High Temperature Mechanical Behavior of Ceramic-Matrix Composites [1 ed.] 3527349030, 9783527349036

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High Temperature Mechanical Behavior of Ceramic-Matrix Composites

High Temperature Mechanical Behavior of Ceramic-Matrix Composites Longbiao Li

Nanjing Univ. of Aeronautics & Astronaut College of Civil Aviation No. 29 Yudao St. 210016 Nanjing China

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Contents Preface xiii Acknowledgments xv 1 1.1 1.2 1.3 1.4 1.5

Introduction 1 Tensile Behavior of CMCs at Elevated Temperature 2 Fatigue Behavior of CMCs at Elevated Temperature 6 Stress Rupture Behavior of CMCs at Elevated Temperature 7 Vibration Behavior of CMCs at Elevated Temperature 9 Conclusion 10 References 10

2

First Matrix Cracking of Ceramic-Matrix Composites at Elevated Temperature 19 Introduction 19 Temperature-Dependent Matrix Cracking Stress of C/SiC Composites 20 Theoretical Models 20 Results and Discussion 21 Temperature-Dependent Matrix Cracking Stress of C/SiC Composite for Different Fiber Volumes 23 Temperature-Dependent Matrix Cracking Stress of C/SiC Composite for Different Interface Shear Stress 24 Temperature-Dependent Matrix Cracking Stress of C/SiC Composite for Different Fiber/Matrix Interface Frictional Coefficients 25 Temperature-Dependent Matrix Cracking Stress of C/SiC Composite for Different Interface Debonding Energies 26 Effect of Matrix Fracture Energy on Temperature-Dependent Matrix Cracking Stress of C/SiC Composite 27 Experimental Comparisons 28 Temperature-Dependent Matrix Cracking Stress of SiC/SiC Composite 29

2.1 2.2 2.2.1 2.2.2 2.2.2.1 2.2.2.2 2.2.2.3 2.2.2.4 2.2.2.5 2.2.3 2.3

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2.3.1 2.3.1.1 2.3.1.2 2.3.1.3 2.3.1.4 2.3.1.5 2.3.2 2.4 2.4.1 2.4.2 2.4.2.1 2.4.2.2 2.4.2.3 2.4.2.4 2.4.2.5 2.4.3 2.5 2.5.1 2.5.1.1 2.5.1.2 2.5.1.3 2.5.1.4 2.5.2 2.6

3 3.1 3.2

Results and Discussion 30 Temperature-Dependent Matrix Cracking Stress of SiC/SiC Composite for Different Fiber Volumes 30 Temperature-Dependent Matrix Cracking Stress of SiC/SiC Composite for Different Interface Shear Stress 30 Temperature-Dependent Matrix Cracking Stress of SiC/SiC Composite for Different Interface Frictional Coefficients 33 Temperature-Dependent Matrix Cracking Stress of SiC/SiC Composite for Different Interface Debonding Energies 34 Temperature-Dependent Matrix Cracking Stress of SiC/SiC Composite for Different Matrix Fracture Energies 34 Experimental Comparisons 36 Time-Dependent Matrix Cracking Stress of C/SiC Composites 39 Theoretical Models 39 Results and Discussion 41 Time-Dependent Matrix Cracking Stress of C/SiC Composite for Different Fiber Volumes 42 Time-Dependent Matrix Cracking Stress of C/SiC Composite for Different Interface Shear Stress 42 Time-Dependent Matrix Cracking Stress of C/SiC Composite for Different Interface Frictional Coefficients 50 Time-Dependent Matrix Cracking Stress of C/SiC Composite for Different Interface Debonding Energies 53 Time-Dependent Matrix Cracking Stress of C/SiC Composite for Different Matrix Fracture Energies 56 Experimental Comparisons 59 Time-Dependent Matrix Cracking Stress of Si/SiC Composites 59 Results and Discussion 59 Time-Dependent Matrix Cracking Stress of SiC/SiC Composite for Different Fiber Volumes 60 Time-Dependent Matrix Cracking Stress of SiC/SiC Composite for Different Interface Shear Stress 62 Time-Dependent Matrix Cracking Stress of SiC/SiC Composite for Different Interface Debonding Energies 66 Time-Dependent Matrix Cracking Stress of SiC/SiC Composite for Different Matrix Fracture Energies 68 Experimental Comparisons 68 Conclusion 71 References 71 Matrix Multiple Cracking Evolution of Fiber-Reinforced Ceramic-Matrix Composites at Elevated Temperature 75 Introduction 75 Temperature-Dependent Matrix Multiple Cracking Evolution of C/SiC Composites 76

Contents

3.2.1 3.2.1.1 3.2.1.2 3.2.1.3 3.2.2 3.2.2.1 3.2.2.2 3.2.2.3 3.2.3 3.3 3.3.1 3.3.1.1 3.3.1.2 3.3.1.3 3.3.1.4 3.3.1.5 3.3.2 3.4 3.4.1 3.4.1.1 3.4.1.2 3.4.1.3 3.4.2 3.4.2.1 3.4.2.2 3.4.2.3 3.4.2.4 3.4.3 3.5 3.5.1

Theoretical Models 77 Temperature-Dependent Stress Analysis 77 Temperature-Dependent Interface Debonding 78 Temperature-Dependent Matrix Multiple Cracking 79 Results and Discussion 80 Temperature-Dependent Matrix Multiple Cracking of C/SiC Composite for Different Interface Shear Stress 82 Temperature-Dependent Matrix Multiple Cracking of C/SiC Composite for Different Interface Debonding Energies 84 Temperature-Dependent Matrix Multiple Cracking of C/SiC Composite for Different Matrix Fracture Energies 85 Experimental Comparisons 88 Temperature-Dependent Matrix Multiple Cracking Evolution of SiC/SiC Composites 89 Results and Discussion 90 Temperature-Dependent Matrix Multiple Cracking of SiC/SiC Composite for Different Fiber Volumes 90 Temperature-Dependent Matrix Multiple Cracking of SiC/SiC Composite for Different Interface Shear Stress 92 Temperature-Dependent Matrix Multiple Cracking of SiC/SiC Composite for Different Interface Frictional Coefficients 93 Temperature-Dependent Matrix Multiple Cracking of SiC/SiC Composite for Different Interface Debonding Energies 95 Temperature-Dependent Matrix Multiple Cracking of SiC/SiC Composite for Different Matrix Fracture Energies 98 Experimental Comparisons 100 Time-Dependent Matrix Multiple Cracking Evolution of C/SiC Composites 101 Theoretical Models 102 Time-Dependent Stress Analysis 102 Time-Dependent Interface Debonding 103 Time-Dependent Matrix Multiple Cracking 105 Results and Discussion 106 Time-Dependent Matrix Multiple Cracking of C/SiC Composite for Different Interface Shear Stress 106 Time-Dependent Matrix Multiple Cracking of C/SiC Composite for Different Interface Frictional Coefficients 108 Time-Dependent Matrix Multiple Cracking of C/SiC Composite for Different Interface Debonding Energies 111 Time-Dependent Matrix Multiple Cracking of C/SiC Composite for Different Matrix Fracture Energies 113 Experimental Comparisons 114 Time-Dependent Matrix Multiple Cracking Evolution of SiC/SiC Composites 116 Results and Discussion 117

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Contents

3.5.1.1 3.5.1.2 3.5.1.3 3.5.1.4 3.5.1.5 3.5.2 3.5.2.1 3.5.2.2 3.6

4 4.1 4.2 4.3 4.3.1 4.3.2 4.3.3 4.3.4 4.3.5 4.3.6 4.3.7 4.3.8 4.3.9 4.3.10 4.4 4.4.1 4.4.2

Time-Dependent Matrix Multiple Cracking of SiC/SiC Composite for Different Fiber Volumes 117 Time-Dependent Matrix Multiple Cracking of SiC/SiC Composite for Different Interface Shear Stress 120 Time-Dependent Matrix Multiple Cracking of SiC/SiC Composite for Different Interface Frictional Coefficients 127 Time-Dependent Matrix Multiple Cracking of SiC/SiC Composite for Different Interface Debonding Energies 130 Time-Dependent Matrix Cracking Stress of SiC/SiC Composite for Different Matrix Fracture Energies 133 Experimental Comparisons 136 Unidirectional SiC/SiC Composite 136 SiC/SiC Minicomposite 139 Conclusion 139 References 140

Time-Dependent Tensile Behavior of Ceramic-Matrix Composites 145 Introduction 145 Theoretical Analysis 148 Results and Discussion 149 Time-Dependent Tensile Behavior of SiC/SiC Composite for Different Fiber Volumes 149 Time-Dependent Tensile Behavior of SiC/SiC Composite for Different Fiber Radii 149 Time-Dependent Tensile Behavior of SiC/SiC Composite for Different Matrix Weibull Moduli 152 Time-Dependent Tensile Behavior of SiC/SiC Composite for Different Matrix Cracking Characteristic Strengths 152 Time-Dependent Tensile Behavior of SiC/SiC Composite for Different Matrix Cracking Saturation Spacings 155 Time-Dependent Tensile Behavior of SiC/SiC Composite for Different Interface Shear Stress 155 Time-Dependent Tensile Behavior of SiC/SiC Composite for Different Interface Debonding Energies 155 Time-Dependent Tensile Behavior of SiC/SiC Composite for Different Fiber Strengths 159 Time-Dependent Tensile Behavior of SiC/SiC Composite for Different Fiber Weibull Moduli 160 Time-Dependent Tensile Behavior of SiC/SiC Composite for Different Oxidation Durations 160 Experimental Comparisons 161 Time-Dependent Tensile Behavior of SiC/SiC Composite 161 Time-Dependent Tensile Behavior of C/SiC Composite 173

Contents

4.5

Conclusion 179 References 181

5

Fatigue Behavior of Ceramic-Matrix Composites at Elevated Temperature 187 Introduction 187 Theoretical Analysis 189 Experimental Comparisons 191 2.5D Woven Hi-NicalonTM SiC/[Si-B-C] at 600 ∘ C in Air Atmosphere 191 2.5D Woven Hi-NicalonTM SiC/[Si-B-C] at 1200 ∘ C in Air Atmosphere 193 2D Woven Self-Healing Hi-NicalonTM SiC/[SiC-B C] at 1200 ∘ C in Air and

5.1 5.2 5.3 5.3.1 5.3.2 5.3.3 5.3.4 5.4

6 6.1 6.2 6.2.1 6.2.2 6.2.2.1 6.2.2.2 6.2.2.3 6.2.2.4 6.2.2.5 6.2.2.6 6.2.3 6.3 6.3.1 6.3.1.1 6.3.1.2

4

in Steam Atmospheres 199 Discussion 203 Conclusion 206 References 206 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature 211 Introduction 211 Stress Rupture of Ceramic-Matrix Composites Under Constant Stress at Intermediate Temperature 213 Theoretical Models 214 Results and Discussion 215 Stress Rupture of SiC/SiC Composite for Different Fiber Volumes 215 Stress Rupture of SiC/SiC Composite for Different Peak Stress Levels 218 Stress Rupture of SiC/SiC Composite for Different Saturation Spaces Between Matrix Cracking 221 Stress Rupture of SiC/SiC Composite for Different Interface Shear Stress 221 Stress Rupture of SiC/SiC Composite for Different Fiber Weibull Modulus 227 Stress Rupture of SiC/SiC Composite for Different Environmental Temperatures 229 Experimental Comparisons 230 Stress Rupture of Ceramic-Matrix Composites Under Stochastic Loading Stress and Time at Intermediate Temperature 234 Results and Discussion 236 Stress Rupture of SiC/SiC Composite Under Stochastic Loading for Different Stochastic Stress Levels 236 Stress Rupture of SiC/SiC Composite Under Stochastic Loading for Different Stochastic Loading Time Intervals 240

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6.3.1.3 6.3.1.4 6.3.1.5 6.3.1.6 6.3.2 6.3.2.1 6.3.2.2 6.3.2.3 6.3.2.4 6.4 6.4.1 6.4.1.1 6.4.1.2 6.4.1.3 6.4.1.4 6.4.2 6.5

7 7.1 7.2 7.2.1 7.2.2 7.2.2.1 7.2.2.2 7.2.2.3 7.2.2.4 7.2.2.5 7.2.3

Stress Rupture of SiC/SiC Composite Under Stochastic Loading for Different Fiber Volumes 247 Stress Rupture of SiC/SiC Composite Under Stochastic Loading for Different Matrix Crack Spacings 251 Stress Rupture of SiC/SiC Composite Under Stochastic Loading for Different Interface Shear Stress 253 Stress Rupture of SiC/SiC Composite Under Stochastic Loading for Different Environmental Temperatures 261 Experimental Comparisons 264 𝜎 = 80 MPa and 𝜎 s = 90 MPa with Δt = 7.2, 10.8, and 14.4 ks 267 𝜎 = 100 MPa and 𝜎 s = 110 MPa with Δt = 7.2 ks 267 𝜎 = 120 MPa and 𝜎 s = 130 and 140 MPa with Δt = 7.2 ks 271 Discussion 271 Stress Rupture of Ceramic-Matrix Composites Under Multiple Load Sequence at Intermediate Temperature 274 Results and Discussion 274 Stress Rupture of SiC/SiC Composite Under Multiple Loading Sequence for Different Fiber Volumes 275 Stress Rupture of SiC/SiC Composite Under Multiple Loading Sequence for Different Matrix Crack Spacings 280 Stress Rupture of SiC/SiC Composite Under Multiple Loading Sequence for Different Interface Shear Stress 285 Stress Rupture of SiC/SiC Composite Under Multiple Loading Sequence for Different Environment Temperatures 292 Experimental Comparisons 295 Conclusion 302 References 302 Vibration Damping of Ceramic-Matrix Composites at Elevated Temperature 307 Introduction 307 Temperature-Dependent Vibration Damping of CMCs 308 Theoretical Models 308 Results and Discussion 310 Effect of Fiber Volume on Temperature-Dependent Vibration Damping of SiC/SiC Composite 310 Effect of Matrix Crack Spacing on Temperature-Dependent Vibration Damping of SiC/SiC Composite 314 Effect of Interface Debonding Energy on Temperature-Dependent Vibration Damping of SiC/SiC Composite 317 Effect of Steady-State Interface Shear Stress on Temperature-Dependent Vibration Damping of SiC/SiC Composite 321 Effect of Interface Frictional Coefficient on Temperature-Dependent Vibration Damping of SiC/SiC Composite 325 Experimental Comparisons 329

Contents

7.3 7.3.1 7.3.2 7.3.2.1 7.3.2.2 7.3.2.3 7.3.2.4 7.3.2.5 7.3.3 7.3.3.1 7.3.3.2 7.3.3.3 7.3.3.4 7.4

Time-Dependent Vibration Damping of CMCs 329 Theoretical Models 329 Results and Discussion 331 Effect of Fiber Volume on Time-Dependent Vibration Damping of C/SiC Composite 331 Effect of Vibration Stress on Time-Dependent Vibration Damping of C/SiC Composite 334 Effect of Matrix Crack Spacing on Time-Dependent Vibration Damping of C/SiC Composite 337 Effect of Interface Shear Stress on Time-Dependent Vibration Damping of C/SiC Composite 340 Effect of Temperature on Time-Dependent Vibration Damping of C/SiC Composite 343 Experimental Comparisons 343 t = 2 hours at T = 700, 1000, and 1300 ∘ C 346 t = 5 hours at T = 700, 1000, and 1300 ∘ C 346 t = 10 hours at T = 700, 1000, and 1300 ∘ C 351 Discussion 354 Conclusion 356 References 356 Index 359

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Preface Monolithic ceramic is a kind of brittle material. The properties of the material will be greatly reduced by microdefects, which limit the practical application of ceramics in many fields. However, the inherent brittleness of ceramic materials can be improved by a continuous or discontinuous ceramic fiber or carbon fiber reinforcement, namely, ceramic-matrix composites (CMCs). This dispersed second phase can improve the fracture toughness of ceramic materials. The main mechanism is that the crack bridging effect in the process of fracture can make the matrix materials connect with each other, disperse the fracture energy by the way of fiber debonding, and fiber pulling out to prevent the fracture of the material. Compared with the monolithic ceramic, the mechanical behavior of CMCs has many different characteristics. Understanding the failure mechanisms and internal damage evolution represents an important step to ensure reliability and safety of CMCs. This book focuses on the high-temperature mechanical behavior of CMCs as follows: (1) Temperature- and time-dependent first matrix cracking stress of fiber-reinforced CMCs is investigated using the energy balance approach. The temperaturedependent micromechanical parameters are incorporated into the analysis of the microstress analysis, interface debonding criterion, and energy balance approach. Relationships between the first matrix cracking stress, interface debonding, temperature, and time are established. Effects of fiber volume, interface shear stress, interface frictional coefficient, interface debonding energy, matrix fracture energy, temperature, and time on the first matrix cracking stress and interface debonding length are discussed. Experimental first matrix cracking stress and interface debonding of C/SiC and SiC/SiC composites at elevated temperature are predicted. (2) Temperature- and time-dependent matrix cracking evolution of fiber-reinforced CMCs is investigated using the critical matrix strain energy (CMSE) criterion. Temperature-dependent interface shear stress, Young’s modulus of the fibers and the matrix, matrix fracture energy, and the interface debonding energy are considered in the microstress field analysis, interface debonding criterion, and matrix multiple cracking evolution model. Effects of fiber volume, interface shear stress, interface debonding energy, matrix fracture energy, temperature, and time on matrix multiple cracking evolution and interface debonding are discussed. Experimental matrix multiple cracking evolution and fiber/matrix interface debonding of C/SiC and SiC/SiC composite at elevated temperatures are predicted. (3) Time-dependent tensile damage and fracture of fiber-reinforced CMCs are investigated considering the interface and fiber oxidation. Time-dependent damage mechanisms of matrix cracking, interface debonding, fiber failure, and interface and fiber oxidation are considered in the analysis of the tensile

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stress–strain curve. Experimental time-dependent tensile stress–strain curves, matrix cracking, interface debonding, and fibers failure of different SiC/SiC and C/SiC composites are predicted for different oxidation durations. (4) Cyclic-dependent damage development in self-healing 2.5D woven Hi-NicalonTM SiC/[Si-B-C] and 2D woven Hi-Nicalon SiC/[SiC-B4 C] composites at T = 600 and 1200 ∘ C is investigated. Cyclic-dependent damage parameters of internal friction, dissipated energy, Kachanov’s damage parameter, and broken fiber fraction are obtained to analyze damage development in self-healing CMCs. Relationships between cyclic-dependent damage parameters and multiple fatigue damage mechanisms are established. Experimental fatigue damage evolution of self-healing Hi-Nicalon SiC/[Si-B-C] and Hi-Nicalon SiC/[SiC-B4 C] composites are predicted. Effects of fatigue peak stress, testing environment, and loading frequencies on the evolution of internal damage and final fracture are analyzed. (5) Time-dependent deformation, damage, and fracture of fiber-reinforced CMCs that were exposed to stress rupture loading at intermediate environmental temperatures are investigated. The composite microstress field and tensile constitutive relationship of the damaged CMCs were examined to characterize their time-dependent damage mechanisms. Relationships between stress rupture lifetime, peak stress level, time-dependent composite deformation, and evolution of internal damages are established. Effects of composite material properties, composite damage state, and environmental temperature on stress rupture lifetime, time-dependent composite deformation, and evolution of the internal damages of SiC/SiC are analyzed. Experimental stress rupture lifetime, time-dependent composite deformation, and composite internal damage evolution of SiC/SiC composite subjected to the stress rupture loading are evaluated. (6) A micromechanical temperature-dependent vibration damping model of fiber-reinforced CMCs is developed. Temperature-dependent damage mechanisms of matrix cracking, interface debonding and slip, and fiber fracture contribute to the vibration damping of damaged CMCs. Temperature-dependent fiber and matrix strain energy and dissipated energy density are formulated of composite constituent properties and damage-related microparameters of matrix crack spacing, interface debonding and slip length, and broken fibers fraction. Relationships between temperature-dependent composite damping, temperature-dependent damage mechanisms, temperature, and oxidation duration are established. Effects of composite constituent properties and composite damage state on temperature-dependent composite vibration damping of SiC/SiC and C/SiC composites are analyzed. Experimental temperature-dependent composite vibration damping of 2D SiC/SiC and C/SiC composites are predicted. I hope this book can help the material scientists and engineering designers to understand and master the high-temperature mechanical behavior of CMCs. 30 November 2020 Nanjing, PR China

Longbiao Li

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Acknowledgments I am grateful to my wife Li Peng and my son Sheng-Ning Li for their encouragement. A special thanks to Dr. Shaoyu Qian and Katherine Wong for their help with my original manuscript. I am also grateful to the team at Wiley for their professional assistance.

1

1 Introduction Monolithic ceramic is a kind of brittle material. The properties of the material will be greatly reduced by microdefects, which limit the practical application of ceramics in many fields. However, the inherent brittleness of ceramic materials can be improved by continuous or discontinuous ceramic fiber or carbon fiber reinforcement, namely, ceramic-matrix composites (CMCs). This dispersed second phase can improve the fracture toughness of ceramic materials. The main mechanism is that the crack bridging effect in the process of fracture can make the matrix materials connect with each other, disperse the fracture energy by the way of fiber debonding, and fiber pulling out to prevent the fracture of the material [1, 2]. Compared with the superalloy, fiber-reinforced CMCs can withstand higher temperature, reduce cooling air flow, and improve the turbine efficiency. The density of fiber-reinforced CMCs is 2.0–2.5 g/cm3 , which is only 1/4–1/3 of superalloy. CMCs have already been applied to aeroengine combustion chambers, nozzle flaps, turbine vanes, and blades. For example, the CMC nozzle flaps and seals manufactured by SNECMA have already been used for more than 10 years in the M88 and M53 aeroengines. The CMC tail nozzle designed by SAFRAN Group of France passed the commercial flight certification of European Union Aviation Safety Agency (EASA) and completed its first commercial flight on the CFM56-5B aeroengine on 16 June 2015. National Aeronautics and Space Administration (NASA) has prepared and tested the CMC turbine guide vanes and turbine blade disc components in the Ultra-Efficient Engine Technology (UEET) project. General Electric (GE) tested the CMC combustor and high-pressure turbine components in the ground test of GEnx aeroengine. The CMC turbine blades were successfully tested on the F414 engine, which are planned to be used in GE90 series aeroengines. The engine weight is expected to be reduced by 455 kg, accounting for about 6% of the total weight of GE90-115 aeroengine. The LEAP (Leading Edge Aviation Propulsion) series aeroengine developed by CFM company adopts CMC components. The LEAP-1A, 1B, and 1C aeroengine provides power for Airbus A320, Boeing 737MAX, and COMAC C919, respectively. Compared with polymer matrix composites (PMCs), CMCs have some similarities, including anisotropy, braided structure, high strength/high modulus fibers, manufacturing process sensitivity, and diversity, but there are also differences, High Temperature Mechanical Behavior of Ceramic-Matrix Composites, First Edition. Longbiao Li. © 2021 WILEY-VCH GmbH. Published 2021 by WILEY-VCH GmbH.

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1 Introduction

such as high operation temperature (>500 ∘ C), diversity of material constituents (i.e. oxide matrix, nonoxidized matrix, carbide matrix, silicon nitride matrix, carbon matrix, etc.) and processing methods (i.e. reaction bonding [RB], hot pressing sintering [HPS], precursor infiltration and pyrolysis [PIP], reactive melt infiltration [RMI], chemical vapor infiltration [CVI], slurry infiltration and hot pressing [SIPH], CVI-PIP, CVI-RMI, and PIP-HP), low matrix failure strain, complex degradation/damage/failure mechanisms at elevated temperature, difficult connection of structures in high-temperature environment, and high requirement of nondestructive testing and repair technology. To ensure the reliability and safety of their use in aircraft and aeroengine structures, it is necessary to investigate the tensile, fatigue, stress rupture, and vibration behavior of CMCs at elevated temperature.

