa 50.0 Temperature change, DT Ž . K 500 Fiber volume fraction, f 0.20 Altered parameters are indicated in the corresponding figure. 4.1. Composite fracture criterion Considering the plane–strain condition for a Mode-I crack lying in the plane of transverse isotropy, Budiansky and Cui 1994 obtained the Ž . following expression of critical Mode-I stress intensity factor, K , on the basis of energy balance IC arguments: 2 bEŽ . 1ynm KIC mc sK ) Ž . Ž. 1yf , 13 2 EmŽ . 1yn where Kmc is the critical Mode-I stress intensity factor for monolithic matrix. A criterion K sK is I IC used to simulate crack growth, where K is the stress I intensity factor calculated in Section 2. 4.2. Matrix crack growth analysis Fig. 13 shows the flow chart for crack growth simulation carried out in the present study. Firstly, cracking initiation stress, s0 , for a given initial flaw size, c , is calculated as s sK rŽ . 2'crp , where 0 0 IC Fig. 7. Influences of G on the distribution of COD. ic Fig. 8. Influences of G on the distribution of bridging stress. ic K is obtained from Eq. 13 . Then, a crack growth Ž . IC length of Dc is extended, leaving intact fibers behind to bridge the matrix crack. Owing to the fiber bridging, the value of KI is reduced dŽ . KrdcF0 so that gradually increasing loading is required to satisfy K sK . The newly obtained loading stress, I IC s , is used to describe fracture process of fiber-reinforced ceramics in the form of R-curve or syc curve, where csc qDc. The R-curve approach is 0 utilized in the current study and fracture resistance, K , is defined as: R KRs2s'crp . 14 Ž . Cracking extension becomes unstable under the following two situations while the relation KI IC sK is satisfied. One is due to initiation of fiber fracture at the largest fiber stress obtained for each loading increment and this leads to next fiber fracture, and then catastrophic cracking follows. Fiber fracture is Fig. 9. Influences of G on the distribution of debond length. ic
l18 Y-F. Liu et al./ Mechanics of Materials 29(1998)111-121 10.oFoGic-200N/m 100△T=100MPa G:=12.N/m ·t=50MPa O T=lOMPa 100.0 1000150.0 Fig. 10. Influences of Gis on the stress intensity factors Fig. 12. Influences of T on the stress intensity factors assumed to occur, once the following equation is through matrix cracking alone without occurrence of fiber fracture. The maximum loading stress which KI=KIc satisfies is used to calculate Kr(Eq (14)) (15) where o is the maximum fiber bridging stress and Tu is fiber tensile strength. Loading increment con- tinues until Eq. (15)is satisfied The other case for unstable cracking ensues from large-scale matrix cracking when the whole crack is Calculate fairly long. For a further infinitesimal crack exten Kic /(2Nc/r sion, fiber bridging effect remains small and the relation dKv/dc>0 holds. This relation implies that C=c+△c it is no longer possible to reach K=Kic even with imposition of further larger loading stress, o. In other words, as long as the loading level at a former Solve distrbuted spring model to obtain o steady stage is provided, matrix cracking continues because the following equation is al ways satisfied G=a+△a K1>K1 (16) Under such a situation, the composite fractures 3.0 Calculate fiber stres 32.0 d=100.0MPa Fiber fracture Matrix cracking Normalized position, x/c Fig. 13. Flow chart for simulating R-curve. Various steps shown Fig. 11. Influences of T on the distribution of bridging stress. in the flow chart are detailed in the text
118 Y.-F. Liu et al.rMechanics of Materials 29 1998 111–121 ( ) Fig. 10. Influences of G on the stress intensity factors. ic assumed to occur, once the following equation is satisfied: s Gs , 15 Ž . f fu where sf is the maximum fiber bridging stress and s is fiber tensile strength. Loading increment con- fu tinues until Eq. 15 is satisfied. Ž . The other case for unstable cracking ensues from large-scale matrix cracking when the whole crack is fairly long. For a further infinitesimal crack extension, fiber bridging effect remains small and the relation dK rdc)0 holds. This relation implies that I it is no longer possible to reach KI IC sK even with imposition of further larger loading stress, s . In other words, as long as the loading level at a former steady stage is provided, matrix cracking continues because the following equation is always satisfied: K )K . 16 Ž . I IC Under such a situation, the composite fractures Fig. 11. Influences of t on the distribution of bridging stress. Fig. 12. Influences of t on the stress intensity factors. through matrix cracking alone without occurrence of fiber fracture. The maximum loading stress which K sK satisfies is used to calculate K Ž Ž .. Eq. 14 . I IC R Fig. 13. Flow chart for simulating R-curve. Various steps shown in the flow chart are detailed in the text
