June 2000 Rate of strength Decrease of Fiber-Reinforced Ceramic-Matrix Composites during Fatigue APPENDIX AR. F. Cooper and K. Chyung, "Structure and Chemistry of Fiber-Matrix Interfaces in Silicon Carbide Fiber-Reinforced Glass-Ceramic Composites: An By the of a simple micromechanical model T can be muted from the hysteresis modulus, E, or the stress-strain data Reinforced Glass-Ceramics: A High-Resolution Scanning Transmission Electron ter, the following procedure is used. For a given strain incre ure of Silicon Carbide Fiber-Reinforced rom the unloaded state, Ae, the corresponding stress increment Glass-Ceramics, J. Mater. Sci., 22, 2695-701(1987) is determined. The hysteresis modulus at that stress level is 'L M. Butkus, L. P Zawada, and G. A Hartman, "Room Temperature Tensile then e= Ao/. this value is then used in the model. however and Fatigue Properties of Silicon Carbide Fiber-Reinforced Ceramic Matrix Com- different equations are valid for different states of interfacial slip If the slip length is smaller than s/2 the composite experiences J. w. Holmes and S. F. Shuler, Temperature Rise During Fatigue of Fiber- artial slip. Else, the composite experiences full slip. The transition Reinforced Ceramics, J. Mater. Sci. Lett, 9, 1290-91(1990) tween partial and full slip occurs when the hysteresis modul J. w. Holmes and C. Cho,"Experimental Observations of Frictional Heating in Fiber-Reinforced Ceramics, J. Am. Ceram. Soc., 75, 929-38(1992)- L P. Zawada, L M. Butkus, and G. A. Hartman, "Room Temperature Tensile E (A-1) Ceram. Eng. Sci. Proc., 11, 1592-606(1990) D. Rouby and P. Reynaud, "Fatigue Behavior Related to the Interface Modifi- E where Em is the Youngs modulus of the matrix If E>Eps-s 2J. W. Holmes, X. Wu, and B. F. Sorensen,"Frequency Dependency of Fatigue the composite is in partial slip, and T can be calculated from- Life and Internal Heating of a Fiber-Reinforced Ceramic Matrix Composite,J.Am Cera. Soc,7,3284-86(1994). Er△a/Em1 Zok, and R. M. MeMeeking, ""Fatigue of Ceramic Matrix Composites, Acta Metall. Mater, 43, 859-75(1995) EsEr(E。vr E 2808-18 (1990)5 es rom nictona Heating Measurements, 3. mt ceram 2E. Y. Luh and A. G. Evans, "High-Temperature Mechanical Properties of i where Er denotes the Young's modulus of the fiber and r is the Ceramic Matrix Composite,J. Am. Ceram Soc., 70, 466-69(1987). fiber radius. If full slip applies(E Ens-s, then T can be 24K. M. Prewo, B. Johnson, and S. Starrett, "Silicon Carbide Fibre-Reinforced calculated from Glass-Ceramic Composite Tensile Behaviour at Elevated Temperature, "J.Mater. Sa.,24,137y-79(1989 R. T, Bhatt, " Oxidation Effects on the Mechanical Properties of Sic-Fiber (A-3) 406-12(1992) 2J F. Mandell, D. H. Grande, and J. Jacobs, " Tensile Behaviour of Glass/Ceramic When E was determined from the stress-strain data of the residual Composite Materials at Elevated Temperatures,J. Eng Turbines Power: 109 strength test, the matrix J. W. Holmes, "Influence of Stress Ratio on the Elevated-Temperature Fatigue of (i. e, before the tensile tes Silicon Carbide Fiber-Reinforced Silicon Nitride Composite, J. Am. Ceram. Soc., 74, cracking is assumed to 639-45(1991) stress is below the maximum 2J. W. Holmes, "A Technique for Tensile Fatigue and Creep Testing of the cycling. 