1.1 Tensile Behavior of CMCs at Elevated Temperature The stress–strain curve of CMCs under tensile load appears obviously nonlinear. The tensile stress–strain curve can be divided into three regions, i.e. elastic region, nonlinear region, and secondary linear region before failure. In region I, there is no damage in the material during initial loading, and the tensile stress–strain curve is linear. With the increase of stress, microcracks appear in the matrix-rich area or matrix defects. The initial matrix cracking stress is defined as 𝜎 mc . These microcracks can only be detected by means of acoustic emission (AE), microscopic observation of specimen surface, and temperature measurement of sample surface. When the stress reaches the proportional limit stress, the accumulation of matrix cracks makes the stress–strain curve deflect, and the stress–strain curve is nonlinear, which marks the beginning of region II. In region II, the matrix cracks propagate along the vertical load direction while the number of matrix cracks increases. When the cracks extend to the fiber/matrix interface, the cracks deflect along the fiber/matrix interface, and debonding occurs at the fiber/matrix interface. With the increase of stress, when the slip zones of adjacent matrix cracks overlap each other, the matrix cracks reach saturation. The saturated stress of matrix cracks is defined as 𝜎 sat . When the matrix crack is saturated, the region III of the stress–strain curve starts, and the external load is mainly borne by the fiber. The tangent modulus of the stress–strain curve is about V f Ef (V f is the volume content of the fiber and Ef is the elastic modulus of the fiber). With the increase of the stress, some fibers fail, and the failed fibers continue to carry through the shear stress at the fiber/matrix interface. When the fibers broken fraction reaches the critical value, the composite fracture occurs. The tensile stress–strain behavior reflects the strength of the composite material to resist the damage of external tensile loading. The tensile properties (i.e. proportional limit stress, matrix crack spacing, tensile strength, and fracture strain) can be obtained from the tensile stress–strain curves and can be used for component design [3–5]. Jia et al. [6] investigated the relationship between the

1.1 Tensile Behavior of CMCs at Elevated Temperature

interphase and tensile strength of SiC fiber monofilament. The tensile strength of the SiC fiber monofilament decreases with the increasing coating layers. The SiC fibers with single boron nitride (BN) coating have the high monofilament strength retention of about 70%, 42.1% with two BN coatings, and 32.3% with four BN coatings. The minicomposite comprises one single fiber tow, interphase, and matrix and can be used to optimize the fiber–matrix interfacial zone and to generate micromechanical data necessary for modeling the mechanical behavior [7]. Almansour [8], Sauder et al. [9], and Yang et al. [10] performed investigations on the tensile behavior of SiC/SiC minicomposites with different fiber types and interface properties. Shi et al. [11] performed an investigation on the variability in tensile behavior of SiC/SiC minicomposite. The tensile strength of the SiC/SiC minicomposite satisfied the Weibull distribution. He et al. [12] performed an investigation on the tensile behavior of SiC/SiC minicomposites with different interphase thickness. The tensile strength and fiber pullout length increase with the interphase thickness. Chateau et al. [13] investigated the damage evolution and final fracture in SiC/SiC minicomposite using the in situ X-ray microtomography under tensile loading. Zeng et al. [14] performed experimental and theoretical investigations on the tensile damage evolution of unidirectional C/SiC composite at room temperature. Ma et al. [15], Wang et al. [16], Liang and Jiao [17], and Hu et al. [18] performed investigations on the tensile damage and fracture of 2.5D and 3D CMCs. The nonlinearity appears in the tensile curves along both the warp and weft directions. Under tensile loading, the matrix cracking first occurs because of the local stress concentration of the pores inside of the composite, and the transverse cracks and longitudinal interlaminar cracks result in the final brittle fracture of the composite. The acoustic emission technique is used to monitor the damage evolution of a 3D needled C/SiC composite [19]. The damage signal contained three main frequencies of 240, 370, and 455 kHz corresponding to the damage mechanisms of the interface damage, matrix damage, and fiber fracture, respectively. Wang et al. [20] compared the tensile behavior of C/SiC composites with different fiber preforms. The minicomposite has the largest strength, modulus, and strain energy density to failure in contrast to the lowest values of the 2D composite and the intermediate properties of the 3D composite. The tensile behavior of CMCs is affected by temperature [21–23]. For the unidirectional C/SiC composite at 1300 ∘ C, the composite tensile strength was 𝜎 UTS = 374 MPa and the composite tensile modulus was Ec = 134 GPa, and at 1450 ∘ C, the composite tensile strength was 𝜎 UTS = 338 MPa and the composite tensile modulus was Ec = 116 GPa. For the 2D SiC/SiC composite, the fracture strain at 1200 ∘ C is higher than that at room temperature because of the interface oxidation. For the 3D C/SiC composite, when the temperature increases from room temperature to 1500 ∘ C, the composite elastic modulus and the strain for saturation matrix cracking remained unchanged; the first matrix cracking stress, matrix cracking saturation stress, and fracture stress all increased first with temperature to the peak value at the temperature range of 1100–1300 ∘ C and then decreased with temperature. Luo and Qiao [24] investigated the effect of loading rate on tensile behavior of 3D C/SiC composite at room temperature, 1100, and 1500 ∘ C. At room temperature, the fracture stress

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1 Introduction

increased with loading rate; at 1500 ∘ C, the fracture stress decreased with loading rate; and at 1100 ∘ C, the fracture stress remained the same without changing with loading rate. At elevated oxidizing temperature, the applied stress opens the existing cracks and allows for easier ingress of oxygen to the fibers [25, 26]. Under thermal and mechanical load cycling in oxidative environment, the strain is damage dependent and a combination of physical mechanism in the form of matrix microcracking and fiber debonding and chemical mechanism of fiber oxidation. Li et al. [27, 28] and Li [29] developed a micromechanical approach to predict the tensile behavior of CMCs with different fiber preforms considering multiple damage mechanisms. Li [30] predicted the time-dependent proportional limit stress of C/SiC composites with different fiber volumes, interface properties, and matrix damage. Li [31] analyzed matrix multi-cracking of fiber-reinforced CMCs considering the interface oxidation and compared the matrix cracking evolution of C/SiC composite with/without the interface oxidation. Martinez-Fernandez and Morscher [32] investigated the tensile properties of single tow Hi-NicalonTM SiC/PyC/SiC composite at room temperature, 700, 950, and 1200 ∘ C. The elevated temperature stress rupture behavior was dependent on the precrack stress, and the stress rupture life increases with the decreasing precrack stress. Forio et al. [33] investigated the lifetime of SiC multifilament tows under static fatigue in air at a temperature range of 600–700 ∘ C. A slow-crack-growth mechanism is considered in the analysis of delayed failure of SiC/SiC minicomposite under low stress state. Morscher and Cawley [34] investigated the time-dependent strength degradation of woven SiC/BN/SiC composite at intermediate temperatures. The strength degradation is dependent on the kinetics for fusion of fibers to one another, the number of matrix cracks, and the applied stress state. Larochelle and Morscher [35] investigated the tensile stress rupture behavior of woven Sylramic–iBN/BN/SiC composite at 550 and 750 ∘ C in a humid environment. The stress rupture strengths decreased with respect to time with the rate of decrease related to the temperature and the amount of moisture content. Pailler and Lamon [36] developed a micromechanics-based model of fatigue/oxidation for CMCs considering thermally induced residual stresses and kinetics of interphase degradation or crack healing. Santhosh et al. [37, 38] investigated the time-dependent deformation and damage of 2D SiC/SiC composite under multiaxial stress and dwell fatigue at 1204 ∘ C. Morscher et al. [39] investigated the damage evolution and failure mechanisms of 2D Sylramic–iBN SiC/SiC composite under tensile creep and fatigue loading at 1204 ∘ C in air condition. The damage development was the growth of matrix cracks and increasing number of matrix cracks with stress and time. Four dominant failure criterions are present in the literature for modeling matrix crack evolution of CMCs: maximum stress theories, energy balance approach, critical matrix strain energy (CMSE) criterion, and statistical failure approach. The maximum stress criterion assumes that a new matrix crack will form whenever the matrix stress exceeds the ultimate strength of the matrix, which is assumed to be single valued and a known material property [40]. The energy balance failure criteria involve calculation of the energy balance relationship before and after the formation of a single dominant crack as originally proposed by Aveston et al. [41]. The progression

1.1 Tensile Behavior of CMCs at Elevated Temperature

of matrix cracking as determined by the energy criterion is dependent on matrix strain energy release rate. The energy criterion is represented by Zok and Spearing [42] and Zhu and Weitsman [43]. The concept of a CMSE criterion presupposes the existence of an ultimate or critical strain energy limit beyond which the matrix fails. Beyond this, as more energy is placed into the composite, the matrix, unable to support the additional load, continues to fail. As more energy is placed into the system, the matrix fails such that all the additional energy is transferred to the fibers. Failure may consist of the formation of matrix cracks, the propagation of existing cracks, or interface debonding [44]. Statistical failure approach assumes matrix cracking is governed by statistical relations, which relate the size and spatial distribution of matrix flaws to their relative propagation stress [45]. In Chapter 2 “First Matrix Cracking of Ceramic-Matrix Composites at Elevated Temperature,” the temperature- and time-dependent first matrix cracking stress of fiber-reinforced CMCs is investigated using the energy balance approach. The temperature-dependent micromechanical parameters of fiber and matrix modulus, fiber/matrix interface shear stress, interface debonding energy, and matrix fracture energy are incorporated into the analysis of the microstress analysis, fiber/matrix interface debonding criterion, and energy balance approach. Relationships between the first matrix cracking stress, fiber/matrix interface debonding, temperature, and time are established. Effects of fiber volume, interface shear stress, interface frictional coefficient, interface debonding energy, matrix fracture energy, temperature and time on the first matrix cracking stress, and interface debonding length are discussed. Experimental first matrix cracking stress and interface debonding of C/SiC and SiC/SiC composites at elevated temperature are predicted. In Chapter 3, “Matrix Multiple Cracking Evolution of Fiber-Reinforced Ceramic-Matrix Composites at Elevated Temperature,” the temperature- and time-dependent matrix cracking evolution of fiber-reinforced CMCs is investigated using the CMSE criterion. The temperature-dependent fiber/matrix interface shear stress, Young’s modulus of the fibers and the matrix, matrix fracture energy, and the fiber/matrix interface debonding energy are considered in the microstress field analysis, fiber/matrix interface debonding criterion, and matrix multiple cracking evolution model. Effects of fiber volume, fiber/matrix interface shear stress, fiber/matrix interface debonding energy, matrix fracture energy, temperature and time on matrix multiple cracking evolution, and fiber/matrix interface debonding are discussed. Experimental matrix multiple cracking evolution and fiber/matrix interface debonding of C/SiC and SiC/SiC composite at elevated temperatures are predicted. In Chapter 4, “Time-Dependent Tensile Behavior of Ceramic-Matrix Composites,” time-dependent tensile damage and fracture of fiber-reinforced CMCs are investigated, considering the interface and fiber oxidation. Time-dependent damage mechanisms of matrix cracking, interface debonding, fiber failure, and interface and fiber oxidation are considered in the analysis of the tensile stress–strain curve. Experimental time-dependent tensile stress–strain curves, matrix cracking, interface debonding, and fiber failure of different SiC/SiC and C/SiC composites are predicted for different oxidation duration.

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1.2 Fatigue Behavior of CMCs at Elevated Temperature For CMCs, because of the low fracture toughness of the matrix, there is no fatigue damage in the matrix itself. The degradation of interface or fiber properties is the main cause of fatigue damage. The interface degradation is mainly caused by interface layer fracture, wear, thermal residual stress release, and temperature rise at the interface; fiber degradation is caused by defects on the fiber surface during the interface wear process. It is found that the strength of the fiber decreases obviously after fatigue failure. The decrease of interfacial shear stress leads to the increase of interfacial debonding length with the increase of cycle, resulting in the increase of residual strain and decrease of modulus. The characteristic length and load of fiber increase with the decrease of interfacial shear stress. At the same time, the strength of fiber decreases because of interface wear, which further increases the probability of fiber fracture. When the percentage of fiber fracture reaches the critical value, the modulus of the composite decreases sharply, and the fatigue failure occurs. At elevated temperature, oxidation is the key factor to limit the application of CMCs on hot section load-carrying components of aeroengine. Combining carbides deposited by CVI process with specific sequences, a new generation of SiC/SiC composite with self-healing matrix has been developed to improve the oxidation resistance [46, 47]. The self-sealing matrix forms a glass with oxygen at high temperature and consequently prevents oxygen diffusion inside the material. At low temperature of 650–1000 ∘ C in dry and wet oxygen atmosphere, the self-healing 2.5D NicalonTM NL202 SiC/[Si-B-C] with a pyrolytic carbon (PyC) interphase exhibits a better oxidation resistance compared to SiC/SiC with PyC because of the presence of boron compounds [48]. The fatigue lifetime duration in air atmosphere at intermediate and high temperature is considerably reduced beyond the elastic yield point. For the Nicalon SiC/[Si-B-C] composite, the elastic yield point is about 𝜎 = 80 MPa. The lifetime duration was about t = 10–20 hours at T = 873 K and less than t = 1 hour at T = 1123 K under 𝜎 max = 120 MPa. For the self-healing Hi-NicalonTM SiC/SiC composite, a duration of t = 1000 hours without failure is reached at 𝜎 max = 170 MPa, and a duration higher than t = 100 hours at 𝜎 max = 200 MPa at T = 873 K [49]. For the self-healing Hi-Nicalon SiC/[SiC-B4 C] composite, at 1200 ∘ C, there was little influence on fatigue performance at f = 1.0 Hz but noticeably degraded fatigue lifetime at f = 0.1 Hz with the presence of steam [50, 51]. Increase in temperature from T = 1200 to 1300 ∘ C slightly degrades fatigue performance in air atmosphere but not in steam atmosphere [52]. The crack growth in the SiC fiber controls the fatigue lifetime of self-healing Hi-Nicalon SiC/[Si-B-C] at T = 873 K, and the fiber creep controls the fatigue lifetime of self-healing SiC/[Si-B-C] at T = 1200 ∘ C [53]. The typical cyclic fatigue behavior of a self-healing Hi-Nicalon SiC/[Si-B-C] composite involves an initial decrease of effective modulus to a minimum value, followed by a stiffening, and the time-to-the minimum modulus is in inverse proportion to the loading frequency [54]. The initial cracks within the longitudinal tows caused by interphase oxidation contribute to the initial decrease of modulus. The glass produced by the oxidation of self-healing matrix may contribute to the stiffening of the composite either by sealing the cracks or by bonding the fiber to the matrix [55]. The damage evolution of

1.3 Stress Rupture Behavior of CMCs at Elevated Temperature

self-healing Hi-Nicalon SiC/[Si-B-C] composite at elevated temperature can be monitored using acoustic emission (AE) [56, 57]. The relationship between interface oxidation and AE energy under static fatigue loading at elevated temperature has been developed [58]. However, at high temperature above 1000 ∘ C, AE cannot be applied for cyclic fatigue damage monitoring. The complex fatigue damage mechanisms of self-healing CMCs affect damage evolution and lifetime. Hysteresis loops related with cyclic-dependent fatigue damage mechanisms [59–61]. The damage parameters derived from hysteresis loops have already been applied for analyzing fatigue damage and fracture of different nonoxide CMCs at elevated temperatures [62–65]. However, the cyclic-dependent damage evolution and accumulation of self-healing CMCs are much different from previous analysis results especially at elevated temperatures. In Chapter 5, “Fatigue Behavior of Ceramic-Matrix Composites at Elevated Temperature,” cyclic-dependent damage development in self-healing 2.5D woven Hi-Nicalon SiC/[Si-B-C] and 2D woven Hi-Nicalon SiC/[SiC-B4 C] composites at T = 600 and 1200 ∘ C are investigated. Cyclic-dependent damage parameters of internal friction, dissipated energy, Kachanov’s damage parameter, and broken fiber fraction are obtained to analyze damage development in self-healing CMCs. Relationships between cyclic-dependent damage parameters and multiple fatigue damage mechanisms are established. Experimental fatigue damage evolution of self-healing Hi-Nicalon SiC/[Si-B-C] and Hi-Nicalon SiC/[SiC-B4 C] composites are predicted. Effects of fatigue peak stress, testing environment, and loading frequencies on the evolution of internal damage and final fracture are analyzed.

1.3 Stress Rupture Behavior of CMCs at Elevated Temperature At the intermediate temperature, SiC/SiC composite lifetime is decreased because of interface oxidation and lowered fiber content following exposure to stress rupture loading [66]. At elevated temperature range of 700–1200 ∘ C, the SiC/SiC minicomposite with the PyC interphase exhibited severe damages of oxidation embrittlement, and the SiC/SiC minicomposite with the BN interphase showed only mild degradation subjected to the stress rupture loading [67]. Martinez-Fernandez and Morscher [32] investigated the monotonic tensile, stress rupture under constant load, and low-cycle fatigue of single tow Hi-Nicalon, PyC interphase, and CVI SiC matrix minicomposites at room temperature, 700, 950, and 1200 ∘ C in air atmosphere. The stress rupture behavior at elevated temperature depended on the precracking stress level. However, for the 2D woven melt-infiltrated (MI) Hi-Nicalon SiC/SiC composite, the stress rupture properties were obvious worse than the SiC/SiC minicomposite properties under similar testing conditions because of complex fiber preform and damage evolution mechanisms [68]. Verrilli et al. [69] investigated the lifetime of C/SiC composite at elevated temperatures of 600 and 1200 ∘ C subjected to the stress rupture loading in different environments. Stress rupture lives in air and in steam containing environments were similar at a low

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stress level of 69 MPa at an elevated temperature of 1200 ∘ C. The fiber oxidation rate correlated with the composite stress rupture lifetime in the various environments. In the theoretical research area, Marshall et al. [70] and Zok and Spearing [42] applied the fracture mechanics approach to explore nonsteady first matrix cracking stress and multiple matrix cracking in fiber-reinforced CMCs. The energy balance relationship before and after the matrix cracking is established considering the mutual inference factors of the stress field between the adjacent matrix cracks. Curtin [45] investigated multiple matrix cracking in CMCs in the presence of matrix internal flaws. Evans [71] reported a method to predict design and life problems in fiber-reinforced CMCs. In addition, a connection between the macro-mechanical behavior and constituent properties of CMCs was established based on these predictions. McNulty and Zok [72] investigated the low-cycle fatigue damage mechanism and reported predictive damage models to describe the low-cycle CMC fatigue life. The degradation of the interface properties and fiber strength controls the fatigue life of CMCs. Lara-Curzio [73] established a micromechanical model for fiber-reinforced CMC reliability and time-to-failure estimation, particularly following the application of stresses greater than the first matrix cracking stress. The relationship between internal damage mechanisms and lifetime was established. In addition, the stress and temperature influences on the fiber-reinforced CMCs were investigated. Halverson and Curtin [74] developed a micromechanically based model for composite strength, and stress rupture lifetime of oxide/oxide fiber-reinforced CMCs considering the degradation of the fiber, matrix damage, and fiber pullout. Casas and Martinez-Esnaola [75] produced a fiber-reinforced CMC micromechanical creep-oxidation model, which was used to characterize oxidation at the CMC interface and the matrix, fiber creep, and fiber degradation with respect to time. Pailler and Lamon [36] developed a micromechanics-based model for the thermomechanical behavior of minicomposites based on multi-matrix cracking and fiber failure, which was derived from a fracture statistics-based model. Dassios et al. [76] analyzed the micromechanical behavior and micromechanics of crack growth resistance and bridging laws. The contributions of intact and pulled-out fibers on the bridging strain were discussed. Baranger [77] developed a reduced constitutive law to characterize the complex material behavior and applied the constitutive law to the mechanical modeling of SiC/SiC composites. Li [78–80] developed micromechanical damage models and constitutive relationship of cross-ply CMCs subjected to the dwell-fatigue loading at elevated temperature. Li et al. [27, 28] and Li [29] developed a micromechanical constitutive relationship to predict the damage and fracture of different fiber-reinforced CMCs subjected to tensile loading considering multiple damage mechanisms. Nonetheless, the above research did not consider fiber-reinforced CMC time-dependent deformation, damage, and fracture following the application of stress rupture loading at intermediate environmental temperatures. In Chapter 6, “Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature,” time-dependent deformation, damage, and fracture of fiber-reinforced CMCs that were exposed to stress rupture loading at intermediate environmental temperatures are investigated. The composite microstress field and tensile

1.4 Vibration Behavior of CMCs at Elevated Temperature

constitutive relationship of the damaged CMCs were examined to characterize their time-dependent damage mechanisms. Relationships between stress rupture lifetime, peak stress level, time-dependent composite deformation, and evolution of internal damages are established. Effects of composite material properties, composite damage state, and environmental temperature on stress rupture lifetime, time-dependent composite deformation, and evolution of the internal damages of SiC/SiC are analyzed. Experimental stress rupture lifetime, time-dependent composite deformation, and composite internal damage evolution of SiC/SiC composite subjected to the stress rupture loading are evaluated.

1.4 Vibration Behavior of CMCs at Elevated Temperature Failure analysis shows that approximately two-thirds of the failures are related to vibration and noise, leading to reduced operational control accuracy, structural fatigue damage, and shortened safety life. Therefore, studying the damping performance of fiber-reinforced CMCs and improving their reliability in the service environment is an important guarantee for the safe service of CMCs in various fields. Compared with metals and alloys, CMCs have many unique damping mechanisms because of internal structure and complex damage mechanisms [81–85]. Hao et al. [86] performed computational and experimental analysis on the modal parameters and vibration response of C/SiC bolted fastenings. The composite vibration damping affects the vibration amplitude of CMC components. Zhang [87] investigated the vibration characteristics of a CMC panel subjected to high temperature and large gradient thermal environment and performed damping measurement experiments of CMC panels in the thermal environment. The effect of thermal environment on the natural frequency and vibration damping of CMCs was analyzed. Huang and Wu [88] performed natural frequencies and acoustic emission testing of 2D SiC/SiC composite subjected to tensile loading. The natural frequencies decrease with increasing tensile stress because of internal damages. Natural frequency and vibration damping are affected by the damage state of CMCs [89]. Energy dissipation of frictional sliding between matrix cracks is the main mechanism for the damping of unidirectional or cross-ply CMCs at room temperature [90]. The damping of CMCs is also affected by the fabrication method [91], property of the fiber [91], internal damages inside of CMCs [82, 92], and interphase thickness [93]. At elevated temperature, the temperature-dependent damage mechanisms affect the mechanical behavior and vibration damping of CMCs [94–99]. In Chapter 7 “Vibration Damping of Ceramic-Matrix Composites at Elevated Temperature,” a micromechanical temperature-dependent vibration damping model of fiber-reinforced CMCs is developed. Temperature-dependent damage mechanisms of matrix cracking, interface debonding and slip, and fiber fracture contribute to the vibration damping of damaged CMCs. Temperature-dependent fiber and matrix strain energy and dissipated energy density are formulated of composite constituent properties and damage-related microparameters of matrix crack spacing, interface debonding and slip length, and broken fiber

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fraction. Relationships between temperature-dependent composite damping, temperature-dependent damage mechanisms, temperature, and oxidation duration are established. Effects of composite constituent properties and composite damage state on temperature-dependent composite vibration damping of SiC/SiC and C/SiC composites are analyzed. Experimental temperature-dependent composite vibration damping of 2D SiC/SiC and C/SiC composites is predicted.

1.5 Conclusion In this chapter, tensile, fatigue, stress rupture, and vibration behavior of fiber-reinforced CMCs are briefly introduced. In the following chapters, detailed formation about the matrix cracking, matrix multiple cracking, tensile, fatigue, stress rupture, and vibration behavior of fiber-reinforced CMCs are given: ●

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Chapter 2: First Matrix Cracking of Ceramic-Matrix Composites at Elevated Temperature. Chapter 3: Matrix Multiple Cracking Evolution of Fiber-Reinforced Ceramic-Matrix Composites at Elevated Temperature. Chapter 4: Time-Dependent Tensile Behavior of Ceramic-Matrix Composites. Chapter 5: Fatigue Behavior of Ceramic-Matrix Composites at Elevated Temperature. Chapter 6: Stress Rupture of Ceramic-Matrix Composites at Elevated Intemperature. Chapter 7: Vibration Damping of Ceramic-Matrix composites at Elevated Temperature.

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21 Cao, Y., Zhang, C., Zhou, X., and Chen, C. (2001). Tensile properties of UD-C/SiC ceramic matrix composites at elevated temperature. Acta Mater. Compos. Sin. 18: 82–84. 22 Qiao, S., Luo, G., Du, S., and Li, M. (2004). Tensile performance of 3D-C/SiC composites at high temperature. Mech. Sci. Technol. 23: 335–338. 23 Chen, M., Chen, X., Zhang, D. et al. (2019). Tensile behavior and failure mechanisms of plain weave SiC/SiC composites at room and high temperatures. J. Shanghai Jiaotong Univ. Sci. 53: 11–18. 24 Luo, G. and Qiao, S. (2003). Influence of loading rates on 3D-C/SiC tensile properties at different temperature. Mater. Eng. 10: 9–10. 25 Halbig, M.C., Brewer, D.N., and Eckel, A.J. (2000). Degradation of continuous fiber ceramic matrix composites under constant-load conditions. NASA/TM-2000-209681. 26 Mei, H., Cheng, L., Zhang, L., and Xu, Y. (2007). Modeling the effects of thermal and mechanical load cycling on a C/SiC composite in oxygen/argon mixtures. Carbon 45: 2195–2204. https://doi.org/10.1016/j.carbon.2007.06.051. 27 Li, L.B., Song, Y., and Sun, Y. Modeling the tensile behavior of unidirectional C/SiC ceramic-matrix composites. Mech. Compos. Mater. 203 (49): 659–672. https://doi.org/10.1007/s11029-013-9382-y. 28 Li, L.B., Song, Y., and Sun, Y. (2015). Modeling the tensile behavior of cross-ply C/SiC ceramic-matrix composites. Mech. Compos. Mater. 51: 359–376. https://doi .org/10.1007/s11029-015-9507-6. 29 Li, L.B. (2018). Modeling the monotonic and cyclic tensile stress–strain behavior of 2D and 2.5D woven C/SiC ceramic-matrix composites. Mech. Compos. Mater. 54: 165–178. https://doi.org/10.1007/s11029-018-9729-5. 30 Li, L. (2019). Time-dependent proportional limit stress of carbon fiber-reinforced silicon carbide-matrix composites considering interface oxidation. J. Ceram. Soc. Jpn. 127: 279–287. https://doi.org/10.2109/jcersj2.18176. 31 Li, L.B. (2018). Effect of interface oxidation on matrix multi-cracking evolution of fiber-reinforced ceramic-matrix composites at elevated temperatures. J. Ceram. Sci. Technol. 9: 397–410. 32 Martinez-Fernandez, J. and Morscher, G.N. (2000). Room and elevated temperature tensile properties of single tow Hi-Nicalon, carbon interphase, CVI SiC matrix minicomposites. J. Eur. Ceram. Soc. 20: 2627–2636. https://doi.org/10 .1016/S0955-2219(00)00138-2. 33 Forio, P., Lavaire, F., and Lamon, J. (2004). Delayed failure at intermediate temperatures (600 ∘ C–700 ∘ C) in air in silicon carbide multifilament tows. J. Am. Ceram. Soc. 87: 888–893. https://doi.org/10.1111/j.1551-2916.2004.00888.x. 34 Morscher, G.N. and Cawley, J.D. (2002). Intermediate temperature strength degradation in SiC/SiC composites. J. Eur. Ceram. Soc. 22: 2777–2787. https://doi .org/10.1016/S0955-2219(02)00144-9. 35 Larochelle, K.J. and Morscher, G.N. (2006). Tensile stress rupture behavior of a woven ceramic matrix composite in humid environments at intermediate temperature – Part I. Appl. Compos. Mater. 13: 147–172. https://doi.org/10.1007/ s10443-006-9009-8.