Y-F. Liu et al./ Mechanics of Materials 29(1998)111-121 During the above simulation process, partic care was taken regarding selection of Ac.Exces- Maximum K ively large step sizes should be avoided to catch Initial slopes of R-curves ding stresses that satisfy Ki=Kic, while fairly small ones may hinder efficient computation. In the 29.0 current study, Ac=Co/12 was used at initial incre- ments of loading stress, however, a finer cracking increment, Ac=Co/60, was taken to obtain accurate loading stresses leading to matrix cracking(Eq (16) latrix cracking Fiber fracture 43. R-curve results Fig. 14 shows typical R-curves for three different values of Gic. In the case of large Gic, initial slope Fig. 15. Influences of G on the maximum fracture resistance and of the r-curve is large, however fiber failure occurs nitial R-curve slope. at an early stage and the maximum resistance cannot be expected. Meanwhile, for small fiber failure or matrix cracking. Fig. 15 reveals that case of weakly bonding composite, the composite the larger the debonding toughness, the larger the fracture pattern is matrix cracking alone and only a initial slope, however the maximum Kg possesses a small maximum resistance is achieved. It is an inter- peak value at an intermedium debonding toughness medium value of interfacial debonding toughness The maximum Kg increases initially, then decreases that yields the largest fracture resistance, while an with the increase of Gic. Accompanied with the appropriate initial slope of the R-curve is obtained as increase of Gic, the underlying fracture mechanism well. Thus, the toughest composite is obtained in of the composite is dominated by matrix cracking at such a case. In addition to the results shown in Fis first but then changed to fiber fracture(Figs. 14 and 14, interfacial toughness was changed over a wide 15). The analytical results demonstrate that there investigate its influence on the R-curve and exists an appropriate debonding toughness to achieve slopes of the curve. Fig. 15 shows the ob- maximum fracture resistance. Thus, G results. In this figure, the initial slope is important role in determining fracture resistance, defined as slope of the line connecting the first two crack growth stability and fracture mechanism transi- points of crack extension in Fig. 14, with the maxi- tions. As G may be changed by heat treatment mum fracture resistance obtained as the larger one at conditions of the composite and introduction of the point of composite unstable cracking, namely, terphases between matrix and fiber using interlayers of different types, control of processing conditions, on para phases( Gupta et al., 1993), such a R-curve tendency Fiher fracture Fiber fracture身 shown in Fig. 14 has important significance in mate A→ Matrix cracking Altering interfacial frictional stress, T, the ob- shown in Fig. 16. Like the debonding toughness, larger T results in a larger initial slope of R-curve but a maximum fracture resistance appears at an G;=12.0Nm Kg=20vc/T intermediate level of T. Although not shown here. it has been revealed that smaller residual stresses and larger fiber volume fraction yield the same effect on the initial slope of R-curve and maximum fracture Fig. 14. R-curve results for three different values of G resistance as that of larger G and T(Liu. 1995). All
Y.-F. Liu et al.rMechanics of Materials 29 1998 111–121 ( ) 119 During the above simulation process, particular care was taken regarding selection of Dc. Excessively large step sizes should be avoided to catch loading stresses that satisfy KI IC sK , while fairly small ones may hinder efficient computation. In the current study, Dcsc r12 was used at initial incre- 0 ments of loading stress, however, a finer cracking increment, Dcsc r60, was taken to obtain accurate 0 loading stresses leading to matrix cracking Eq. 16 . Ž Ž .. 4.3. R-curÕe results Fig. 14 shows typical R-curves for three different values of G . In the case of large G , initial slope ic ic of the R-curve is large, however fiber failure occurs at an early stage and the maximum resistance cannot be expected. Meanwhile, for small G , i.e., in the ic case of weakly bonding composite, the composite fracture pattern is matrix cracking alone and only a small maximum resistance is achieved. It is an intermedium value of interfacial debonding toughness that yields the largest fracture resistance, while an appropriate initial slope of the R-curve is obtained as well. Thus, the toughest composite is obtained in such a case. In addition to the results shown in Fig. 14, interfacial