HSUEH and BECHER INTERFACIAL SHEAR DEBONDING 5 b/a=100 o>o=9N目oz 10 Youngs modulus ratio, E/E Fig. 4. The normalized effective defect length, h/a, as a function of EEm for vr=Vm=0.25 and debond stress increases with the increase in Ed Em. Also, the curve in Fig. 5 becomes flatter as b/a For convenience, the ratio of aa/ts to oa/(eTi/a) Increases is defined as z. Assuming that the strength-based and the energy-based criteria predict the same in- the effective defect length 6. CONCLUDING REMARKS can be related to z, such that Both the strength-based and the energy-based cri- teria have been adopted to analyze the required (35) loading stress on the fiber, ad, to initiate mode II nterfacial debonding during fiber pullout. The The solution of equation(35) yields interfacial shear strength, ts, and the interface debond energy, Ti. are used in the strength-based + (36) and the energy-based criteria, respectively,to characterize interfacial debonding. The relationship etween ts and Ti is derived in the present study The normalized effective defect length, h/a, as a (equation (29)] by adopting the approach of function of Er/Em is shown in Fig. 4 at different Griffith theory. However, instead of using an exist- ing defect in Griffith theory for a monolithic cer- increase in either Er/ Em or b/a. When the fiber and amic. an effective circumferential defect at the the matrix have similar Youngs modulus, the effec- nterface which extends from the loading surface to tive defect length is in the order of the fiber radius a depth, h [Fig. 1(b)], is assumed in the present ha≈ I when ErEm≈1) study. It is noted that the effective defect is not a Using equation (34), the diagram of interfacial real defect. Instead, it is introduced to account for debonding vs fiber fracture is constructed in Fig. 5, the stress intensity due to the presence of the fiber n which the Dundurs'parameter, z, is defined in the matrix and the fiber-pullout geometry. The as[26] effective defect length. h. can be determined Er(1-=)-Em( equation(36) by equating the initial debond stres- Er(1-=)+Em(1 (37) ses derived from the two debonding criteria. When the fiber and the matrix have the similar young It is noted that both the relative defect size cla, in modulus (e.g. for ceramic composites), h is in the the fiber and the defect-geometry factor, i, are order of the fiber radius. The normalized effective olved in defining the diagram. Interfacial defect length, h/a, increases with the increase in onding and fiber fracture occur when c T /alr either the Young's modulus ratio of fiber to matrix below and above the curve, respectively, in Fig. 5. Er Em, or the radius ratio of matrix to fiber b/a This critical ratio decreases with the increase in a.(Fig. 4). For the material design, the usefulness ofdebond stress increases with the increase in Ef/Em. For convenience, the ratio of sd/ts to sd/(E*Gi/a) 1/2 is de®ned as w. Assuming that the strength-based and the energy-based criteria predict the same in￾itial debond stress, sd, the e€ective defect length can be related to w, such that h2 a2 ‡ h ab 1=2 ˆ w: …35† The solution of equation (35) yields h a ˆ ÿ 1 2b ‡  1 4b2 ‡ w2 s : …36† The normalized e€ective defect length, h/a, as a function of Ef/Em is shown in Fig. 4 at di€erent ratios of b/a. It is shown that h/a increases with the increase in either Ef/Em or b/a. When the ®ber and the matrix have similar Young's modulus, the e€ec￾tive defect length is in the order of the ®ber radius (i.e. h/a11 when Ef/Em11). Using equation (34), the diagram of interfacial debonding vs ®ber fracture is constructed in Fig. 5, in which the Dundurs' parameter, a, is de®ned as [26] a ˆ Ef…1 ÿ 2 m† ÿ Em…1 ÿ 2 f † Ef…1 ÿ 2 m† ‡ Em…1 ÿ 2 f † : …37† It is noted that both the relative defect size, c/a, in the ®ber and the defect-geometry factor, l, are involved in de®ning the diagram. Interfacial debonding and ®ber fracture occur when cl2 Gi/aGf is below and above the curve, respectively, in Fig. 5. This critical ratio decreases with the increase in a. Also, the curve in Fig. 5 becomes ¯atter as b/a increases. 6. CONCLUDING REMARKS Both the strength-based and the energy-based cri￾teria have been adopted to analyze the required loading stress on the ®ber, sd, to initiate mode II interfacial debonding during ®ber pullout. The interfacial shear strength, ts, and the interface debond energy, Gi, are used in the strength-based and the energy-based criteria, respectively, to characterize interfacial debonding. The relationship between ts and Gi is derived in the present study [equation (29)] by adopting the approach of Grith theory. However, instead of using an exist￾ing defect in Grith theory for a monolithic cer￾amic, an e€ective circumferential defect at the interface which extends from the loading surface to a depth, h [Fig. 1(b)], is assumed in the present study. It is noted that the e€ective defect is not a real defect. Instead, it is introduced to account for the stress intensity due to the presence of the ®ber in the matrix and the ®ber-pullout geometry. The e€ective defect length, h, can be determined [equation (36)] by equating the initial debond stres￾ses derived from the two debonding criteria. When the ®ber and the matrix have the similar Young's modulus (e.g. for ceramic composites), h is in the order of the ®ber radius. The normalized e€ective defect length, h/a, increases with the increase in either the Young's modulus ratio of ®ber to matrix, Ef/Em, or the radius ratio of matrix to ®ber, b/a (Fig. 4). For the material design, the usefulness of Fig. 4. The normalized e€ective defect length, h/a, as a function of Ef/Em for nf=nm=0.25 and t/a 4 1. HSUEH and BECHER: INTERFACIAL SHEAR DEBONDING 3243