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