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50 Ruggles-Wrenn, M.B., Delapasse, J., Chamberlain, A.L. et al. (2012). Fatigue behavior of a Hi-NicalonTM /SiC-B4 C composite at 1200 ∘ C in air and in steam. Mater. Sci. Eng., A 534: 119–128. https://doi.org/10.1016/j.msea.2011.11.049. 51 Ruggles-Wrenn, M.B. and Kurtz, G. (2013). Notch sensitivity of fatigue behavior of a Hi-NicalonTM /SiC-B4 C composite at 1200 ∘ C in air and in steam. Appl. Compos. Mater. 20: 891–905. https://doi.org/10.1007/s10443-012-9277-4. 52 Ruggles-Wrenn, M.B. and Lee, M.D. (2016). Fatigue behavior of an advanced SiC/SiC composite with a self-healing matrix at 1300 ∘ C in air and in steam. Mater. Sci. Eng., A 677: 438–445. https://doi.org/10.1016/j.msea.2016.09.076. 53 Reynaud, P., Rouby, D., and Fantozzi, G. (2005). Cyclic fatigue behaviour at high temperature of a self-healing ceramic matrix composite. Ann. Chim. Sci. Mater. 30: 649–658. 54 Carrere, P. and Lamon, J. (1999). Fatigue behavior at high temperature in air of a 2D woven SiC/SiBC with a self healing matrix. Key Eng. Mater. 164–165: 321–324. https://doi.org/10.4028/www.scientific.net/KEM.164-165.321. 55 Forio, P. and Lamon, J. (2001). Fatigue behavior at high temperatures in air of a 2D SiC/Si-B-C composite with a self-healing multilayered matrix. In: Advances in Ceramic Matrix Composites VII (eds. N.P. Bansal, J.P. Singh and H.-T. Lin), 127–141. The American Ceramic Society. https://doi.org/10.1002/9781118380925 .ch10. 56 Simon, C., Rebillat, F., and Camus, G. (2017). Electrical resistivity monitoring of a SiC/[Si-B-C] composite under oxidizing environments. Acta Mater. 132: 586–597. https://doi.org/10.1016/j.actamat.2017.04.070. 57 Simon, C., Rebillat, F., Herb, V., and Camus, G. (2017). Monitoring damage evolution of SiCf /[Si-B-C]m composites using electrical resistivity: crack density-based electromechanical modeling. Acta Mater. 124: 579–587. https://doi .org/10.1016/j.actamat.2016.11.036. 58 Moevus, M., Reynaud, P., R’Mili, M. et al. (2006). Static fatigue of a 2.5D SiC/[Si-B-C] composite at intermediate temperature under air. Adv. Sci. Technol. 50: 141–146. https://doi.org/10.4028/www.scientific.net/AST.50.141. 59 Reynaud, P. (1996). Cyclic fatigue of ceramic-matrix composites at ambient and elevated temperatures. Compos. Sci. Technol. 56: 809–814. https://doi.org/10 .1016/0266-3538(96)00025-5. 60 Dalmaz, A., Reynaud, P., and Rouby, D. (1998). Fantozzi, Abbe F. mechanical behavior and damage development during cyclic fatigue at high-temperature of a 2.5D carbon/sic composite. Compos. Sci. Technol. 58: 693–699. https://doi.org/ 10.1016/S0266-3538(97)00150-4. 61 Fantozzi, G. and Reynaud, P. (2009). Mechanical hysteresis in ceramic matrix composites. Mater. Sci. Eng., A 521–522: 18–23. https://doi.org/10.1016/j.msea .2008.09.128. 62 Li, L.B. (2015). A hysteresis dissipated energy-based damage parameter for life prediction of carbon fiber-reinforced ceramic-matrix composites under fatigue loading. Compos. Part B: Eng. 82: 108–128. https://doi.org/10.1016/j.compositesb .2015.08.026.

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76 Dassios, K.G., Galiotis, C., Kostopoulos, V., and Steen, M. (2003). Direct in situ measurements of bridging stresses in CFCCs. Acta Mater. 51: 5359–5373. https:// doi.org/10.1016/S1359-6454(03)00393-8. 77 Baranger, E. (2013, 2013). Building of a reduced constitutive law for ceramic matrix composites. Int. J. Damage Mech. 22: 1222–1238. https://doi.org/10.1177/ 1056789513482338. 78 Li, L.B. (2017). Synergistic effects of temperature, oxidation, loading frequency and stress-rupture on damage evolution of cross-ply ceramic-matrix composites under cyclic fatigue loading at elevated temperatures in oxidizing atmosphere. Eng. Fract. Mech. 175: 15–30. https://doi.org/10.1016/j.engfracmech.2017.03.013. 79 Li, L.B. (2017). Synergistic effects of stress-rupture and cyclic loading on strain response of fiber-reinforced ceramic-matrix composites at elevated temperature in oxidizing atmosphere. Materials 10: 182. https://doi.org/10.3390/ma10020182. 80 Li, L.B. (2017). Damage evolution of cross-ply ceramic-matrix composites under stress-rupture and cyclic loading at elevated temperatures in oxidizing atmosphere. Mater. Sci. Eng., A 688: 315–321. https://doi.org/10.1016/j.msea.2017.02 .012. 81 Zhang, J., Perez, R.J., and Lavernia, E.J. (1993). Documentation of damping capacity of metallic ceramic and metal-matrix composite materials. J. Mater. Sci. 28: 2395–2404. https://doi.org/10.1007/BF01151671. 82 Li, L.B. (2020). A micromechanical temperature-dependent vibration damping model of fiber-reinforced ceramic-matrix composites. Compos. Struct. https://doi .org/10.1016/j.compstruct.2020.113297. 83 Li, L.B. (2020). A cyclic-dependent vibration damping model of fiber-reinforced ceramic-matrix composites. Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. https://doi .org/10.1177/0954406220971665. 84 Li, L.B. (2020). A micromechanical vibration damping model of fiber-reinforced ceramic-matrix composites considering interface debonding. Proc. Inst. Mech. Eng. Pt. L. J. Mater. Des. Appl. https://doi.org/10.1177/1464420720969711. 85 Li, L.B. (2020). A time-dependent vibration damping model of fiber-reinforced ceramic-matrix composites at elevated temperature. Ceram. Int. 46: 27031–27045. https://doi.org/10.1016/j.ceramint.2020.07.180. 86 Hao, B., Yin, X., Liu, X. et al. (2014). Vibration response characteristics and looseness – proof performance of C/SiC ceramic matrix composite bolted fastenings. Acta Mater. Compos. Sin. 31: 653–660. 87 Zhang, X. (2015). Thermal vibration analysis and model updating of ceramic matrix thermal protection/insulation structure. PhD thesis. Harbin Institute of Technology, Harbin, China. 88 Huang, D. and Wu, J. (2020). Natural frequency and acoustic emission test analysis of ceramic matrix composites under monotonic tensile. Open J. Acoust. Vib. 8: 17–25. https://doi.org/10.12677/ojav.2020.81003. 89 Li, H. (2019). Analysis of vibration and damage test of ceramic matrix composites structure. Master thesis. Nanchang Hangkong University, Nanjing, China.

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90 Birman, V. and Byrd, L.W. (2002). Effect of matrix cracks on damping in unidirectional and cross-ply ceramic matrix composites. J. Compos. Mater. 36: 1859–1877. https://doi.org/10.1177/0021998302036015247. 91 Sato, S., Serizawa, H., Araki, H. et al. (2003). Temperature dependence of internal friction and elastic modulus of SiC/SiC composites. J. Alloys Compd. 355: 142–147. https://doi.org/10.1016/S0925-8388(03)00252-4. 92 Wang, W., Cheng, L.F., Zhang, L.T. et al. (2006). Study on damping capacity of two dimensional carbon fiber reinforced silicon carbide (2D C/SiC) composites. J. Solid Rocket Technol. 29: 455–459. 93 Zhang, Q., Cheng, L.F., Wang, W. et al. (2007). Effect of interphase thickness on damping behavior of 2D C/SiC composites. Mater. Sci. Forum 546–549: 1531–1534. 94 Hong, Z.L., Cheng, L.F., Zhao, C.N. et al. (2013). Effect of oxidation on internal friction behavior of C/SC composites. Acta Mater. Compos. Sin. 30: 93–100. 95 Li, L.B. (2019). Modeling matrix multicracking development of fiber-reinforced ceramic-matrix composites considering fiber debonding. Int. J. Appl. Ceram. Technol. 16: 97–107. https://doi.org/10.1111/ijac.13068. 96 Li, L.B. (2019). Time-dependent damage and fracture of fiber-reinforced ceramic-matrix composites at elevated temperatures. Compos. Interfaces 26: 963–988. https://doi.org/10.1080/09276440.2019.1569397. 97 Reynaud, P., Douby, D., and Fantozzi, G. (1998). Effects of temperature and of oxidation on the interfacial shear stress between fibers and matrix in ceramic-matrix composites. Acta Mater. 46: 2461–2469. https://doi.org/10.1016/ S1359-6454(98)80029-3. 98 Snead, L.L., Nozawa, T., Katoh, Y. et al. (2007). Handbook of SiC properties for fuel performance modeling. J. Nucl. Mater. 371: 329–377. https://doi.org/10.1016/ j.jnucmat.2007.05.016. 99 Wang, R.Z., Li, W.G., Li, D.Y., and Fang, D.N. (2015). A new temperature dependent fracture strength model for the ZrB2 -SiC composites. J. Eur. Ceram. Soc. 35: 2957–2962. https://doi.org/10.1016/j.jeurceramsoc.2015.03.025.

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2 First Matrix Cracking of Ceramic-Matrix Composites at Elevated Temperature 2.1 Introduction With the development of aerospace industry, the requirements for high temperature, high specific strength, and high specific modulus materials are getting higher. Ultra-high temperature, long-life, lightweight thermal structural materials are the key prerequisites for the future development of aerospace engines to high performance, lightweight, low emissions, and low noise. Ceramic-matrix composites (CMCs) possess the advantages of high specific strength, high specific modulus, low density, good wear resistance, and chemical resistance at elevated temperatures, making them the material of choice for replacing high-temperature alloys in high thrust-to-weight ratio aeroengines [1–5]. The mechanical properties of CMCs are much different from those of single-phase ceramics. In single-phase ceramics, the failure of materials is caused by the initiation and propagation of main cracks. The elastic modulus of the whole material does not change during this process. However, when the CMC is subjected to stress, there are many microscopic failure mechanisms generated inside of composite, i.e. matrix cracking, fiber/matrix interface debonding, and fiber fracture, leading to the quasi-ductile behavior in tensile stress–strain curves [6–10]. Many researchers performed experimental and theoretical investigations on matrix cracking in fiber-reinforced CMCs. Energy balance approach can be used to determine the steady-state first matrix cracking stress, including the ACK model [11], AK model [12], BHE model [13], Kuo–Chou model [14], Sutcu–Hilling model [15], Chiang model [16], and Li model [17–21], and the stress intensity factor method is adopted to determine the short matrix cracking stress, including the MCE model [22], MC model [23], McCartney model [24], Chiang–Wang–Chou model [25], Danchaivijit–Shetty model [26], and Thouless–Evans model [27]. Kim and Pagano [28] and Dutton et al. [29] investigated the first matrix cracking in CMCs using the acoustic emission (AE), optical microscopy, and scanning electronic microscopy (SEM). It was found that the experimental first matrix cracking stress is much lower than the theoretical results predicted by the ACK model [11]. The micromatrix cracking appears first in the matrix-rich region, and with increasing applied stress, these micromatrix cracks propagate and stop at the fiber/matrix interface. In fact, these micromatrix cracks do not affect the macrostrain and High Temperature Mechanical Behavior of Ceramic-Matrix Composites, First Edition. Longbiao Li. © 2021 WILEY-VCH GmbH. Published 2021 by WILEY-VCH GmbH.

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stiffness of CMCs [30]; however, at higher applied stress, these micromatrix cracks evolve first into the short matrix cracking defined by the MCE model [22] and then the steady-state matrix cracking defined by the ACK model [11]. The steady-state matrix cracking model can be used to predict the proportional limit stress (PLS). In this chapter, the temperature- and time-dependent first matrix cracking stress of fiber-reinforced CMCs is investigated using the energy balance approach. The temperature-dependent micromechanical parameters of fiber and matrix modulus, fiber/matrix interface shear stress, interface debonding energy, and matrix fracture energy are incorporated into the analysis of the microstress analysis, fiber/matrix interface debonding criterion, and energy balance approach. Relationships between the first matrix cracking stress, fiber/matrix interface debonding, temperature, and time are established. Effects of fiber volume, interface shear stress, interface frictional coefficient, interface debonding energy, matrix fracture energy, temperature, and time on the first matrix cracking stress and interface debonding length are discussed. Experimental first matrix cracking stress and interface debonding of C/SiC and SiC/SiC composites at elevated temperature are predicted.

2.2 Temperature-Dependent Matrix Cracking Stress of C/SiC Composites In this section, the temperature-dependent matrix cracking stress of C/SiC composite is investigated using the energy balance approach. Temperature-dependent micromechanical parameters of the fiber and matrix modulus, fiber/matrix interface shear stress, interface debonding energy, and matrix fracture energy are incorporated into the analysis of the microstress analysis, fiber/matrix interface debonding criterion, and energy balance approach. Relationships between matrix cracking stress, interface debonding, and environmental temperature are established. Effects of fiber volume, interface shear stress, interface frictional coefficient, interface debonding energy, and matrix fracture energy on matrix cracking stress and interface debonding length versus environmental temperature are discussed. Experimental matrix cracking stress of 2D C/SiC composite at elevated temperature is predicted.

2.2.1

Theoretical Models

The energy balance relationship to evaluate the matrix cracking stress of fiber-reinforced CMCs is [13] } ∞{ Vf Vm 1 2 2 [𝜎 (T) − 𝜎fd (T)] + [𝜎 (T) − 𝜎md (T)] dx 2 ∫−∞ Ef (T) fu Em (T) mu ] [ ld (T) R rf 𝜏i (x, T) 1 2𝜋r dr dx (2.1) + r 2𝜋R2 Gm (T) ∫−ld (T) ∫rf 4V l (T) = Vm Γm (T) + f d Γd (T) rf

2.2 Temperature-Dependent Matrix Cracking Stress of C/SiC Composites

where V f and V m denote the fiber and matrix volume, respectively, Ef (T) and Em (T) denote the temperature-dependent fiber and matrix elastic modulus, respectively, 𝜎 fu (T) and 𝜎 mu (T) denote the fiber and matrix axial stress distribution in the matrix cracking upstream region, respectively, and 𝜎 fd (T) and 𝜎 md (T) denote the fiber and matrix axial stress distribution in the matrix cracking downstream region, respectively. Γm (T) and Γd (T) denote the temperature-dependent matrix fracture energy and interface deboning energy, respectively. Ef (T) 𝜎 Ec (T) E (T) 𝜎 𝜎mu (T) = m Ec (T)

𝜎fu (T) =

⎧𝜎 2𝜏i (T) ⎪ Vf − rf x, x ∈ [[0, ld (T)] ] 𝜎fd (x, T) = ⎨ Ef (T) l (T) x ∈ ld (T), c 2 ⎪ Ec (T) 𝜎, ⎩ ⎧ Vf 𝜏i (T) ⎪2 Vm rf x, x ∈ [[0, ld (T)] ] 𝜎md (x, T) = ⎨ Em (T) l (T) x ∈ ld (T), c 2 ⎪ Ec (T) 𝜎, ⎩ √ rf Vm Ef (T)𝛾d (T) rf Vm Em (T)𝜎 ld (T) = − 2Vf Ec (T)𝜏i (T) Ec (T)𝜏i2 (T)

(2.2) (2.3)

(2.4)

(2.5)

(2.6)

where [31] |𝛼rf (T) − 𝛼rm (T)|(Tm − T) (2.7) A Substituting the upstream and downstream temperature-dependent fiber and matrix axial stresses of Eqs. (2.2–2.5), and the temperature-dependent interface debonding length of Eq. (2.6) into Eq. (2.1), the energy balance equation leads to the following equation. 𝜏i (T) = 𝜏0 + 𝜇

𝛼𝜎 2 + 𝛽𝜎 + 𝛾 = 0

(2.8)

where Vm Em (T)ld (T) Vf Ef (T)Ec (T) 2𝜏 (T) 2 l (T) 𝛽=− i rf Ef (T) d ]2 [ 4V Γ (T) Vf Ec (T) 4 𝜏i (T) l3 (T) − f d − Vm Γm (T) 𝛾= 3 rf Vm Ef (T)Em (T) d rf 𝛼=

2.2.2

(2.9) (2.10) (2.11)

Results and Discussion

The ceramic composite system of C/SiC is used for the case study and its material properties are given by V f = 30%, r f = 3.5 μm, Γm = 25 J/m2 (at room temperature), and Γd = 0.1 J/m2 (at room temperature).

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Temperature-dependent carbon fiber elastic modulus Ef (T) is [32] )] [ ( T , T < 2273 K (2.12) Ef (T) = 230 1 − 2.86 × 10−4 exp 324 Temperature-dependent SiC matrix elastic modulus Em (T) is [33] [ ( )] 962 350 460 − 0.04T exp − , T ∈ [300 K 1773 K] (2.13) Em (T) = 460 T Temperature-dependent carbon fiber axial and radial thermal expansion coefficient 𝛼 lf (T) and 𝛼 rf (T) are [34] 𝛼lf (T) = 2.529 × 10−2 − 1.569 × 10−4 T + 2.228 × 10−7 T 2 [ ] − 1.877 × 10−11 T 3 − 1.288 × 10−14 T 4 , T ∈ 300 K 2500 K

(2.14)

𝛼rf (T) = −1.86 × 10−1 + 5.85 × 10−4 T − 1.36 × 10−8 T 2 [ ] + 1.06 × 10−22 T 3 , T ∈ 300 K 2500 K

(2.15)

Temperature-dependent SiC matrix axial and radial thermal expansion coefficient of 𝛼 lm (T) and 𝛼 rm (T) are [33] −5 2 ⎧−1.8276 + 0.0178T − 1.5544 × 10 T [ ] ⎪ 𝛼lm (T) = 𝛼rm (T) = ⎨ +4.5246 × 10−9 T 3 , T ∈ 125 K 1273 K ⎪ ⎩ 5.0 × 10−6 K−1 , T > 1273 K (2.16)

Temperature-dependent interface debonding energy Γd (T) and matrix fracture energy Γm (T) are [35] T ⎡ ∫T CP (T) dT ⎤ ⎥ Γd (T) = Γdo ⎢1 − To m ⎥ ⎢ ∫ C (T) dT P ⎦ ⎣ To

(2.17)

T ⎡ ∫T CP (T) dT ⎤ ⎢ ⎥ Γm (T) = Γmo 1 − To m ⎢ ⎥ ∫ C (T) dT P ⎣ ⎦ To

(2.18)

where T o denotes the reference temperature, T m denotes the fabricated temperature, Γdo and Γmo denote the interface debonding energy and matrix fracture energy at the reference temperature of T o , respectively, and CP (T) is CP (T) = 76.337 + 109.039 × 10−3 T − 6.535 × 105 T −2 − 27.083 × 10−6 T 2 (2.19) Effects of fiber volume, interface shear stress, interface frictional coefficient, interface debonding energy, and matrix fracture energy on temperature-dependent matrix cracking stress of C/SiC composite are discussed.

2.2 Temperature-Dependent Matrix Cracking Stress of C/SiC Composites

(a)

(b)

Figure 2.1 Effect of fiber volume (i.e. V f = 30% and 35%) on (a) the matrix cracking stress versus temperature curves and (b) the interface debonding length versus temperature curves of C/SiC composite.

2.2.2.1 Temperature-Dependent Matrix Cracking Stress of C/SiC Composite for Different Fiber Volumes

Figure 2.1 shows the matrix cracking stress 𝜎 mc and interface debonding length ld /r f versus environmental temperature curves for different fiber volumes. When the fiber volume is V f = 30%, the matrix cracking stress increases from 𝜎 mc = 48 MPa at T = 973 K to 𝜎 mc = 103 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 0.68 to ld /r f = 4.5. When the fiber volume is V f = 35%, the matrix cracking stress increases from 𝜎 mc = 47 MPa at T = 973 K to 𝜎 mc = 113 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 0.08 to ld /r f = 3.9.

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(a)

(b)

Figure 2.2 Effect of interface shear stress (i.e. 𝜏 0 = 30 and 40 MPa) on (a) the matrix cracking stress versus environmental temperature curves and (b) the interface debonding length versus environmental temperature curves of C/SiC composite.

2.2.2.2 Temperature-Dependent Matrix Cracking Stress of C/SiC Composite for Different Interface Shear Stress

Figure 2.2 shows the matrix cracking stress 𝜎 mc and the interface debonding length ld /r f versus environmental temperature curves for different interface shear stress. When the interface shear stress is 𝜏 0 = 30 MPa, the matrix cracking stress increases from 𝜎 mc = 65 MPa at T = 973 K to 𝜎 mc = 115 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 1.4 to ld /r f = 4.2. When the interface shear stress is 𝜏 0 = 40 MPa, the matrix cracking stress increases from 𝜎 mc = 93 MPa at T = 973 K to 𝜎 mc = 134 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 2.1 to ld /r f = 3.7.

2.2 Temperature-Dependent Matrix Cracking Stress of C/SiC Composites

(a)

(b)

Figure 2.3 Effect of interface frictional coefficient (i.e. 𝜇 = 0.03 and 0.05) on (a) the matrix cracking stress versus environmental temperature curves and (b) the interface debonding length versus environmental temperature curves of C/SiC composite.

2.2.2.3 Temperature-Dependent Matrix Cracking Stress of C/SiC Composite for Different Fiber/Matrix Interface Frictional Coefficients

Figure 2.3 shows the matrix cracking stress 𝜎 mc and the fiber/matrix interface debonding length ld /r f versus environmental temperature curves for different interface frictional coefficient. When the fiber/matrix interface frictional coefficient is 𝜇 = 0.03, the matrix cracking stress increases from 𝜎 mc = 84 MPa at T = 973 K to 𝜎 mc = 131 MPa at T = 1273 K, and the fiber/matrix interface debonding length increases from ld /r f = 1.9 to ld /r f = 3.8. When the fiber/matrix interface frictional coefficient is 𝜇 = 0.05, the matrix cracking stress increases from 𝜎 mc = 65 MPa at T = 973 K to 𝜎 mc = 127 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 1.5 to ld /r f = 3.9.

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(a)

(b)

Figure 2.4 Effect of interface debonding energy (i.e. Γd = 0.3 and 0.5 J/m2 ) on (a) the matrix cracking stress versus environmental temperature curves and (b) the interface debonding length versus temperature curves of C/SiC composite.

2.2.2.4 Temperature-Dependent Matrix Cracking Stress of C/SiC Composite for Different Interface Debonding Energies

Figure 2.4 shows the matrix cracking stress 𝜎 mc and the interface debonding length ld /r f versus environmental temperature curves for different interface debonding energy. When the interface debonding energy is Γd = 0.3 J/m2 , the matrix cracking stress increases from 𝜎 mc = 102 MPa at T = 973 K to 𝜎 mc = 139 MPa at T = 1273 K, and the fiber/matrix interface debonding length increases from ld /r f = 1.3 to ld /r f = 3.3. When the fiber/matrix interface debonding energy is Γd = 0.5 J/m2 , the matrix cracking stress increases from 𝜎 mc = 110 MPa at T = 973 K to 𝜎 mc = 143 MPa at T = 1273 K, and the fiber/matrix interface debonding length increases from ld /r f = 0.9 to ld /r f = 3.0.

2.2 Temperature-Dependent Matrix Cracking Stress of C/SiC Composites

(a)

(b)

Figure 2.5 Effect of interface debonding energy (i.e. Γm = 20 and 30 J/m2 ) on (a) the matrix cracking stress versus environmental temperature curves and (b) the interface debonding length versus environmental temperature curves of C/SiC composite.

2.2.2.5 Effect of Matrix Fracture Energy on Temperature-Dependent Matrix Cracking Stress of C/SiC Composite

Figure 2.5 shows the matrix cracking stress 𝜎 mc and the interface debonding length ld /r f versus environmental temperature curves for different matrix fracture energies. When the matrix fracture energy is Γm = 20 J/m2 , the matrix cracking stress increases from 𝜎 mc = 49 MPa at T = 973 K to 𝜎 mc = 102 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 0.59 to ld /r f = 3.6. When the matrix fracture energy is Γm = 30 J/m2 , the matrix cracking stress increases from 𝜎 mc = 79 MPa at T = 973 K to 𝜎 mc = 126 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 2.2 to ld /r f = 4.7.

27

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2 First Matrix Cracking of Ceramic-Matrix Composites at Elevated Temperature

(a)

(b)

Figure 2.6 Tensile stress–strain curves of 2D C/SiC composite at (a) T = 973 K and (b) T = 1273 K.

