Journal J.Am. Cera. Soc.84D]1565-7(2001) Multiple Cracking and Tensile Behavior for an Orthogonal 3-D Woven Si-Ti-C-O Fiber/Si-Ti-C-O Matrix Composite Toshio Ogasawara and Takashi Ishikawa National Aerospace Laboratory of Japan, Mitaka, Tokyo, 181-0015, Japan Hiroshi Ito and naoyuki Watanabe Aerospace Systems Department, Tokyo Metropolitan Institute of Technology, Hino, Tokyo, 191-0065, Japan an. Davies Advanced Fibro Science, Kyoto Institute of Technology, Sakyo-ku, Kyoto, 606-8585, Japan This paper presents experimental results for the multiple It is now well understood that unidirectional (UD)CMCs microcracking and tensile behavior of an orthogonal 3-d exhibit nonlinear stress-strain behavior under unidirectional ten- woven Si-Ti-C-O fiber (Tyranno M Lox-M)/Si-Ti-C-O matrix sile loading as a result of multiple microcracking and fiber pullout composite with a nanoscale carbon fiber/matrix interphase within the longitudinal(0%)fiber bundles. An overview of CMC Ind processed using a polymer impregnation and pyrolysis mechanical properties has been proviede by Evans and Zok'and oute. Based on microscopie observations and unidirectional Evans.The change in stiffness due to multiple matrix cracking tensile tests. it is revealed that the inelastic tensile stress/strain has been estimated by two different approaches: (1)elastic ehavior is governed by matrix cracking in transverse (90o) analysis based on the Lame problem"and(2)shear-lag analysis o fiber bundles between 65 and 180 MPa, matrix cracking in In cross-ply composites, this change also involves initial cracking longitudinal (0%)fiber bundles between 180 and 300 MPa, and in the transverse(90%) plies as tunneling cracks. Subse- fiber fragmentation above 300 MPa. A methodology for esti- quently, transverse cracks penetrate the longitudinal plies as the mation of unidirectional tensile behavior in orthogonal 3-D load is increased. Based on the energy criterion and finite element composites has been established by the use and modification of analysis, transverse crack propagation in cross-ply brittle-matrix existing theory. A good correlation was obtained between the composites has been analyzed. Shear-lag analysis is often used to predicted and measured composite strain using this procedure. estimate transverse crack propagation within polymer-matrix com- and this method has also been applied to cross-ply CMCs. 0, I3 Following matrix crack saturation, estimates for the L. Introduction fiber pullout length, pullout work, and the ultimate tensile strength of unidirectional CMCs have been derived by Curtin, ,based on T IS well known that monolithic ceramics do not possess a level the statistical analysis of fiber strength and stress redistribution due of damage tolerance that is sufficient for aerospace applications to fiber fragmentation processes For this reason, a great deal of effort has been devoted to the This paper presents experimental results for the multiple micro development of continuous-ceramic-fiber-reinforced ceramic- cracking and tensile behavior of an orthogonal 3-D woven Si-Ti matrix composites(CMCs or CFCCs) for jet engine comp C-O fiber/Si-Ti-C-O matrix composite(NUSK-CMC)fabricated nents, rocket engine nozzles, and the thermal protection systems (TPS)of future space transportation vehicles. -The National were estimated from hysteresis loop analysis of loading/unloading Aerospace Laboratory of Japan, Ube Industries Ltd, Shikib cycles. A methodology for estimating the unidirectional tensile Ltd, and Kawasaki Heavy Industries Ltd. have conducted a behavior of orthogonal 3-D composites has been provided by using joint program in order to develop and evaluate a continuous- and modifying the above-mentioned theories fiber-reinforced CMC. The composite contains Tyranno M Lox M fiber (Si 54%, Ti 2%, C 32%, 0 12%(mass%))with an additional surface modification process. The fibers are woven Il. Experimental Procedure into an orthogonal 3-D structure that has advantages from the (0 Material and Specimens rocessing,and improved delamination resistance and tensile The composite under investigation contained Tyranno Lox-M rength. Using these technologies. the composite exhibits volume fractions of 19%, 19%, and 2% in the duration with fiber fibers woven into an orthogonal 3-D confi excellent tensile stre at room temperature", and creep and- directions trength at elevated temperature. The composite is referred to respectively. Optical micrographs in Fig. I illustrate the fiber as"NUSK-CMC" from the initials of the collaborating partners architecture of the present composites with each fiber bundle atmosphere, resulting in the formation of a 10 nm Sio- rich layer surrounding an inner 40 nm carbon-rich layer at the fiber surface B. N. Cox--contributing editor The nanoscale carbon- rich layer is believed to result in an interphase with desirable properties between the fiber and the matrix.Polytitanocarbosilane was used as the matrix precursor with eight impregnation and pyrolysis cycles, the average com- ipt No. 188832. Received January 3, 2000: approved March 6, 2001 posite bulk density was 2.20 g/cm". Tensile specimens were Currently with Nagoya Aerospace Systems, Mitsubishi Heavy Industries Ltd machined from the composite plates such that the loading direction Nagoya, Japan. was parallel to the y-axis. The specimen surfaces were also ground
Multiple Cracking and Tensile Behavior for an Orthogonal 3-D Woven Si-Ti-C-O Fiber/Si-Ti-C-O Matrix Composite Toshio Ogasawara* and Takashi Ishikawa National Aerospace Laboratory of Japan, Mitaka, Tokyo, 181-0015, Japan Hiroshi Ito† and Naoyuki Watanabe Aerospace Systems Department, Tokyo Metropolitan Institute of Technology, Hino, Tokyo, 191-0065, Japan Ian J. Davies* Advanced Fibro Science, Kyoto Institute of Technology, Sakyo-ku, Kyoto, 606-8585, Japan This paper presents experimental results for the multiple microcracking and tensile behavior of an orthogonal 3-D woven Si-Ti-C-O fiber (Tyranno™ Lox-M)/Si-Ti-C-O matrix composite with a nanoscale carbon fiber/matrix interphase and processed using a polymer impregnation and pyrolysis route. Based on microscopic observations and unidirectional tensile tests, it is revealed that the inelastic tensile stress/strain behavior is governed by matrix cracking in transverse (90°) fiber bundles between 65 and 180 MPa, matrix cracking in longitudinal (0°) fiber bundles between 180 and 300 MPa, and fiber fragmentation above 300 MPa. A methodology for estimation of unidirectional tensile behavior in orthogonal 3-D composites has been established by the use and modification of existing theory. A good correlation was obtained between the predicted and measured composite strain using this procedure. I. Introduction I T IS well known that monolithic ceramics do not possess a level of damage tolerance that is sufficient for aerospace applications. For this reason, a great deal of effort has been devoted to the development of continuous-ceramic-fiber-reinforced ceramicmatrix composites (CMCs or CFCCs) for jet engine components, rocket engine nozzles, and the thermal protection systems (TPS) of future space transportation vehicles.1–3 The National Aerospace Laboratory of Japan, Ube Industries Ltd., Shikibo Ltd., and Kawasaki Heavy Industries Ltd. have conducted a joint program in order to develop and evaluate a continuousfiber-reinforced CMC. The composite contains Tyranno™ Lox-M fiber (Si 54%, Ti 2%, C 32%, O 12% (mass%)) with an additional surface modification process. The fibers are woven into an orthogonal 3-D structure that has advantages from the points of view of the