2410 Journal of the American Ceramic SocieryMorvan and Baste Vol 81. No 9 Inter-Bundle Inelastic Strains IntraBundle Inelastic Strains Fig 8. Evolution of the inelastic strains of a 2D SiC-SiC composite the interbundle matrix microcrack and with it, the value at saturation of the transverse crack pattern with respect inelastic strain appears. The inelastic strain exhibits two very to the two-scale effect different increases related to the scale at which they occur: an increment of strain at constant stress(AEhs), which starts at-80 MPa and stops at-160 MPa, and a strain that requires an increase in stress and begins at 120 MPa. Pr\S%2-S% Thus, Eq.(22) can be improved by using the following relation. elastic Ebs.8 where ebs represents the inelastic strain that grows at constant stress and es is the inelastic strain due to an increment of stress Iv. Identification of the model The experimental variation of the compliances gives access to the constitutive law of the transverse crack densities using (Fig. 7; in this figure, the symbols represent the values dentified from Eq (1)using the compliance variations and the traight lines represent the interpolated evolution laws ). Be cause of the two-scale effect of the microcracking, the identi fication is not straightforward. Actually, the values of the com- liances of the so-called uncracked material must be adjusted For the interbundle microcracking. the values of the compli ances at 80 MPa (the damage threshold) have been used whereas, for the intrabundle scale, the values at 120 MPa(th onset of intrabundle cracking) were considered. As a result both the interbundle and intrabundle transverse crack densities (A) can be described with a linear function. Up to 80 MPa, the crack density is equal to zero, because the damage has not begun. The interbundle crack density saturates at-120 MPa, at hich point the intrabundle crack density begins After the transverse crack densities are known, Eq (5)gives access to the inelastic strain variation at both scales of the 添 composite. These variations are depicted in Fig. 8, where the symbols represent the experimental data and the straight lines represent the predictions The two-scale effect in the 2D SiC-SiC composite must also be taken into account in Eq.(20). Because this relation has EE been written at the fiber scale. it must be modified at the bundle scale. Assuming that the bundles are elliptic, the radius of the fiber R in Eq. (20)must be replaced by the term [2m Mf/(m2+ MI where M and m are the major and minor semi-axe respectively, of the bundle. The Young's modulus of the fiber, Er must then become E, which is the Youngs modulus of the bundle 34 The compliance variations are sensitive not only to the pres- ence of cracks but also to their size. thus. the debonding length la can be estimated using the transverse crack density and the variation of S22(this variation is dependent upon Mode II ), acking). Assuming that la is zero when the material is ur cracked and is equal to one-half the intercrack distance when the matrix microcracking is at saturation, it becomes possible, using a simple rule of mixtures, to estimate la from the values of the compliances S22, of the uncracked material, and S%2, the scale(Aubard35)the interbundle matrix microcracking begins, and with it, the inelastic strain appears. The inelastic strain exhibits two very different increases related to the scale at which they occur: an increment of strain at constant stress (D«bs), which starts at ∼80 MPa and stops at ∼160 MPa, and a strain that requires an increase in stress and begins at 120 MPa. Thus, Eq. (22) can be improved by using the following relation: «inelastic = «bs + «fs (23) where «bs represents the inelastic strain that grows at constant stress and «fs is the inelastic strain due to an increment of stress. IV. Identification of the Model The experimental variation of the compliances gives access to the constitutive law of the transverse crack densities using Eq. (1) (Fig. 7; in this figure, the symbols represent the values identified from Eq. (1) using the compliance variations and the straight lines represent the interpolated evolution laws). Be￾cause of the two-scale effect of the microcracking, the identi￾fication is not straightforward. Actually, the values of the com￾pliances of the so-called uncracked material must be adjusted. For the interbundle microcracking, the values of the compli￾ances at 80 MPa (the damage threshold) have been used, whereas, for the intrabundle scale, the values at 120 MPa (the onset of intrabundle cracking) were considered. As a result, both the interbundle and intrabundle transverse crack densities can be described with a linear function. Up to 80 MPa, the crack density is equal to zero, because the damage has not begun. The interbundle crack density saturates at ∼120 MPa, at which point the intrabundle crack density begins. After the transverse crack densities are known, Eq. (5) gives access to the inelastic strain variation at both scales of the composite. These variations are depicted in Fig. 8, where the symbols represent the experimental data and the straight lines represent the predictions. The two-scale effect in the 2D SiC–SiC composite must also be taken into account in Eq. (20). Because this relation has been written at the fiber scale, it must be modified at the bundle scale. Assuming that the bundles are elliptic, the radius of the fiber R in Eq. (20) must be replaced by the term [2m2 M2 /(m2 + M2 )]1/2, where M and m are the major and minor semi-axes, respectively, of the bundle. The Young’s modulus of the fiber, Ef , must then become Et , which is the Young’s modulus of the bundle.34 The compliance variations are sensitive not only to the pres￾ence of cracks but also to their size. Thus, the debonding length ld can be estimated using the transverse crack density and the variation of S22 (this variation is dependent upon Mode II cracking). Assuming that ld is zero when the material is un￾cracked and is equal to one-half the intercrack distance when the matrix microcracking is at saturation, it becomes possible, using a simple rule of mixtures, to estimate ld from the values of the compliances S° 22, of the uncracked material, and Ss 22, the value at saturation of the transverse crack pattern with respect to the two-scale effect: ld = a bT S S22 − S° 22 S22 s − S° 22D (24) Fig. 8. Evolution of the inelastic strains of a 2D SiC–SiC composite. Fig. 9. Micrographs of the transverse cracking of a 2D SiC–SiC composite ((A) interbundle scale (Guillaumat34) and (B) intrabundle scale (Aubard35)). 2410 Journal of the American Ceramic Society—Morvan and Baste Vol. 81, No. 9