2.2.3

Experimental Comparisons

Yang et al. [36] investigated the tensile behavior of 2D T300TM -C/SiC composite at elevated temperature. The C/SiC composite was fabricated using the chemical vapor infiltration (CVI) method with the pyrolytic carbon (PyC) interphase of approximately 1.5–2.0 μm. The fiber volume fraction is approximately 40%. Tensile tests were performed under the displacement control and the loading speed was 0.3 mm/min. The tensile stress–strain curves of 2D C/SiC composite at elevated temperatures of T = 973 and 1273 K are shown in Figure 2.6. The tensile stress–strain response of 2D C/SiC composite exhibits obviously nonlinearly. At T = 973 K, the composite matrix cracking stress is approximately 𝜎 mc = 50 MPa, and the composite tensile strength is approximately 𝜎 UTS = 232 MPa with the failure strain 𝜀f = 0.25%; at T = 1273 K, the composite matrix cracking stress is approximately 𝜎 mc = 80 MPa, and the composite tensile strength is approximately 𝜎 UTS = 271 MPa with the failure

2.3 Temperature-Dependent Matrix Cracking Stress of SiC/SiC Composite

(a)

(b)

Figure 2.7 (a) Experimental and theoretical matrix cracking stress versus environmental temperature curves and (b) the interface debonded length versus environmental temperature curves of 2D C/SiC composite.

strain 𝜀f = 0.33%. Experimental and theoretical predicted PLS and the fiber/matrix interface debonding length versus environmental temperature curves are shown in Figure 2.7. With increasing temperature, the matrix cracking stress of 2D C/SiC composite increased from 𝜎 mc = 48 MPa at T = 973 K to 𝜎 mc = 82 MPa at T = 1273 K, and the interface debonded length increases from ld /r f = 2.7 to ld /r f = 6.3.

2.3 Temperature-Dependent Matrix Cracking Stress of SiC/SiC Composite In this section, temperature-dependent matrix cracking stress of SiC/SiC composite is investigated using the micromechanical approach. The matrix cracking

29

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2 First Matrix Cracking of Ceramic-Matrix Composites at Elevated Temperature

stress of SiC/SiC composite is predicted using the energy balance approach considering the effect of environmental temperature. The relationship between environmental temperature, matrix cracking stress, and composite damage state is established. Effects of fiber volume, interface properties, and matrix properties on the temperature-dependent matrix crack stress and composite damage state of SiC/SiC composite are analyzed. Experimental matrix cracking stress and interface debonding length of 2D SiC/SiC composites with the PyC and boron nitride (BN) interphase at elevated temperatures are predicted.

2.3.1

Results and Discussion

The ceramic composite system of SiC/SiC is used for the case study and its material properties are given by [37] V f = 30%, r f = 7.5 μm, Γm = 25 J/m2 (at room temperature), Γd = 0.1 J/m2 (at room temperature), 𝛼 rf = 2.9 × 10−6 K−1 , and 𝛼 lf = 3.9 × 10−6 K−1 . Effects of fiber volume, interface properties, and matrix properties on the temperature-dependent matrix cracking stress of SiC/SiC composite are discussed. 2.3.1.1 Temperature-Dependent Matrix Cracking Stress of SiC/SiC Composite for Different Fiber Volumes

Figure 2.8 shows the matrix cracking stress and the interface debonding length versus environmental temperature curves for different fiber volumes (i.e. V f = 25%, 30%, and 35%). When temperature increases from T = 873 to 1273 K, the matrix cracking stress and fiber/matrix interface debonding length decrease with increasing temperature. At the same temperature, the matrix cracking stress increases with the fiber volume, and the fiber/matrix interface debonding length decreases with the fiber volume. When the fiber volume increases, the stress transfer between the fiber and the matrix increases, the stress carried by the matrix increases, leading to the increase of the matrix cracking stress and the decrease of the fiber/matrix interface debonding length. When the fiber volume is V f = 25%, the matrix cracking stress decreases from 𝜎 mc = 103 MPa at T = 873 K to 𝜎 mc = 87 MPa at T = 1273 K, and the fiber/matrix interface debonding length decreases from ld /r f = 6.1 at 𝜎 mc = 103 MPa to ld /r f = 5.0 at 𝜎 mc = 87 MPa. When the fiber volume is V f = 30%, the matrix cracking stress decreases from 𝜎 mc = 117 MPa at T = 873 K to 𝜎 mc = 99 MPa at T = 1273 K, and the fiber/matrix interface debonding length decreases from ld /r f = 5.3 at 𝜎 mc = 117 MPa to ld /r f = 4.4 at 𝜎 mc = 99 MPa. When the fiber volume is V f = 35%, the matrix cracking stress decreases from 𝜎 mc = 130 MPa at T = 873 K to 𝜎 mc = 110 MPa at T = 1273 K, and the interface debonding length decreases from ld /r f = 4.6 at 𝜎 mc = 130 MPa to ld /r f = 3.9 at 𝜎 mc = 110 MPa. 2.3.1.2 Temperature-Dependent Matrix Cracking Stress of SiC/SiC Composite for Different Interface Shear Stress

Figure 2.9 shows the matrix cracking stress and the interface debonding length versus environmental temperature curves for different interface shear stress (i.e.

2.3 Temperature-Dependent Matrix Cracking Stress of SiC/SiC Composite

(a)

(b)

Figure 2.8 Effect of fiber volume on (a) the matrix cracking stress versus temperature curves and (b) the fiber/matrix interface debonding length versus temperature curves of SiC/SiC composite.

𝜏 i = 15, 20, and 25 MPa). When environmental temperature increases from T = 873 to 1273 K, the matrix cracking stress and fiber/matrix interface debonding length decrease with the increasing temperature. The matrix cracking stress increases with the interface shear stress, and the fiber/matrix interface debonding length decreases with the interface shear stress at the same temperature. When the interface shear stress increases, the stress transfer between the fiber and the matrix also increases, leading to the increase of the matrix cracking stress and the decrease of the interface debonding length. When 𝜏 i = 15 MPa, the matrix cracking stress decreases from 𝜎 mc = 89 MPa at T = 873 K to 𝜎 mc = 81 MPa at T = 1273 K, and the interface debonding length

31

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2 First Matrix Cracking of Ceramic-Matrix Composites at Elevated Temperature

(a)

(b)

Figure 2.9 Effect of interface shear stress on (a) the matrix cracking stress versus temperature curves and (b) the fiber/matrix interface debonding length versus temperature curves of SiC/SiC composite.

decreases from ld /r f = 6.8 at 𝜎 mc = 89 MPa to ld /r f = 5.9 at 𝜎 mc = 81 MPa. When 𝜏 i = 20 MPa, the matrix cracking stress decreases from 𝜎 mc = 104 MPa at T = 873 K to 𝜎 mc = 90 MPa at T = 1273 K, and the interface debonding length decreases from ld /r f = 5.9 at 𝜎 mc = 104 MPa to ld /r f = 5 at 𝜎 mc = 90 MPa. When 𝜏 i = 25 MPa, the matrix cracking stress decreases from 𝜎 mc = 117 MPa at T = 873 K to 𝜎 mc = 99 MPa at T = 1273 K, and the interface debonding length decreases from ld /r f = 5.3 at 𝜎 mc = 117 MPa to ld /r f = 4.4 at 𝜎 mc = 99 MPa.

2.3 Temperature-Dependent Matrix Cracking Stress of SiC/SiC Composite

(a)

(b)

Figure 2.10 Effect of interface frictional coefficient on (a) the matrix cracking stress versus temperature curves and (b) the interface debonding length versus temperature curves of SiC/SiC composite.

2.3.1.3 Temperature-Dependent Matrix Cracking Stress of SiC/SiC Composite for Different Interface Frictional Coefficients

Figure 2.10 shows the matrix cracking stress and the interface debonding length versus environmental temperature curves for different interface frictional coefficients (i.e. 𝜇 = 0.02, 0.03, and 0.04). When temperature increases from T = 873 to 1273 K, the matrix cracking stress and fiber/matrix interface debonding length decrease with the increasing temperature. At the same temperature, the matrix cracking stress decreases with the interface frictional coefficient, and the interface debonding length increases with the interface frictional coefficient. When the interface frictional coefficient increases, the interface shear stress decreases at the

33

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2 First Matrix Cracking of Ceramic-Matrix Composites at Elevated Temperature

same temperature, leading to the decrease of the matrix cracking stress and the increase of the interface debonding length. When 𝜇 = 0.02, the matrix cracking stress decreases from 𝜎 mc = 117 MPa at T = 873 K to 𝜎 mc = 99 MPa at T = 1273 K, and the interface debonding length decreases from ld /r f = 5.3 at 𝜎 mc = 117 MPa to ld /r f = 4.4 at 𝜎 mc = 99 MPa. When 𝜇 = 0.03, the matrix cracking stress decreases from 𝜎 mc = 113 MPa at T = 873 K to 𝜎 mc = 98 MPa at T = 1273 K, and the interface debonding length decreases from ld /r f = 5.5 at 𝜎 mc = 113 MPa to ld /r f = 4.4 at 𝜎 mc = 98 MPa. When 𝜇 = 0.04, the matrix cracking stress decreases from 𝜎 mc = 109 MPa at T = 873 K to 𝜎 mc = 97 MPa at T = 1273 K, and the interface debonding length decreases from ld /r f = 5.7 at 𝜎 mc = 109 MPa to ld /r f = 4.5 at 𝜎 mc = 97 MPa. 2.3.1.4 Temperature-Dependent Matrix Cracking Stress of SiC/SiC Composite for Different Interface Debonding Energies

Figure 2.11 shows the matrix cracking stress and the interface debonding length versus environmental temperature curves for different interface debonding energies (i.e. Γd = 0.1, 0.2, and 0.3 J/m2 ). When environmental temperature increases from T = 873 to 1273 K, the matrix cracking stress and interface debonding length decrease with the increasing temperature. At the same temperature, the matrix cracking stress increases with the interface debonding energy, and the interface debonding length decreases with the interface debonding energy. When the interface debonding energy increases, the resistance for the interface debonding also increases, leading to the increase of matrix cracking stress and the decrease of interface debonding length. When Γd = 0.1 J/m2 , the matrix cracking stress decreases from 𝜎 mc = 117 MPa at T = 873 K to 𝜎 mc = 99 MPa at T = 1273 K, and the interface debonding length decreases from ld /r f = 5.3 at 𝜎 mc = 117 MPa to ld /r f = 4.4 at 𝜎 mc = 99 MPa. When Γd = 0.2 J/m2 , the matrix cracking stress decreases from 𝜎 mc = 121 MPa at T = 873 K to 𝜎 mc = 101 MPa at T = 1273 K, and the interface debonding length decreases from ld /r f = 4.8 at 𝜎 mc = 121 MPa to ld /r f = 4.1 at 𝜎 mc = 101 MPa. When Γd = 0.3 J/m2 , the matrix cracking stress decreases from 𝜎 mc = 125 MPa at T = 873 K to 𝜎 mc = 103 MPa at T = 1273 K, and the interface debonding length decreases from ld /r f = 4.5 at 𝜎 mc = 125 MPa to ld /r f = 3.9 at 𝜎 mc = 103 MPa. 2.3.1.5 Temperature-Dependent Matrix Cracking Stress of SiC/SiC Composite for Different Matrix Fracture Energies

Figure 2.12 shows the matrix cracking stress and the interface debonding length versus environmental temperature curves for different matrix fracture energies (i.e. Γm = 15, 20, and 25 J/m2 ). When environmental temperature increases from T = 873 to 1273 K, the matrix cracking stress and fiber/matrix interface debonding length decrease with the increasing temperature. At the same temperature, the matrix cracking stress and the fiber/matrix interface debonding length increase with the matrix fracture energy. When the matrix fracture energy increases, the energy needed for the matrix cracking increases, leading to the increase of the matrix cracking stress and the interface debonding length.

2.3 Temperature-Dependent Matrix Cracking Stress of SiC/SiC Composite

(a)

(b)

Figure 2.11 Effect of the interface debonding energy on (a) the matrix cracking stress versus temperature curves and (b) the fiber/matrix interface debonding length versus temperature curves of SiC/SiC composite.

When Γm = 15 J/m2 , the matrix cracking stress decreases from 𝜎 mc = 117 MPa at T = 873 K to 𝜎 mc = 99 MPa at T = 1273 K, and the interface debonding length decreases from ld /r f = 5.3 at 𝜎 mc = 117 MPa to ld /r f = 4.4 at 𝜎 mc = 99 MPa. When the matrix fracture energy is Γm = 20 J/m2 , the matrix cracking stress decreases from 𝜎 mc = 132 MPa at T = 873 K to 𝜎 mc = 110 MPa at T = 1273 K, and the interface debonding length decreases from ld /r f = 6.2 at 𝜎 mc = 132 MPa to ld /r f = 5 at 𝜎 mc = 110 MPa. When the matrix fracture energy is Γm = 25 J/m2 , the matrix cracking stress decreases from 𝜎 PLS = 145 MPa at T = 873 K to 𝜎 mc = 119 MPa at T = 1273 K, and the fiber/matrix interface debonding length decreases from ld /r f = 6.9 at 𝜎 mc = 145 MPa to ld /r f = 5.5 at 𝜎 mc = 119 MPa.

35

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2 First Matrix Cracking of Ceramic-Matrix Composites at Elevated Temperature

(a)

(b)

Figure 2.12 Effect of matrix fracture energy on (a) the matrix cracking stress versus temperature curves and (b) the fiber/matrix interface debonding length versus temperature curves of SiC/SiC composite.

2.3.2

Experimental Comparisons

Guo and Kagawa [38] investigated the tensile behavior of 2D SiC/SiC composites with the PyC and BN interphase at elevated temperature. Experimental tensile stress–strain curves of NicalonTM SiC/PyC/SiC and Hi-NicalonTM SiC/BN/SiC composites at room and elevated temperatures are shown in Figures 2.13 and 2.14. For the Nicalon SiC/PyC/SiC composite, the matrix cracking stress decreases from 𝜎 mc = 65 MPa at T = 298 K to 𝜎 mc = 33 MPa at T = 1200 K, and for the Hi-Nicalon SiC/BN/SiC composite, the matrix cracking stress decreases from 𝜎 mc = 75 MPa at T = 298 K to 𝜎 mc = 45 MPa at T = 1400 K. Experimental and predicted matrix

2.3 Temperature-Dependent Matrix Cracking Stress of SiC/SiC Composite

Figure 2.13 Experimental tensile stress–strain curves of NicalonTM SiC/PyC/SiC at (a) T = 298 K, (b) T = 800 K, and (c) T = 1200 K.

(a)

(b)

(c)

37

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2 First Matrix Cracking of Ceramic-Matrix Composites at Elevated Temperature

Figure 2.14 Experimental tensile stress–strain curves of NicalonTM SiC/BN/SiC at (a) T = 298 K, (b) T = 1200 K, and (c) T = 1400 K.

(a)

(b)

cracking stress versus temperature curves of Nicalon SiC/PyC/SiC and Hi-Nicalon SiC/BN/SiC are shown in Figure 2.15. For the 2D Nicalon SiC/SiC composite with the PyC interphase, matrix cracking stress decreases from 𝜎 mc = 65 MPa at T = 298 K to 𝜎 mc = 33 MPa at T = 1200 K, and the fiber/matrix interface debonding length decreases from ld /r f = 12.8 at 𝜎 mc = 65 MPa to ld /r f = 7.3 at 𝜎 mc = 33 MPa. For the 2D Hi-Nicalon SiC/SiC composite with the BN interphase, the interface shear stress of SiC/BN/SiC at room and elevated temperatures are much lower than those of the SiC/C/SiC composite because of the better oxidation resistance of BN-coating on the Hi-Nicalon fiber surface than C-coating on Nicalon fiber surface [38]. The matrix cracking stress decreases from 𝜎 mc = 75 MPa at T = 298 K to 𝜎 mc = 45 MPa at T = 1400 K, and the fiber/matrix interface debonding length decreases from ld /r f = 7.3 at 𝜎 mc = 75 MPa to ld /r f = 2.9 at 𝜎 mc = 45 MPa.

2.4 Time-Dependent Matrix Cracking Stress of C/SiC Composites

(a)

(b)

Figure 2.15 Experimental and predicted matrix cracking stress versus the temperature curves of (a) SiC/SiC composite with the PyC interphase and (b) SiC/SiC composite with the BN interphase.

2.4 Time-Dependent Matrix Cracking Stress of C/SiC Composites In this section, the time-dependent matrix cracking stress of C/SiC composite is investigated. Effects of fiber volume, interface properties, and matrix fracture energy on time-dependent matrix cracking stress and interface debonding of C/SiC composite are discussed. Experimental time-dependent matrix cracking stresses of C/SiC composites corresponding to different oxidation times are predicted.

2.4.1

Theoretical Models

Because of a mismatch between the axial thermal expansion coefficients of the carbon fiber and silicon carbide matrix, unavoidable microcracks were created in

39

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2 First Matrix Cracking of Ceramic-Matrix Composites at Elevated Temperature

the matrix when the composite was cooled down from a high fabricated temperature to ambient temperature. These processing-induced microcracks, which were present on the surface of the material, served as avenues for the ingress of environmental atmosphere into the composite at elevated temperature. Time-dependent fiber/matrix interface oxidation length 𝜁 is [39] ]} { [ 𝜑 (T)t (2.20) 𝜁(t, T) = 𝜑1 (T) 1 − exp − 2 b where b is the delay factor considering the deceleration of reduced oxygen activity, t denotes the oxidation time, and 𝜑1 (T) and 𝜑2 (T) are the temperature-dependent parameters described using the Arrhenius law [39]. ) ( 8231 (2.21) 𝜑1 (T) = 7.021 × 10−3 × exp T ) ( 17 090 (2.22) 𝜑2 (T) = 227.1 × exp − T Temperature-dependent fiber/matrix interface shear stress in the oxidation region and debonded region are [31] 𝜏i (T) = 𝜏0 + 𝜇

|𝛼rf (T) − 𝛼rm (T)|(Tm − T) A

(2.23)

𝜏f (T) = 𝜏s + 𝜇

|𝛼rf (T) − 𝛼rm (T)|(Tm − T) A

(2.24)

where 𝜏 0 and 𝜏 s denote the initial interface shear stress, 𝜇 denotes the interface frictional coefficient, 𝛼 rf and 𝛼 rm denote the fiber and matrix radial thermal expansion coefficient, respectively, T m and T denote the processing temperature and testing temperature, respectively, and A is a constant dependent on the elastic properties of the matrix and fibers. The energy balance relationship for evaluating the matrix cracking stress of fiber-reinforced CMCs is [13] { } Vf 1 ∞ 2 + Vm [𝜎 (T) − 𝜎 (T)]2 dx ∫ [𝜎 (T) − 𝜎 (T)] fd md 2 −∞ Ef (T) fu E (T) mu [ ] m ld (T) R rf 𝜏i (x,T) 1 + 2𝜋R2 G (T) ∫−l (T) ∫r 2𝜋r dr dx r m

= Vm Γm (T) +

d

f

4Vf ld (T) Γd (T) rf

(2.25)

where 𝜎fu (T) = 𝜎mu (T) =

Ef (T) 𝜎 Ec (T)

(2.26)

Em (T) 𝜎 Ec (T)

⎧ 𝜎 − 2𝜏f (T) x, ⎪ Vf rf 𝜎fd (x) = ⎨ 2𝜏f (T) 𝜎 ⎪ V − r 𝜁(t, T) − f ⎩ f

(2.27) x ∈ [0, 𝜁(t, T)] 2𝜏i (T) [x rf

− 𝜁(t, T)], x ∈ [𝜁(t, T), ld (t, T)]

(2.28)

2.4 Time-Dependent Matrix Cracking Stress of C/SiC Composites

⎧ 2 Vf 𝜏f (T) x, ⎪ Vm rf 𝜎md (z) = ⎨ V 𝜏 (T) V ⎪2 V f fr 𝜁(t, T) + 2 V f m ⎩ m f

x ∈ [0, 𝜁(t, T)] 𝜏i (t,T) [x rf

− 𝜁(t, T)], x ∈ [𝜁(t, T), ld (t, T)] (2.29)

√ ] [ rf Vm Em (T)𝜎 𝜏f (T) rf Vm Em (T)Ef (T) 𝜁(t, T) + − Γd (T) ld (t, T) = 1 − 𝜏i (T) 2 Vf Ec (T)𝜏i (T) Ec (T)𝜏i2 (T) (2.30) Substituting the upstream and downstream temperature-dependent fiber and matrix axial stresses of Eqs. (2.26–2.29), and the temperature-dependent interface debonding length of Eq. (2.30) into Eq. (2.1), the energy balance equation leads to the following equation: 𝛼𝜎 2 + 𝛽𝜎 + 𝛾 = 0

(2.31)

where Vm Em (T)ld (t, T) (2.32) Vf Ef (T)Ec (T) { } 𝜏 (T) 2𝜏 (T) [ld (t, T) − 𝜁(t, T)]2 + f (2.33) 𝛽=− i 𝜁(t, T)[2ld (t, T) − 𝜁(t, T)] rf Ef (T) 𝜏i (T) } [ [ ]2 { ]2 V Ec (T) 𝜏i (T) 3 + 𝜏f (T) 𝜁 3 (t, T) [l (t, T) − 𝜁(t, T)] 𝛾 = 43 V E f(T)E d (T) r 𝜏 (T) 𝛼=

m f

m

f

i

4Vf Ec (T) 𝜏f (T)𝜏i (T) 𝜁(t, T)[ld (t, T) + V E (T)E rf2 m f m (T)

{

[ × ld (t, T) − 1 −

2.4.2

𝜏f (T) 𝜏i (T)

]

− 𝜁(t, T)] } 𝜁(t, T) − Vm Γm (T) −

4Vf ld (t,T) Γd (T) rf

(2.34)

Results and Discussion

The ceramic composite system of C/SiC is used as a case study and its material properties are given by V f = 30%, r f = 3.5 μm, Γm = 25 J/m2 (at room temperature), and Γd = 0.1 J/m2 (at room temperature). When oxidation temperature is below 1000 ∘ C, the oxidation of C/SiC composite is controlled by the carbon/oxygen reaction, as shown in Eq. (2.35), and when the temperature is higher than 1000 ∘ C, the oxidation of C/SiC composite is controlled by the SiC/oxygen reaction, as shown in Eq. (2.36) [40, 41]. In the present analysis, an oxidation temperature below 1000 ∘ C is considered, but the effects of the physical properties of SiO2 on the fiber/matrix interface shear stress and energy and matrix fracture energy have not been considered. C + O2 → 2CO

(2.35)

SiC + 2O2 → SiO2 + CO2

(2.36)

Effects of fiber volume, interface shear stress, interface debonding energy, and matrix fracture energy on time-dependent matrix cracking stress and interface debonding are discussed.

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2 First Matrix Cracking of Ceramic-Matrix Composites at Elevated Temperature

2.4.2.1 Time-Dependent Matrix Cracking Stress of C/SiC Composite for Different Fiber Volumes

Figure 2.16 shows the matrix cracking stress and interface debonding length versus temperature curves of C/SiC composite corresponding to different fiber volumes V f = 25%, 30%, and 35% and oxidation duration t = 0, 1, and 2 hours. When V f = 25% and without oxidation, the matrix cracking stress increases from 𝜎 mc = 48.5 MPa at T = 973 K to 𝜎 mc = 93.1 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 1.5 at T = 973 K to ld /r f = 5.3 at T = 1273 K; when the oxidation time is t = 1 hour, the matrix cracking stress increases from 𝜎 mc = 45.9 MPa at T = 973 K to the peak value 𝜎 mc = 73.9 MPa at T = 1253 K and then decreases to 𝜎 mc = 73.4 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 1.5 at T = 973 K to ld /r f = 5.7 at T = 1273 K, and when oxidation time is t = 2 hour, the matrix cracking stress increases from 𝜎 mc = 43.5 MPa at T = 973 K to the peak value 𝜎 mc = 63.3 MPa at T = 1233 K and then decreases to 𝜎 mc = 63 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 1.5 at T = 973 K to ld /r f = 6.9 at T = 1273 K. When V f = 30% and without oxidation, the matrix cracking stress increases from 𝜎 mc = 48.1 MPa at T = 973 K to 𝜎 mc = 103.6 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 0.6 at T = 973 K to ld /r f = 4.5 at T = 1273 K; when oxidation time is t = 1 hour, the matrix cracking stress increases from 𝜎 mc = 44.8 MPa at T = 973 K to the peak value 𝜎 mc = 79.8 MPa and then decreases to 𝜎 mc = 79.4 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 0.68 at T = 973 K to ld /r f = 5.0 at T = 1273 K, and when oxidation time is t = 2 hours, the matrix cracking stress increases from 𝜎 mc = 41.8 MPa at T = 973 K to 𝜎 mc = 67.5 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 0.7 at T = 973 K to ld /r f = 6.2 at T = 1273 K. When V f = 35% and without oxidation, the matrix cracking stress increases from 𝜎 mc = 47.4 MPa at T = 973 K to 𝜎 mc = 113.8 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 0.08 at T = 973 K to ld /r f = 3.9 at T = 1273 K; when oxidation time is t = 1 hour, the matrix cracking stress increases from 𝜎 mc = 43.4 MPa at T = 973 K to the peak value 𝜎 mc = 85.1 MPa and then decreases to 𝜎 mc = 84.9 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 0.08 at T = 973 K to ld /r f = 4.47 at T = 1273 K, and when oxidation time is t = 2 hours, the matrix cracking stress increases from 𝜎 mc = 39.7 MPa at T = 973 K to 𝜎 mc = 71.8 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 0.1 at T = 973 K to ld /r f = 5.7 at T = 1273 K. 2.4.2.2 Time-Dependent Matrix Cracking Stress of C/SiC Composite for Different Interface Shear Stress

Figure 2.17 shows the matrix cracking stress and interface debonding length versus temperature curves of C/SiC composite corresponding to different interface shear stress 𝜏 0 = 30, 35, and 40 MPa and oxidation duration t = 0, 1, and 2 hours. When 𝜏 0 = 30 MPa and without oxidation, the matrix cracking stress increases from 𝜎 mc = 65 MPa at T = 973 K to 𝜎 mc = 115 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 1.47 at T = 973 K to ld /r f = 4.23

2.4 Time-Dependent Matrix Cracking Stress of C/SiC Composites

Figure 2.16 (a) The matrix cracking stress versus temperature curves when V f = 25% and t = 0, 1, and 2 hours; (b) the interface debonding length versus temperature curves when V f = 25% and t = 0, 1, and 2 hours; (c) the matrix cracking stress versus temperature curves when V f = 30% and t = 0, 1, and 2 hours; (d) the interface debonding length versus temperature curves when V f = 30% and t = 0, 1, and 2 hours; (e) the matrix cracking stress versus temperature curves when V f = 35% and t = 0, 1, and 2 hours; and (f) the interface debonding length versus temperature curves when V f = 35% and t = 0, 1, and 2 hours.