polymer impregnation and pyrolysis (PIP) processing, and improved delamination resistance and tensile strength. Using these technologies, the composite exhibits excellent tensile strength at room temperature4,5 and creep strength at elevated temperature.6 The composite is referred to as “NUSK-CMC” from the initials of the collaborating partners. It is now well understood that unidirectional (UD) CMCs exhibit nonlinear stress–strain behavior under unidirectional tensile loading as a result of multiple microcracking and fiber pullout within the longitudinal (0°) fiber bundles. An overview of CMC mechanical properties has been proviede by Evans and Zok7 and Evans.8 The change in stiffness due to multiple matrix cracking has been estimated by two different approaches: (1) elastic analysis based on the Lame problem9 and (2) shear-lag analysis.10 In cross-ply composites, this change also involves initial cracking in the transverse (90°) plies as tunneling cracks.10–13 Subsequently, transverse cracks penetrate the longitudinal plies as the load is increased. Based on the energy criterion and finite element analysis, transverse crack propagation in cross-ply brittle-matrix composites has been analyzed.14 Shear-lag analysis is often used to estimate transverse crack propagation within polymer-matrix composites,15,16 and this method has also been applied to cross-ply CMCs.10,13 Following matrix crack saturation, estimates for the fiber pullout length, pullout work, and the ultimate tensile strength of unidirectional CMCs have been derived by Curtin,17,18 based on the statistical analysis of fiber strength and stress redistribution due to fiber fragmentation processes. This paper presents experimental results for the multiple microcracking and tensile behavior of an orthogonal 3-D woven Si-TiC-O fiber/Si-Ti-C-O matrix composite (NUSK-CMC) fabricated using the PIP method. Constituent properties of the composite were estimated from hysteresis loop analysis of loading/unloading cycles. A methodology for estimating the unidirectional tensile behavior of orthogonal 3-D composites has been provided by using and modifying the above-mentioned theories. II. Experimental Procedure (1) Material and Specimens The composite under investigation contained Tyranno Lox-M fibers woven into an orthogonal 3-D configuration with fiber volume fractions of 19%, 19%, and 2% in the x, y, and z directions, respectively. Optical micrographs in Fig. 1 illustrate the fiber architecture of the present composites with each fiber bundle containing 1600 fibers. The composite preform plate (240 mm 3 120 mm 3 6 mm) was treated at elevated temperature in a CO atmosphere, resulting in the formation of a 10 nm SiOx-rich layer surrounding an inner 40 nm carbon-rich layer at the fiber surface.4 The nanoscale carbon-rich layer is believed to result in an interphase with desirable properties between the fiber and the matrix.5 Polytitanocarbosilane was used as the matrix precursor with eight impregnation and pyrolysis cycles; the average composite bulk density was 2.20 g/cm3 . Tensile specimens were machined from the composite plates such that the loading direction was parallel to the y-axis. The specimen surfaces were also ground B. N. Cox—contributing editor Manuscript No. 188832. Received January 3, 2000; approved March 6, 2001. *Member, American Ceramic Society. † Currently with Nagoya Aerospace Systems, Mitsubishi Heavy Industries Ltd., Nagoya, Japan. J. Am. Ceram. Soc., 84 [7] 1565–74 (2001) 1565 journal
1566 Journal of the American Ceramic Sociery-0 ol.84.No.7 Japan)between 100 and 1000C. The UD composite was esti- mated to contain 36.3 vol% fiber and 25.2 vol% porosity Nano-indentation tests(Model ENT-1 100, Elionix, Japan)were conducted in order to estimate the matrix elastic modulus with the basic theory for this method being explained elsewhere. Essen- tially, the elastic modulus of a material may be estimated using the load/displacement curve obtained from the nano-indentation test In this study, a relative com of elastic moduli between the matrix and fiber component 2n Il. Experimental Results (1 Stress/Strain Behavior and Multiple Microcracking Stress/strain curves and hysteresis loops obtained during load- ing/unloading tensile cycles are summarized in Fig. 3, while optical micrographs of the replica films, illustrating matrix crack ing within the longitudinal (0%)and transverse(90%)fiber bundles, are shown in Fig. 4. The micrographs labelled(a) to(e)in Fig. 4 correspond to the loading stages(a) to(e)in Fig 3 The following damage processes for each load stage are under a) The stress/strain curve is linear with an initial elastic modulus, E 141 GPa, and no microscopic damage is observed 2mm up to a tensile stress of 65 MPa.(b) Propagation of matrix cracks within transverse fiber bundles is observed above 65 MPa. In this paper, the term transverse crack is used to indicate matrix crack Fig. 1. Optical microphotographs of Si-Ti-C-O fiber/Si-Ti-C-O matrix composite (NUSK-CMC) illustrating the orthogonal 3-D woven fiber within transverse(90%) fiber bundles The onset and evolu- tion of matrix cracks in longitudinal fiber bundles is observed above 180 MPa Matrix cracks that originate in the transverse fiber bundles only partially penetrate the longitudinal fiber bundles. In this paper, the term matrix crack is used to describe matrix cracks to a flat finish such that the interlacing loops shown in Fig. I were in the longitudinal(0%)fiber bundles.(d) The matrix crack not present in the final specimens. density in longitudinal fiber bundles increases with the applied load up to 300 MPa. A small amount of transverse crack propa (2) Tensile Tests gation is also observed. (e) Matrix crack densities in both Tensile testing was conducted on a servo-hydraulic testing rig transverse and longitudinal fiber bundles are saturated above a tress level of 300 MPa Model 8501, Instron, USA)at room temperature in air using a onded to the specimen end regions with the load being applied (2 Transverse and Matrit Crack Densities using hydraulic wedge grips. A clip gauge-type extensometer Crack ity measurements for the composite are shown in (gauge length 25 mm; Model 632.11C-20, MTS, USA)was used Fig. 5, and indicate that matrix cracking initiates at Ume =180 to measure the longitudinal strain. Matrix cracking characteristics MPa, and is saturated by o= 300 MPa with a crack saturation were investigated using the replica film method with surface pacing of s= 45.4 um. In contrast to this, the onset of transverse replicas being taken under load at various stages of the loading crack propagation is 65 MPa with the crack density first cycle. Tensile tests were conducted with a number of loading/ rapidly up to 120 MPa, then more slowly above 200 MPa, and is unloading cycles applied to each specimen. saturated beyond 300 MPa. In this final stage, oblique transverse () Thermal Expansion and Nano-indentation Tests The thermal expansion behavior of the UD and 3-D woven 500 composites was investigated in order to estimate the coefficient of qs=423.8[MPa] thermal expansion(CTE)of the matrix and fiber components using a thermal mechanical analyzer (TMA-6300, Seiko Instruments, 9/8 0002040.60.81.01.21.4 Strain [% Units: mm Fig. 3. Stress/strain curve and hyster agonal 3-D Fig. 2. Specimen configuration and dimensions used for unidirectional
