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 andV. Discussion (1) Initial Elastic Modulus The thermoelastic properties of the composite under investiga￾tion 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 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. 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 com￾posite 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 trans￾verse 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