1568 Journal of the American Ceramic Sociery-Ogasanwara et al Vol. 84. No. 7 respectively. In contrast to this, the Cte of the 3-D woven oun=2oh,(br bF)bF(h,+h, (1b) omposite was 3.98 X 10-K. The experimental results thus uggest that the cte of the matrix is almost the same as that of the uG)=20h, (br+ br)/bph fiber, i.e., approximately 4.0X 10(K- )up to 1000C The unloading curve for the nano-indentation tests followed a At fi based on ROM-type equations ar power law. The contact stiffness is defined by the slope of the applied to each of the labeled E, F, and G regions prior to the onset unloading curve, which is proportional to the composite elastic of transverse crack propagation, i.e response of the indenter and test material. The contact stiffness of fiber was estimated to be 377 kN/m. while that of the matrix was hEn+h ex 396 kN/m, and it thus appears that the elastic modulus of the (+ matrix is similar to that of the fiber. Such a result might be expected as the chemical compositions of the matrix and fiber h,E+ hey 2bF+ br basically similar h,+h,+ h/2 2(bF+ br According to microscopic observations, no significant amount of porosity was found to be present in the intrabundle regions. The elastic modulus of each fiber bundle may thus be estimated h=2(b2+b uming a void-free unidirectional composite, and is an important or for the analysis provided in the next section where EEh Eury and Eug are the elastic moduli in the x direction of the e, F, and G regions, respectively. E, and E, are the elastic moduli in the longitudinal and transverse directions for a unidi IV. Analysis of Stress/Strain Behavior rectional composite. Following the initiation of transverse crack ing, the in-plane elastic modulus, E, for each laminate is calcu The inelastic tensile stress/strain behavior is known to be lated using a shear-lag model as described in the next section. governed by transverse cracking, matrix cracking, fiber debonding, Assuming a serial linkage of the three regions in Fig. 7(c) based on interfacial sliding, and fiber fragmentation. In this work, the a Reuss-type estimation, the total average modulus in the x stress/strain behavior of an orthogonal 3-D woven composite is direction, E, can be written as the following estimated by utilizing and modifying existing theories found in the literature 2(6+ br) JEKE+ 2bEun+ by/Eyg Orthogonal 3-D Wo (2) Transverse Crack Density and Stiffness Changes Due The parallel-serial approach(PSA)pr d by ishikawa et al Transverse Cracking was used to estimate the orthogonal 3-D woven composite in-pla By modifying the shear. odel proposed by Park and elastic modulus. The accuracy of this approach ted to be McManus, the CMC transverse crack density was calculated proved several percent when compared to the standard rule-of- The basic model for this is illustrated in Fig. 8, where E, a, and t mixtures(ROM) theory. A schematic representation of the ort onal 3-D composite re unit cell is illustrated in Fig h hog wenh t th sebastio ms u ausd ze indica ind te thicks es ge plies. The unit cell is divided into 12 subregions as shown in Figs. 7(a) spectively. If a new crack forms, then strain energy stored within and (b). The fill and warp bundles are assumed to have the same the laminate will be released with a shear- lag model being used to width, br, and thickness, h,, while the translaminar fiber bundle calculate this energy. The energy release rate, ,, is expressed by width is denoted by br. In the present CMC, the Si-Ti-C-O matrix the difference between the strain energies per unit area of new derived from PIP is not fully infiltrated into the pocket region, and. crack surface. The hypothetical crack will form under the condi- in order to simplify the model, it is assumed that the pocket region tion of 1 2ic, where c is the critical energy release rate onsists entirely of porosity. Using this assumption, the 3-D mode Thus, the following equation is obtained can be redrawn as Fig. 7(c) with equivalent fiber bundle thick nesses,h, =[(br beyb h and h. =(2b/by)h,. In this model a,EE, equivalent stresses, U1() UuB, and oug) are also defined for the 8EKE,G+ E,l,) tanh(2E1/1,)-2 tanh(E1/12)] laminates labeled E. F. and G under a unidirectional stress. g oue=20h (br+ br)/br(h,+ h2) Using the definitions h Fig. 7. Unit cell models of the orthogonal 3-D composite: (a)unit cell, (b) procedure of linkage for the parallel-serial approach(PSA) solution, and (c)three laminated composite models equivalent to the stiffness of the orthogonal 3-D compositesrespectively. In contrast to this, the CTE of the 3-D woven composite was 3.98 3 1026 K21 . The experimental results thus suggest that the CTE of the matrix is almost the same as that of the fiber, i.e., approximately 4.0 3 1026 (K21 ) up to 1000°C. The