to a flat finish such that the interlacing loops shown in Fig. 1 were not present in the final specimens. (2) Tensile Tests Tensile testing was conducted on a servo-hydraulic testing rig (Model 8501, Instron, USA) at room temperature in air using a specimen geometry as shown in Fig. 2. Cardboard tabs were bonded to the specimen end regions with the load being applied using hydraulic wedge grips. A clip gauge-type extensometer (gauge length 25 mm; Model 632.11C-20, MTS, USA) was used to measure the longitudinal strain. Matrix cracking characteristics were investigated using the replica film method with surface replicas being taken under load at various stages of the loading cycle. Tensile tests were conducted with a number of loading/ unloading cycles applied to each specimen. (3) Thermal Expansion and Nano-indentation Tests The thermal expansion behavior of the UD and 3-D woven composites was investigated in order to estimate the coefficient of thermal expansion (CTE) of the matrix and fiber components using a thermal mechanical analyzer (TMA-6300, Seiko Instruments, Japan) between 100° and 1000°C. The UD composite was estimated to contain 36.3 vol% fiber and 25.2 vol% porosity. Nano-indentation tests (Model ENT-1100, Elionix, Japan) were conducted in order to estimate the matrix elastic modulus with the basic theory for this method being explained elsewhere.19 Essentially, the elastic modulus of a material may be estimated using the load/displacement curve obtained from the nano-indentation test. In this study, a relative comparison of elastic moduli between the matrix and fiber components was conducted. III. Experimental Results (1) Stress/Strain Behavior and Multiple Microcracking Stress/strain curves and hysteresis loops obtained during loading/unloading tensile cycles are summarized in Fig. 3, while optical micrographs of the replica films, illustrating matrix cracking within the longitudinal (0°) and transverse (90°) fiber bundles, are shown in Fig. 4. The micrographs labelled (a) to (e) in Fig. 4 correspond to the loading stages (a) to (e) in Fig. 3, respectively. The following damage processes for each load stage are understood: (a) The stress/strain curve is linear with an initial elastic modulus, E ' 141 GPa, and no microscopic damage is observed up to a tensile stress of 65 MPa. (b) Propagation of matrix cracks within transverse fiber bundles is observed above 65 MPa. In this paper, the term transverse crack is used to indicate matrix crack within transverse (90°) fiber bundles. (c) The onset and evolution of matrix cracks in longitudinal fiber bundles is observed above 180 MPa. Matrix cracks that originate in the transverse fiber bundles only partially penetrate the longitudinal fiber bundles.12 In this paper, the term matrix crack is used to describe matrix cracks in the longitudinal (0°) fiber bundles. (d) The matrix crack density in longitudinal fiber bundles increases with the applied load up to 300 MPa. A small amount of transverse crack propagation is also observed. (e) Matrix crack densities in both the transverse and longitudinal fiber bundles are saturated above a stress level of 300 MPa. (2) Transverse and Matrix Crack Densities Crack density measurements for the composite are shown in Fig. 5, and indicate that matrix cracking initiates at s# mc 5 180 MPa, and is saturated by s# s 5 300 MPa with a crack saturation spacing of l # s 5 45.4 mm. In contrast to this, the onset of transverse crack propagation is 65 MPa with the crack density first increasing rapidly up to 120 MPa, then more slowly above 200 MPa, and is saturated beyond 300 MPa. In this final stage, oblique transverse Fig. 1. Optical microphotographs of Si-Ti-C-O fiber/Si-Ti-C-O matrix composite (NUSK-CMC) illustrating the orthogonal 3-D woven fiber architecture. Fig. 2. Specimen configuration and dimensions used for unidirectional tensile testing. Fig. 3. Stress/strain curve and hysteresis loops for the orthogonal 3-D woven Si-Ti-C-O fiber/Si-Ti-C-O matrix composite (NUSK-CMC) under loading/unloading testing. 1566 Journal of the American Ceramic Society—Ogasawara et al. Vol. 84, No. 7
July 2001 Multiple Cracking and Tensile behavior 1567 (a)64.0MPa (b)183.4MPa (c)2423MPa (d)301.4MPa (e)331.5MPa Fig. 4. Optical microphotographs of surface replica films showing matrix cracks in the longitudinal(0%)and transverse(90%)fiber bundles at different stress evels width at a stress equal to half the peak value, designated as dem Transverse Crac and the permanent strain, Eo, are plotted in Fig. 6 as a function of Matrix Crack the peak stress, G. The hysteresis loop width shows significant increase above 180 MPa, together with a saturation point of 300 MPa, which corresponds to the increasing matrix crack density On Predic Matrix Crack the other hand, the permanent strain increases for peak applied stresses above 65 MPa, and permanent strain increases are ob- served beyond the matrix crack density saturation point. (4 Thermal Expansion and Elastic Modulus of the Fiber The Cte of the UD composite was 4.07 X 10(K)and 4.01 x 10(K)for the longitudinal and transverse directions 0.16 0.14 lent Strain 050100150200250300350400 ○ Loop width,e Stress [MPa Fig. 5. Matrix crack densities in the longitudinal (0)fiber bundles, and transverse crack densities in the transverse (90)fiber bundles, as function of applied stress, a 89 cracks(indicated by the white arrows in Fig. 4(d)) are noted occur. Since the composite contains translaminar fiber bundles, e,: fibers, no delamination is observed at the interlaminar regions between the 0 and 90 fiber bundles. Thus, these cracks a002 interlansgested to propagate because of shear stresses in the (3) Hysteresis Measurements Peak Applied Stress [MPa] The presence of hysteresis is evident in Fig. 3 with permanent strains, implying a major contribution to the inelastic g. 6. Variation of permanent strain and maximum hysteresis loop width under loading/unloading testing Inction of peak applied stress, train from interfacial debonding and sliding. The hysteresis loop together with the predicted loop
cracks (indicated by the white arrows in Fig. 4(d)) are noted to occur. Since the composite contains translaminar fiber bundles, i.e., z fibers, no delamination is observed at the interlaminar regions between the 0° and 90° fiber bundles. Thus, these cracks are suggested to propagate because of shear stresses in the interlaminar regions. (3) Hysteresis Measurements The presence of hysteresis is evident in Fig. 3 with appreciable permanent strains, implying a major contribution to the inelastic strain from interfacial debonding and sliding. The hysteresis loop width at a stress equal to half the peak value, designated as dεmax, and the permanent strain, ε0, are plotted in Fig. 6 as a function of the peak stress, s# p. The hysteresis loop width shows significant increase above 180 MPa, together with a saturation point of 300 MPa, which corresponds to the increasing matrix crack density. On the other hand, the permanent strain increases for peak applied stresses above 65 MPa, and permanent strain increases are observed beyond the matrix crack density saturation point. (4) Thermal Expansion and Elastic Modulus of the Fiber and Matrix The CTE of the UD composite was 4.07 3 1026 (K21 ) and 4.01 3 1026 (K21 ) for the longitudinal and transverse directions, Fig. 4. Optical microphotographs of surface replica films showing matrix cracks in the longitudinal (0°) and transverse (90°) fiber bundles at different stress levels. Fig. 5. Matrix crack densities in the longitudinal (0°) fiber bundles, and transverse crack densities in the transverse (90°) fiber bundles, as a function of applied stress, s#. Fig. 6. Variation of permanent strain and maximum hysteresis loop width under loading/unloading testing as a function of peak applied stress, s# p, together with the predicted loop width for t 5 14 MPa. July 2001 Multiple Cracking and Tensile Behavior 1567
