150 Journal of the American Ceramic Society-Morscher and Martines-Fernandes Vol. 82. Ne The relative loop width, de/o2, was plotted versus the rela- e stress,(o/op -omin/opx(1-olap), as shown in Fig9.The slope is 29, and T can be determined if the crack spacing is known(Eq (2)). T was determined from the hysteresis loop just prior to failure, where the matrix-crack spacing was assumed to be that of the failed minicomposite. Table II lists the average crack spacing and T for the three different minicomposites The T value that was determined for the Syl-PBN minicom 1 mm po a lower in the range of 65-175 MPa. The sample that failec (a) Syl-PBN minicomposite that failed at the highest stress(Fig. 1) had a relatively high shear stress(170 MPa). This apparent stress-dependent T behavior was not observed in the other two (2) Stress-Strain Behavior of 0--mm Different Fiber Minicomposites If T is assumed to be constant throughout the minicomposite which is probably not the case for the Syl-PBN minicomposite and will be discussed below ) the minicomposite stress-strain curves can be modeled as a function of e and T if the amount of damage is known. The displacement that occurs around the cracks(AL1) can be determined in a manner similar to that by Lissart and Lamon 18 (4 where a is a material parameter(Er/E and Is is the sliding length and can be approximated from the relation (b) R Fig. 7. Examples of transverse matrix cracks in(a)Hi-Nicalon and (b) Sylramic minicomposites Ur is the stress on the fibers(o). The strain of the ked matrix regions is simply ee o/Ee. The total mInI- he fiber volume fraction, T the interfacial shear stress, and e from the elastic properties of the composite constituents: 2bed N△L1+(lg-2)e E E where Le is the gauge length and N is the number of cracks in Em[+(I-2vEd b2=(+vE[(1+p)E+(1-v)E was determined from knowing the number of cracks at failure and estimating the number of cracks at a given stress Where v is the Poisson's ratio(considered to be the same for the from the extent of AE activity at that stress(see Appendix)by er and the matrix and equal to 0.2)and the subscripts m, f, he relationship and c respectively refer to the matrix, fiber, and composite EA(o) N(o)=Nould (7) where N(o)and N(ouly are the number of cracks for the same sample estimated at the peak hysteresis-loop stress of interest Syl-PBN avg.=034 minimum=0.03 mm 12 HN-PBN avg=0.58 mm Hi-Nic-PBN: tF 20 MPa SyPBN: T=180 MP Crack Spacing, mm 0 Fig 8. Cumulative distribution of matrix-crack spacing for HN-PBN 00050101502025 and Syl-PBN; the total length of the minicomposite over which these values were determined was -70 mm for both types of minicompos- Fig 9. Interfacial shear-stress determination from hysteresis loopsthe fiber volume fraction, t the interfacial shear stress, and Em the matrix modulus. The parameters a1 and b2 are determined from the elastic properties of the composite constituents:20 a1 = Ef E (3a) b2 = ~1 + n! Em@Ef + ~1 − 2n!Ec# Ef@~1 + n!Ef + ~1 − n!Ec# (3b) where n is the Poisson’s ratio (considered to be the same for the fiber and the matrix and equal to 0.2) and the subscripts m, f, and c respectively refer to the matrix, fiber, and composite. The relative loop width, d«/s2 p, was plotted versus the rela￾tive stress, (s/sp − smin/sp)(1 − s/sp), as shown in Fig. 9. The slope is 2+, and t can be determined if the crack spacing is known (Eq. (2)). t was determined from the hysteresis loop just prior to failure, where the matrix-crack spacing was assumed to be that of the failed minicomposite. Table II lists the average crack spacing and t for the three different minicomposites. The t value that was determined for the Syl-PBN minicom￾posites was in the range of 65–175 MPa. The sample that failed at a lower stress near the epoxy had a lower shear stress. The Syl-PBN minicomposite that failed at the highest stress (Fig. 1) had a relatively high shear stress (∼170 MPa). This apparent stress-dependent t behavior was not observed in the other two minicomposite systems. (2) Stress–Strain Behavior of Different Fiber Minicomposites If t is assumed to be constant throughout the minicomposite (which is probably not the case for the Syl-PBN minicomposite and will be discussed below), the minicomposite stress–strain curves can be modeled as a function of Ef and t if the amount of damage is known. The displacement that occurs around the cracks (DL1) can be determined in a manner similar to that by Lissart and Lamon:18 DL1 = S 2ls Ef DSsc f DS2 + a 1 + aD (4) where a is a material parameter (Ef/Ec) and ls is the sliding length and can be approximated from the relation ls = sf Rf 2t (5) where sf is the stress on the fibers (sc /f ). The strain of the uncracked matrix regions is simply «c 4 sc /Ec. The total mini￾composite strain can then be determined from the relation « = NDL1 + ~Lg − 2lsN!«c Lg (6) where Lg is the gauge length and N is the number of cracks in the gauge section. The only variable that needs to be deter￾mined is N as a function of stress. N was determined from knowing the number of cracks at failure and estimating the number of cracks at a given stress from the extent of AE activity at that stress (see Appendix) by the relationship N~s! = N~sult! S EAE~s! EAE~sult! D (7) where N(s) and N(sult) are the number of cracks for the same sample estimated at the peak hysteresis-loop stress of interest Fig. 7. Examples of transverse matrix cracks in (a) Hi-Nicalon and (b) Sylramic minicomposites. Fig. 8. Cumulative distribution of matrix-crack spacing for HN-PBN and Syl-PBN; the total length of the minicomposite over which these values were determined was ∼70 mm for both types of minicompos￾ites. Fig. 9. Interfacial shear-stress determination from hysteresis loops. 150 Journal of the American Ceramic Society—Morscher and Martinez-Fernandez Vol. 82, No. 1