1570 Journal of the amer nic Society-O Vol. 84. No. 7 Table L. Summary of Constants Given by Hutchinson and matrix. In this case, the load is partially carried by those fibers with Jensen for the Case of Identical Fiber and Matrix Elastic no breaks within the effective pullout of the matrix crack The average stress per fiber is o/ p for the case of a unidirectional stress,Uu, and depends only on the stress, S, carried by the unbroken fibers at the matrix crack plane, i.e (1-p)(1+v) s[1-1 poch oc// (20) (1-p)(1+v) It is possible to consider fiber fracture through use of Eq (20). The a4=-2(1-v2) effective stress for unidirectional plies, ou can be calculated as pS Th e avera composite strain is the same as the average of the intact fibers. Therefore. the stress/strain behavior matrix crack saturation is obtained by substituting G Eq(14),(15),or(18). Furthermore, by maximizing Ea stained respect to S, the ultimate tensile strength, Outs, can be ob 2 (1+v) Outs-po c\m+2 m+2 (22) Davies et al.estimated in situ fiber strength parameters for the where ld is the debond length given by the following equation composite using observations of fiber fracture mirrors. Values of IoR=(1-P)(O-U)/2c3Tp (16) och and m obtained following tensile testing at room temperature (16) were 3.09 MPa, and 4. 19, respectivel where G, is the average stress at initiation of debonding. By using the critical energy release rate for the debond crack, I the following equation is obtained (6) Hysteresis Loop Analysis Methodology Hysteresis loop analysis has provided a methodology for 0:=1/C1VEmTIR-CcIEmE evaluation of constituent properties in unidirectional CMCs, 2-2 and cross-ply CMCs through use of a load-partitioning factor, A. where b2, C1, C2, and c3 are the parameters given by Hutchinson Domergue et al rovided schematic forms for the inverse (B) Karandikar and Chou Theory: Karandikar and Choulo tangent moduli(ITM)of unidirectional large debond energy and Jensen dopted a shear-lag model in order to approximate the shear stress between the linear region and plateau is indicative of an inherent distribution at the interface in a bonded region In contrast to this, nterfacial resistance to debonding. Reverse slip is arrested at the the matrix stress in the bonded region increases linearly with debond tip when the stress upon unloading reaches a transition distance from the matrix crack due to the presence of a constant stress, du given by sliding stress, T. From this, the composite stress/strain relation is found to be Gu「,,Em(1-p)24,2la E7)+ar-a)△T where u, is the peak stress and o, is the debond stress and is related to the interfacial debond energy by eq. (17). Symmetrically, upon 2 Em(1-p) reloading, sliding again stops at the debonded region at a stress u, C.+ (24) T E ERl Our reverse slip stops at the bond end and the reload strain es linear, such that the ITM is constant and given by where ar and am are the Cte of the fiber and the matri respectively, and Luo is the initial elastic modulus of a microcrack lEe=l/E*+2on(on≤n) ree UD composite. The shear-lag constant B is written as 1/Ere= 4( +1/ (G1 B REEO-D where E. is the reloading elastic modulus, and E. is the elast LG+Gmo-pr modulus of the composite with matrix cracks, respectively. An inelastic strain index, is obtained that is related to T by where G and G are the shear moduli of the fiber and the matrix. respectively sP= b2(I-a1p)R 4lp-TEm Stresses above Crack Saturation Following matrix crack saturation, Curtins theory can be where a, and b, are coefficients provied by Hutchinson and utilized for the calculation of stresses within the com Jensen°I possible to evaluate from the maximum width xample, Okabe et al. reported that Curtins model of the hy It is also loop, eMax For LDE materials, the relationship good correlation with experimental data in a Hi-Nica for s also depends on a;when3/4≤alG,≤1 matrix composite. At a stress o >Us, the crack density and a sliding shear stress transfer exists between the fiber and thewhere ld is the debond length given by the following equation: ld/R 5 ~1 2 r!~s#lu 2 s#i !