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 composites
respectively. 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-ofmixtures (ROM) theory. A schematic representation of the orthogonal 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 thicknesses, 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 unidirectional composite. Following the initiation of transverse cracking, the in-plane elastic modulus, El , for each laminate is calculated 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 condition 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
July 2001 Multiple Cracking and Tensile Behavior 1569 Existing Cracks Hypothetical New Crack (4 Stiffness Change Due to Matrix Cracks in the Longitudinal fiber bundle A partial fiber/matrix interfacial debond model has been ana E10° lyzed through use of a concentric cylinder with fiber radius, R, as shown in Fig 9.The problem can be solved by two different Er shear-lag analysis. In this paper, both of the theories were applied The average applied stress, o,u, in the longitudinal fiber bundles under a composite stress, o, was approximated by E LL+LL where A is the load-partitioning factor Fig. 8. Schematic drawing of transverse crack calculations for cross-ply (11) ated composites using the shear-lag model. Here, E and a denote the modulus and the CTE of the plies, with subscripts I and 2 indicating and E and Elu are the elastic moduli of the composite and the 0° plies and90° plies, respectively longitudinal unidirectional plies in each laminate, respectively. For the composite, the resulting values are A= 1. 44 below 50 MPa, A=2.68 at omc(180 MPa), and A= 2.75 above o(300 MPa).A =E11+E2)(21EE2 d approximation can be obtained through linear interpolation ψ=-2k/t(a2-a1)△T-{1+2(E1)}l (A) Hutchinson and Jensen Theory: Hutchinson and Jensen found a solution for an axisymmetric cylindrical model with a where AT is the change in temperature, o, is the average stress single matrix crack using the Lame problem. The axial stress, of (equivalent stress)in the laminate, K is an effective shear stiffnes radial stress, o, and axial strain, Ef, for a fiber in the bonded of the bonding layer, and /, is the transverse crack spacing. The egion( denoted by a superscript + under an average applied dimensionless parameter S is the ratio between the shear and axial stress,Gu in a longitudinal fiber bundle are given by the stiffnesses of the shear-lag model. The average stress, Olu in 0. following plies along the x-axis is given by the following o=a-a,EmE (12a) 1+l2 L,Ex cosh(2Ex'1r sh(El/t,) cosh(2Er'l12) E,+e (ax2-a)△71 cosh(E//t,) where a, to as are parameters provied by Hutchinson and Jensen The first and second terms are due to the applied load and thermal and listed in Table I for isotopic materials. e is the mismatch load, respectively. Thus, the elastic modulus, E, for each laminate strain between fiber and matrix, due to thermal stresses. The stress is obtained using the following equation crack tip are given by A d△er=Er El1E1(E11+E22) the following equation is obtained with Ao as the free variable in (1 1+12LEIEI+ RE] tanh(EI/i2) the solution △Er=b2△o/Em ( Matrix Crack Density in the Longitudinal Fiber bundles is difficult to estimate the matrix crack density within 0 plies Full contact over the debonded region occurs with a constan using a theoretical approach. Therefore, an empirical equation sliding stress, T, and the fiber stress has a maximum value, on/p, roposed by Evans et al. has been directly applied in order to at the matrix crack surface account for the 3-D composite matrix crack der 7≈7G/m-1)/(G/。-1) where p is the fiber volume fraction and I is the matrix crack where omc, o and s are the matrix crack onset stress, the crack spacing. The total strain is then given by aturation stress, and the saturated crack spacing, respectively. The olid line shown in Fig. 5 obtained using omc 180 MPa,o 300 MPa, and 1s =45.4 mm agrees well with the experimental (15) Matrix Crack Debonding Region R Fibe o Fig. 9. Partial interfacial debonding model for a unidirectional composite
j 5 ÎKt2~E1t1 1 E2t2!/~2t1E1E2! (5) c 5 22K/t2@~a2 2 a1!DT 2 $~t1 1 t2!/~E1t1!%s#l # (6) where DT is the change in temperature, s#l is the average stress (equivalent stress) in the laminate, K is an effective shear stiffness of the bonding layer, and lL is the transverse crack spacing. The dimensionless parameter j is the ratio between the shear and axial stiffnesses of the shear-lag model. The average stress, s#lu, in 0° plies along the x-axis is given by the following: s# lu 5 t1 1 t2 E1t1 1 E2t2 SE1 1 t2E2 t1 cosh ~2jx9/t2! cosh ~jlL/t2! Ds#l 1 t2E1E2 E1t1 1 E2t2 ~a2 2 a1!DTF1 2 cosh ~2jx9/t2! cosh ~jlL/t2! G (7) The first and second terms are due to the applied load and thermal load, respectively. Thus, the elastic modulus, El , for each laminate (E, F, G) is obtained using the following equation: El 5 jlLt1E1~E1t1 1 E2t2! ~t1 1 t2!@lLE1jt1 1 t2 2 E2 tanh ~jlL/t2!# (8) (3) Matrix Crack Density in the Longitudinal Fiber Bundles It is difficult to estimate the matrix crack density within 0° plies using a theoretical approach. Therefore, an empirical equation proposed by Evans et al. 7 has been directly applied in order to account for the 3-D composite matrix crack density. l # < l # s~s# s/s# mc 2 1!/~s#/s# mc 2 1! (9) where s# mc, s# s, and l # s are the matrix crack onset stress, the crack saturation stress, and the saturated crack spacing, respectively. The solid line shown in Fig. 5 obtained using s# mc 5 180 MPa, s# s 5 300 MPa, and l # s 5 45.4 mm agrees well with the experimental data. (4) Stiffness Change Due to Matrix Cracks in the Longitudinal Fiber Bundle A partial fiber/matrix interfacial debond model has been analyzed through use of a concentric cylinder with fiber radius, R, as shown in Fig. 9.9,13 The problem can be solved by two different approaches: (1) elastic analysis of the Lame problem, and (2) shear-lag analysis. In this paper, both of the theories were applied. The average applied stress, s#lu, in the longitudinal fiber bundles under a composite stress, s#, was approximated by s# lu 5 ls# (10) where l is the load-partitioning factor l 5 Elu/E (11) and E and Elu are the elastic moduli of the composite and the longitudinal unidirectional plies in each laminate, respectively. For the composite, the resulting values are l 5 1.44 below 50 MPa, l 5 2.68 at s# mc (180 MPa), and l 5 2.75 above s# s (300 MPa). A good approximation can be obtained through linear interpolation between s# mc and s# s. 12 (A) Hutchinson and Jensen Theory: Hutchinson and Jensen found a solution for an axisymmetric cylindrical model with a single matrix crack using the Lame problem.9 The axial stress, sf 1, radial stress, sr 1, and axial strain, εf 1, for a fiber in the bonded region (denoted by a superscript 1) under an average applied stress, s#lu, in a longitudinal fiber bundle are given by the following: sf 1 5 a1s#lu 2 a2EmεT (12a) sr 1 5 a3 2 a4EmεT (sr at r 5 R) (12b) εf 1 5 a5~s#lu/Em! 1 a6εT (12c) where a1 to a6 are parameters provied by Hutchinson and Jensen9 and listed in Table I for isotopic materials. εT is the mismatch strain between fiber and matrix, due to thermal stresses. The stress and strain differences in the fiber above and below the debond crack tip are given by Dsf 5 sf 2 sf 1 and Dεf 5 εf 2 εf 1. Thus, the following equation is obtained with Dsf as the free variable in the solution: Dεf 5 b2Dsf/Em (13) Full contact over the debonded region occurs with a constant sliding stress, t, and the fiber stress has a maximum value, s#lu/r, at the matrix crack surface. sf 5 s#lu/r 2 2t~l/2 2 x!/R (14) where r is the fiber volume fraction and l is the matrix crack spacing. The total strain is then given by εcu 5 εf 1 1 2 l E l/22ld l/ 2 Dε dx (15) Fig. 8. Schematic drawing of transverse crack calculations for cross-ply laminated composites using the shear-lag model. Here, E and a denote the elastic modulus and the CTE of the plies, with subscripts 1 and 2 indicating 0° plies and 90° plies, respectively. Fig. 9. Partial interfacial debonding model for a unidirectional composite. July 2001 Multiple Cracking and Tensile Behavior 1569