toughness was changed over a wide range to investigate its influence on the R-curve and initial slopes of the curve. Fig. 15 shows the obtained results. In this figure, the initial slope is defined as slope of the line connecting the first two points of crack extension in Fig. 14, with the maximum fracture resistance obtained as the larger one at the point of composite unstable cracking, namely, Fig. 14. R-curve results for three different values of G . ic Fig. 15. Influences of Gic on the maximum fracture resistance and initial R-curve slope. fiber failure or matrix cracking. Fig. 15 reveals that the larger the debonding toughness, the larger the initial slope, however the maximum K possesses a R peak value at an intermedium debonding toughness. The maximum K increases initially, then decreases R with the increase of G . Accompanied with the ic increase of Gic , the underlying fracture mechanism of the composite is dominated by matrix cracking at first but then changed to fiber fracture Figs. 14 and Ž 15 . The analytical results demonstrate that there . exists an appropriate debonding toughness to achieve maximum fracture resistance. Thus, Gic plays an important role in determining fracture resistance, crack growth stability and fracture mechanism transitions. As G may be changed by heat treatment ic conditions of the composite and introduction of interphases between matrix and fiber using interlayers of different types, control of processing conditions, and also varying deposition parameters for interphases Gupta et al., 1993 , such a Ž . R-curve tendency shown in Fig. 14 has important significance in material design and application. Altering interfacial frictional stress, t , the obtained relationship between K and crack length, is R shown in Fig. 16. Like the debonding toughness, larger t results in a larger initial slope of R-curve but a maximum fracture resistance appears at an intermediate level of t . Although not shown here, it has been revealed that smaller residual stresses and larger fiber volume fraction yield the same effect on the initial slope of R-curve and maximum fracture resistance as that of larger G and t Ž . Liu, 1995 . All ic
Y-F. Liu et al./ Mechanics of Materials 29(1998)111-121 It is known that an optimal interface design Matrix cracking maximize the toughness enhancement exists because Matrix cracking of the energy dissipation balance through interfacial pullout due to scattered fib failure(Thouless and Evans, 1988). Through the present simulation, it is revealed that even without pullout energy dissipation after fiber K,=20vc/t failure, there exists an optimal interfacial toughness T=IOMP to achieve a maximum fracture resistance of the composite. This is clear from the fracture mechanism transition of the composite from matrix cracking to Fig. 16. R-curve results for three different values of T fiber fracture (F 16) in th changing microstructural parameters. The maximum fracture resistance is important in some particular the results correspond to the bridging effect analysis than large-scale matrix cracking s arorcoess rather applications where relatively high toughn in Section 3 For weakly bonded composites(small G),ma- The above R-curve results are for a fixed initial trix cracking starting from a penny-shaped defect, is crack length. The initial crack length is important to the dominant fracture pattern accompanied with determine crack initiation stress and the absolute large-scale interface debonding. However, intact values of subsequent fracture resistances presented in fibers left behind may sustain additional loading, Figs. 14-16 are subject to changes for different with the subsequent fracture pattern being either initial crack lengths as in ductile materials. However, single or multiple matrix cracking. The ultimate he relative tendency of Kg associated with various strength of the composite will depend on various microstructural parameters is universal because the parameters such as fiber strength and initial defect eneral tendency of the bridging effect analysis re- length, among others. This subject has been treated ported in Section 3 is unchanged in(Aveston et al.. 1971: Cox and Marshall. 