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) 3.0116 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, al￾though discretizing and solution procedure were dif￾ferent 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 mate￾rial 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 pa￾rameters 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 demon￾strate 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 distri￾bution 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 ten￾dency as that of G . Effects of other microstructural ic parameters, such as fiber volume fraction and resid￾ual 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 cri￾teria 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