Journal of the American Ceramic Society-Carelli et al. Vol. 85. No. 3 In terms engineering elastic constants of an The fitting parameters, s, in Eqs.(A-3)and(A-4) are taken to be nidirecti lamina. In the second, the lamina pi elated to operties of the fibers and the matrix and The matrix modulus was inferred from the measured modulus orientation through established composite models. The results Orgo via Eqs.(A-1)and(A-3), along with the known constituent from these two steps are then combined with other(known) properties (Er= 260 GPa, vm S v S V12 8 0.2(Ref. 13)) constituent properties to obtain the matrix modulus. The pertinent Similarly, it was inferred from the measured modulus Eas via eqs esults in each of the two main steps are summarized below. (A-2)and(A-4)and the same constituent properties. The results of The unidirectional lamina is treated as being transversely these calculations are plotted in Fig. 7(b) isotropic. The relevant in-plane engineering elastic constants are denoted E1, Ex, V12 and G12, where E denotes Youngs modulus, v is Poissons ratio, G is the shear modulus, and the subscripts I and 2 refer to the directions parallel and perpendicular to the fibers, Acknowledgment espectively. From laminate theory, Youngs modulus of the acknowledge Professor Carlos G. Levi for his invaluable contribu- laminate can be expressed as tions in the de 2v12 +E 1+E1/E E E (A-1) References R. L. Bannister, N.S. Ceruvu, D. A. Little, and G. McQuiggan, "Developmen Requirements for an Advanced Gas Turbine System, Trans. ASME, 117, 724-33 For typical values of the various engineering elastic properties of W. P. Parks, R. R Ramey, D. C. Rawlins, J. R. Price, and M. Van Roode is very close to unity (within about 0. 4%]). Consequently, for most Gas nuri es Picter, 13 S// oz 4 (r991 cases only the term in brackets,[,I is required. 3C. G. Levi, J. Y. Yang, B. J. Dalgleish, F. W. Zok, and A G. Evans, "Process 9. Similarly, for in-plane uniaxial loading at 45 to the two fiber and Performance of an All-Oxide Ceramic Composite,"J. Am. Ceram Soc., 81 (8 erections, Youngs modulus Eas of the laminate is given by 4J. A Heathcote, X.-Y. Gong, J. Yang, U. Ramamurty, and F. w. Zok, "In-Plane 111-v2(E2/E1) Mechanical Properties of an All-Oxide Ceramic Composite, "J. Am Ceram. Soc., (A-2) 012721-30(199 toni. and J. P. A. Lofvande ble Porous Matrices for All-Oxide Ceramic Compos- FA i N SC I A i o) o ou Mis Ccmt The engineering elastic constants E, and vu are related to the fiber ites, "Z Metallkd, 90[12]1037-47(1999) and matrix properties using the rule of mixtures. The other constants F W. Zok and E, and G12, are calculated using the Tsai-Halpin equation 1+ SEmE/ Turbines Power,122[2]202-205(2000) University Press, Cambridge, U.K., 1981 9w. A. Curtin, "Theory of the Mechanical Properties of Ceramic Composites, Er/Em-1 J. Am Ceram.Soc,74[2837-45(1991) E/En SE (A-3b) IoM. Ibnabdeljalil and w. A, Curtin, "Strength and Reliability of Fiber-Reinforced Composites: Localized Load-Sharing and Associated Size Effects, "Int.J. Solids Struct,34,2649-68(1997) J. Lu and J. w. Hutchinson,"Thermal Conductivity and G (A-4a) Cross-Ply Composites with Matrix Cracks,J. Mech Phys. Solids 12D. Wilson, Statistical Tensile Strength of Nextel M 610 and Nextel M720 G/Gm+ Eg (A-4b)M.Bari.AMsEngieringMrdsReremeBoo2ndEd;pp195-264in terms of the engineering elastic constants of an individual (unidirectional) lamina. In the second, the lamina properties are related to the properties of the fibers and the matrix and the fiber orientation through established composite models. The results from these two steps are then combined with other (known) constituent properties to obtain the matrix modulus. The pertinent results in each of the two main steps are summarized below. The unidirectional lamina is treated as being transversely isotropic. The relevant in-plane engineering elastic constants are denoted E1, E2, 12 and G12, where E denotes Young’s modulus, is Poisson’s ratio, G is the shear modulus, and the subscripts 1 and 2 refer to the directions parallel and perpendicular to the fibers, respectively. From laminate theory,8 Young’s modulus of the laminate can be expressed as E0/90