7 A. w. Pryce and P. A Smith, "Matrix Cracking i Ceramic matrix W. P Keith and K. T. Ke "The Stress-Strain Behavior of a porous The impetus for this study arose from discussions with Dr. Xin Wu. Unidirectional Ceramic matrix s,"Model of the Mechanical behavior o Continuous Fiber Reinforced Ceramic Matrix Composites During Cyclic Loading, "J. References Eur Ceran Soc. submitted T. Mackin and F. W. Zok,"Fiber Bundle Pushout: A Technique for the J. W. Holmes and B. F. Sorensen, "Fatigue Behavior of Continuous-Fiber- Measurement of Interfacial Sliding Properties, J. Am. Ceram. Soc., 75, 3169-71 (1992) havior of Ceramic Matrix Composites. Edited by S. V. Nair and K. Jakus. E. Lara-Curzio and M. K, Feber, "Methodology for the Determination of the Butterworth Heineman, Newton, MA, 1995. Interfacial Properties of Brittle Matrix Composites, J. Mater. Sci, 29, 6152-58 Karandikar and T -w. Chou, "Damage Development and Moduli Reduction (1994) icalon-Calcium Aluminosilicate Com under Static Fatigue and Cyclic B. F, Sorensen and J. w. Holmes,"Effect of Loading Rate on the Monotonic Fatigue, J. Am. Ceram. Soc 76, 1720-28(1993 S. Beyerle, S. M. Spearing, F. w. Zok, and A. G. Evans, "Damage and Failur Ceram. Soc., 79, 313-20(1996 Cracking of a Fiber-Reinforced Ceramic, "J.Am. Tensile behavior 3SR. W. Goettler and K. T. Faber. ""Interfacial Shear Stress in Fiber-Reinforced Glasses,Compos. Sci. TechnoL, 37, 129-47(1989) B. F. Sorensen and R. Talreja, "Analysis of Damage in Ceramic Matrix Holmes, "Fatigue of Continuous Fiber-Reinforced i IntJ. Damage M Ceramic Matrix Composites: Review of Mechanisms and Models": pp. 487-500 in B. Budiansky, J, W, Hutchinson, and A. G. Evans, "Matrix Fracture in Fiber- Reinforced Ceramics, J. Mech. Phys. Solids, 34, 167-78(1986) Proceedings of the International Symposium on Fatigue under Thermal and Mechan- rittle- MatrIx Fiber- ical Loading. Edited by J. Bressers and L. Remy, Kluwer Academic Publishers Reinforced Composites, Proc. R. Soc. London, A409, 329-50(1987 Dordrecht, Netherlands, 1996 man and H. Zhu, "Multi-fracture of Ceramic Composites, J. Mech. S. M. Spearing, F. M. Zok, and A. G. Evans, "Stress Corrosion Cr Sd,4.351-88199 w.A. Curtin, ""Theory of Mechanical Properties of Ceramic Matrix Composites, 3*B F. Sorensen and J.w. Holmes, "Influence of Stress oc, 77, 562- Unidirectional Ceramic-Matrix Composite, J. Am. Cera J.Am. Ceran.Soe,74,2837-45(1991) J. w. Holmes and x. Wu, "Elevated Temperature Creep Behavior of Continuous ated Temperature Behavior of J. I. Eldridge. "Environmental Effects on Fiber Debonding and Sliding in an amic Matrix Composites. Edited by S. V, Nair and K. Jakus. Butterworth SCS-6 SIC C Mater.,32,1085-89(1995) Camus, R. Naslain, and J. Thebault, "Oxidation Mechanisms and 41J dge, R. T. Bhatt, and N. P, Bansal, "Investigation of Fiber /Matrix Kinetics of ID-SiC/C/SiC Composite Materials: 1, An Experimental Approach," Interfacial Mechanical Behavior in Ceramic Matrix Composites by Cyclic