unloading curve for the nano-indentation tests followed a power law. The contact stiffness is defined by the slope of the unloading curve, which is proportional to the composite elastic response of the indenter and test material.19 The contact stiffness of fiber was estimated to be 377 kN/m, while that of the matrix was 396 kN/m, and it thus appears that the elastic modulus of the matrix is similar to that of the fiber. Such a result might be expected as the chemical compositions of the matrix and fiber are basically similar. According to microscopic observations, no significant amount of porosity was found to be present in the intrabundle regions.6 The elastic modulus of each fiber bundle may thus be estimated assuming a void-free unidirectional composite, and is an important factor for the analysis provided in the next section. IV. Analysis of Stress/Strain Behavior The inelastic tensile stress/strain behavior is known to be governed by transverse cracking, matrix cracking, fiber debonding, interfacial sliding, and fiber fragmentation. In this work, the stress/strain behavior of an orthogonal 3-D woven composite is estimated by utilizing and modifying existing theories found in the literature. (1) Estimation of In-Plane Elastic Modulus for an Orthogonal 3-D Woven Composite The parallel-serial approach (PSA) proposed by Ishikawa et al. 4 was used to estimate the orthogonal 3-D woven composite in-plane elastic modulus. The accuracy of this approach is expected to be improved several percent when compared to the standard rule-of￾mixtures (ROM) theory. A schematic representation of the orthog￾onal 3-D composite repeating unit cell is illustrated in Fig. 7(a). The unit cell is divided into 12 subregions as shown in Figs. 7(a) and (b). The fill and warp bundles are assumed to have the same width, bF, and thickness, ht , while the translaminar fiber bundle width is denoted by bT. In the present CMC, the Si-Ti-C-O matrix derived from PIP is not fully infiltrated into the pocket region, and, in order to simplify the model, it is assumed that the pocket region consists entirely of porosity. Using this assumption, the 3-D model can be redrawn as Fig. 7(c) with equivalent fiber bundle thick￾nesses, hy 5 [(bT 1 bF)/bF]ht , and hz 5 (2bT/bF)ht . In this model, equivalent stresses, s#l(E), s#l(F), and s#l(G) are also defined for the laminates labeled E, F, and G under a unidirectional stress, s#. s# l(E) 5 2s# ht ~bT 1 bF!/bF~ht 1 hz! (1a) s# l(F) 5 2s# ht ~bT 1 bF!/bF~ht 1 hy! (1b) s# l(G) 5 2s# ht ~bT 1 bF!/bFht (1c) At first, parallel averages based on ROM-type equations are applied to each of the labeled E, F, and G regions prior to the onset of transverse crack propagation, i.e., El(E) 5 ht E1 1 hzE2 2ht 1 hz 5 S bF 1 2bT 2~bF 1 bT! DE1 (2a) El(F) 5 ht E1 1 hzE2 ht 1 hy 1 hz/2 5 S 2bF 1 bT 2~bF 1 bT! DE1 (2b) El(G) 5 ht E1 2ht 1 hz 5 S bF 2~bF 1 bT! DE1 (2c) where El(E), El(F), and El(G) are the elastic moduli in the x direction of the E, F, and G regions, respectively. E1 and E2 are the elastic moduli in the longitudinal and transverse directions for a unidi￾rectional composite. Following the initiation of transverse crack￾ing, the in-plane elastic modulus, El , for each laminate is calcu￾lated using a shear-lag model as described in the next section. Assuming a serial linkage of the three regions in Fig. 7(c) based on a Reuss-type estimation, the total average modulus in the x direction, E, can be written as the following: E 5 2~bf 1 bT! bT/El(E) 1 2bf/El(F) 1 bT/El(G) (3) (2) Transverse Crack Density and Stiffness Changes Due to Transverse Cracking By modifying the shear-lag model proposed by Park and McManus,20 the CMC transverse crack density was calculated. The basic model for this is illustrated in Fig. 8, where E, a, and t denote the elastic modulus, the CTE, and the thickness of the plies, with the subscripts 1 and 2 indicating 0° plies and 90° plies, respectively. If a new crack forms, then strain energy stored within the laminate will be released with a shear-lag model being used to calculate this energy. The energy release rate, &I , is expressed by the difference between the strain energies per unit area of new crack surface. The hypothetical crack will form under the condi￾tion of &I $ &IC, where &IC is the critical energy release rate. Thus, the following equation is obtained: &IC 5 2 c2 t2 3 t1E1E2 8jK2 ~E1t1 1 E2t2! @tanh ~2jlL/t2! 2 2 tanh ~jlL/t2!# (4) Using the definitions Fig. 7. Unit cell models of the orthogonal 3-D composite: (a) unit cell, (b) procedure of linkage for the parallel-serial approach (PSA) solution, and (c) three laminated composite models equivalent to the stiffness of the orthogonal 3-D composites. 1568 Journal of the American Ceramic Society—Ogasawara et al. Vol. 84, No. 7