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 re￾ported 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 interfa￾cial sliding stress and debonding toughness, analysis of a bridging model in unidirectionally aligned fiber-reinforced ceramics with an initial penny￾shaped crack was carried out by treating fiber bridg￾ing as an equivalent traction–crack opening displace￾ment law distributed spring model . Particular atten- Ž . tion was given to the fracture resistance curve ac￾companying 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-