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 the
where 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 microcrackfree 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
July 2001 Multiple Cracking and Tensile behavior 1571 Table Il. Thermoelastic Properties of the Composite under Investigation (NUSK-CMC) Fiber modulus, Er 185 GPa 0.2 Fiber shear modulus. G 77.1 GPa Fiber thermal expansion coefficient, a 4.0×10-6K-1 Matrix Youngs modulus, E 185 GPa Matrix Poisson's ratio 0.2 Matrix shear modulus, G 77.lG Matrix thermal expansion coefficient, a 4.0×10=6K-1 Fiber radius. r 4.25um Fiber volume fraction, p(%) 41.3(x=19.6,y=19.6,==2.1) Fixed Slip Variabl Reverse Slip lengt h I/Ew Reload 4UE Cksure 4L Closure ←1E Fig. 10. Schematic drawing of the expected inverse tangent moduli (ITMs)in large debond energy(LDE)composites by Domergue et al. (I Initial Elastic Modulus 120 The thermoelastic properties of the composite under investiga- on are listed in Table Il. Fiber bundle tons In resent 3-D composite were determined by optic scopy to be bF= 110 04 mm, br =0.40 mm, and h, =0 initial elastic modulus estimated by the PSA solution is 121 GPa, hich is 9% smaller compared to the experimental result(141 GPa). The difference between the estimated and experimental values may be attributed to the approximation of the pocket region In Fig. 7(a) 2) Transverse Crack Densin Since the stiffness change due to transverse cracking is consid- erable in the low crack density region, Ic and K in Eq(4)were calculated from the data set at the onset of cracking. Througl correlating the estimated curve and experimental data, gIc and K 20 vere determined to be gc =8.0 J/m, and K=698 GPa/mm, Transverse Crack Density [mm-1 e 1. 24 MPam"by using the relationship gc=KIdE(I- v)for Fig. 11. Relationship between elastic modulus for the orthogonal 3-D plane strain conditions. Since the fracture toughness of the matrix composite and transverse crack density in the transverse(90)plies corresponds to that of the fiber(1.0 MPa m), the value of gc estimated from transverse crack propagation is reasonable when considering the fracture toughness of the Si-Tl-C-O matrix (3) Hysteresis Analysis The estimated relationship between transverse crack density stress is compared with the experimental data in Fig. 5. Deviation The debond stresses, G, were estimated using the transition between the estimated curve and experimental data gradually stress,Our during unloading and the peak stress, ap in the ncreases with increasing transverse crack density (above 150 hysteresis curve. Estimated values of o, are plotted in Fig. 12, and MPa). Figure 1 l shows the effect of transverse crack density on the composite elastic modulus as calculated from Eq (8). The initial Gp. This result is similar to that observed in other CMCs. 12, 242g stages of transverse crack propagation rapidly decrease the com The normalized stress, 2,=o,, has a constant value of0. 78. By posite elastic modulus, but thereafter decrease more gradually with utilizing the reload ITMs, o and o, the interface friction index, almost no decrease being observed above a crack density of 10 was calculated as shown in Fig. 13. s varies approximately m. Thus, the estimated values of gic and k provide a good inearly with the peak stress, suggesting that the interface slip approximation for calculating the stiffness change due to trans- with a constant friction stress up to 300 MPa. This phenomenon verse cracking may be due to the offsetting effects of Poisson contraction and
V. Discussion (1) Initial Elastic Modulus The thermoelastic properties of the composite under investigation are listed in Table II. Fiber bundle dimensions in the present 3-D composite were determined by optical microscopy to be bF 5 1.04 mm, bT 5 0.40 mm, and ht 5 0.13 mm, respectively. The initial elastic modulus estimated by the PSA solution is 121 GPa, which is 9% smaller compared to the experimental result (141 GPa). The difference between the estimated and experimental values may be attributed to the approximation of the pocket region in Fig. 7(a). (2) Transverse Crack Density Since the stiffness change due to transverse cracking is considerable in the low crack density region, &IC and K in Eq. (4) were calculated from the data set at the onset of cracking. Through correlating the estimated curve and experimental data, &IC and K were determined to be &IC 5 8.0 J/m2 , and K 5 698 GPa/mm, respectively. The matrix fracture toughness, KIC, was estimated to be 1.24 MPazm1/2 by using the relationship &IC 5 KIC/E(1 2 n) for plane strain conditions. Since the fracture toughness of the matrix corresponds to that of the fiber (1.0 MPazm1/2),5 the value of &IC estimated from transverse crack propagation is reasonable when considering the fracture toughness of the Si-Ti-C-O matrix. The estimated relationship between transverse crack density and stress is compared with the experimental data in Fig. 5. Deviation between the estimated curve and experimental data gradually increases with increasing transverse crack density (above 150 MPa). Figure 11 shows the effect of transverse crack density on the composite elastic modulus as calculated from Eq. (8). The initial stages of transverse crack propagation rapidly decrease the composite elastic modulus, but thereafter decrease more gradually with almost no decrease being observed above a crack density of 10 mm21 . Thus, the estimated values of &IC and K provide a good approximation for calculating the stiffness change due to transverse cracking. (3) Hysteresis Analysis The debond stresses, s#i , were estimated using the transition stress, s#tu, during unloading and the peak stress, s# p, in the hysteresis curve. Estimated values of s#i are plotted in Fig. 12, and reveal that s#i is not constant but instead increases with increasing s# p. This result is similar to that observed in other CMCs.12,24,25 The normalized stress, Si 5 si /sp, has a constant value of 0.78. By utilizing the reload ITMs, s# p and s#i , the interface friction index, +, was calculated as shown in Fig. 13. + varies approximately linearly with the peak stress, suggesting that the interface slips with a constant friction stress up to 300 MPa. This phenomenon may be due to the offsetting effects of Poisson contraction and Table II. Thermoelastic Properties of the Composite under Investigation (NUSK-CMC) Fiber modulus, Ef 185 GPa Fiber Poisson’s ratio, nf 0.2 Fiber shear modulus, Gf 77.1 GPa Fiber thermal expansion coefficient, af 4.0 3 1026 K21 Matrix Young’s modulus, Em 185 GPa Matrix Poisson’s ratio, nm 0.2 Matrix shear modulus, Gm 77.1 GPa Matrix thermal expansion coefficient, am 4.0 3 1026 K21 Fiber radius, R 4.25 mm Fiber volume fraction, r (%) 41.3 (x 5 19.6, y 5 19.6, z 5 2.1) Fig. 10. Schematic drawing of the expected inverse tangent moduli (ITMs) in large debond energy (LDE) composites by Domergue et al. 12 Fig. 11. Relationship between elastic modulus for the orthogonal 3-D composite and transverse crack density in the transverse (90°) plies. July 2001 Multiple Cracking and Tensile Behavior 1571