E1 E2 2 1  2 12 1 E1/E2  2 1 12 2 E2 E1  (A-1) For typical values of the various engineering elastic properties of the lamina, the term in braces, {. . . }, on the right side of Eq. (A-1) is very close to unity (within about 0.4%). Consequently, for most cases only the term in square brackets, [. . . ], is required. Similarly, for in-plane uniaxial loading at 45° to the two fiber directions, Young’s modulus E45 of the laminate is given by 1 E45
1 4G12 1 12 2 E2/E1 E1 E21 2 12 (A-2) The engineering elastic constants E1 and 12 are related to the fiber and matrix properties using the rule of mixtures. The other constants, E2 and G12, are calculated using the Tsai–Halpin equations:11 E2
Em 1 EE f 1 E f  (A-3a) E
Ef /Em 1 Ef /Em E (A-3b) G12
Gm 1 GG f 1 G f  (A-4a) G
Gf /Gm 1 Gf /Gm G (A-4b) The fitting parameters, , in Eqs. (A-3) and (A-4) are taken to be G  1 and E  2.8,11 The matrix modulus was inferred from the measured modulus E0/90 via Eqs. (A-1) and (A-3), along with the known constituent properties (Ef  260 GPa;12 m f 12 0.2 (Ref. 13)). Similarly, it was inferred from the measured modulus E45 via Eqs. (A-2) and (A-4) and the same constituent properties. The results of these calculations are plotted in Fig. 7(b). Acknowledgment We gratefully acknowledge Professor Carlos G. Levi for his invaluable contribu￾tions in the design and implementation of the porous-matrix concept utilized in this study. References 1 R. L. Bannister, N. S. Ceruvu, D. A. Little, and G. McQuiggan, “Development Requirements for an Advanced Gas Turbine System,” Trans. ASME, 117, 724–33 (1995). 2 W. P. Parks, R. R. Ramey, D. C. Rawlins, J. R. Price, and M. Van Roode, “Potential Applications of Structural Ceramic Composites in Gas Turbines,” J. Eng. Gas Turbines Power, 113 [4] 628–34 (1991). 3 C. G. Levi, J. Y. Yang, B. J. Dalgleish, F. W. Zok, and A. G. Evans, “Processing and Performance of an All-Oxide Ceramic Composite,” J. Am. Ceram. Soc., 81 [8] 2077–86 (1998). 4 J. A. Heathcote, X.-Y. Gong, J. Yang, U. Ramamurty, and F. W. Zok, “In-Plane Mechanical Properties of an All-Oxide Ceramic Composite,” J. Am. Ceram. Soc., 82 [10] 2721–30 (1999). 5 C. G. Levi, F. W. Zok, J.-Y. Yang, M. Mattoni, and J. P. A. Lo¨fvander, “Microstructural Design of Stable Porous Matrices for All-Oxide Ceramic Compos￾ites,” Z. Metallkd., 90 [12] 1037–47 (1999). 6 F. W. Zok and C. G. Levi, “Mechanical Properties of Porous Matrix Ceramic Composites,” Adv. Eng. Mater., 3 [1–2] 15–23 (2001). 7 R. A. Jurf and S. C. Butner, “Advances in Oxide–Oxide CMC,” J. Eng. Gas Turbines Power, 122 [2] 202–205 (2000). 8 D. Hull, An Introduction to Composite Materials; pp. 81–124. Cambridge University Press, Cambridge, U.K., 1981. 9 W. A. Curtin, “Theory of the Mechanical Properties of Ceramic Composites,” J. Am. Ceram. Soc., 74 [11] 2837–45 (1991). 10M. Ibnabdeljalil and W. A. Curtin, “Strength and Reliability of Fiber-Reinforced Composites: Localized Load-Sharing and Associated Size Effects,” Int. J. Solids Struct., 34, 2649–68 (1997). 11T. J. Lu and J. W. Hutchinson, “Thermal Conductivity and Expansion of Cross-Ply Composites with Matrix Cracks,” J. Mech. Phys. Solids, 43, 1175–98 (1995). 12D. Wilson, “Statistical Tensile Strength of NextelTM 610 and NextelTM 720 Fibers,” J. Mater. Sci, 32, 2532–42 (1997). 13M. Bauccio, AMS Engineering Materials Reference Book, 2nd Ed.; pp. 195–264. ASM International, Materials Park, OH, 1994.
602 Journal of the American Ceramic Society—Carelli et al. Vol. 85, No. 3