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 experimen￾tal 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 microstruc￾ture 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 orthog￾onal 3-D woven Si-Ti-C-O fiber/Si-Ti-C-O matrix composite was investigated using microscopic observations. Constituent proper￾ties 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 unidirec￾tional tensile behavior of orthogonal 3-D composites was con￾ducted by using and modifying established theories. A good correlation between the predicted and measured strains was ob￾tained 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,” Compos￾ites: 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 Microcrack￾ing 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