1572 Journal of the American Ceramic Sociery--Ogasaara et al Vol. 84. No. 7 400 003 350 ∑;=0.78 Experimental a30 002 250 200 0.01 Predicted line (t=14MPa) 0 050100150200250300350400 150200250300350400450 Peak Stress [MPa] Stress [MPa Fig. 12. Relationship between debond stress, d and peak stress, d. The Fig. 14. Relationship between the debond length, Ia, calculated with 2 normalized stress, 2i=oyap, has a constant value of 0. 78 o/o =0.78 and T= 14 MPa. The debond length, Id, at one is approximately one half that of the saturated matrix crack spacing, /,(45.4 E 400 006 300 T=18MPa 200 0.002 100 Karandikar, Chou Hutchinson Jensen 200220240260280300320 00020.40.60.81.01.214 Fig. 13. Relationship between inelastic strain index, y, and Strain [ dp. Predicted lines for friction stresses of T=10, 14, and 18 MPa have ig. 15. Comparison between predicted and experimental stress/strain been superimposed onto the data. nduced fiber clamping stresses that accompany roughne ing, as reported by Parthasarathy and Kerans.2 Predicted possess suitable interface properties with the low sidis a / SK for friction stresses of t=10.14. and 18 MPa are CMC 5 superposed onto Fig. 13, and indicate that T is approximately 14 MPa for the present composite. The predicted trend in hysteresis (4) Debond Length and Debond Energ loop width, assuming a frictional stress of T s= 14 MPa, is shown Figure 14 shows the debond length as a function of applied in Fig. 6, and indicates good agreement with the experimental data stress,o, calculated from Eq(16)with 2i=o 0,=0 By measuring the fiber fracture mirror size and pull-out length, Davies et al. estimated the sliding stress of the present composite of adjacent debond regions. The debond length, Id, at Gms is one using Curtin's theory to be t=4.94 MPa, which is approximately half that of the saturated matrix crack spacing, Is, (45. 4 um)in Fig. one third of the sliding stress, T=14 MPa, estimated by hysteresis 14, and is in agreement with the experimental results. The debond nalysis. The difference may be attributed to the stress level at energy obtained from Eq (17)is found to be 3.5-8.0 J/m and to which T is estimated, in that the matrix crack density is not be similar to that of other SiC/SiC composites. 2 saturated for the estimation of r by the hysteresis analysis, whereas Curtin's theory can only be utilized following matrix crack (5) Simulation of the Stress/Strain Behavior Simulated stress/strain curves based on the model proposed ontraction of the fiber and/or wear of the interface roughness above are shown in Fig. 15 between the fiber and the matrix may contribute to a decrease in data, indicating estimation of stress partitioning to be fairly good sliding stress throughout the entire stress/strain regime. The predicted curve The range of sliding stress in SiC fiber/SiC matrix composites using the Hutchinson and Jensen theory is almost the same as that is reported to vary widely between I and 200 MPa. 227-29 In spite by the Karandikar and Chou theory. o The stress partitioning of the coating free interface, the present composite appears to actor,A, is plotted in Fig. 16 as a function of the applied stress. It
induced fiber clamping stresses that accompany roughness unseating, as reported by Parthasarathy and Kerans.26 Predicted trends for friction stresses of t 5 10, 14, and 18 MPa are shown superposed onto Fig. 13, and indicate that t is approximately 14 MPa for the present composite. The predicted trend in hysteresis loop width, assuming a frictional stress of t ' 14 MPa, is shown in Fig. 6, and indicates good agreement with the experimental data. By measuring the fiber fracture mirror size and pull-out length, Davies et al. 5 estimated the sliding stress of the present composite using Curtin’s theory to be t 5 4.94 MPa, which is approximately one third of the sliding stress, t 5 14 MPa, estimated by hysteresis analysis. The difference may be attributed to the stress level at which t is estimated, in that the matrix crack density is not saturated for the estimation of t by the hysteresis analysis, whereas Curtin’s theory can only be utilized following matrix crack saturation. In the latter region, it is conceivable that Poisson’s contraction of the fiber and/or wear of the interface roughness between the fiber and the matrix may contribute to a decrease in sliding stress. The range of sliding stress in SiC fiber/SiC matrix composites is reported to vary widely between 1 and 200 MPa.12,27–29 In spite of the coating free interface, the present composite appears to possess suitable interface properties with the low sliding stress resulting in excellent mechanical properties for this “NUSK” CMC.5 (4) Debond Length and Debond Energy Figure 14 shows the debond length as a function of applied stress, s#, calculated from Eq. (16) with Si 5 si /sp 5 0.78 and t 5 14 MPa. The matrix crack density saturation is due to the linking of adjacent debond regions. The debond length, ld, at s# mc is one half that of the saturated matrix crack spacing, ls, (45.4 mm) in Fig. 14, and is in agreement with the experimental results. The debond energy obtained from Eq. (17) is found to be 3.5–8.0 J/m2 and to be similar to that of other SiC/SiC composites.12 (5) Simulation of the Stress/Strain Behavior Simulated stress/strain curves based on the model proposed above are shown in Fig. 15, and agree well with the experimental data, indicating estimation of stress partitioning to be fairly good throughout the entire stress/strain regime. The predicted curve using the Hutchinson and Jensen theory9 is almost the same as that by the Karandikar and Chou theory.10 The stress partitioning factor, l, is plotted in Fig. 16 as a function of the applied stress. It Fig. 14. Relationship between the debond length, ld, calculated with Si 5 si /sp 5 0.78 and t 5 14 MPa. The debond length, ld, at s# mc is approximately one half that of the saturated matrix crack spacing, ls (45.4 mm). Fig. 15. Comparison between predicted and experimental stress/strain behavior for the orthogonal 3-D composite. Fig. 12. Relationship between debond stress, s#i , and peak stress, s# p. The normalized stress, Si 5 si /sp, has a constant value of 0.78. Fig. 13. Relationship between inelastic strain index, +, and peak stress, s# p. Predicted lines for friction stresses of t 5 10, 14, and 18 MPa have been superimposed onto the data. 1572 Journal of the American Ceramic Society—Ogasawara et al. Vol. 84, No. 7