(a)

3 2

(b)

(c)

43

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2 First Matrix Cracking of Ceramic-Matrix Composites at Elevated Temperature

Figure 2.16

(d)

(e)

(f)

(Continued)

2.4 Time-Dependent Matrix Cracking Stress of C/SiC Composites

Figure 2.17 (a) The matrix cracking stress versus temperature curves when 𝜏 0 = 30 MPa and t = 0, 1, and 2 hours; (b) the interface debonding length versus temperature curves when 𝜏 0 = 30 MPa and t = 0, 1, and 2 hours; (c) the matrix cracking stress versus temperature curves when 𝜏 0 = 35 MPa and t = 0, 1, and 2 hours; (d) the interface debonding length versus temperature curves when 𝜏 0 = 35 MPa and t = 0, 1, and 2 hours; (e) the matrix cracking stress versus temperature curves when 𝜏 0, = 40 MPa and t = 0, 1, and 2 hours; and (f) the interface debonding length versus temperature curves when 𝜏 0 = 40 MPa and t = 0, 1, and 2 hours.

(a)

(b)

(c)

45

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2 First Matrix Cracking of Ceramic-Matrix Composites at Elevated Temperature

Figure 2.17

(d)

(e)

(f)

(Continued)

2.4 Time-Dependent Matrix Cracking Stress of C/SiC Composites

at T = 1273 K; when t = 1 hour, the matrix cracking stress increases from 𝜎 mc = 60.5 MPa at T = 973 K to the peak value 𝜎 mc = 84.9 MPa at T = 1213 K and then decreases to 𝜎 mc = 83.3 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 1.47 at T = 973 K to ld /r f = 4.87 at T = 1273 K, and when t = 2 hours, the matrix cracking stress increases from 𝜎 mc = 56.1 MPa at T = 973 K to the peak value 𝜎 mc = 70 MPa at T = 1173 K and then decreases to 𝜎 mc = 68.7 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 1.5 at T = 973 K to ld /r f = 6.3 at T = 1273 K. When 𝜏 0 = 35 MPa and without oxidation, the matrix cracking stress increases from 𝜎 mc = 79.8 MPa at T = 973 K to 𝜎 mc = 125 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 1.88 at T = 973 K to ld /r f = 3.98 at T = 1273 K; when t = 1 hour, the matrix cracking stress increases from 𝜎 mc = 73.9 MPa at T = 973 K to the peak value 𝜎 mc = 90.8 MPa at T = 1173 K and then decreases to 𝜎 mc = 86.3 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 1.89 at T = 973 K to ld /r f = 4.72 at T = 1273 K, and when t = 2 hours, the matrix cracking stress increases from 𝜎 mc = 68.3 MPa at T = 973 K to the peak value 𝜎 mc = 75.3 MPa at T = 1093 K and then decreases to 𝜎 mc = 69.7 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 1.9 at T = 973 K to ld /r f = 6.3 at T = 1273 K. When 𝜏 0 = 40 MPa and without oxidation, the matrix cracking stress increases from 𝜎 mc = 92.9 MPa at T = 973 K to 𝜎 mc = 134 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 2.1 at T = 973 K to ld /r f = 3.7 at T = 1273 K; when t = 1 hour, the matrix cracking stress increases from 𝜎 mc = 85.7 MPa at T = 973 K to the peak value 𝜎 mc = 97.2 MPa at T = 1133 K and then decreases to 𝜎 mc = 88.7 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 2.1 at T = 973 K to ld /r f = 4.6 at T = 1273 K, and when t = 2 hours, the matrix cracking stress increases from 𝜎 mc = 78.9 MPa at T = 973 K to the peak value 𝜎 mc = 82.1 MPa at T = 1053 K and then decreases to 𝜎 mc = 70.4 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 2.1 at T = 973 K to ld /r f = 6.3 at T = 1273 K. Figure 2.18 shows the matrix cracking stress and interface debonding length versus temperature curves of C/SiC composite corresponding to different interface shear stress 𝜏 f = 1, 5, and 10 MPa and oxidation duration t = 0, 1, and 2 hours. When 𝜏 f = 1 MPa and without oxidation, the matrix cracking stress increases from 𝜎 mc = 65 MPa at T = 973 K to 𝜎 mc = 114.9 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 1.4 at T = 973 K to ld /r f = 4.2 at T = 1273 K; when t = 1 hour, the matrix cracking stress increases from 𝜎 mc = 58 MPa at T = 973 K to the peak value 𝜎 mc = 74 MPa at T = 1153 K and then decreases to 𝜎 mc = 67.3 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 1.4 at T = 973 K to ld /r f = 5.1 at T = 1273 K, and when t = 2 hours, the matrix cracking stress increases from 𝜎 mc = 51.3 MPa at T = 973 K to the peak value 𝜎 mc = 56.6 MPa at T = 1073 K and then decreases to 𝜎 mc = 42.2 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 1.5 at T = 973 K to ld /r f = 7.0 at T = 1273 K. When 𝜏 f = 5 MPa and without oxidation, the matrix cracking stress increases from 𝜎 mc = 65 MPa at T = 973 K to 𝜎 mc = 114.9 MPa at T = 1273 K, and the interface

47

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2 First Matrix Cracking of Ceramic-Matrix Composites at Elevated Temperature

(a)

(b)

(c)

Figure 2.18 (a) The matrix cracking stress versus temperature curves when 𝜏 f = 1 MPa and t = 0, 1, and 2 hours; (b) the interface debonding length versus temperature curves when 𝜏 f = 1 MPa and t = 0, 1, and 2 hours; (c) the matrix cracking stress versus temperature curves when 𝜏 f = 5 MPa and t = 0, 1, and 2 hours; (d) the interface debonding length versus temperature curves when 𝜏 f = 5 MPa and t = 0, 1, and 2 hours; (e) the matrix cracking stress versus temperature curves when 𝜏 f = 10 MPa and t = 0, 1, and 2 hours; and (f) the interface debonding length versus temperature curves when 𝜏 f = 10 MPa and t = 0, 1, and 2 hours.

2.4 Time-Dependent Matrix Cracking Stress of C/SiC Composites

Figure 2.18

(Continued)

(d)

(e)

(f)

49

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2 First Matrix Cracking of Ceramic-Matrix Composites at Elevated Temperature

debonding length increases from ld /r f = 1.4 at T = 973 K to ld /r f = 4.2 at T = 1273 K; when t = 1 hour, the matrix cracking stress increases from 𝜎 mc = 59.1 MPa at T = 973 K to the peak value 𝜎 mc = 78.5 MPa at T = 1173 K and then decreases to 𝜎 mc = 74.4 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 1.4 at T = 973 K to ld /r f = 5.0 at T = 1273 K, and when t = 2 hours, the matrix cracking stress increases from 𝜎 mc = 53.5 MPa at T = 973 K to the peak value 𝜎 mc = 61.7 MPa at T = 1113 K and then decreases to 𝜎 mc = 54.3 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 1.5 at T = 973 K to ld /r f = 6.7 at T = 1273 K. When 𝜏 f = 10 MPa and without oxidation, the matrix cracking stress increases from 𝜎 mc = 65 MPa at T = 973 K to 𝜎 mc = 114.9 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 1.4 at T = 973 K to ld /r f = 4.2 at T = 1273 K; when t = 1 hour, the matrix cracking stress increases from 𝜎 mc = 60.5 MPa at T = 973 K to the peak value 𝜎 mc = 84.9 MPa at T = 1213 K and then decreases to 𝜎 mc = 83.2 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 1.4 at T = 973 K to ld /r f = 4.8 at T = 1273 K, and when t = 2 hours, the matrix cracking stress increases from 𝜎 mc = 56.1 MPa at T = 973 K to the peak value 𝜎 mc = 70 MPa at T = 1173 K and then decreases to 𝜎 mc = 68.7 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 1.5 at T = 973 K to ld /r f = 6.3 at T = 1273 K. 2.4.2.3 Time-Dependent Matrix Cracking Stress of C/SiC Composite for Different Interface Frictional Coefficients

Figure 2.19 shows the matrix cracking stress and interface debonding length versus temperature curves of C/SiC composite corresponding to different interface frictional coefficients 𝜇 = 0.03, 0.05, and 0.07 and oxidation time t = 0, 1, and 2 hours. When 𝜇 = 0.03 and without oxidation, the matrix cracking stress increases from 𝜎 mc = 108.8 MPa at T = 973 K to 𝜎 mc = 148.2 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 2.2 at T = 973 K to ld /r f = 3.4 at T = 1273 K; when t = 1 hour, the matrix cracking stress increases from 𝜎 mc = 99.7 MPa at T = 973 K to the peak value 𝜎 mc = 107.4 MPa at T = 1093 K and then decreases to 𝜎 mc = 92.1 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 2.2 at T = 973 K to ld /r f = 4.4 at T = 1273 K, and when t = 2 hours, the matrix cracking stress increases from 𝜎 mc = 91.4 MPa at T = 973 K to the peak value 𝜎 mc = 92.4 MPa at T = 1013 K and then decreases to 𝜎 mc = 71.3 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 2.3 at T = 973 K to ld /r f = 6.3 at T = 1273 K. When 𝜇 = 0.05 and without oxidation, the matrix cracking stress increases from 𝜎 mc = 93.6 MPa at T = 973 K to 𝜎 mc = 144.3 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 2.1 at T = 973 K to ld /r f = 3.5 at T = 1273 K; when t = 1 hour, the matrix cracking stress increases from 𝜎 mc = 86.2 MPa at T = 973 K to the peak value 𝜎 mc = 101.6 MPa at T = 1133 K and then decreases to 𝜎 mc = 91.3 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 2.1 at T = 973 K to ld /r f = 4.4 at T = 1273 K, and when t = 2 hours, the matrix cracking stress increases from 𝜎 mc = 79.4 MPa at T = 973 K to the peak value

2.4 Time-Dependent Matrix Cracking Stress of C/SiC Composites

Figure 2.19 (a) The matrix cracking stress versus temperature curves when 𝜇 = 0.03 and t = 0, 1, and 2 hours; (b) the interface debonding length versus temperature curves when 𝜇 = 0.03 and t = 0, 1, and 2 hours; (c) the matrix cracking stress versus temperature curves when 𝜇 = 0.05 and t = 0, 1, and 2 hours; (d) the interface debonding length versus temperature curves when 𝜇 = 0.05 and t = 0, 1, and 2 hours; (e) the matrix cracking stress versus temperature curves when 𝜇 = 0.07 and t = 0, 1, and 2 hours; and (f) the interface debonding length versus temperature curves when 𝜇 = 0.07 and t = 0, 1, and 2 hours.

(a)

(b)

(c)

51

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2 First Matrix Cracking of Ceramic-Matrix Composites at Elevated Temperature

Figure 2.19

(d)

(e)

(f)

(Continued)

2.4 Time-Dependent Matrix Cracking Stress of C/SiC Composites

𝜎 mc = 84.7 MPa at T = 1073 K and then decreases to 𝜎 mc = 71.1 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 2.1 at T = 973 K to ld /r f = 6.3 at T = 1273 K. When 𝜇 = 0.07 and without oxidation, the matrix cracking stress increases from 𝜎 mc = 76.3 MPa at T = 973 K to 𝜎 mc = 140.3 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 1.8 at T = 973 K to ld /r f = 3.6 at T = 1273 K; when t = 1 hour, the matrix cracking stress increases from 𝜎 mc = 70.7 MPa at T = 973 K to the peak value 𝜎 mc = 97.1 MPa at T = 1173 K and then decreases to 𝜎 mc = 90.3 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 1.8 at T = 973 K to ld /r f = 4.5 at T = 1273 K, and when t = 2 hours, the matrix cracking stress increases from 𝜎 mc = 65.5 MPa at T = 973 K to the peak value 𝜎 mc = 79.1 MPa at T = 1113 K and then decreases to 𝜎 mc = 70.8 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 1.8 at T = 973 K to ld /r f = 6.3 at T = 1273 K. 2.4.2.4 Time-Dependent Matrix Cracking Stress of C/SiC Composite for Different Interface Debonding Energies

Figure 2.20 shows the matrix cracking stress and interface debonding length versus temperature curves of C/SiC composite corresponding to different interface debonding energies Γd = 0.3, 0.5, and 0.7 J/m2 and oxidation durations t = 0, 1, and 2 hours. When Γd = 0.3 J/m2 and without oxidation, the matrix cracking stress increases from 𝜎 mc = 75.2 MPa at T = 973 K to 𝜎 mc = 120.5 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 0.48 at T = 973 K to ld /r f = 3.6 at T = 1273 K; when t = 1 hour, the matrix cracking stress increases from 𝜎 mc = 70.6 MPa at T = 973 K to the peak value 𝜎 mc = 91.7 MPa at T = 1173 K and then decreases to 𝜎 mc = 88.9 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 0.48 at T = 973 K to ld /r f = 4.3 at T = 1273 K, and when t = 2 hours, the matrix cracking stress increases from 𝜎 mc = 66.2 MPa at T = 973 K to the peak value 𝜎 mc = 77.5 MPa at T = 1153 K and then decreases to 𝜎 mc = 74.8 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 0.5 at T = 973 K to ld /r f = 5.7 at T = 1273 K. When Γd = 0.5 J/m2 and without oxidation, the matrix cracking stress increases from 𝜎 mc = 84.2 MPa at T = 973 K to 𝜎 mc = 125.5 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 0 at T = 973 K to ld /r f = 3.3 at T = 1273 K; when t = 1 hour, the matrix cracking stress increases from 𝜎 mc = 79.6 MPa at T = 973 K to the peak value 𝜎 mc = 98 MPa at T = 1173 K and then decreases to 𝜎 mc = 94.2 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 0 at T = 973 K to ld /r f = 3.9 at T = 1273 K, and when t = 2 hours, the matrix cracking stress increases from 𝜎 mc = 75.3 MPa at T = 973 K to the peak value 𝜎 mc = 84.7 MPa at T = 1133 K and then decreases to 𝜎 mc = 80.3 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 0 at T = 973 K to ld /r f = 5.4 at T = 1273 K. When Γd = 0.7 J/m2 and without oxidation, the matrix cracking stress increases from 𝜎 mc = 92.7 MPa at T = 973 K to 𝜎 mc = 130.3 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 0 at T = 973 K to ld /r f = 3.0 at T = 1273 K;

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2 First Matrix Cracking of Ceramic-Matrix Composites at Elevated Temperature

(a)

(b)

(c)

Figure 2.20 (a) The matrix cracking stress versus temperature curves when Γd = 0.3 J/m2 and t = 0, 1, and 2 hours; (b) the interface debonding length versus temperature curves when Γd = 0.3 J/m2 and t = 0, 1, and 2 hours; (c) the matrix cracking stress versus temperature curves when Γd = 0.5 J/m2 and t = 0, 1, and 2 hours; (d) the interface debonding length versus temperature curves when Γd = 0.5 J/m2 and t = 0, 1, and 2 hours; (e) the matrix cracking stress versus temperature curves when Γd = 0.7 J/m2 and t = 0, 1, and 2 hours; and (f) the interface debonding length versus temperature curves when Γd = 0.7 J/m2 and t = 0, 1, and 2 hours.

2.4 Time-Dependent Matrix Cracking Stress of C/SiC Composites

Figure 2.20

(Continued)

(d)

(e)

(f)

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2 First Matrix Cracking of Ceramic-Matrix Composites at Elevated Temperature

when t = 1 hour, the matrix cracking stress increases from 𝜎 mc = 88.1 MPa at T = 973 K to the peak value 𝜎 mc = 104.2 MPa at T = 1173 K and then decreases to 𝜎 mc = 99 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 0 at T = 973 K to ld /r f = 3.7 at T = 1273 K, and when t = 2 hours, the matrix cracking stress increases from 𝜎 mc = 83.8 MPa at T = 973 K to the peak value 𝜎 mc = 91.5 MPa at T = 1133 K and then decreases to 𝜎 mc = 85.5 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 0 at T = 973 K to ld /r f = 5.2 at T = 1273 K.

2.4.2.5 Time-Dependent Matrix Cracking Stress of C/SiC Composite for Different Matrix Fracture Energies

Figure 2.21 shows the matrix cracking stress and interface debonding length versus temperature curves of C/SiC composite corresponding to different matrix fracture energies Γm = 20, 25, and 30 J/m2 and oxidation duration t = 0, 1, and 2 hours. When Γm = 20 J/m2 and without oxidation, the matrix cracking stress increases from 𝜎 mc = 49.2 MPa at T = 973 K to 𝜎 mc = 102.3 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 0.59 at T = 973 K to ld /r f = 3.6 at T = 1273 K; when t = 1 hour, the matrix cracking stress increases from 𝜎 mc = 44.6 MPa at T = 973 K to the peak value 𝜎 mc = 72.1 MPa at T = 1233 K and then decreases to 𝜎 mc = 71.5 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 0.6 at T = 973 K to ld /r f = 4.3 at T = 1273 K, and when t = 2 hours, the matrix cracking stress increases from 𝜎 mc = 40.3 MPa at T = 973 K to 𝜎 mc = 58.3 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 0.62 at T = 973 K to ld /r f = 5.8 at T = 1273 K. When Γm = 25 J/m2 and without oxidation, the matrix cracking stress increases from 𝜎 mc = 65 MPa at T = 973 K to 𝜎 mc = 114.9 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 1.47 at T = 973 K to ld /r f = 4.2 at T = 1273 K; when t = 1 hour, the matrix cracking stress increases from 𝜎 mc = 60.5 MPa at T = 973 K to the peak value 𝜎 mc = 84.9 MPa at T = 1213 K and then decreases to 𝜎 mc = 83.2 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 1.47 at T = 973 K to ld /r f = 4.87 at T = 1273 K, and when t = 2 hours, the matrix cracking stress increases from 𝜎 mc = 56.1 MPa at T = 973 K to the peak value 𝜎 mc = 70 MPa at T = 1173 K and then decreases to 𝜎 mc = 68.7 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 1.5 at T = 973 K to ld /r f = 6.3 at T = 1273 K. When Γm = 30 J/m2 and without oxidation, the matrix cracking stress increases from 𝜎 mc = 79 MPa at T = 973 K to 𝜎 mc = 125.9 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 2.2 at T = 973 K to ld /r f = 4.7 at T = 1273 K; when t = 1 hour, the matrix cracking stress increases from 𝜎 mc = 74.4 MPa at T = 973 K to the peak value 𝜎 mc = 96.6 MPa at T = 1193 K and then decreases to 𝜎 mc = 93.6 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 2.24 at T = 973 K to ld /r f = 5.35 at T = 1273 K, and when t = 2 hours, the matrix cracking stress increases from 𝜎 mc = 70 MPa at T = 973 K to the peak value 𝜎 mc = 81.9 MPa at T = 1153 K and then decreases to 𝜎 mc = 78.1 MPa at T = 1273 K,

2.4 Time-Dependent Matrix Cracking Stress of C/SiC Composites

Figure 2.21 (a) The matrix cracking stress versus temperature curves when Γm = 20 J/m2 and t = 0, 1, and 2 hours; (b) the interface debonding length versus temperature curves when Γm = 20 J/m2 and t = 0, 1, and 2 hours; (c) the matrix cracking stress versus temperature curves when Γm = 25 J/m2 and t = 0, 1, and 2 hours; (d) the interface debonding length versus temperature curves when Γm = 25 J/m2 and t = 0, 1, and 2 hours; (e) the matrix cracking stress versus temperature curves when Γm = 30 J/m2 and t = 0, 1, and 2 hours, and (f) the interface debonding length versus temperature curves when Γm = 30 J/m2 and t = 0, 1, and 2 hours.

(a)

(b)

(c)

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2 First Matrix Cracking of Ceramic-Matrix Composites at Elevated Temperature

Figure 2.21

(d)

(e)

(f)

(Continued)

2.5 Time-Dependent Matrix Cracking Stress of Si/SiC Composites

and the interface debonding length increases from ld /r f = 2.2 at T = 973 K to ld /r f = 6.7 at T = 1273 K.

2.4.3

Experimental Comparisons

Yang et al. [36] investigated the tensile behavior of 2D C/SiC composite at elevated temperatures. The T300-C/SiC composite was fabricated using the CVI method with a pyrolytic carbon interphase of 1.5–2.0 μm. The fiber volume fraction was approximately 40%, and the composite density was approximately 2.0 g/cm3 . Tensile tests were performed under displacement control at a loading speed of 0.3 mm/min. Tensile stress–strain curves of 2D C/SiC composite at T = 973 and 1273 K are shown in Figure 2.6. At T = 973 K, the composite matrix cracking stress was 𝜎 mc = 50 MPa and the tensile strength was 𝜎 UTS = 232 MPa with a failure strain of 0.25%, and at T = 1273 K, the matrix cracking stress was 𝜎 PLS = 80 MPa and the tensile strength was 𝜎 UTS = 271 MPa with a failure strain of 0.33%. Experimental and theoretical time-dependent matrix cracking stress and interface debonding length of 2D C/SiC composite are shown as functions of temperature with oxidation times of 0, 1, and 2 hours, respectively (Figure 2.22). For the case of without oxidation, when the temperature increases, the matrix cracking stress of 2D C/SiC composite increases from 𝜎 mc = 48 MPa at T = 973 K to 𝜎 mc = 82 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 2.7 to ld /r f = 6.3; for the case of oxidation time t = 1 hour, the matrix cracking stress increases from 𝜎 mc = 46.3 MPa at T = 973 K to the peak value 𝜎 mc = 67.3 MPa at T = 1233 K and then decreases to 𝜎 mc = 66.5 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 2.7 to ld /r f = 6.7, and for the case of oxidation time t = 2 hours, the matrix cracking stress increases from 𝜎 mc = 44.4 MPa at T = 973 K to the peak value 𝜎 mc = 58.8 MPa at T = 1233 K and then decreases to 𝜎 mc = 57.7 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 2.7 to ld /r f = 7.8.

2.5 Time-Dependent Matrix Cracking Stress of Si/SiC Composites In this section, synergistic effects of temperature and time on matrix cracking stress of SiC/SiC composites are investigated. Temperature-dependent constituent properties and time-dependent interface oxidation are considered to determine matrix cracking stress. Effects of fiber volume, interface shear stress, interface debonding energy, and matrix fracture energy on time-dependent matrix cracking stress and interface debonding length of SiC/SiC composite are discussed. Experimental matrix cracking stress of SiC/SiC composite corresponding to different temperatures and times are predicted.

2.5.1

Results and Discussion

The ceramic composite system of SiC/SiC is used for the case study and its material properties are given by V f = 30%, r f = 7.5 μm, Γm = 25 J/m2 (at room temperature),

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2 First Matrix Cracking of Ceramic-Matrix Composites at Elevated Temperature

(a)

(b)

Figure 2.22 (a) Experimental and predicted matrix cracking stress versus temperature curves when t = 0, 1, and 2 hours and (b) the interface debonding length versus temperature curves when t = 0, 1, and 2 hours of C/SiC composite.

Γd = 0.1 J/m2 (at room temperature), 𝛼 rf = 2.9 × 10−6 K−1 , and 𝛼 lf = 3.9 × 10−6 K−1 . Effects of fiber volume, interface shear stress, interface frictional coefficient, interface debonding energy, and matrix fracture energy on time-dependent matrix cracking stress and interface debonding are discussed. 2.5.1.1 Time-Dependent Matrix Cracking Stress of SiC/SiC Composite for Different Fiber Volumes

Figure 2.23 shows the matrix cracking stress and interface debonding length versus temperature curves of SiC/SiC composite corresponding to different fiber volumes V f = 25% and 30% and oxidation time t = 0, 1, and 3 hours.

2.5 Time-Dependent Matrix Cracking Stress of Si/SiC Composites

Figure 2.23 (a) The matrix cracking stress versus temperature curves when V f = 25% and t = 0, 1, and 3 hours; (b) the interface debonding length versus temperature curves when V f = 25% and t = 0, 1, and 3 hours; (c) the matrix cracking stress versus temperature curves when V f = 30% and t = 0, 1, and 3 hours; and (d) the interface debonding length versus temperature curves when V f = 30% and t = 0, 1, and 3 hours.

