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 trac￾tion–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 trans￾versely isotropic body and making use of Cas￾Fig. 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, respec￾tively; b is an orthotropic factor of the composite, which is near ;1.0 for most engineering combina￾tions 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 mi￾crostructural parameters including the interface fric￾tion and debonding toughness. Eqs. 1 and 2 may Ž. Ž. be rewritten, respectively, for descriptive conve￾nience 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 discretiz￾ing the equivalent bridging stress, s Ž . x , over posi￾tion, 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 bridg￾ing stress, s Ž . x , is discretized over normalized co￾ordinates, 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