1994 Budiansky and Cui, 1994; Cui, 1995)and will not be 4.4. Discussion According to J-integral analysis, the steady-state ughness increment by fiber bridging behind the matrix crack tip, AGs, is derived as(Rice, 1968) 5. Conclusions △G=2 F(NdI With a new bridging law including both interfa cial sliding stress and debonding toughness, analysis where lo is the crack opening at the end of the of a bridging model in unidirectionally bridging zone where the bridging stress is equal to fiber-reinforced ceramics with an initial the fiber tensile strength, and o(u) is the distributed shaped crack was carried out by treating fiber bridging stress along the bridging zone. According to ing as an equivalent traction-crack opening displace the analytical results of Figs. 14 and 16, Eq. (17)is ment law(distributed spring model). Particular atten- obviously not applicable to the case of composite tion was given to the fracture resistance curve ac fracture due to matrix cracking. Because uo at the companying the initial flaw growth. Effects of major stage of fiber failure is larger than that at matrix microstructural parameters involved in the model on cracking, application of Eq(17)may lead to overes- the crack propagation process were quantified and timate of bridging effect and consideration of actual discussed with regards to toughening mechanisms fracture process and fracture mechanism is important Numerical examples presented in this study revealed In such a case the general tendencies of how these parameters influ-
120 Y.-F. Liu et al.rMechanics of Materials 29 1998 111–121 ( ) Fig. 16. R-curve results for three different values of t . the results correspond to the bridging effect analysis in Section 3. The above R-curve results are for a fixed initial crack length. The initial crack length is important to determine crack initiation stress and the absolute values of subsequent fracture resistances presented in Figs. 14–16 are subject to changes for different initial crack lengths as in ductile materials. However, the relative tendency of K associated with various R microstructural parameters is universal because the general tendency of the bridging effect analysis reported in Section 3 is unchanged. 4.4. Discussion According to J-integral analysis, the steady-state toughness increment by fiber bridging behind the matrix crack tip, DG , is derived as Rice, 1968 : Ž . ss u0 DGsss2 fH s Ž. Ž . u du, 17 0 where u is the crack opening at the end of the 0 bridging zone where the bridging stress is equal to the fiber tensile strength, and s Ž . u is the distributed bridging stress along the bridging zone. According to the analytical results of Figs. 14 and 16, Eq. 17 is Ž . obviously not applicable to the case of composite fracture due to matrix cracking. Because u at the 0 stage of fiber failure is larger than that at matrix cracking, application of Eq. 17 may lead to overes- Ž . timate of bridging effect and consideration of actual fracture process and fracture mechanism is important in such a case. It is known that an optimal interface design to maximize the toughness enhancement exists because of the energy dissipation balance through interfacial debonding and fiber pullout due to scattered fiber failure Thouless and Evans, 1988 . Through the Ž . present simulation, it is revealed that even without considering fiber pullout energy dissipation after fiber failure, there exists an optimal interfacial toughness to achieve a maximum fracture resistance of the composite. This is clear from the fracture mechanism transition of the composite from matrix cracking to fiber fracture Figs. 14 and 16 in the process of Ž . changing microstructural parameters. The maximum fracture resistance is important in some particular applications where relatively high toughness rather than large-scale matrix cracking is favored. For weakly bonded composites small Ž . G , ma- ic trix cracking starting from a penny-shaped defect, is the dominant fracture pattern accompanied with large-scale interface debonding. However, intact fibers left behind may sustain additional loading, with the subsequent fracture pattern being either single or multiple matrix cracking. The ultimate strength of the composite will depend on various parameters such as fiber strength and initial defect length, among others. This subject has been treated in Aveston et al., 1971; Cox and Marshall, 1994; Ž Budiansky and Cui, 1994; Cui, 1995 and will not be . pursued here. 5. Conclusions With a new bridging law including both interfacial sliding stress and debonding toughness, analysis of a bridging model in unidirectionally aligned fiber-reinforced ceramics with an initial pennyshaped crack was carried out by treating fiber bridging as an equivalent traction–crack opening displacement law distributed spring model . Particular atten- Ž . tion was given to the fracture resistance curve accompanying the initial flaw growth. Effects of major microstructural parameters involved in the model on the crack propagation process were quantified and discussed with regards to toughening mechanisms. Numerical examples presented in this study revealed the general tendencies of how these parameters influ-