T. Ogasawara et al. Composites Science and Technology 65(2005)2541-2549 transverse cracking. However, Whitney [22] reported that out-of-plane shear modulus transverse cracking in 90 layers. These results indicate that Gry decreases due to the accumulation of transverse cracks in the 90 fiber bundles and matrix cracks in the 0o fiber bundles, while Gex is affected only by matrix cracks in o fiber bundles Finally, the degradation of in-plane shear stiffness of the orthogonal 3 D woven CMC as a function of trans- verse crack density is predicted using the shear-lag model. It is assumed that: (1)z-bundles are neglected here because of their low volume fraction, and orthog 2468 onal 3D woven composites are treated as laminates with multiple0°/90° ply blocks (i.e.,([09/90°1) lami nsity(mm nates), and(2)analysis of [0 /90] laminates can be Fig. 16. Relationship between the shear modulus (Grr) and the matic drawing of the shear-lag model of a cross-ply corresponding initial value (Gmg/ar modulus is normalized by the applied to([0%/90%1) laminates. Fig. 15 shows sche transverse crack density. Each sh laminated composite. The relationship between the shear stresses and the through-the-thickness average in-plane displacements of each layer is written by the laminate as a function of transverse crack density p is gi- following equation ven by H1u01「a2-l 1+ t? 2p tanh G10 where u; and v; are average in-plane displacements of the B2(1+) 0(=l)and 90(i= 2)layers in the x and y directions GLt It2 HI parameters are often dealt with as empirical parameters. When Eq (7)is used for the shear-lag parameter, Eq ( 8) The parameters are also determined by assuming the predicts the lower limit of the shear stiffness reduction fur nction l;, Ui) in each [2] example, Tsai and Daniel [ll], Nuismer and Tan [23] Using Eqs. (7) and( 8)the in-plane shear modulus proposed the following equations as the shear-lag was calculated as a function of transverse crack den- sity. The numerical result is plotted by the solid line in Fig. 16. The parameters used for the calculation HIL 3GLT GTT BGLT GI H ) were GLT=453 GPa, GTT=29.3 GPa and 11=t2 t?GLT+hI GTT t, GLT t ?G 0. 16 mm, respectively. The shear moduli, GLT, GTT, where GLt and Grr are longitudinal and transverse of a lamina were assumed by the reuss type approxi- shear stifness of unidirectional composites(or fiber bun- mation based on the experimental results. The effect dles), respectively, and I, and f2 are thickness of 0 and of the 3D woven architecture including voids(pocket 90 layers. The in-plane shear modulus of the whole region) was not explicitly considered as described re, and incorporated into these shear modulus values. Although it is known that Eq(8)predicts the lower limit, the numerical curve almost agreed with the experimental data. This result also suggests that the in- plane shear stifness is degraded by matrix crack (i,+1) ing in 0 fiber bundles as well as transverse cracking in 90° fiber bundles. i" ply T +1+2) 5. Conclusion i+1ply=(+1+2) Shear moduli c、 and G=r of an orthogonal 3D x woven SiC/SiC composite were estimated by torsional tests of rectangular cross-section specimens. In-plane shear modulus Gxy from the torsional test agreed with laminated composite that measured from +45 off-axis tensile test. FEMtransverse cracking. However, Whitney [22] reported that out-of-plane shear modulus Gzx is insensitive to transverse cracking in 90
layers. These results indicate that Gxy decreases due to the accumulation of transverse cracks in the 90
fiber bundles and matrix cracks in the 0
fiber bundles, while Gzx is affected only by matrix cracks in 0
fiber bundles. Finally, the degradation of in-plane shear stiffness of the orthogonal 3 D woven CMC as a function of trans￾verse crack density is predicted using the shear-lag model. It is assumed that: (1) z-bundles are neglected here because of their low volume fraction, and orthog￾onal 3D woven composites are treated as laminates with multiple 0
/90
ply blocks (i.e., ([0
/90
]s)n lami￾nates), and (2) analysis of [0
/90
]s laminates can be applied to ([0
/90
]s)n laminates. Fig. 15 shows sche￾matic drawing of the shear-lag model of a cross-ply laminated composite. The relationship between the shear stresses and the through-the-thickness average in-plane displacements of each layer is written by the following equation: sxz syz   ¼ H11 0 0 H22   u2
u1 v2
v1  ; ð6Þ where ui and vi are average in-plane displacements of the 0
(i = 1) and 90
(i = 2) layers in the x and y directions. H11 and H22 are the shear-lag parameters. The shear-lag parameters are often dealt with as empirical parameters. The parameters are also determined by assuming the function of displacement (ui, vi) in each layer. For example, Tsai and Daniel [11], Nuismer and Tan [23] proposed the following equations as the shear-lag parameters: H11 ¼ 3GLTGTT t2GLT þ t1GTT ; H22 ¼ 3GLTGTT t1GLT þ t2GTT ; ð7Þ where GLT and GTT are longitudinal and transverse shear stiffness of unidirectional composites (or fiber bun￾dles), respectively, and t1 and t2 are thickness of 0
and 90
layers. The in-plane shear modulus of the whole laminate as a function of transverse crack density q is gi￾ven by: Gxy ðqÞ Gxy0 ¼ 1 þ t2 t1 2q a tanh a 2q


1 ; a ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H22 GLT 1 t1 þ 1 t2 s
. ð8Þ When Eq. (7) is used for the shear-lag parameter, Eq. (8) predicts the lower limit of the shear stiffness reduction [12]. Using Eqs. (7) and (8) the in-plane shear modulus was calculated as a function of transverse crack den￾sity. The numerical result is plotted by the solid line in Fig. 16. The parameters used for the calculation were GLT = 45.3 GPa, GTT = 29.3 GPa and t1 = t2 = 0.16 mm, respectively. The shear moduli, GLT, GTT, of a lamina were assumed by the Reuss type approxi￾mation based on the experimental results. The effect of the 3D woven architecture including voids (pocket region) was not explicitly considered as described above, and incorporated into these shear modulus values. Although it is known that Eq. (8) predicts the lower limit, the numerical curve almost agreed with the experimental data. This result also suggests that the in-plane shear stiffness is degraded by matrix crack￾ing in 0
fiber bundles as well as transverse cracking in 90
fiber bundles. 5. Conclusion Shear moduli Gxy and Gzx of an orthogonal 3D woven SiC/SiC composite were estimated by torsional tests of rectangular cross-section specimens. In-plane shear modulus Gxy from the torsional test agreed with that measured from ±45
off-axis tensile test. FEM Fig. 15. Schematic drawing of a shear-lag model of a cross-ply laminated composite. 0 2 4 6 8 10 0 0. 2 0. 4 0. 6 0. 8 1 Transverse crack density (mm-1) Gxy /Gxy0 Fig. 16. Relationship between the shear modulus (Gxy) and the transverse crack density. Each shear modulus is normalized by the corresponding initial value (Gxy0). 2548 T. Ogasawara et al. / Composites Science and Technology 65 (2005) 2541–2549