July 2001 Multiple Cracking and Tensile Behavior 1573 2.8 2 tanh(1E/t)-tanh(2/E/1)=sl (A-1) Constant s4 is represented by a function of o as follows 2.75 2.75 231E(E11+E22) (a2-a147-/4+h (A-2) 22 Matⅸ Crack Stress ore Then, the following cubic equation can be obtained from Eq (A-1) 2 p3-(4/2)p2-2=0 (A-3) Crack Saturation 8 Stress o. p=tanh(/2)(0≤≤1) (A-4) 1.44 1.4 Stress [MPa Equation(A-3)is arranged by using s and sd Fig. 16. Stress partitioning factor, A, of the orthogonal 3-D composite as a function of applied stress, q3-(s2/12)q-13/108-/2=0 According to a general solution of cubic equation x'+ mx=n, the ollowing for can be understood that the effect of transverse cracking is the most important factor for stiffness reduction in the orthogonal 3-D q=15[n+(n2+4m2/27)2] omposite. Following matrix cracking saturation, the numerical results agree well with the stress/strain curve in addition to the ultimate tensile strength of the composite according to Curtins [n-(n2+4m3127) (A-7) theoryand the in situ fiber strength data. 5 The simulated stress/strain curves show a significant stiffness decrease at 70 MPa in comparison with the experimental results, as where a result of a rapid increase in transverse crack density from th m=-s42/12.n=s43/108+s4/2 calculations. Xia et al.reported a similar numerical result. The critical energy release rate(IC=8.0 J/m2)was assumed to be Therefore, it is possible to calculate p as a function of sl from Eqs constant for the simulation and the deviation from the (A-6)and(A-4). The transverse crack spacing is calculated from tal results may be due to this approximation. In reality, the origins Eq(A-3)as a function of applied stress, o of transverse cracking may be porosity, matrix-rich regions, and the weak interface between fiber and matrix. Therefore, gIc may be expected to possess considerable scatter. Furthermore, the Acknowledgments effect of stress concentrations due to the nonuniform microstruc- Or M ture cannot be ignored for the distribution of transverse cracking and T. Tanamura of Shikibo Ltd, and 3. Gotoh of Kawasaki Heavy Industries Ltdfor stresses. A detailed discussion regarding the criterion for matrix their dedication in the research and development of NUSK-CMC racking has been discussed elsewhere References L. Conclusion P. J. Lamicq and J. F. Jamet, "Thermostructural CMCs: An Overview of the rench Experience, Ceram. Trans., 57, I-II(19 The multiple microcracking and tensile behavior of an orthog -H. Ohnabe. S. Masaki. M ka, K. Miyahara, and T. Sasa,"Potential onal 3-D woven Si-Ti-C-O fiber/Si-Ti-C-O matrix composite was Application of Ceramic Matrix Composites to Aero-Engine Components, " Compos- investigated using microscopic observations. Constituent proper- ies: Part d,30,489-%6(1999) ties of the composite were estimated using hysteresis loop analysis M. Muta and J Gotoh,"Development of High Temperature Materials Including CMCs for Space Application," Key Eng Mater, 164-165, 439-44(1999). during loading/u inelastic tensile stress/strain behavior is governed by transverse Behavior of SiC-Matrix Composites cracking between 65 and 180 MPa, longitudinal matrix cracking Elastic Modulus, "Compos. Sci. Technol, 58, 51-63(1998 etween 180 and 300 MPa, and fiber fragmentation above 300 nterfacial Properties Measured in Situ for a 3D Woven SiC/SiC-Based Composite MPa. A mechanical prediction model for estimating the unidirec tional tensile behavior of orthogonal 3-D composites was cor T Ogasawara, T. Ishikawa, N. Suzuki, I J Davies, M. Suzuki, J Gotoh, and T, ducted by using and modifying established theories. A good Hirokawa, "Tensile Creep Behavior of 3-D Woven Si-Ti-C-O Fiber/SiC Based Matrix orrelation between the predicted and measured strains was ob Composite with Glass Sealant, .. Mater. Sci, 38, 1-9(2000) 7A. G. Evans and F. W. Zok, "The Physics and Mechanics of Fibre-Reinforced tained using this procedure ittle Matrix Composites, J.Mater. Sci, 29, 3857-96(1994) A. G. Evans, Ceramics and Ceramic Composites as High-Temperature Structural Materials: Challenges and Opportunities, Philos. Trans. R Soc. London, A, 351 APPENDIX 93. W. Hutchinson and H. M. Jensen, "Models of Fiber Debonding and Pullout in Caleulation of transverse Crack density as a Function of Applied Stress Karandikar and T.-w. Chou, "Characterization and Modeling of Microcrack- ing and Elastic Modulo Changes in Nicalon/CAS Composites, Compos. Sci. echnol.,46,253-63(1993 ity and applied ID S. Beyerle, S M. Spearing, and A. G, Evans, "Damage Mechanisms and the tress is given by Eq. (5).I he equation for Mechanical Properties of Laminated O/90 Ceramic/Matrix Composite,JAm Ceram. alculating the transverse d stress. o. as Soc,75[2]3321-30(1992) the free variable in the solution. based imple energy 12J.-M. Domergue, F. E. Heredia, and A. G. Evans, "Hysteresis Loops and the Inelastic Deformation of 0/90 Ceramic Matrix Composites, JAm Ceram. Soc., 79 criterion, the following equation can be obtained from Eq. (5) 161-70(1996)
can be understood that the effect of transverse cracking is the most important factor for stiffness reduction in the orthogonal 3-D composite. Following matrix cracking saturation, the numerical results agree well with the stress/strain curve in addition to the ultimate tensile strength of the composite according to Curtin’s theory7 and the in situ fiber strength data.5 The simulated stress/strain curves show a significant stiffness decrease at 70 MPa in comparison with the experimental results, as a result of a rapid increase in transverse crack density from the calculations. Xia et al. 19 reported a similar numerical result. The critical energy release rate (&IC 5 8.0 J/m2 ) was assumed to be constant for the simulation, and the deviation from the experimental results may be due to this approximation. In reality, the origins of transverse cracking may be porosity, matrix-rich regions, and the weak interface between fiber and matrix. Therefore, &IC may be expected to possess considerable scatter. Furthermore, the effect of stress concentrations due to the nonuniform microstructure cannot be ignored for the distribution of transverse cracking stresses. A detailed discussion regarding the criterion for matrix cracking has been discussed elsewhere.7 VI. Conclusion The multiple microcracking and tensile behavior of an orthogonal 3-D woven Si-Ti-C-O fiber/Si-Ti-C-O matrix composite was investigated using microscopic observations. Constituent properties of the composite were estimated using hysteresis loop analysis during loading/unloading testing. The results reveal that the inelastic tensile stress/strain behavior is governed by transverse cracking between 65 and 180 MPa, longitudinal matrix cracking between 180 and 300 MPa, and fiber fragmentation above 300 MPa. A mechanical prediction model for estimating the unidirectional tensile behavior of orthogonal 3-D composites was conducted by using and modifying established theories. A good correlation between the predicted and measured strains was obtained using this procedure. APPENDIX Calculation of Transverse Crack Density as a Function of Applied Stress The relationship between transverse crack density and applied stress is given by Eq. (5). It is convenient to show the equation for calculating the transverse crack density with applied stress, s#, as the free variable in the solution. Based on a simple energy criterion, the following equation can be obtained from Eq. (5). 2 tanh ~lj/t2! 2 tanh ~2lj/t2! 5 ! (A-1) ! is represented by a function of s# as follows: ! 5 2&1cj~E1t1 1 E2t2! t1t2E1E2 H~a2 2 a1!DT 2 S t1 1 t2 E1t1 Ds#J 22 (A-2) Then, the following cubic equation can be obtained from Eq. (A-1) because l 1 p2 . 0: p3 2 ~!/ 2! p2 2 !/ 2 5 0 (A-3) with p 5 tanh ~lj/t2! ~0 # t # 1! (A-4) Using the definitions q 5 p 2 !/6 (A-5) Equation (A-3) is arranged by using s and !, q3 2 ~!2 /12!q 2 !3 /108 2 !/ 2 5 0 (A-6) According to a general solution of cubic equation x3 1 mx 5 n, the following formula can be obtained: q 5 H 1 2 @n 1 ~n2 1 4m3 /27! 1/ 2#J 1/3 1 H 1 2 @n 2 ~n2 1 4m3 /27! 1/ 2#J 1/3 (A-7) where m 5 2!2 /12, n 5 !3 /108 1 !/2 (A-8) Therefore, it is possible to calculate p as a function of ! from Eqs. (A-6) and (A-4). The transverse crack spacing is calculated from Eq. (A-3) as a function of applied stress, s#. Acknowledgments We wish to sincerely thank Dr. M. Shibuya of Ube Industries Ltd., T. Hirokawa and T. Tanamura of Shikibo Ltd., and J. Gotoh of Kawasaki Heavy Industries Ltd. for their dedication in the research and development of NUSK-CMC. References 1 P. J. Lamicq and J. F. Jamet, “Thermostructural CMCs: An Overview of the French Experience,” Ceram. Trans., 57, 1–11 (1995). 