(a)

(b)

(c)

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2 First Matrix Cracking of Ceramic-Matrix Composites at Elevated Temperature

Figure 2.23

(Continued)

(d)

When V f = 25% and without oxidation, the matrix cracking stress increases from 𝜎 mc = 100 MPa at T = 973 K to 𝜎 mc = 87 MPa at T = 1273 K, and the interface debonding length decreases from ld /r f = 5.8 at T = 973 K to ld /r f = 5 at T = 1273 K; when t = 1 hour, the matrix cracking stress decreases from 𝜎 mc = 98 MPa at T = 973 K to 𝜎 mc = 72 MPa at T = 1273 K, and the interface debonding length decreases from ld /r f = 5.8 at T = 973 K to ld /r f = 5.4 at T = 1273 K, and when t = 3 hours, the matrix cracking stress decreases from 𝜎 mc = 94 MPa at T = 973 K to 𝜎 mc = 62 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 5.9 at T = 973 K to ld /r f = 7.6 at T = 1273 K. When V f = 30% and without oxidation, the matrix cracking stress increases from 𝜎 mc = 114 MPa at T = 973 K to 𝜎 mc = 99 MPa at T = 1273 K, and the interface debonding length decreases from ld /r f = 5.1 at T = 973 K to ld /r f = 4.4 at T = 1273 K; when t = 1 hour, the matrix cracking stress decreases from 𝜎 mc = 111 MPa at T = 973 K to 𝜎 mc = 80 MPa at T = 1273 K, and the interface debonding length decreases from ld /r f = 5.1 at T = 973 K to ld /r f = 4.9 at T = 1273 K, and when t = 3 hours, the matrix cracking stress decreases from 𝜎 mc = 105 MPa at T = 973 K to 𝜎 mc = 70 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 5.1 at T = 973 K to ld /r f = 7.2 at T = 1273 K. 2.5.1.2 Time-Dependent Matrix Cracking Stress of SiC/SiC Composite for Different Interface Shear Stress

Figure 2.24 shows the matrix cracking stress and interface debonding length versus temperature curves of SiC/SiC composite corresponding to different interface shear stress 𝜏 0 = 15 and 20 MPa and oxidation time t = 0, 1, and 3 hours. When 𝜏 0 = 15 MPa and without oxidation, the matrix cracking stress decreases from 𝜎 mc = 88.3 MPa at T = 973 K to 𝜎 mc = 80.9 MPa at T = 1273 K, and the interface debonding length decreases from ld /r f = 6.6 at T = 973 K to ld /r f = 5.9 at T = 1273 K; when t = 1 hour, the matrix cracking stress decreases from 𝜎 mc = 87.5 MPa at T = 973 K to 𝜎 mc = 74.6 MPa at T = 1273 K, and the interface debonding length decreases from ld /r f = 6.6 at T = 973 K to ld /r f = 6.1 at T = 1273 K, and when

2.5 Time-Dependent Matrix Cracking Stress of Si/SiC Composites

Figure 2.24 (a) The matrix cracking stress versus temperature curves when 𝜏 0 = 15 MPa and t = 0, 1, and 3 hours; (b) the interface debonding length versus the temperature curves when 𝜏 0 = 15 MPa and t = 0, 1, and 3 hours; (c) the matrix cracking stress versus temperature curves when 𝜏 0 = 20 MPa and t = 0, 1, and 3 hours; and (d) the interface debonding length versus temperature curves when 𝜏 0 = 20 MPa and t = 0, 1, and 3 hours.

(a)

(b)

(c)

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2 First Matrix Cracking of Ceramic-Matrix Composites at Elevated Temperature

Figure 2.24

(Continued)

(d)

t = 2 hours, the matrix cracking stress decreases from 𝜎 mc = 86.1 MPa at T = 973 K to 𝜎 mc = 70.2 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 6.6 at T = 973 K to ld /r f = 7.2 at T = 1273 K. When 𝜏 0 = 20 MPa and without oxidation, the matrix cracking stress decreases from 𝜎 mc = 102 MPa at T = 973 K to 𝜎 mc = 90 MPa at T = 1273 K, and the interface debonding length decreases from ld /r f = 5.7 at T = 973 K to ld /r f = 5.0 at T = 1273 K; when t = 1 hour, the matrix cracking stress decreases from 𝜎 mc = 100.5 MPa at T = 973 K to 𝜎 mc = 78 MPa at T = 1273 K, and the interface debonding length decreases from ld /r f = 5.7 at T = 973 K to ld /r f = 5.3 at T = 1273 K, and when t = 2 hours, the matrix cracking stress decreases from 𝜎 mc = 97.2 MPa at T = 973 K to 𝜎 mc = 70.4 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 5.7 at T = 973 K to ld /r f = 7.2 at T = 1273 K. Figure 2.25 shows the matrix cracking stress and interface debonding length versus temperature curves of SiC/SiC composite corresponding to different interface shear stress 𝜏 s = 10 and 15 MPa and oxidation time t = 0, 1, and 3 hours. When 𝜏 s = 10 MPa and without oxidation, the matrix cracking stress decreases from 𝜎 mc = 114 MPa at T = 973 K to 𝜎 mc = 99 MPa at T = 1273 K, and the interface debonding length decreases from ld /r f = 5.1 at T = 973 K to ld /r f = 4.4 at T = 1273 K; when t = 1 hour, the matrix cracking stress decreases from 𝜎 mc = 111 MPa at T = 973 K to 𝜎 mc = 80 MPa at T = 1273 K, and the interface debonding length decreases from ld /r f = 5.1 at T = 973 K to ld /r f = 4.9 at T = 1273 K, and when t = 2 hours, the matrix cracking stress decreases from 𝜎 mc = 106 MPa at T = 973 K to 𝜎 mc = 70 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 5.1 at T = 973 K to ld /r f = 7.2 at T = 1273 K. When 𝜏 s = 15 MPa and without oxidation, the matrix cracking stress decreases from 𝜎 mc = 114 MPa at T = 973 K to 𝜎 mc = 99 MPa at T = 1273 K, and the interface debonding length decreases from ld /r f = 5.1 at T = 973 K to ld /r f = 4.4 at T = 1273 K; when t = 1 hour, the matrix cracking stress decreases from 𝜎 mc = 112 MPa at T = 973 K to 𝜎 mc = 87 MPa at T = 1273 K, and the interface debonding length decreases from ld /r f = 5.1 at T = 973 K to ld /r f = 4.7 at T = 1273 K, and when

2.5 Time-Dependent Matrix Cracking Stress of Si/SiC Composites

Figure 2.25 (a) The matrix cracking stress versus temperature curves when 𝜏 s = 10 MPa and t = 0, 1, and 3 hours; (b) the interface debonding length versus temperature curves when 𝜏 s = 10 MPa and t = 0, 1, and 3 hours; (c) the matrix cracking stress versus temperature curves when 𝜏 s = 15 MPa and t = 0, 1, and 3 hours; and (d) the interface debonding length versus temperature curves when 𝜏 0 = 15 MPa and t = 0, 1, and 3 hours.

(a)

(b)

(c)

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2 First Matrix Cracking of Ceramic-Matrix Composites at Elevated Temperature

Figure 2.25

(Continued)

(d)

t = 2 hours, the matrix cracking stress decreases from 𝜎 mc = 108 MPa at T = 973 K to 𝜎 mc = 82 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 5.1 at T = 973 K to ld /r f = 6.3 at T = 1273 K.

2.5.1.3 Time-Dependent Matrix Cracking Stress of SiC/SiC Composite for Different Interface Debonding Energies

Figure 2.26 shows the matrix cracking stress and interface debonding length versus temperature curves of SiC/SiC composite corresponding to different interface debonding energies Γd = 0.1 and 0.3 J/m2 and oxidation time t = 0, 1, and 2 hours. When Γd = 0.1 J/m2 and without oxidation, the matrix cracking stress decreases from 𝜎 mc = 114 MPa at T = 973 K to 𝜎 mc = 99 MPa at T = 1273 K, and the interface debonding length decreases from ld /r f = 5.1 at T = 973 K to ld /r f = 4.4 at T = 1273 K; when t = 1 hour, the matrix cracking stress decreases from 𝜎 mc = 111 MPa at T = 973 K to 𝜎 mc = 80 MPa at T = 1273 K, and the interface debonding length decreases from ld /r f = 5.1 at T = 973 K to ld /r f = 4.9 at T = 1273 K, and when t = 2 hours, the matrix cracking stress decreases from 𝜎 mc = 105 MPa at T = 973 K to 𝜎 mc = 70 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 5.1 at T = 973 K to ld /r f = 7.2 at T = 1273 K. When Γd = 0.3 J/m2 and without oxidation, the matrix cracking stress decreases from 𝜎 mc = 121 MPa at T = 973 K to 𝜎 mc = 103 MPa at T = 1273 K, and the interface debonding length decreases from ld /r f = 4.4 at T = 973 K to ld /r f = 3.9 at T = 1273 K; when t = 1 hour, the matrix cracking stress decreases from 𝜎 mc = 118 MPa at T = 973 K to 𝜎 mc = 84 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 4.3 at T = 973 K to ld /r f = 4.4 at T = 1273 K, and when t = 2 hours, the matrix cracking stress decreases from 𝜎 mc = 113 MPa at T = 973 K to 𝜎 mc = 75 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 4.4 at T = 973 K to ld /r f = 6.8 at T = 1273 K.

2.5 Time-Dependent Matrix Cracking Stress of Si/SiC Composites

Figure 2.26 (a) The matrix cracking stress versus temperature curves when Γd = 0.1 J/m2 and t = 0, 1, and 3 hours; (b) the interface debonding length versus temperature curves when Γd = 0.1 J/m2 and t = 0, 1, and 3 hours; (c) the matrix cracking stress versus temperature curves when Γd = 0.3 J/m2 and t = 0, 1, and 3 hours; and (d) the interface debonding length versus temperature curves when Γd = 0.3 J/m2 and t = 0, 1, and 3 hours.

(a)

(b)

(c)

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2 First Matrix Cracking of Ceramic-Matrix Composites at Elevated Temperature

Figure 2.26

(Continued)

(d)

2.5.1.4 Time-Dependent Matrix Cracking Stress of SiC/SiC Composite for Different Matrix Fracture Energies

Figure 2.27 shows the matrix cracking stress and interface debonding length versus temperature curves of C/SiC composite corresponding to different matrix fracture energies Γm = 15 and 20 J/m2 and oxidation durations t = 0, 1, and 2 hours. When Γm = 15 J/m2 and without oxidation, the matrix cracking stress decreases from 𝜎 mc = 114 MPa at T = 973 K to 𝜎 mc = 99 MPa at T = 1273 K, and the interface debonding length decreases from ld /r f = 5.1 at T = 973 K to ld /r f = 4.4 at T = 1273 K; when t = 1 hour, the matrix cracking stress decreases from 𝜎 mc = 111 MPa at T = 973 K to 𝜎 mc = 80 MPa at T = 1273 K, and the interface debonding length decreases from ld /r f = 5.1 at T = 973 K to ld /r f = 4.9 at T = 1273 K, and when t = 2 hours, the matrix cracking stress decreases from 𝜎 mc = 106 MPa at T = 973 K to 𝜎 mc = 70 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 5.1 at T = 973 K to ld /r f = 7.2 at T = 1273 K. When Γm = 20 J/m2 and without oxidation, the matrix cracking stress decreases from 𝜎 mc = 128 MPa at T = 973 K to 𝜎 mc = 110 MPa at T = 1273 K, and the interface debonding length decreases from ld /r f = 5.9 at T = 973 K to ld /r f = 5.0 at T = 1273 K; when t = 1 hour, the matrix cracking stress decreases from 𝜎 mc = 125 MPa at T = 973 K to 𝜎 mc = 91 MPa at T = 1273 K, and the interface debonding length decreases from ld /r f = 5.9 at T = 973 K to ld /r f = 5.4 at T = 1273 K, and when t = 2 hours, the matrix cracking stress decreases from 𝜎 mc = 120 MPa at T = 973 K to 𝜎 mc = 79 MPa at T = 1273 K, and the interface debonding length increases from ld /r f = 6.0 at T = 973 K to ld /r f = 7.7 at T = 1273 K.

2.5.2

Experimental Comparisons

Guo and Kagawa [38] investigated tensile behavior of 2D SiC/SiC composites with PyC and BN interphase at elevated temperature. Figure 2.28 shows the experimental and predicted matrix cracking stress versus temperature curves. For 2D SiC/SiC composite with the PyC interphase without oxidation, the experimental matrix cracking stress decreases from 𝜎 mc = 65 MPa at T = 298 K

2.5 Time-Dependent Matrix Cracking Stress of Si/SiC Composites

Figure 2.27 (a) The matrix cracking stress versus temperature curves when Γm = 15 J/m2 and t = 0, 1, and 3 hours; (b) the interface debonding length versus temperature curves when Γm = 15 J/m2 and t = 0, 1, and 3 hours; (c) the matrix cracking stress versus temperature curves when Γm = 20 J/m2 and t = 0, 1, and 3 hours; and (d) the interface debonding length versus temperature curves when Γm = 20 J/m2 and t = 0, 1, and 3 hours.

(a)

(b)

(c)

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2 First Matrix Cracking of Ceramic-Matrix Composites at Elevated Temperature

Figure 2.27

(Continued)

(d)

(a)

(b)

Figure 2.28 Experimental and predicted matrix cracking stress versus temperature curves of (a) SiC/SiC composite with PyC interphase and (b) SiC/SiC composite with BN interphase.

References

to 𝜎 mc = 33 MPa at T = 1200 K, the predicted matrix cracking stress decreases from 𝜎 mc = 63 MPa at T = 298 K to 𝜎 mc = 35 MPa at T = 1200 K, and the interface debonding length decreases from ld /r f = 12.8 to ld /r f = 7.3; when t = 1 hour, the matrix cracking stress decreases from 𝜎 mc = 63 MPa at T = 298 K to 𝜎 mc = 31 MPa at T = 1200 K, and the interface debonding length decreases from ld /r f = 12.8 to ld /r f = 7.7; when t = 3 hours, the matrix cracking stress decreases from 𝜎 mc = 63 MPa at T = 298 K to 𝜎 mc = 28 MPa at T = 1200 K, and the interface debonding length decreases from ld /r f = 12.8 to ld /r f = 9.8. For 2D SiC/SiC composite with the BN interphase without oxidation, the experimental matrix cracking stress decreases from 𝜎 mc = 75 MPa at T = 298 K to 𝜎 mc = 45 MPa at T = 1400 K, the predicted matrix cracking stress decreases from 𝜎 mc = 84 MPa at T = 298 K to 𝜎 mc = 33 MPa at T = 1400 K, and the interface debonding length decreases from ld /r f = 7.3 to ld /r f = 2.9; when t = 1 hour, the matrix cracking stress decreases from 𝜎 mc = 84 MPa at T = 298 K to 𝜎 mc = 25 MPa at T = 1400 K, and the interface debonding length decreases from ld /r f = 7.3 to ld /r f = 3.8; when t = 3 hours, the matrix cracking stress decreases from 𝜎 mc = 84 MPa at T = 298 K to 𝜎 mc = 23 MPa at T = 1400 K, and the interface debonding length decreases from ld /r f = 7.3 to ld /r f = 7.1.

2.6 Conclusion In this chapter, first matrix cracking of fiber-reinforced CMCs was investigated using the micromechanical approach. Temperature- and time-dependent micromechanical parameters were incorporated into analysis of the microstress analysis, interface debonding, and matrix cracking. Relationships between the first matrix cracking stress, interface debonding, temperature, and time were established. Experimental first matrix cracking stress of C/SiC and SiC/SiC composites at elevated temperature was predicted. ●

●

●

●

With increasing temperature, the first matrix cracking stress of C/SiC composite increased, and the first matrix cracking stress of SiC/SiC composite decreased. When the fiber volume, interface shear stress, and interface debonding energy increased, the matrix cracking stress at the same temperature increased, and the interface debonding length decreased. When matrix fracture energy increases, the matrix cracking stress and the interface debonding length increased at the same temperature. When oxidation time increased, the matrix cracking stress decreased, and the interface debonding length increased.

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30 Barsoum, M.W., Kangutkar, P., and Wang, A.S.D. (1992). Matrix crack initiation in ceramic matrix composites Part I: experiments and test results. Compos. Sci. Technol. 44: 257–269. https://doi.org/10.1016/0266-3538(92)90016-V. 31 Reynaud, P., Rouby, D., and Fantozzi, G. (1994). Effects of interfacial evolution on the mechanical behavior of ceramic matrix composites during cyclic fatigue. Scr. Metal. Mater. 31: 1061–1066. https://doi.org/10.1016/0956-716X(94)90527-4. 32 Sauder, C., Lamon, J., and Pailler, R. (2004). The tensile behavior of carbon fibers at high temperatures up to 2400 ∘ C. Carbon 42: 715–725. https://doi.org/10 .1016/j.carbon.2003.11.020. 33 Snead, L.L., Nozawa, T., Katoh, Y. et al. (2007). Handbook of SiC properties for fuel performance modeling. J. Nucl. Mater. 371: 329–377. https://doi.org/10.1016/ j.jnucmat.2007.05.016. 34 Pradere, C. and Sauder, C. (2008). Transverse and longitudinal coefficient of thermal expansion of carbon fibers at high temperatures (300–2500 K). Carbon 46: 1874–1884. https://doi.org/10.1016/j.carbon.2008.07.035. 35 Wang, R.Z., Li, W.G., Li, D.Y., and Fang, D.N. (2015). A new temperature dependent fracture strength model for the ZrB2 -SiC composites. J. Eur. Ceram. Soc. 35: 2957–2962. https://doi.org/10.1016/j.jeurceramsoc.2015.03.025. 36 Yang, C., Zhang, L., Wang, B. et al. (2017). Tensile behavior of 2D-C/SiC composites at elevated temperatures: experiment and modeling. J. Eur. Ceram. Soc. 37: 1281–1290. https://doi.org/10.1016/j.jeurceramsoc.2016.11.011. 37 Fantozzi, G., Reynaud, P., and Rouby, D. (2001). Thermomechanical behavior of long fibers ceramic–ceramic composites. Silic. Indus. 66: 109–119. 38 Guo, S. and Kagawa, Y. (2002). Tensile fracture behavior of continuous SiC fiber-reinforced SiC matrix composites at elevated temperatures and correlation to in situ constituent properties. J. Eur. Ceram. Soc. 22: 2349–2356. https://doi .org/10.1016/S0955-2219(02)00028-6. 39 Casas, L. and Martinez-Esnaola, J.M. (2003). Modelling the effect of oxidation on the creep behavior of fiber-reinforced ceramic matrix composites. Acta Mater. 51: 3745–3757. https://doi.org/10.1016/S1359-6454(03)00189-7. 40 Lamourous, F., Camus, G., and Thebault, J. (1994). Kinetics and mechanisms of oxidation of 2D woven C/SiC composites: I, Experimental approach. J. Am. Ceram. Soc. 77: 2049–2057. https://doi.org/10.1111/j.1151-2916.1994.tb07096.x. 41 Lamoroux, F., Naslain, R., and Jouin, J.-M. (1994). Kinetics and mechanisms of oxidation of 2D woven C/SiC composites: II, Theoretical approach. J. Am. Ceram. Soc. 77: 2058–2068. https://doi.org/10.1111/j.1151-2916.1994.tb07097.x.

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3 Matrix Multiple Cracking Evolution of Fiber-Reinforced Ceramic-Matrix Composites at Elevated Temperature 3.1 Introduction Ceramic-matrix composites (CMCs) not only possess higher mechanical properties, wear, and corrosion resistance than metal materials at elevated temperature but also have better strength and toughness than ceramics [1]. In recent years, the CMCs have already been successfully applied on some aeroengine components [2]. The SAFRAN designed, built, and ground tested the CMC exhaust cone demonstrator, which was certified on 22 April 2015 by the European Aviation Safety Agency (EASA) for use on commercial flights after carrying out its initial test on an A320 in 2012. On 16 June 2015, the CMC exhaust cone designed by SAFRAN company Herakles made its first commercial flight on a CFM56-5B engine powering an Air France Airbus A320 jetliner, which is the first time in the world that a CMC part has flown on a jetliner in commercial service. The application of CMCs on aeroengines has the following advantages (i) as the specific strength of CMCs is higher than that of nickel-based alloys and the weight of turbine components made of nickel-based alloys can be reduced by 61% if using CMCs; (ii) due to the high temperature resistance of CMCs, the turbine temperature can be increased up to 1650 ∘ C, which is beneficial to increase the thrust–weight ratio of the aeroengine; (iii) compared with ceramics, the strain tolerance of CMCs is greatly improved, which will not cause catastrophic damage, and it is possible to detect the mechanical degradation before materials failure and improve the reliability of life prediction; and (iv) replacing the existing high-temperature metal materials with CMCs can reduce the weight, pollution emission, and noise level of the aeroengine. At elevated temperatures, matrix cracking is an important failure mode of CMCs [3–7]. The generation and propagation of matrix cracking consume the energy inside of CMCs, which slows down or prevent the further matrix cracking propagation and achieves the toughness behavior [8–12]. The nonlinearity of stress–strain curves of CMCs is mainly caused by matrix cracking and propagation [13–21]. The damage stage from first matrix cracking to saturation of matrix cracking affects the deformation characteristics of CMCs. For the multiple matrix cracking problem, the maximum stress criterion is firstly used to predict the generation of matrix multiple cracking; however, the predicted matrix cracking approaches to saturation at single applied stress, leading to the step behavior on the predicted stress–strain curve High Temperature Mechanical Behavior of Ceramic-Matrix Composites, First Edition. Longbiao Li. © 2021 WILEY-VCH GmbH. Published 2021 by WILEY-VCH GmbH.

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3 Matrix Multiple Cracking Evolution of Fiber-Reinforced CMCs at Elevated Temperature

[22, 23]; the matrix interaction cracking model involving calculation of the steady-state strain energy release rate considered the energy change before and after steady-state matrix cracking and matrix flaw density [24]; the energy balance approach is also used to predict the matrix multiple cracking by establishing the energy balance relationship between two different matrix cracking spaces or configurations [25–28]; the critical matrix strain energy (CMSE) criterion presumes the critical value of the matrix strain energy, when the matrix strain energy is higher than the critical value, the additional matrix strain energy is dissipated through new cracking and fiber/matrix interface debonding [29, 30]; the matrix statistical cracking model considered the matrix flaw distribution using the Weibull model and divided the matrix cracking space into three cases, i.e. long matrix crack spacing, medium matrix crack spacing, and short matrix crack spacing, with increasing applied stress, the long and medium matrix crack spacing continually transferred to the short matrix crack spacing because of the interaction effect between neighboring cracks, when all the matrix crack spacings are less than the fiber/matrix interface debonded length, and the matrix cracking approaches to the saturation [31, 32]. However, most of the existing theoretical matrix cracking models for the fiber-reinforced CMCs can only be used at normal temperature ranges [33, 34], and there are few experimental investigations on matrix cracking at elevated temperature [35, 36]. The temperature dependence of the fiber/matrix interface shear stress, Young’s modulus of matrix and fibers, and the matrix fracture energy and fiber/matrix interface debonding energy in fiber-reinforced CMCs affects the matrix multi-cracking evolution [37–42]. The principal objective of the present study is to develop a temperature-dependent matrix multiple cracking model for the fiber-reinforced CMCs. In this chapter, temperature-dependent matrix multiple cracking evolution of fiber-reinforced CMCs is investigated using the CMSE criterion. The temperature-dependent fiber/matrix interface shear stress, Young’s modulus of the fibers and the matrix, the matrix fracture energy, and the fiber/matrix interface debonding energy are considered in the microstress field analysis, fiber/matrix interface debonding criterion, and matrix multiple cracking evolution model. Effects of fiber volume, fiber/matrix interface shear stress, fiber/matrix interface debonding energy, matrix fracture energy, temperature and duration on matrix multiple cracking evolution, and fiber/matrix interface debonding are discussed. Experimental matrix multiple cracking evolution and fiber/matrix interface debonding of unidirectional C/SiC composite at elevated temperatures are predicted.

3.2 Temperature-Dependent Matrix Multiple Cracking Evolution of C/SiC Composites In this section, temperature-dependent matrix multiple cracking evolution of fiber-reinforced CMCs is investigated using the CMSE criterion. Temperaturedependent fiber/matrix interface shear stress, Young’s modulus of the fibers and the

3.2 Temperature-Dependent Matrix Multiple Cracking Evolution of C/SiC Composites

matrix, matrix fracture energy, and the interface debonding energy are considered in the microstress field analysis, interface debonding criterion, and matrix multiple cracking evolution model. Effects of interface shear stress, interface debonding energy, matrix fracture energy, and temperature on temperature-dependent matrix multiple cracking evolution and interface debonding are discussed. Experimental matrix multiple cracking evolution and interface debonding of unidirectional C/SiC composite at elevated temperatures are predicted.