/2c3tr (16) where s#i is the average stress at initiation of debonding. By using the critical energy release rate for the debond crack, Gi , the following equation is obtained: s# i 5 1/c1 ÎEmGi /R 2 c2/c1EmεT (17) where b2, c1, c2, and c3 are the parameters given by Hutchinson and Jensen.9 (B) Karandikar and Chou Theory: Karandikar and Chou10 adopted a shear-lag model in order to approximate the shear stress distribution at the interface in a bonded region. In contrast to this, the matrix stress in the bonded region increases linearly with distance from the matrix crack due to the presence of a constant sliding stress, t. From this, the composite stress/strain relation is found to be εcu 5 s#lu Elu0 H1 1 Em~1 2 r! Efr 2ld l J 1 2ld l ~af 2 alu!DT 1 2 bl tanh bS l 2 2 ldDSEm~1 2 r! Elu0Efr s#lu 1 ~af 2 alu!DT 2 2 t Ef ld RD 2 2tld 2 EfRl (18) where af and am are the CTE of the fiber and the matrix, respectively, and Elu0 is the initial elastic modulus of a microcrack￾free UD composite. The shear-lag constant b is written as b2 5 8Elu0 R2 EfEm~1 2 r! 3 F 1 Gf 1 1 Gm H 2 ~1 2 r!2 ln S 1 rD 2 3 2 2r 1 2 rJG21 (19) where Gf and Gm are the shear moduli of the fiber and the matrix, respectively. (5) Stresses above Crack Saturation Following matrix crack saturation, Curtin’s theory17 can be utilized for the calculation of stresses within the composite. For example, Okabe et al. 20 reported that Curtin’s model provides a good correlation with experimental data in a Hi-Nicalon/glass matrix composite. At a stress s.ss, the crack density is saturated and a sliding shear stress transfer exists between the fiber and the matrix. In this case, the load is partially carried by those fibers with no breaks within the effective pullout length of the matrix crack. The average stress per fiber is s#lu/r for the case of a unidirectional stress, s#lu, and depends only on the stress, S, carried by the unbroken fibers at the matrix crack plane, i.e., s# lu rsch 5 S sch F1 2 1 2 S S schD m11 G (20) It is possible to consider fiber fracture through use of Eq. (20). The effective stress for unidirectional plies, s# *lu, can be calculated as follows: s# *lu 5 rS (21) The average composite strain is the same as the average strain, ε, of the intact fibers. Therefore, the stress/strain behavior following matrix crack saturation is obtained by substituting s# *lu for s#lu in Eq. (14), (15), or (18). Furthermore, by maximizing Eq. (20) with respect to S, the ultimate tensile strength, suts, can be obtained: suts 5 rschS 2 m 1 2D 1/~m11! S m 1 1 m 1 2D (22) Davies et al. 5 estimated in situ fiber strength parameters for the composite using observations of fiber fracture mirrors. Values of sch and m obtained following tensile testing at room temperature were 3.09 MPa, and 4.19, respectively. (6) Hysteresis Loop Analysis Methodology Hysteresis loop analysis has provided a methodology for the evaluation of constituent properties in unidirectional CMCs,21–23 and cross-ply CMCs through use of a load-partitioning factor, l. 12 Domergue et al. 12 provided schematic forms for the inverse tangent moduli (ITM) of unidirectional large debond energy (LDE) composites as shown in Fig. 10. The stress at the transition between the linear region and plateau is indicative of an inherent interfacial resistance to debonding. Reverse slip is arrested at the debond tip when the stress upon unloading reaches a transition stress, s#tu, given by s# tu 5 2s#i 2 s# p (23) where s# p is the peak stress and s#i is the debond stress and is related to the interfacial debond energy by Eq. (17). Symmetrically, upon reloading, sliding again stops at the debonded region at a stress s#tr: s# tr 5 2~s# p 2 s#i ! (24) Above s#tr, reverse slip stops at the bond end and the reload strain becomes linear, such that the ITM is constant and given by 1/Ere 5 1/E* 1 2+sre ~sre # str! (25) 1/Ere 5 4+~sp 2 si ! 1 1/E* ~str # sre! (26) where Ere is the reloading elastic modulus, and E* is the elastic modulus of the composite with matrix cracks, respectively. An inelastic strain index, +, is obtained that is related to t by + 5 b2~1 2 a1r!2 R 4l #r2 tEm (27) where a1 and b2 are coefficients provied by Hutchinson and Jensen.9 It is also possible to evaluate + from the maximum width of the hysteresis loop, dεmax. For LDE materials, the relationship for + also depends on s#i when 3/4 # s#i /s# p # 1, dεmax 5 4+~s# p 2 s#i ) 2 (28) Table I. Summary of Constants Given by Hutchinson and Jensen for the Case of Identical Fiber and Matrix Elastic Properties a1 5 1 a2 5 ~1 2 r!~1 1 n! 1 2 n2 a3 5 0 a4 5 ~1 2 r!~1 1 n! 2~1 2 n2 ! a5 5 1 a6 5 r b2 5 1 2 n2 c1 5 @~1 2 n2 !~1 2 r!#1/2 2r c2 5 1 2 S 1 2 r 1 2 n2D 1/2 ~1 1 n! c3 5 1 1570 Journal of the American Ceramic Society—Ogasawara et al. Vol. 84, No. 7