Fiber Push-in Testin 266-78(1996 A. G. Evans, F. W. Zok, R. M. McMeeking, and Z. Z. Du, "Models of 2w. A. Thomas and J. M. Sanchez, "Influence of Inter facial Sliding Stress on High-Temperature, Environmentally Assisted Embrittlement in Ceramic Matrix Fatigue Behavior of Oxidized Nicalon/Calcium Aluminosilicate Composites, " J.Am Composites,J. Am. Ceram Soc., 79, 2345-52(1996). Ceram.Sor,79,2659-65(1996,APPENDIX By the use of a simple micromechanical model t can be computed from the hysteresis modulus, E#, or the stress–strain data from a monotonic tensile test of precycled specimens. For the latter, the following procedure is used. For a given strain increment from the unloaded state, Dε, the corresponding stress increment, Ds, is determined. The hysteresis modulus at that stress level is then E# 5 Ds/D«. This value is then used in the model. However, different equations are valid for different states of interfacial slip. If the slip length is smaller than s/2 the composite experiences partial slip. Else, the composite experiences full slip. The transition between partial and full slip occurs when the hysteresis modulus is31 E# ps2fs 5 Ec 1 1 1 2 vf vf Em Ec (A-1) where Em is the Young’s modulus of the matrix. If E# . E# ps2fs the composite is in partial slip, and t can be calculated from29–31 t 5 E# 1 2 E# Ec r s Ds Ef S Em Ec 1 2 vf vf D 2 (A-2) where Ef denotes the Young’s modulus of the fiber and r is the fiber radius. If full slip applies (E# , E# ps2fs), then t can be calculated from30,31 t 5 E# 2 vfEc vfE# r s Ds (A-3) When E# was determined from the stress–strain data of the residual strength test, the matrix crack spacing, s, recorded after cycling (i.e., before the tensile test) was used, since no additional matrix cracking is assumed to occur as long as the maximum applied stress is below the maximum stress level that was applied during the cycling.17 Acknowledgment The impetus for this study arose from discussions with Dr. Xin Wu. References 1 J. W. Holmes and B. F. Sørensen, “Fatigue Behavior of Continuous-Fiber￾Reinforced Ceramic Matrix Composites”; pp. 261–326 in Elevated Temperature Behavior of Ceramic Matrix Composites. Edited by S. V. Nair and K. Jakus. Butterworth Heineman, Newton, MA, 1995. 2 P. Karandikar and T.-W. Chou, “Damage Development and Moduli Reductions in Nicalon–Calcium Aluminosilicate Composites under Static Fatigue and Cyclic Fatigue,” J. Am. Ceram. Soc., 76, 1720–28 (1993). 3 D. S. Beyerle, S. M. Spearing, F. W. Zok, and A. G. Evans, “Damage and Failure in Unidirectional Ceramic-Matrix Composites,” J. Am. Ceram. Soc., 75, 2719–25 (1992). 4 B. F. Sørensen and R. Talreja, “Analysis of Damage in Ceramic Matrix Composites,” Int. J. Damage Mech., 2, 246–71 (1993). 5 B. Budiansky, J. W. Hutchinson, and A. G. Evans, “Matrix Fracture in Fiber￾Reinforced Ceramics,” J. Mech. Phys. Solids, 34, 167–78 (1986). 6 L. N. McCartney, “Mechanics of Matrix Cracking in Brittle-Matrix Fiber￾Reinforced Composites,” Proc. R. Soc. London, A409, 329–50 (1987). 7 Y. J. Weitsman and H. Zhu, “Multi-fracture of Ceramic Composites,” J. Mech. Phys. Solids, 41, 351–88 (1993). 