3.2.1

Theoretical Models

The temperature affects the constituent properties and damage behavior of fiber-reinforced CMCs, i.e. matrix multi-cracking, fiber/matrix interface debonding, and thermal residual stress. If the radial thermal expansion coefficient of the matrix is higher than the coefficient of the fibers, at a testing temperature lower than the processing temperature, the radial thermal residual stresses are compressive stresses. The temperature-dependent fiber/matrix interface shear stress (𝜏 i (T)) can be described using the following equation [43]. |𝛼 (T) − 𝛼rm (T)|(Tm − T) (3.1) 𝜏i (T) = 𝜏0 + 𝜇 rf A where 𝜏 0 denotes the steady-state fiber/matrix interface shear stress, 𝜇 denotes the fiber/matrix interface frictional coefficient, 𝛼 rf and 𝛼 rm denote the temperaturedependent fiber and matrix radial thermal expansion coefficient, respectively, T m and T denote the processing temperature and testing temperature, and A is a constant depending on the elastic properties of the matrix and fibers. 3.2.1.1 Temperature-Dependent Stress Analysis

The shear-lag model is adopted to perform temperature-dependent stress and strain calculations in the interface debonding region (x ∈ [0, ld (T)]) and interface bonding region (x ∈ [ld (T), lc (T)/2]). Temperature-dependent fiber axial stress 𝜎 f (x, T), matrix axial stress 𝜎 m (x, T), and the interface shear stress 𝜏 i (x, T) are ⎧ 𝜎 − 2𝜏i (T) x, x ∈ [0, ld (T)] rf ⎪ Vf [ ] [ ] [ ] 𝜎f (x, T) = ⎨ l (T) x−l (T) l (T) V , x ∈ ld (T), c 2 ⎪𝜎fo + Vm 𝜎mo − 2 dr 𝜏i (T) exp −𝜌 rd f f f ⎩ (3.2) ⎧2𝜏 (T) Vf x , Vm rf ⎪ i [ 𝜎m (x, T) = ⎨ V ⎪𝜎mo − 𝜎mo − 2𝜏i (T) V f m ⎩

ld (T) rf

]

x ∈ [0, ld (T)] [ ] [ ] 𝜌(x−l (T)) l (T) exp − rd , x ld (T), c 2 f

(3.3) ⎧𝜏 (T), x ∈ [0, ld (T)] ⎪ i ] [ ] [ ] 𝜏i (x, T) = ⎨ 𝜌 [ V ld (T) 𝜌(x−ld (T)) l (T) m , x ∈ ld (T), c 2 ⎪ 2 Vf 𝜎mo − 2𝜏i (T) rf exp − rf ⎩

(3.4)

77

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3 Matrix Multiple Cracking Evolution of Fiber-Reinforced CMCs at Elevated Temperature

where 𝜎 denotes the applied stress, V f and V m denote the fiber and matrix volume, respectively, r f denotes the fiber radius, 𝜏 i (T) denotes the temperature-dependent interface shear stress, ld and lc denote the temperature-dependent interface debonding length and matrix crack spacing, 𝜌 denotes the shear-lag model parameter, and 𝜎 fo and 𝜎 mo denote the temperature-dependent fiber and matrix axial stress in the interface bonding region, respectively. 𝜎fo (T) = 𝜎mo (T) =

Ef (T) 𝜎 + Ef (T)[𝛼lc (T) − 𝛼lf (T)]ΔT Ec (T)

(3.5)

Em (T) 𝜎 + Em (T)[𝛼lc (T) − 𝛼lm (T)]ΔT Ec (T)

(3.6)

where Ef (T), Em (T), and Ec (T) denote the temperature-dependent fiber, matrix, and composite elastic modulus, respectively, 𝛼 lf (T), 𝛼 lm (T), and 𝛼 lc (T) denote the temperature-dependent fiber, matrix, and composite axial thermal expansion coefficient, respectively, and ΔT denotes the temperature difference between testing and fabricated temperature. 3.2.1.2 Temperature-Dependent Interface Debonding

When matrix cracking propagates to the fiber/matrix interface, it will deflect along the interface. The fracture mechanics approach is adopted to determine the temperature-dependent fiber/matrix interface debonding length. The fiber/matrix interface debonding criterion can be described using the following equation [44]. Γd (T) =

ld 𝜕v(T) F 𝜕wf (T) 1 − 𝜏 (T) dx 4𝜋rf 𝜕ld 2 ∫0 i 𝜕ld

(3.7)

where F(= 𝜋rf2 𝜎∕Vf ) denotes the fiber load at the matrix cracking plane, wf (T) denotes the temperature-dependent fiber axial displacement at the matrix cracking plane, and v(T) denotes the temperature-dependent relative displacement between the fiber and the matrix. The temperature-dependent axial displacements of the fiber and matrix, i.e. wf (x, T) and wm (x, T), are wf (x, T) = =

lc 2

∫x

𝜎f (T) dx Ef (T)

𝜏 (T) 2 𝜎 (l − x) − i (l − x2 ) Vf Ef (T) d rf Ef (T) d r V E (T) 2𝜏 (T) 𝜎 ld + f m m 𝜎+ (l ∕2 − ld ) − i 𝜌Ef (T) 𝜌Vf Ef (T)Ec (T) Ec (T) c lc 2

(3.8)

𝜎m (T) dx Em (T) ( ) lc Vf 𝜏i (T) 2Vf 𝜏i (T) rf 𝜎 (l2d − x2 ) + ld − 𝜎+ − ld = rf Vm Em (T) 𝜌Vm Em (T) 𝜌Ec (T) Ec (T) 2 (3.9)

wm (x) =

∫x

3.2 Temperature-Dependent Matrix Multiple Cracking Evolution of C/SiC Composites

Temperature-dependent relative displacement between the fiber and the matrix, i.e. v(x, T), are 𝜏i (T)Ec (T) 𝜎 (ld − x) − (l2 − x2 ) Vf Ef (T) Vm Em (T)Ef (T)rf d 2𝜏i (T)Ec (T)ld rf + 𝜎 (3.10) − 𝜌Vm Em (T)Ef (T) 𝜌Vf Ef (T)

v(x, T) = |wf (x, T) − wm (x, T)| =

Substituting wf (x = 0, T) and v(x, T) into Eq. (3.7), it leads to ] [ Ec (T)𝜏i2 (T) 2 Ec (T)𝜏i2 (T) 𝜏i (T)𝜎 − l l + Vm Em (T)Ef (T)rf d 𝜌Vm Em (T)Ef (T) Vf Ef (T) d +

rf Vm Em (T)𝜎 2 4Vf 2 Ef (T)Ec (T)

−

rf 𝜏i (T) 𝜎 − 𝛾d (T) = 0 2𝜌Vf Ef (T)

(3.11)

Solving Eq. (3.11), temperature-dependent fiber/matrix interface debonding length ld (T) is √ [ ] √ √( r )2 r V E (T)E (T) rf Vm Em (T)𝜎 1 Γd − −√ f + f m m 2 f (3.12) ld (T) = 2 Vf Ec (T)𝜏i (T) 𝜌 2𝜌 Ec (T)𝜏i (T) 3.2.1.3 Temperature-Dependent Matrix Multiple Cracking

Solti et al. [29] developed the CMSE criterion to predict the matrix multi-cracking in fiber-reinforced CMCs. In the present analysis, the effects of temperature on the constituent properties and internal damage in fiber-reinforced CMCs are considered in the CMSE criterion. Temperature-dependent matrix strain energy can be described using the following equation. Um (T) =

1 2Em (T) ∫Am ∫0

lc (T)

2 𝜎m (T) dx dAm

(3.13)

where Am is the cross-sectional area of matrix in the unit cell. Substituting the temperature-dependent matrix axial stresses in Eq. (3.3) into Eq. (3.13), the matrix strain energy considering matrix multi-cracking and fiber/matrix interface partially debonding, can be described using the following equation. { [ ]2 [ ] l (T) Am 4 Vf 𝜏i (T) 2 Um (T) = − ld (T) ld (T) ld (T) + 𝜎mo (T) c Em (T) 3 Vm rf 2 [ )] ][ ( l (T)∕2 − ld (T) r 𝜎 (T) 2Vf 𝜏i (T)ld (T) − 𝜎mo (T) 1−exp −𝜌 c +2 f mo 𝜌 Vm rf rf [ ( )]} ]2 [ lc (T)∕2 − ld (T) rf 2Vf 𝜏i (T)ld (T) 1 − exp −2𝜌 − 𝜎mo (T) + 2𝜌 Vm rf rf (3.14) When the fiber/matrix interface completely debonds, the matrix strain energy can be described using the following equation. [ ] Am l3c (T) V𝜏i (T)f 2 (3.15) Um (T) = 6Em (T) rf Vm

79

80

3 Matrix Multiple Cracking Evolution of Fiber-Reinforced CMCs at Elevated Temperature

Evaluating matrix strain energy at a critical stress 𝜎 cr , the CMSE of U mcr can be obtained. The CMSE can be described using the following equation. Umcr (T) =

𝜎 2 (T) 1 kAm l0 mocr 2 Em (T)

(3.16)

where k (k ∈ [0,1]) is the CMSE parameter, l0 is the initial matrix crack spacing, and 𝜎 mocr (T) can be described using the following equation. 𝜎mocr (T) =

Em (T) 𝜎 (T) + Em (T)[𝛼lc (T) − 𝛼lm (T)]ΔT Ec (T) cr

(3.17)

where 𝜎 cr (T) denotes the temperature-dependent critical stress corresponding to composite’s proportional limit stress, i.e. the stress at which the stress–strain curve starts to deviate from linearity because of the damage accumulation of matrix cracking [13]. The critical stress is defined to be ACK matrix cracking stress [38], which was determined using the energy balance criterion, involving the calculation of energy balance relationship before and after the formation of a single dominant cracking [38]. [ 2 ]1 6Vf Ef (T)Ec2 (T)𝜏i (T)Γm (T) 3 − Ec (T)[𝛼lc (T) − 𝛼lm (T)]ΔT (3.18) 𝜎cr (T) = 2 rf Vm Em (T) where Γm (T) denotes the temperature-dependent matrix fracture energy. Matrix multiple cracking evolution can be determined using the following equation. Um (𝜎 > 𝜎cr , T) = Ucrm (𝜎cr , T)

(3.19)

Matrix cracking density is determined using the following equation. 𝜓=

1000 lc

(3.20)

The interface debonding ratio is defined as 𝜂=

3.2.2

2ld lc

(3.21)

Results and Discussion

Under tensile loading of fiber-reinforced CMCs, matrix multiple cracking and fiber/matrix interface debonding occur with increasing applied stress. The CMSE criterion is adopted in the present analysis to determine the matrix crack spacing for different applied stress levels. By combining Eqs. (3.1, 3.12, 3.14, 3.16, 3.19), the matrix multiple cracking evolution when the fiber/matrix interface partially debonding can be determined; combining Eqs. (3.1, 3.15, 3.16, 3.19), the matrix multiple cracking evolution when the fiber/matrix interface completely debonding can be determined. The temperature-dependent matrix elastic modulus, matrix axial, and radial thermal expansion coefficient, fiber/matrix interface debonded energy, and matrix fracture energy are given in the following equations.

3.2 Temperature-Dependent Matrix Multiple Cracking Evolution of C/SiC Composites

The temperature-dependent carbon fiber elastic modulus of Ef (T) can be described using the following equation [45]. )] [ ( T , T < 2273 K (3.22) Ef (T) = 230 1 − 2.86 × 10−4 exp 324 The temperature-dependent SiC matrix elastic modulus of Em (T) can be described using the following equation [46]. [ ( )] 962 350 460 − 0.04T exp − , T ∈ [300 K 1773 K] (3.23) Em (T) = 460 T The temperature-dependent carbon fiber axial and radial thermal expansion coefficient of 𝛼 lf (T) and 𝛼 rf (T) can be described using the following equation, respectively [47]. 𝛼lf (T) = 2.529 × 10−2 − 1.569 × 10−4 T + 2.228 × 10−7 T 2 [ ] − 1.877 × 10−11 T 3 − 1.288 × 10−14 T 4 , T ∈ 300 K 2500 K

(3.24)

𝛼rf (T) = − 1.86 × 10−1 + 5.85 × 10−4 T − 1.36 × 10−8 T 2 [ ] + 1.06 × 10−22 T 3 , T ∈ 300 K 2500 K

(3.25)

The temperature-dependent SiC matrix axial and radial thermal expansion coefficient of 𝛼 lm (T) and 𝛼 rm (T) can be described using the following equation, respectively [46]. ⎧−1.8276 + 0.0178T − 1.5544 T ∈ [125 K 1273K] ⎪ 𝛼lm (T) = 𝛼rm (T) = ⎨ ×10−5 T 2 + 4.5246 × 10−9 T 3 , ⎪5.0 × 10−6 K−1 , T > 1273 K ⎩ (3.26) The temperature-dependent fiber/matrix interface debonding energy Γd (T) and the matrix fracture energy Γm (T) can be described using the following equation, respectively [48]. T ⎡ ∫T CP (T) dT ⎤ ⎥ ⎢ Γd (T) = Γdo 1 − To m ⎥ ⎢ ∫ C (T) dT P ⎦ ⎣ To

(3.27)

T ⎡ ∫T CP (T) dT ⎤ ⎥ ⎢ Γm (T) = Γmo 1 − To m ⎥ ⎢ ∫ C (T) dT P ⎦ ⎣ To

(3.28)

where T o denotes the reference temperature, T m denotes the fabricated temperature, Γdo and Γmo denote the fiber/matrix interface debonded energy and matrix fracture energy at the reference temperature of T o , respectively, and CP (T) can be described using the following equation. CP (T) = 76.337 + 109.039 × 10−3 T − 6.535 × 105 T −2 − 27.083 × 10−6 T 2 (3.29) The ceramic composite system of C/SiC is used for the case study, and its material properties are given by V f = 30%, r f = 7.5 μm, Γm = 6 J/m2 (at room temperature),

81

3 Matrix Multiple Cracking Evolution of Fiber-Reinforced CMCs at Elevated Temperature

Γd = 0.4 J/m2 (at room temperature), and 𝜏 i = 20 MPa (at room temperature). Effects of interface shear stress, interface debonding energy, matrix fracture energy, and temperature on the temperature-dependent matrix multiple cracking evolution and fiber/matrix interface debonding are discussed. 3.2.2.1 Temperature-Dependent Matrix Multiple Cracking of C/SiC Composite for Different Interface Shear Stress

Figure 3.1 shows the matrix cracking density and fiber/matrix interface debonding ratio of C/SiC composite at T = 773, 873, 973, and 1073 K for different interface shear stress (i.e. 𝜏 0 = 30 and 35 MPa). For 𝜏 0 = 30 MPa, when T = 773 K, matrix cracking density increases from 𝜓 = 0.08 mm−1 at the first matrix cracking stress 𝜎 mc = 53 MPa to 𝜓 = 1.2 mm−1 at

Matrix cracking density (mm–1)

2.5

2.0

1.5

1.0 1

0.5

0.0

0

2

50

1: T = 773 K 2: T = 873 K 3: T = 973 K 4: T = 1073 K

4

3

100

(a)

150

200

250

300

Stress (MPa) 0.7 0.6

Interface debonding ratio

82

0.5 0.4 0.3 1

0.2

2

3

1: T = 773 K 2: T = 873 K 3: T = 973 K 4: T = 1073 K

4

0.1 0.0

(b)

0

50

100

150

200

250

300

Stress (MPa)

Figure 3.1 (a) The matrix cracking density versus applied stress curves when 𝜏 0 = 30 MPa; (b) the fiber/matrix interface debonding ratio versus applied stress curves when 𝜏 0 = 30 MPa; (c) the matrix cracking density versus applied stress curves when 𝜏 0 = 35 MPa; and (d) the fiber/matrix interface debonding ratio versus applied stress curves when 𝜏 0 = 35 MPa of C/SiC composite at elevated temperatures of T = 773, 873, 973, and 1073 K.

3.2 Temperature-Dependent Matrix Multiple Cracking Evolution of C/SiC Composites

Matrix cracking density (mm–1)

3.0 2.5 2.0 1.5 1: T = 773 K 2: T = 873 K 3: T = 973 K 4: T = 1073 K

1.0 1 2

3

4

0.5 0.0

0

50

100

(c)

150

200

250

300

350

Stress (MPa) 0.8

Interface debonding ratio

0.7

4

0.6

3

0.5

2

0.4

1

0.3

1: T = 773 K 2: T = 873 K 3: T = 973 K 4: T = 1073 K

0.2 0.1 0.0

0

50

100

(d)

Figure 3.1

150

200

250

300

350

Stress (MPa)

(Continued)

the matrix cracking saturation stress 𝜎 sat = 107 MPa, and the interface debonding ratio increases from 𝜂 = 0.9% to 30.2%; when T = 873 K, the matrix cracking density increases from 𝜓 = 0.07 mm−1 at the first matrix cracking stress 𝜎 mc = 77 MPa to 𝜓 = 1.5 mm−1 at the matrix cracking saturation stress 𝜎 sat = 154 MPa, and the interface debonding ratio increases from 𝜂 = 0.9% to 40.6%; when T = 973 K, the matrix cracking density increases from 𝜓 = 0.07 mm−1 at the first matrix cracking stress 𝜎 mc = 104 MPa to 𝜓 = 1.9 mm−1 at the matrix cracking saturation stress 𝜎 sat = 209 MPa, and the interface debonding ratio increases from 𝜂 = 0.98% to 51.9%, and when T = 1073 K, the matrix cracking density increases from 𝜓 = 0.08 mm−1 at the first matrix cracking stress 𝜎 mc = 134 MPa to 𝜓 = 2.2 mm−1 at the matrix cracking saturation stress 𝜎 sat = 268 MPa, and the interface debonding ratio increases from 𝜂 = 1% to 62.8%. For 𝜏 0 = 35 MPa, when T = 773 K, the matrix cracking density increases from 𝜓 = 0.09 mm−1 at the first matrix cracking stress 𝜎 mc = 73 MPa to 𝜓 = 1.7 mm−1

83

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3 Matrix Multiple Cracking Evolution of Fiber-Reinforced CMCs at Elevated Temperature

at the matrix cracking saturation stress 𝜎 sat = 146 MPa, and the interface debonding length increases from 𝜂 = 0.9% to 36.6%; when T = 873 K, the matrix cracking density increases from 𝜓 = 0.09 mm−1 at the first matrix cracking stress 𝜎 mc = 95 MPa to 𝜓 = 2.0 mm−1 at the matrix cracking saturation stress 𝜎 sat = 189 MPa, and the interface debonding ratio increases from 𝜂 = 0.9% to 45.3%; when T = 973 K, the matrix cracking density increases from 𝜓 = 0.09 mm−1 at the first matrix cracking stress 𝜎 mc = 121 MPa to 𝜓 = 2.3 mm−1 at the matrix cracking saturation stress 𝜎 sat = 242 MPa, and the interface debonding ratio increases from 𝜂 = 0.98% to 55.3%, and when T = 1073 K, the matrix cracking density increases from 𝜓 = 0.09 mm−1 at the first matrix cracking stress 𝜎 mc = 149 MPa to 𝜓 = 2.7 mm−1 at the matrix cracking saturation stress 𝜎 sat = 298 MPa, and the interface debonding ratio increases from 𝜂 = 1% to 65.3%. 3.2.2.2 Temperature-Dependent Matrix Multiple Cracking of C/SiC Composite for Different Interface Debonding Energies

Figure 3.2 shows the matrix cracking density and interface debonding ratio of C/SiC composite at T = 773, 873, 973, and 1073 K for different fiber/matrix interface debonding energies (i.e. Γd = 0.5 and 1.0 J/m2 ). For Γd = 0.5 J/m2 , when T = 773 K, matrix cracking density increases from 𝜓 = 0.1 mm−1 at the first matrix cracking stress 𝜎 mc = 88 MPa to 𝜓 = 2.2 mm−1 at the matrix cracking saturation stress 𝜎 sat = 176 MPa, and the interface debonding ratio increases from 𝜂 = 0.93% to 40.5%; when T = 873 K, matrix cracking density increases from 𝜓 = 0.1 mm−1 at the first matrix cracking stress 𝜎 mc = 109 MPa to 𝜓 = 2.5 mm−1 at matrix cracking saturation stress 𝜎 sat = 218 MPa, and the interface debonding ratio increases from 𝜂 = 0.95% to 48.4%; when T = 973 K, matrix cracking density increases from 𝜓 = 0.1 mm−1 at the first matrix cracking stress 𝜎 mc = 134 MPa to 𝜓 = 2.8 mm−1 at matrix cracking saturation stress 𝜎 sat = 269 MPa, and the interface debonding ratio increases from 𝜂 = 0.98% to 57.5%, and when T = 1073 K, matrix cracking density increases from 𝜓 = 0.1 mm−1 at the first matrix cracking stress 𝜎 mc = 162 MPa to 𝜓 = 3.1 mm−1 at the matrix cracking saturation stress 𝜎 sat = 324 MPa, and the interface debonding ratio increases from 𝜂 = 1% to 66.7%. For Γd = 1.0 J/m2 , when T = 773 K, the matrix cracking density increases from 𝜓 = 0.12 mm−1 at the first matrix cracking stress 𝜎 mc = 88.4 MPa to 𝜓 = 2.3 mm−1 at the matrix cracking saturation stress 𝜎 sat = 176.8 MPa, and the interface debonding ratio increases from 𝜂 = 0.9% to 40.1%; when T = 873 K, the matrix cracking density increases from 𝜓 = 0.12 mm−1 at the first matrix cracking stress 𝜎 mc = 109 MPa to 𝜓 = 2.6 mm−1 at the matrix cracking saturation stress 𝜎 sat = 218 MPa, and the interface debonding ratio increases from 𝜂 = 0.93% to 47.8%; when T = 973 K, matrix cracking density increases from 𝜓 = 0.11 mm−1 at the first matrix cracking stress 𝜎 mc = 134 MPa to 𝜓 = 2.9 mm−1 at the matrix cracking saturation stress 𝜎 sat = 269 MPa, and the interface debonding ratio increases from 𝜂 = 0.95% to 56.4%, and when T = 1073 K, the matrix cracking density increases from 𝜓 = 0.12 mm−1 at the first matrix cracking stress 𝜎 mc = 162 MPa to 𝜓 = 3.3 mm−1 at the matrix

3.2 Temperature-Dependent Matrix Multiple Cracking Evolution of C/SiC Composites

Matrix cracking density (mm–1)

3.5 3.0 2.5 2.0 1.5 1.0

1

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4

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Stress (MPa) 0.8

Interface debonding ratio

0.7

4

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0.2 0.1 0.0 50

(b)

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Stress (MPa)

Figure 3.2 (a) The matrix cracking density versus applied stress curves when Γd = 0.5 J/m2 ; (b) the fiber/matrix interface debonding ratio versus applied stress curves when Γd = 0.5 J/m2 ; (c) the matrix cracking density versus applied stress curves when Γd = 1.0 J/m2 ; and (d) the fiber/matrix interface debonding ratio versus applied stress curves when Γd = 1.0 J/m2 of C/SiC composite at elevated temperatures of T = 773, 873, 973, and 1073 K.

cracking saturation stress 𝜎 sat = 324 MPa, and the fiber/matrix interface debonding length increases from 𝜂 = 0.97% to 65%. 3.2.2.3 Temperature-Dependent Matrix Multiple Cracking of C/SiC Composite for Different Matrix Fracture Energies

Figure 3.3 shows the matrix cracking density and interface debonding ratio of C/SiC composite at T = 773, 873, 973, and 1073 K for different matrix fracture energies (i.e. Γm = 20 and 30 J/m2 ). For Γm = 20 J/m2 , when T = 773 K, the matrix cracking density increases from 𝜓 = 0.14 mm−1 at the first matrix cracking stress 𝜎 mc = 78 MPa to 𝜓 = 2.4 mm−1 at the matrix cracking saturation stress 𝜎 sat = 156 MPa, and the interface debonding ratio

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3 Matrix Multiple Cracking Evolution of Fiber-Reinforced CMCs at Elevated Temperature

Matrix cracking density (mm–1)

4.0 3.5 3.0 2.5 2.0 1.5 1.0

1

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1: T = 773 K 2: T = 873 K 3: T = 973 K 4: T = 1073 K

4

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Stress (MPa) 0.8 0.7

Interface debonding ratio

86

4

0.6

3 2

0.5 1

0.4 0.3

1: T = 773 K 2: T = 873 K 3: T = 973 K 4: T = 1073 K

0.2 0.1 0.0 50

100

(d)

Figure 3.2

150

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350

Stress (MPa)

(Continued)

increases from 𝜂 = 0.9% to 37.2%; when T = 873 K, matrix cracking density increases from 𝜓 = 0.13 mm−1 at the first matrix cracking stress 𝜎 mc = 97 MPa to 𝜓 = 2.8 mm−1 at the matrix cracking saturation stress 𝜎 sat = 195 MPa, and the interface debonding ratio increases from 𝜂 = 0.9% to 44.9%; when T = 973 K, matrix cracking density increases from 𝜓 = 0.13 mm−1 at the first matrix cracking stress 𝜎 mc = 122 MPa to 𝜓 = 3.2 mm−1 at the matrix cracking saturation stress 𝜎 sat = 243 MPa, and the interface debonding ratio increases from 𝜂 = 0.93% to 53.8%, and when T = 1073 K, matrix cracking density increases from 𝜓 = 0.13 mm−1 at the first matrix cracking stress 𝜎 mc = 148 MPa to 𝜓 = 3.5 mm−1 at the matrix cracking saturation stress 𝜎 sat = 296 MPa, and the interface debonding ratio increases from 𝜂 = 0.95% to 62.6%. For Γm = 30 J/m2 , when T = 773 K, the matrix cracking density increases from 𝜓 = 0.1 mm−1 at the first matrix cracking stress 𝜎 mc = 97.3 MPa to 𝜓 = 2.2 mm−1 at

3.2 Temperature-Dependent Matrix Multiple Cracking Evolution of C/SiC Composites

Matrix cracking density (mm–1)

4.0 3.5 3.0 2.5 2.0 1.5 1.0

1

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4

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0.6 3 0.5

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0.3 1: T = 773 K 2: T = 873 K 3: T = 973 K 4: T = 1073 K

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(b)

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350

Stress (MPa)

Figure 3.3 (a) The matrix cracking density versus applied stress curves when Γm = 20 J/m2 ; (b) the fiber/matrix interface debonding ratio versus applied stress curves when Γm = 20 J/m2 ; (c) the matrix cracking density versus applied stress curves when Γm = 30 J/m2 ; and (d) the fiber/matrix interface debonding ratio versus applied stress curves when Γm = 30 J/m2 of C/SiC composite at elevated temperatures of T = 773, 873, 973, and 1073 K.

the matrix cracking saturation stress 𝜎 sat = 194 MPa, and the interface debonding ratio increases from 𝜂 = 0.93% to 42.5%; when T = 873 K, matrix cracking density increases from 𝜓 = 0.1 mm−1 at the first matrix cracking stress 𝜎 mc = 119 MPa to 𝜓 = 2.5 mm−1 at the matrix cracking saturation stress 𝜎 sat = 238 MPa, and the fiber/matrix interface debonding ratio increases from 𝜂 = 0.95% to 50.1%; when T = 973 K, matrix cracking density increases from 𝜓 = 0.1 mm−1 at the first matrix cracking stress 𝜎 mc = 146 MPa to 𝜓 = 2.8 mm−1 at the matrix cracking saturation stress 𝜎 sat = 292 MPa, and the fiber/matrix interface debonding ratio increases from 𝜂 = 0.96% to 58.5%, and when T = 1073 K, matrix cracking density increases from 𝜓 = 0.1 mm−1 at the first matrix cracking stress 𝜎 mc = 174 MPa to 𝜓 = 3.1 mm−1 at

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3 Matrix Multiple Cracking Evolution of Fiber-Reinforced CMCs at Elevated Temperature

Matrix cracking density (mm–1)

3.5 3.0 2.5 2.0 1.5 1

1.0

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0.7

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88

4 3

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0.2 0.1 0.0 50

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(d)

200

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400

Stress (MPa)

Figure 3.3

(Continued)

the matrix cracking saturation stress 𝜎 sat = 349 MPa, and the interface debonding ratio increases from 𝜂 = 0.98% to 66.9%.