2 H. Ohnabe, S. Masaki, M. Onozuka, K. Miyahara, and T. Sasa, “Potential Application of Ceramic Matrix Composites to Aero-Engine Components,” Composites: Part A, 30, 489–96 (1999). 3 M. Imuta and J. Gotoh, “Development of High Temperature Materials Including CMCs for Space Application,” Key Eng. Mater., 164–165, 439–44 (1999). 4 T. Ishikawa, K. Bansaku, N. Watanabe, Y. Nomura, M. Shibuya, and T. Hirokawa, “Experimental Stress/Strain Behavior of SiC-Matrix Composites of Matrix Elastic Modulus,” Compos. Sci. Technol., 58, 51–63 (1998). 5 I. J. Davies, T. Ishikawa, M. Shibuya, T. Hirokawa, and J. Gotoh, “Fibre and Interfacial Properties Measured in Situ for a 3D Woven SiC/SiC-Based Composite with Glass Sealant,” Composites: Part A, 30, 587–91 (1999). 6 T. Ogasawara, T. Ishikawa, N. Suzuki, I. J. Davies, M. Suzuki, J. Gotoh, and T. Hirokawa, “Tensile Creep Behavior of 3-D Woven Si-Ti-C-O Fiber/SiC Based Matrix Composite with Glass Sealant,” J. Mater. Sci., 38, 1–9 (2000). 7 A. G. Evans and F. W. Zok, “The Physics and Mechanics of Fibre-Reinforced Brittle Matrix Composites,” J. Mater. Sci., 29, 3857–96 (1994). 8 A. G. Evans, “Ceramics and Ceramic Composites as High-Temperature Structural Materials: Challenges and Opportunities,” Philos. Trans. R. Soc. London, A, 351, 511–27 (1995). 9 J. W. Hutchinson and H. M. Jensen, “Models of Fiber Debonding and Pullout in Brittle Composites with Friction,” Mech. Mater., 9, 139–63 (1990). 10P. Karandikar and T.-W. Chou, “Characterization and Modeling of Microcracking and Elastic Modulo Changes in Nicalon/CAS Composites,” Compos. Sci. Technol., 46, 253–63 (1993). 11D. S. Beyerle, S. M. Spearing, and A. G. Evans, “Damage Mechanisms and the Mechanical Properties of Laminated 0/90 Ceramic/Matrix Composite,” J. Am. Ceram. Soc., 75 [12] 3321–30 (1992). 12J.-M. Domergue, F. E. Heredia, and A. G. Evans, “Hysteresis Loops and the Inelastic Deformation of 0/90 Ceramic Matrix Composites,” J. Am. Ceram. Soc., 79 [1] 161–70 (1996). Fig. 16. Stress partitioning factor, l, of the orthogonal 3-D composite as a function of applied stress, s#. July 2001 Multiple Cracking and Tensile Behavior 1573
1574 Journal of the American Ceramic Sociery-Ogasanwara et al Vol. 84. No. 7 w.-s. Kuo and T.w. Chou, "Multiple Cracking of Unidir and Cross-ply -E. Vagaggini, J.-M. Domergue, and A. G. Evans, ips between Ceramic Matrix Composites, J. Am Ceram Soc., 78]745-55(1995) Z. C. Xia, R. R. Carr, and J. w. Hutchinson, "T se Cracking in Fiber-Reinforced Brittle Matrix, Cross-Ply Laminates, Acta 2365-76(1993) L,1:AmCm802702309单A Hysteresis Measurements and the Constituent Properties of Ceramic Matrix C. H. Park and H. L. McManus,"Thermally Induced Damage in Composite es: ll, Experimental Studies on Unidirectional Materials,JAm. Ceram. Laminates: Predictive Methodology and Experimental Investigation, Compos. Sci ady, F. E Heredia, and A G. Evans, "In-Plane Mechanical Properties of Several Gudmundson and W. Zang,"An Analysis Model for Thermoelastic Properties F. E. Heredia, A. G. Evans, and C. C. Andersson, "The Tensile and Shear Properties of Continuous Fiber-Reinforced SiC/Al2O3 Composites Processed by Melt al Properties of Ceramic-Matrix Compos-02630n, Am四mnA段;381029=80 ites,J. Am. Ceram Soc., 74[11]2837-45(1991). w.A. Curtin,"In Situ Fiber Strength in Ceramic-Matrix Composites from 2043-55(1997). n the Behavior of Selected Ceramic Composites, J. Am. Ceram. Soc., 80 774]1075-78(1994) w. C. Oliver and G. M. Pharr, " An Improved Technique for Determining 27F. Rebillat, J. Lamon, R. Naslain, E. Lara-Curzio, M. K. Ferber, and T. M g Indentation Experiments,J Mater. Res, 7[6]1564-83(1992 2F. Rebillat, J. Lamon, R. Naslain, E Lara-Curzio, M. K. Ferber, and T. 2T. Okabe, J. Komotori, M. Shimizu, and N. Takeda, "Mechanical Behavior Besmann, "Properties of Multilayered Interphases in SiC/SiC Chemical-Vapc SiC Fiber Reinforced Brittle-Matrix Composites, J. Mater. Sci, 34, 3405-12 Infiltrated Composities with"Weak"and"Strong" Interface,J. A Ceram Soc., 81 2A. G. Evans, J-M. Domergue, and E, Vagaggini, "Methodology for Relating the Silicon carbic and G. V. Srinvasan,"Push-Out Tests on a New Tensile Constitutive of Ceramic-Matrix Composites to Constituent Properties, " J.Am Soc,7761425-35(1994) .. Am. Ce
13W.-S. Kuo and T.-W. Chou, “Multiple Cracking of Unidirectional and Cross-ply Ceramic Matrix Composites,” J. Am. Ceram. Soc., 78 [3] 745–55 (1995). 14Z. C. Xia, R. R. Carr, and J. W. Hutchinson, “Transverse Cracking in Fiber-Reinforced Brittle Matrix, Cross-Ply Laminates,” Acta Metall. Mater., 41 [8] 2365–76 (1993). 15C. H. Park and H. L. McManus, “Thermally Induced Damage in Composite Laminates: Predictive Methodology and Experimental Investigation,” Compos. Sci. Technol., 56, 1209–19 (1996). 16P. Gudmundson and W. Zang, “An Analysis Model for Thermoelastic Properties of Composite Laminates Containing Transverse Matrix Cracks,” Int. J. Solids Struct., 30, 3211–31 (1993). 17W. A. Curtin, “Theory of Mechanical Properties of Ceramic-Matrix Composites,” J. Am. Ceram. Soc., 74 [11] 2837–45 (1991). 18W. A. Curtin, “In Situ Fiber Strength in Ceramic-Matrix Composites from Fracture Mirrors,” J. Am. Ceram. Soc., 77 [4] 1075–78 (1994). 19W. C. Oliver and G. M. Pharr, “An Improved Technique for Determining Hardness and Elastic Modulus Using Load and Displacement Sensing Indentation Experiments,” J. Mater. Res., 7 [6] 1564–83 (1992). 20T. Okabe, J. Komotori, M. Shimizu, and N. Takeda, “Mechanical Behavior of SiC Fiber Reinforced Brittle-Matrix Composites,” J. Mater. Sci., 34, 3405–12 (1999). 21A. G. Evans, J.-M. Domergue, and E. Vagaggini, “Methodology for Relating the Tensile Constitutive of Ceramic-Matrix Composites to Constituent Properties,” J. Am. Ceram. Soc., 77 [6] 1425–35 (1994). 22E. Vagaggini, J.-M. Domergue, and A. G. Evans, “Relationships between Hysteresis Measurements and the Constituent Properties of Ceramic Matrix Composites: I, Theory,” J. Am. Ceram. Soc., 78 [10] 2709–20 (1995). 23J.-M. Domergue, E. Vagaggini, and A. G. Evans, “Relationships between Hysteresis Measurements and the Constituent Properties of Ceramic Matrix Composites: II, Experimental Studies on Unidirectional Materials,” J. Am. Ceram. Soc., 78 [10] 2721–31 (1995). 24C. Cady, F. E. Heredia, and A. G. Evans, “In-Plane Mechanical Properties of Several Ceramic-Matrix Composites,” J. Am. Ceram. Soc., 78 [8] 2065–78 (1995). 25F. E. Heredia, A. G. Evans, and C. C. Andersson, “The Tensile and Shear Properties of Continuous Fiber-Reinforced SiC/Al2O3 Composites Processed by Melt Oxidation,” J. Am. Ceram. Soc., 78 [10] 2790–800 (1995). 26T. A. Parthasarathy and R. J. Kerans, “Predicted Effect of Interfacial Roughness on the Behavior of Selected Ceramic Composites,” J. Am. Ceram. Soc., 80 [8] 2043–55 (1997). 27F. Rebillat, J. Lamon, R. Naslain, E. Lara-Curzio, M. K. Ferber, and T. M. Besmann, “Interfacial Bond Strength in SiC/C/SiC Composite Materials, As Studied by Single-Fiber Push-Out Tests,” J. Am. Ceram. Soc., 81 [4] 965–78 (1998). 28F. Rebillat, J. Lamon, R. Naslain, E. Lara-Curzio, M. K. Ferber, and T. M. Besmann, “Properties of Multilayered Interphases in SiC/SiC Chemical-VaporInfiltrated Composities with “Weak” and “Strong” Interface,” J. Am. Ceram. Soc., 81 [9] 2315–26 (1998). 29W. A. Curtin, J. I. Eldridge, and G. V. Srinvasan, “Push-Out Tests on a New Silicon Carbide/Reaction-Bonded Silicon Carbide Ceramic Matrix Composite,” J. Am. Ceram. Soc., 76 [9] 2300–304 (1993). M 1574 Journal of the American Ceramic Society—Ogasawara et al. Vol. 84, No. 7