8 W. A. Curtin, “Theory of Mechanical Properties of Ceramic Matrix Composites,” J. Am. Ceram. Soc., 74, 2837–45 (1991). 9 J. W. Holmes and X. Wu, “Elevated Temperature Creep Behavior of Continuous Fiber-Reinforced Ceramics”; pp. 193–260 in Elevated Temperature Behavior of Ceramic Matrix Composites. Edited by S. V. Nair and K. Jakus. Butterworth Heineman, Newton, MA, 1995. 10L. Filipuzzi, G. Camus, R. Naslain, and J. The´bault, “Oxidation Mechanisms and Kinetics of 1D-SiC/C/SiC Composite Materials: I, An Experimental Approach,” J. Am. Ceram. Soc., 70, 459–66 (1996). 11A. G. Evans, F. W. Zok, R. M. McMeeking, and Z. Z. Du, “Models of High-Temperature, Environmentally Assisted Embrittlement in Ceramic Matrix Composites,” J. Am. Ceram. Soc., 79, 2345–52 (1996). 12R. F. Cooper and K. Chyung, “Structure and Chemistry of Fiber-Matrix Interfaces in Silicon Carbide Fiber-Reinforced Glass-Ceramic Composites: An Electron Microscopy Study,” J. Mater. Sci., 22, 3148–69 (1987). 13L. A. Bonney and R. F. Cooper, “Reaction Layer Interfaces in SiC-Fiber￾Reinforced Glass-Ceramics: A High-Resolution Scanning Transmission Electron Microscopy Analysis,” J. Am. Ceram. Soc., 73, 2916–26 (1990). 14K. M. Prewo, “Fatigue and Stress Rupture of Silicon Carbide Fiber-Reinforced Glass-Ceramics,” J. Mater. Sci., 22, 2695–701 (1987). 15L. M. Butkus, L. P. Zawada, and G. A. Hartman, “Room Temperature Tensile and Fatigue Properties of Silicon Carbide Fiber-Reinforced Ceramic Matrix Com￾posites”; in Aeromat’90, Advanced Aerospace Materials/Processes Conference (Long Beach, CA, May 21–24, 1990), 1990. 16J. W. Holmes and S. F. Shuler, “Temperature Rise During Fatigue of Fiber￾Reinforced Ceramics,” J. Mater. Sci. Lett., 9, 1290–91 (1990). 17J. W. Holmes and C. Cho, “Experimental Observations of Frictional Heating in Fiber-Reinforced Ceramics,” J. Am. Ceram. Soc., 75, 929–38 (1992). 18L. P. Zawada, L. M. Butkus, and G. A. Hartman, “Room Temperature Tensile and Fatigue Properties of Silicon Carbide Fiber-Reinforced Aluminosilicate Glass,” Ceram. Eng. Sci. Proc., 11, 1592–606 (1990). 19D. Rouby and P. Reynaud, “Fatigue Behavior Related to the Interface Modifi￾cation during Load Cycling in Ceramic-Matrix Composites,” Compos. Sci. Technol., 48, 109–18 (1993). 20J. W. Holmes, X. Wu, and B. F. Sørensen, “Frequency Dependency of Fatigue Life and Internal Heating of a Fiber-Reinforced Ceramic Matrix Composite,” J. Am. Ceram. Soc., 77, 3284–86 (1994). 21A. G. Evans, F. W. Zok, and R. M. McMeeking, “Fatigue of Ceramic Matrix Composites,” Acta Metall. Mater., 43, 859–75 (1995). 22C. Cho, J. W. Holmes, and J. R. Barber, “Estimation of Interfacial Shear in Ceramic Composites from Frictional Heating Measurements,” J. Am. Ceram. Soc., 74, 2808–18 (1991). 23E. Y. Luh and A. G. Evans, “High-Temperature Mechanical Properties of a Ceramic Matrix Composite,” J. Am. Ceram. Soc., 70, 466–69 (1987). 24K. M. Prewo, B. Johnson, and S. Starrett, “Silicon Carbide Fibre-Reinforced Glass-Ceramic Composite Tensile Behaviour at Elevated Temperature,” J. Mater. Sci., 24, 1373–79 (1989). 