2542 T Ogasawara et al. Composites Science and Technology 65(2005)2541-2549 were compared with the numerical results based on Tsai- A schematic drawing of a torsion beam is shown in Daniel model[ll], Hashin model [12], and Gudmundson- Fig. 1. A specimen consists of a rectangular cross-sec- Zang model[13], and conservative stiffness degradation as tion beam with dimensions b(width) by h(thickness a function of transverse crack density was predicted by in y and z directions, with L (length) in x direction these models The coordinate x is parallel to a material axis. For an Some researchers experimentally investigated the orthotropic material twisted about an axis parallel to degradation of in-plane shear modulus in CMCs by the material direction(x direction) with torsional mo- off-axis tensile test and V-notched shear(losipescu) test ment M, the torsional rigidity GJ(=M o) is given by 14, 15], and the effect of in-plane shear stress on in-plane [17] shear stiffness degradation has been understood However, the effect of on-axis loading on the in-plane GJ=GrB(c)bi and out-of-plane shear properties of CMCs have not B(c) been revealed yet. S{1-m( k=1,3,5. For evaluating the effect of on-axis loading on shear roperties, the she ear mo dulls of a composite which has microscopic damages caused by on-axis tensile stress should be measured. However it is difficult to make thick Cmc specimens because of difficulty in processing where o is a twist angle per unit length, Ga Standard test methods such as rail-shear method in-plane and out-of-plane shear moduli. As this equa (ASTM-D4255), off-axis tensile test method (ASTM on is based on Saint-Venant torsion, So-called"warp. D3518, V-notched shear(losipescu) method (ASTM ing effects"are neglected Ishikawa et al. [16]investigated the effect of warping C1292)are not applicable for measuring out-of-plane torsion on the torsional rigidity of a unidirectional com- shear modulus of thin CMC specimens A unique test method for measuring out-of plane posite beam, and revealed that torsional rigidity shear modulus has been provided by Ishikawa et al increases under the warping torsion. For an actual [16. They applied a torsional test for estimating out-of experiment, specimen grip areas shown in Fig. I are con plane shear modulus of a unidirectional carbon fiber/ strained for applying torsion moment and for fixing epoxy composite based on Lekhnitskii's torsion theor pecimen with a fixture. Therefore the effect of warping [17]. Tsai et al. also presented a closed-form solution on torsional rigidity was preliminarily investigated by for a composite laminate under torsion in terms of the nite element analysis(FI EA). A commercial FEA code lamination geometry, and the experimental methodol ABAQUS was used for the calculation The numerical results under the condition of ogy to determine the three principal shear moduli by L/H=26.7, and Gr/ G=x=2 are shown in Fig. 2 for measuring surface and edge strains in twisted prismatic coupons [18]. The torsional test is useful to estimate b/h of 1, 2, 4, and 8 On and oa are twist angles per length calculated by FEa and Lekhnitskil's torsion out-of plane shear properties, because a thick specimen theory(Eq(1), respectively. While the grip areas are is not required for the experiment In this study, the in-plane and out-of-plane shear assumed to constrain the warping deformation strictly properties of an orthogonal 3D woven Sic fiber/sic in the calculation, these boundary conditions are much matrix composite were evaluated by torsional test of a stricter than those in an actual experiment. The rectangular cross-section beam. The experimental results were compared with numerical results by finite element analysis(FEA). Furthermore, the effect of on- grip area axis tensile loading on shear modulus degradation of the SiC/SiC composite was also examined 2. Torsional test methodology Based on Lekhnitskii's torsion theory for an ortho- tropic material, Swanson established a torsion theory for composite laminated rectangular rods [19]. However, it is difficult to expand this theory for an orthogonal 3D woven composite. Therefore, Lekhnitskii's torsion the ory is directly applied rthogonal 3D woven composite as a uniform orthotropic material Fig. 1. Specimen configuration and coordinate system for torsional testwere compared with the numerical results based on Tsai– Daniel model [11], Hashin model [12], and Gudmundson– Zang model [13], and conservative stiffness degradation as a function of transverse crack density was predicted by these models. Some researchers experimentally investigated the degradation of in-plane shear modulus in CMCs by off-axis tensile test and V-notched shear (Iosipescu) test [14,15], and the effect of in-plane shear stress on in-plane shear stiffness degradation has been understood. However, the effect of on-axis loading on the in-plane and out-of-plane shear properties of CMCs have not been revealed yet. For evaluating the effect of on-axis loading on shear properties, the shear modulus of a composite which has microscopic damages caused by on-axis tensile stress should be measured. However, it is difficult to make thick CMC specimens because of difficulty in processing. Standard test methods such as rail-shear method (ASTM-D4255), off-axis tensile test method (ASTM D3518), V-notched shear (Iosipescu) method (ASTM C1292) are not applicable for measuring out-of-plane shear modulus of thin CMC specimens. A unique test method for measuring out-of plane shear modulus has been provided by Ishikawa et al. [16]. They applied a torsional test for estimating out-of plane shear modulus of a unidirectional carbon fiber/ epoxy composite based on Lekhnitskii
s torsion theory [17]. Tsai et al. also presented a closed-form solution for a composite laminate under torsion in terms of the lamination geometry, and the experimental methodol￾ogy to determine the three principal shear moduli by measuring surface and edge strains in twisted prismatic coupons [18]. The torsional test is useful to estimate out-of plane shear properties, because a thick specimen is not required for the experiment. In this study, the in-plane and out-of-plane shear properties of an orthogonal 3D woven SiC fiber/SiC matrix composite were evaluated by torsional test of a rectangular cross-section beam. The experimental results were compared with numerical results by finite element analysis (FEA). Furthermore, the effect of on￾axis tensile loading on shear modulus degradation of the SiC/SiC composite was also examined. 2. Torsional test methodology Based on Lekhnitskii
s torsion theory for an ortho￾tropic material, Swanson established a torsion theory for composite laminated rectangular rods [19]. However, it is difficult to expand this theory for an orthogonal 3D woven composite. Therefore, Lekhnitskii
s torsion the￾ory is directly applied, assuming an orthogonal 3D woven composite as a uniform orthotropic material. A schematic drawing of a torsion beam is shown in Fig. 1. A specimen consists of a rectangular cross-sec￾tion beam with dimensions b (width) by h (thickness) in y and z directions, with L (length) in x direction. The coordinate x is parallel to a material axis. For an orthotropic material twisted about an axis parallel to the material direction (x direction) with torsional mo￾ment Mt, the torsional rigidity GJ (=Mt/x) is given by [17]: GJ ¼ GxybðcÞbh3 ; bðcÞ ¼ 32c2 p4 X1 k¼1;3;5... 1
2c kp tanh kp 2c 
; c ¼ b h ffiffiffiffiffiffiffi Gzx Gxy s ; ð1Þ where x is a twist angle per unit length, Gxy and Gzx are in-plane and out-of-plane shear moduli. As this equa￾tion is based on Saint–Venant torsion, so-called ‘‘warp￾ing effects’’ are neglected. Ishikawa et al. [16] investigated the effect of warping￾torsion on the torsional rigidity of a unidirectional com￾posite beam, and revealed that torsional rigidity increases under the warping torsion. For an actual experiment, specimen grip areas shown in Fig. 1 are con￾strained for applying torsion moment and for fixing specimen with a fixture. Therefore, the effect of warping on torsional rigidity was preliminarily investigated by fi- nite element analysis (FEA). A commercial FEA code ABAQUS was used for the calculation. The numerical results under the condition of L/H = 26.7, and Gxy/Gzx = 2 are shown in Fig. 2 for b/h of 1, 2, 4, and 8. xn and xa are twist angles per length calculated by FEA and Lekhnitskii
s torsion theory (Eq. (1)), respectively. While the grip areas are assumed to constrain the warping deformation strictly in the calculation, these boundary conditions are much stricter than those in an actual experiment. The Fig. 1. Specimen configuration and coordinate system for torsional test. 2542 T. Ogasawara et al. / Composites Science and Technology 65 (2005) 2541–2549