25R. T. Bhatt, “Oxidation Effects on the Mechanical Properties of SiC-Fiber￾Reinforced Reaction-Bonded Si3N4-Matrix Composites,” J. Am. Ceram. Soc., 75, 406–12 (1992). 26J. F. Mandell, D. H. Grande, and J. Jacobs, “Tensile Behaviour of Glass/Ceramic Composite Materials at Elevated Temperatures,” J. Eng. Gas Turbines Power, 109, 267–73 (1987). 27J. W. Holmes, “Influence of Stress Ratio on the Elevated-Temperature Fatigue of Silicon Carbide Fiber-Reinforced Silicon Nitride Composite,” J. Am. Ceram. Soc., 74, 1639–45 (1991). 28J. W. Holmes, “A Technique for Tensile Fatigue and Creep Testing of Fiber-Reinforced Ceramics,” J. Compos. Mater., 26, 915–32 (1992). 29A. W. Pryce and P. A. Smith, “Matrix Cracking in Unidirectional Ceramic Matrix Composites under Quasi-Static and Cyclic Loading,” Acta Metall. Mater., 41, 1269–81 (1993). 30W. P. Keith and K. T. Kedward, “The Stress–Strain Behavior of a Porous Unidirectional Ceramic Matrix Composite,” Composites, 26, 163–74 (1995). 31B. F. Sørensen and J. W. Holmes, “Model of the Mechanical Behavior of Continuous Fiber Reinforced Ceramic Matrix Composites During Cyclic Loading,” J. Eur. Ceram. Soc., submitted. 32T. Mackin and F. W. Zok, “Fiber Bundle Pushout: A Technique for the Measurement of Interfacial Sliding Properties,” J. Am. Ceram. Soc., 75, 3169–71 (1992). 33E. Lara-Curzio and M. K. Feber, “Methodology for the Determination of the Interfacial Properties of Brittle Matrix Composites,” J. Mater. Sci., 29, 6152–58 (1994). 34B. F. Sørensen and J. W. Holmes, “Effect of Loading Rate on the Monotonic Tensile Behavior and Matrix Cracking of a Fiber-Reinforced Ceramic,” J. Am. Ceram. Soc., 79, 313–20 (1996). 35R. W. Goettler and K. T. Faber, “Interfacial Shear Stress in Fiber-Reinforced Glasses,” Compos. Sci. Technol., 37, 129–47 (1989). 36B. F. Sørensen and J. W. Holmes, “Fatigue of Continuous Fiber-Reinforced Ceramic Matrix Composites: Review of Mechanisms and Models”; pp. 487–500 in Proceedings of the International Symposium on Fatigue under Thermal and Mechan￾ical Loading. Edited by J. Bressers and L. Remy. Kluwer Academic Publishers, Dordrecht, Netherlands, 1996. 37S. M. Spearing, F. M. Zok, and A. G. Evans, “Stress Corrosion Cracking in a Unidirectional Ceramic-Matrix Composite,” J. Am. Ceram. Soc., 77, 562–70 (1994). 38B. F. Sørensen and J. W. Holmes, “Influence of Stress Ratio on the Fatigue Life of a Continuous Fiber-Reinforced Ceramic Matrix Composite,” unpublished work. 39J. W. Holmes, unpublished research. 40J. I. Eldridge, “Environmental Effects on Fiber Debonding and Sliding in an SCS-6 SiC Fiber Reinforced Reaction-Bonded Si3Ni4 Composite,” Scr. Metall. Mater., 32, 1085–89 (1995). 41J. I. Eldridge, R. T. Bhatt, and N. P. Bansal, “Investigation of Fiber/Matrix Interfacial Mechanical Behavior in Ceramic Matrix Composites by Cyclic Fiber Push-in Testing,” Ceram. Eng. Sci. Proc., 17, 266–78 (1996). 42W. A. Thomas and J. M. Sanchez, “Influence of Interfacial Sliding Stress on Fatigue Behavior of Oxidized Nicalon/Calcium Aluminosilicate Composites,” J. Am. Ceram. Soc., 79, 2659–65 (1996). M June 2000 Rate of Strength Decrease of Fiber-Reinforced Ceramic-Matrix Composites during Fatigue 1475