3.2.3

Experimental Comparisons

Experimental and theoretical predicted matrix cracking density and interface debonding ratio of unidirectional C/SiC composite at room and elevated temperatures T = 773 and 873 K are predicted, as shown in Figure 3.4. At room temperature, matrix multiple cracking evolution starts from 𝜎 mc = 100 MPa and approaches to saturation at 𝜎 sat = 220 MPa; matrix cracking density increases from 𝜓 = 4.2 mm−1 to the saturation value 𝜓 = 9.4 mm−1 . At T = 773 K, matrix cracking density increases from 𝜓 = 1.3 mm−1 at the first matrix cracking stress 𝜎 mc = 140 MPa to 𝜓 = 8.7 mm−1 at the matrix cracking saturation stress 𝜎 sat = 281 MPa, and the interface debonding ratio increases from 𝜂 = 0.5% to 43%,

3.3 Temperature-Dependent Matrix Multiple Cracking Evolution of SiC/SiC Composites 12

Matrix cracking density (mm–1)

Figure 3.4 (a) The experimental and theoretical matrix cracking density versus applied stress curves and (b) the fiber/matrix interface debonding ratio versus applied stress curves of unidirectional C/SiC composite.

10 8 6 Experimental data Room temperature T = 773 K T = 873 K

4 2 0

0

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Stress (MPa)

and at T = 873 K, the matrix cracking density increases from 𝜓 = 1.1 mm−1 at the first matrix cracking stress 𝜎 mc = 182 MPa to 𝜓 = 9.4 mm−1 at the matrix cracking saturation stress 𝜎 sat = 365 MPa, and the interface debonding ratio increases from 𝜂 = 0.6% to 52.2%.

3.3 Temperature-Dependent Matrix Multiple Cracking Evolution of SiC/SiC Composites In this section, matrix multiple cracking of SiC/SiC composite is investigated using the CMSE criterion. Effects of fiber volume, interface shear stress, interface frictional coefficient, interface debonding energy, matrix fracture energy, and temperature on matrix multiple cracking of SiC/SiC composite are discussed. Experimental matrix multiple cracking and interface debonding of unidirectional SiC/SiC composite at elevated temperatures are predicted.

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3 Matrix Multiple Cracking Evolution of Fiber-Reinforced CMCs at Elevated Temperature

3.3.1

Results and Discussion

The ceramic composite system of SiC/SiC is used for the case study and its material properties are given by V f = 30%, r f = 7.5 μm, Ef = 230 GPa, Γm = 15 J/m2 , Γd = 0.4 J/m2 , 𝜏 0 = 10 MPa, 𝛼 rf = 2.9 × 10−6 K−1 , and 𝛼 lf = 3.9 × 10−6 K−1 . 3.3.1.1 Temperature-Dependent Matrix Multiple Cracking of SiC/SiC Composite for Different Fiber Volumes

Figure 3.5 shows the matrix cracking density and interface debonding ratio at T = 773, 873, 973, and 1073 K for different fiber volumes (i.e. V f = 30% and 35%). When the fiber volume fraction increases, the first matrix cracking stress and saturation matrix cracking stress increase, and the matrix cracking evolves with higher applied stress. For V f = 30%, when T = 773 K, the matrix cracking density increases from 𝜓 = 0.4 mm−1 at the first matrix cracking stress 𝜎 mc = 156 MPa to 𝜓 = 13.7 mm−1 at the matrix cracking saturation stress 𝜎 sat = 234 MPa, and the interface debonding ratio increases from 𝜂 = 0.7% to 1.0%; when the temperature is T = 873 K, the

(a)

(b)

Figure 3.5 (a) The matrix cracking density versus applied stress curves when V f = 30%; (b) the fiber/matrix interface debonding ratio versus applied stress curves when V f = 30%; (c) the matrix cracking density versus applied stress curves when V f = 35%; and (d) the fiber/matrix interface debonding ratio versus applied stress curves when V f = 35% of SiC/SiC composite at T = 773, 873, 973, and 1073 K.

3.3 Temperature-Dependent Matrix Multiple Cracking Evolution of SiC/SiC Composites

Figure 3.5

(Continued)

(c)

(d)

matrix cracking density increases from 𝜓 = 0.35 mm−1 at the first matrix cracking stress 𝜎 mc = 146 MPa to 𝜓 = 11.8 mm−1 at the matrix cracking saturation stress 𝜎 sat = 219 MPa, and the interface debonding ratio increases from 𝜂 = 0.8% to 97.2%; when T = 973 K, the matrix cracking density increases from 𝜓 = 0.3 mm−1 at the first matrix cracking stress 𝜎 mc = 134 MPa to 𝜓 = 10.2 mm−1 at the matrix cracking saturation stress 𝜎 sat = 202 MPa, and the interface debonding ratio increases from 𝜂 = 0.9% to 91.9%, and when T = 1073 K, the matrix cracking density increases from 𝜓 = 0.26 mm−1 at the first matrix cracking stress 𝜎 mc = 122 MPa to 𝜓 = 8.9 mm−1 at the matrix cracking saturation stress 𝜎 sat = 183 MPa, and the interface debonding ratio increases from 𝜂 = 0.9% to 87.2%. For V f = 35%, when T = 773 K, the matrix cracking density increases from 𝜓 = 0.36 mm−1 at the first matrix cracking stress 𝜎 mc = 194 MPa to 𝜓 = 12.3 mm−1 at the matrix cracking saturation stress 𝜎 sat = 291 MPa, and the fiber/matrix interface debonding ratio increases from 𝜂 = 0.8% to 1.0%; when T = 873 K, the matrix cracking density increases from 𝜓 = 0.32 mm−1 at the first matrix cracking stress 𝜎 mc = 180 MPa to 𝜓 = 10.9 mm−1 at the matrix cracking saturation stress

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3 Matrix Multiple Cracking Evolution of Fiber-Reinforced CMCs at Elevated Temperature

𝜎 sat = 270 MPa, and the interface debonding ratio increases from 𝜂 = 0.9% to 95.2%; when T = 973 K, the matrix cracking density increases from 𝜓 = 0.28 mm−1 at the first matrix cracking stress 𝜎 mc = 165 MPa to 𝜓 = 9.8 mm−1 at the matrix cracking saturation stress 𝜎 sat = 247 MPa, and the interface debonding ratio increases from 𝜂 = 0.9% to 90.8%, and when T = 1073 K, the matrix cracking density increases from 𝜓 = 0.25 mm−1 at the first matrix cracking stress 𝜎 mc = 148 MPa to 𝜓 = 8.9 mm−1 at the matrix cracking saturation stress 𝜎 sat = 222 MPa, and the interface debonding ratio increases from 𝜂 = 0.9% to 86.7%. 3.3.1.2 Temperature-Dependent Matrix Multiple Cracking of SiC/SiC Composite for Different Interface Shear Stress

Figure 3.6 shows the matrix cracking density and interface debonding ratio at T = 773, 873, 973, and 1073 K for different fiber/matrix interface shear stress (i.e. 𝜏 0 = 15 and 20 MPa). When the interface shear stress increases, the stress transfer between the fiber and the matrix increases, leading to the increase of the first matrix cracking stress, matrix cracking saturation stress, and saturation matrix cracking density, and the matrix cracking evolves with higher applied stress, and the fiber/matrix interface debonded length decreases because more energy dissipated during the process of fiber/matrix interface debonding. For 𝜏 0 = 15 MPa, when T = 773 K, the matrix cracking density increases from 𝜓 = 0.4 mm−1 at the first matrix cracking stress 𝜎 mc = 166 MPa to 𝜓 = 13.7 mm−1 at the matrix cracking saturation stress 𝜎 sat = 249 MPa, and the interface debonding ratio increases from 𝜂 = 0.8% to 98%; when T = 873 K, the matrix cracking density increases from 𝜓 = 0.34 mm−1 at the first matrix cracking stress 𝜎 mc = 156 MPa to 𝜓 = 12 mm−1 at the matrix cracking saturation stress 𝜎 sat = 234 MPa, and the interface debonding ratio increases from 𝜂 = 0.9% to 93.6%; when T = 973 K, the matrix cracking density increases from 𝜓 = 0.3 mm−1 at the first matrix cracking stress 𝜎 mc = 145 MPa to 𝜓 = 10.7 mm−1 at the matrix cracking saturation stress 𝜎 sat = 217 MPa, and the interface debonding ratio increases from 𝜂 = 0.9% to 89.2%, and when T = 1073 K, the matrix cracking density increases from 𝜓 = 0.27 mm−1 at the first matrix cracking stress 𝜎 mc = 132 MPa to 𝜓 = 9.6 mm−1 at the matrix cracking saturation stress 𝜎 sat = 199 MPa, and the interface debonding ratio increases from 𝜂 = 0.9% to 85.4%. For 𝜏 0 = 20 MPa, when T = 773 K, the matrix cracking density increases from 𝜓 = 0.38 mm−1 at the first matrix cracking stress 𝜎 mc = 175 MPa to 𝜓 = 13.9 mm−1 at the matrix cracking saturation stress 𝜎 sat = 262 MPa, and the interface debonding ratio increases from 𝜂 = 0.8% to 95%; when T = 873 K, the matrix cracking density increases from 𝜓 = 0.34 mm−1 at the first matrix cracking stress 𝜎 mc = 165 MPa to 𝜓 = 12.4 mm−1 at the matrix cracking saturation stress 𝜎 sat = 247 MPa, and the interface debonding ratio increases from 𝜂 = 0.92% to 91.3%; when T = 973 K, the matrix cracking density increases from 𝜓 = 0.31 mm−1 at the first matrix cracking stress 𝜎 mc = 154 MPa to 𝜓 = 11.2 mm−1 at the matrix cracking saturation stress 𝜎 sat = 231 MPa, and the interface debonding ratio increases from 𝜂 = 0.96% to 87.5%, and when T = 1073 K, the matrix cracking density increases from 𝜓 = 0.28 mm−1 at the first matrix cracking stress 𝜎 mc = 141 MPa to 𝜓 = 10.3 mm−1 at the matrix

3.3 Temperature-Dependent Matrix Multiple Cracking Evolution of SiC/SiC Composites

Figure 3.6 (a) The matrix cracking density versus applied stress curves when 𝜏 0 = 15 MPa; (b) the fiber/matrix interface debonding ratio versus applied stress curves when 𝜏 0 = 15 MPa; (c) the matrix cracking density versus applied stress curves when 𝜏 0 = 20 MPa; and (d) the fiber/matrix interface debonding ratio versus applied stress curves when 𝜏 0 = 20 MPa of SiC/SiC composite at elevated temperatures of T = 773, 873, 973, and 1073 K.

(a)

(b)

cracking saturation stress 𝜎 sat = 212 MPa, and the fiber/matrix interface debonding ratio increases from 𝜂 = 1% to 84.2%. 3.3.1.3 Temperature-Dependent Matrix Multiple Cracking of SiC/SiC Composite for Different Interface Frictional Coefficients

Figure 3.7 shows the matrix cracking density and interface debonding ratio at T = 773, 873, 973, and 1073 K for different interface frictional coefficients (i.e. 𝜇 = 0.15 and 0.2). When the interface frictional coefficient increases, the interface shear stress increases, leading to the increase of the first matrix cracking stress, matrix cracking saturation stress, and saturation matrix cracking density, and the matrix cracking evolves at higher applied stress, and the fiber/matrix interface debonded length decreases because more energy dissipated during the process of fiber/matrix interface debonding. For 𝜇 = 0.15, when T = 773 K, the matrix cracking density increases from 𝜓 = 0.38 mm−1 at the first matrix cracking stress 𝜎 mc = 172 MPa to 𝜓 = 13.8 mm−1 at the matrix cracking saturation stress 𝜎 sat = 258 MPa, and the interface debonding

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3 Matrix Multiple Cracking Evolution of Fiber-Reinforced CMCs at Elevated Temperature

Figure 3.6

(Continued)

(c)

(d)

ratio increases from 𝜂 = 0.87% to 96.4%; when T = 873 K, the matrix cracking density increases from 𝜓 = 0.34 mm−1 at the first matrix cracking stress 𝜎 mc = 160 MPa to 𝜓 = 12.2 mm−1 at the matrix cracking saturation stress 𝜎 sat = 240 MPa, and the interface debonding ratio increases from 𝜂 = 0.91% to 92.4%; when T = 973 K, the matrix cracking density increases from 𝜓 = 0.3 mm−1 at the first matrix cracking stress 𝜎 mc = 147 MPa to 𝜓 = 10.8 mm−1 at the matrix cracking saturation stress 𝜎 sat = 220 MPa, and the interface debonding ratio increases from 𝜂 = 0.95% to 88.8%, and when T = 1073 K, the matrix cracking density increases from 𝜓 = 0.27 mm−1 at the first matrix cracking stress 𝜎 mc = 132 MPa to 𝜓 = 9.6 mm−1 at the matrix cracking saturation stress 𝜎 sat = 198 MPa, and the interface debonding ratio increases from 𝜂 = 0.9% to 85.4%. For 𝜇 = 0.2, when T = 773 K, the matrix cracking density increases from 𝜓 = 0.38 mm−1 at the first matrix cracking stress 𝜎 mc = 186 MPa to 𝜓 = 14.4 mm−1 at the matrix cracking saturation stress 𝜎 sat = 279 MPa, and the interface debonding ratio increases from 𝜂 = 0.9% to 92.6%; when T = 873 K, the matrix cracking density increases from 𝜓 = 0.35 mm−1 at the first matrix cracking stress 𝜎 mc = 172 MPa

3.3 Temperature-Dependent Matrix Multiple Cracking Evolution of SiC/SiC Composites

Figure 3.7 (a) The matrix cracking density versus applied stress curves when 𝜇 = 0.15; (b) the fiber/matrix interface debonding ratio versus applied stress curves when 𝜇 = 0.15; (c) the matrix cracking density versus applied stress curves when 𝜇 = 0.2; and (d) the fiber/matrix interface debonding ratio versus applied stress curves when 𝜇 = 0.2 of SiC/SiC composite at T = 773, 873, 973, and 1073 K.

(a)

(b)

to 𝜓 = 12.8 mm−1 at the matrix cracking saturation stress 𝜎 sat = 259 MPa, and the interface debonding ratio increases from 𝜂 = 0.94% to 89.7%; when T = 973 K, the matrix cracking density increases from 𝜓 = 0.31 mm−1 at the first matrix cracking stress 𝜎 mc = 157 MPa to 𝜓 = 11.5 mm−1 at the matrix cracking saturation stress 𝜎 sat = 236 MPa, and the interface debonding ratio increases from 𝜂 = 0.97% to 86.9%, and when T = 1073 K, the matrix cracking density increases from 𝜓 = 0.28 mm−1 at the first matrix cracking stress 𝜎 mc = 141 MPa to 𝜓 = 10.2 mm−1 at the matrix cracking saturation stress 𝜎 sat = 211 MPa, and the fiber/matrix interface debonding length increases from 𝜂 = 0.99% to 84.2%. 3.3.1.4 Temperature-Dependent Matrix Multiple Cracking of SiC/SiC Composite for Different Interface Debonding Energies

Figure 3.8 shows the matrix cracking density and interface debonding ratio at T = 773, 873, 973, and 1073 K for different interface debonding energies (i.e. Γd = 0.1 and 0.5 J/m2 ). When the interface debonding energy increases, the

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3 Matrix Multiple Cracking Evolution of Fiber-Reinforced CMCs at Elevated Temperature

Figure 3.7

(Continued)

(c)

(d)

matrix saturation cracking stress decreases, and the saturation matrix cracking density increases, and the rate of matrix cracking development increases because of the decrease of interface debonding ratio. For Γd = 0.1 J/m2 , when T = 773 K, the matrix cracking density increases from 𝜓 = 0.32 mm−1 at the first matrix cracking stress 𝜎 mc = 156 MPa to 𝜓 = 12.6 mm−1 at the matrix cracking saturation stress 𝜎 sat = 218 MPa, and the interface debonding ratio increases from 𝜂 = 1% to 99.5%; when T = 873 K, the matrix cracking density increases from 𝜓 = 0.28 mm−1 at the first matrix cracking stress 𝜎 mc = 146 MPa to 𝜓 = 10.9 mm−1 at the matrix cracking saturation stress 𝜎 sat = 204 MPa, and the interface debonding ratio increases from 𝜂 = 1.1% to 94.7%; when T = 973 K, the matrix cracking density increases from 𝜓 = 0.24 mm−1 at the first matrix cracking stress 𝜎 mc = 134 MPa to 𝜓 = 9.6 mm−1 at the matrix cracking saturation stress 𝜎 sat = 188 MPa, and the interface debonding ratio increases from 𝜂 = 1.1% to 90.6%, and when T = 1073 K, the matrix cracking density increases from 𝜓 = 0.21 mm−1 at the first matrix cracking stress 𝜎 mc = 122 MPa to 𝜓 = 8.4 mm−1 at the matrix cracking

3.3 Temperature-Dependent Matrix Multiple Cracking Evolution of SiC/SiC Composites

Figure 3.8 (a) The matrix cracking density versus applied stress curves when Γd = 0.1 J/m2 ; (b) the fiber/matrix interface debonding ratio versus applied stress curves when Γd = 0.1 J/m2 ; (c) the matrix cracking density versus applied stress curves when Γd = 0.5 J/m2 ; and (d) the fiber/matrix interface debonding ratio versus applied stress curves when Γd = 0.5 J/m2 of SiC/SiC composite at T = 773, 873, 973, and 1073 K.

(a)

(b)

saturation stress 𝜎 sat = 171 MPa, and the fiber/matrix interface debonding length increases from 𝜂 = 1.1% to 86.8%. For Γd = 0.5 J/m2 , when T = 773 K, the matrix cracking density increases from 𝜓 = 0.46 mm−1 at the first matrix cracking stress 𝜎 mc = 156 MPa to 𝜓 = 14.4 mm−1 at the matrix cracking saturation stress 𝜎 sat = 218 MPa, and the interface debonding ratio increases from 𝜂 = 0.7% to 86.4%; when T = 873 K, the matrix cracking density increases from 𝜓 = 0.38 mm−1 at the first matrix cracking stress 𝜎 mc = 146 MPa to 𝜓 = 12.3 mm−1 at the matrix cracking saturation stress 𝜎 sat = 204 MPa, and the interface debonding ratio increases from 𝜂 = 0.8% to 81.6%; when T = 973 K, the matrix cracking density increases from 𝜓 = 0.32 mm−1 at the first matrix cracking stress 𝜎 mc = 134 MPa to 𝜓 = 10.6 mm−1 at the matrix cracking saturation stress 𝜎 sat = 188 MPa, and the interface debonding ratio increases from 𝜂 = 0.85% to 77.6%, and when T = 1073 K, the matrix cracking density increases from 𝜓 = 0.28 mm−1 at the first matrix cracking stress 𝜎 mc = 122 MPa to 𝜓 = 9.2 mm−1 at the matrix cracking saturation stress 𝜎 sat = 171 MPa, and the interface debonding ratio increases from 𝜂 = 0.9% to 74.2%.

97

98

3 Matrix Multiple Cracking Evolution of Fiber-Reinforced CMCs at Elevated Temperature

Figure 3.8

(Continued)

(c)

(d)

3.3.1.5 Temperature-Dependent Matrix Multiple Cracking of SiC/SiC Composite for Different Matrix Fracture Energies

Figure 3.9 shows the matrix cracking density and interface debonding ratio at T = 773, 873, 973, and 1073 K for different matrix fracture energies (i.e. Γm = 10 and 20 J/m2 ). When the matrix fracture energy increases, the first matrix cracking stress and saturation matrix cracking stress increase, and the matrix cracking evolves at higher applied stress. For Γm = 10 J/m2 , when T = 773 K, the matrix cracking density increases from 𝜓 = 0.83 mm−1 at the first matrix cracking stress 𝜎 mc = 134 MPa to 𝜓 = 23.6 mm−1 at the matrix cracking saturation stress 𝜎 sat = 201 MPa, and the interface debonding ratio increases from 𝜂 = 0.3% to 1.0%; when T = 873 K, the matrix cracking density increases from 𝜓 = 0.62 mm−1 at the first matrix cracking stress 𝜎 mc = 125 MPa to 𝜓 = 18.4 mm−1 at the matrix cracking saturation stress 𝜎 sat = 188 MPa, and the interface debonding ratio increases from 𝜂 = 0.5% to 1.0%; when T = 973 K, the matrix cracking density increases from 𝜓 = 0.48 mm−1 at the first matrix cracking stress 𝜎 mc = 116 MPa to 𝜓 = 14.9 mm−1 at the matrix cracking saturation stress

3.3 Temperature-Dependent Matrix Multiple Cracking Evolution of SiC/SiC Composites

Figure 3.9 (a) The matrix cracking density versus applied stress curves when Γm = 10 J/m2 ; (b) the fiber/matrix interface debonding ratio versus applied stress curves when Γm = 10 J/m2 ; (c) the matrix cracking density versus applied stress curves when Γm = 20 J/m2 ; and (d) the fiber/matrix interface debonding ratio versus applied stress curves when Γm = 20 J/m2 of SiC/SiC composite at T = 773, 873, 973, and 1073 K.

(a)

(b)

𝜎 sat = 174 MPa, and the interface debonding ratio increases from 𝜂 = 0.69% to 97.4%, and when T = 1073 K, the matrix cracking density increases from 𝜓 = 0.38 mm−1 at the first matrix cracking stress 𝜎 mc = 106 MPa to 𝜓 = 12.3 mm−1 at the matrix cracking saturation stress 𝜎 sat = 158 MPa, and the fiber/matrix interface debonding ratio increases from 𝜂 = 0.79% to 89.7%. For Γm = 20 J/m2 , when T = 773 K, the matrix cracking density increases from 𝜓 = 0.29 mm−1 at the first matrix cracking stress 𝜎 mc = 173 MPa to 𝜓 = 10.4 mm−1 at the matrix cracking saturation stress 𝜎 sat = 260 MPa, and the interface debonding ratio increases from 𝜂 = 0.9% to 98.3%; when T = 873 K, the matrix cracking density increases from 𝜓 = 0.26 mm−1 at the first matrix cracking stress 𝜎 mc = 162 MPa to 𝜓 = 9.2 mm−1 at the matrix cracking saturation stress 𝜎 sat = 243 MPa, and the interface debonding ratio increases from 𝜂 = 0.98% to 94%; when T = 973 K, the matrix cracking density increases from 𝜓 = 0.23 mm−1 at the first matrix cracking stress 𝜎 mc = 149 MPa to 𝜓 = 8.2 mm−1 at the matrix cracking saturation stress 𝜎 sat = 224 MPa, and the interface debonding ratio increases from 𝜂 = 1% to 90%, and when T = 1073 K, the matrix cracking density increases from 𝜓 = 0.2 mm−1 at the

99

100

3 Matrix Multiple Cracking Evolution of Fiber-Reinforced CMCs at Elevated Temperature

Figure 3.9

(Continued)

(c)

(d)

first matrix cracking stress 𝜎 mc = 135 MPa to 𝜓 = 7.4 mm−1 at the matrix cracking saturation stress 𝜎 sat = 203 MPa, and the interface debonding ratio increases from 𝜂 = 1% to 86.4%.

3.3.2

Experimental Comparisons

Figure 3.10 shows the experimental and theoretical matrix cracking density and interface debonding ratio of unidirectional SiC/SiC composite at room and elevated temperatures of T = 773, 873, 973, and 1073 K. At room temperature, the matrix multiple cracking evolution starts from 𝜎 mc = 240 MPa and approaches to saturation at 𝜎 sat = 320 MPa; the matrix cracking density increases from 𝜓 = 1.1 mm−1 to the saturation value 𝜓 = 13 mm−1 . At T = 773 K, the matrix cracking density increases from 𝜓 = 0.5 mm−1 at the first matrix cracking stress 𝜎 mc = 222 MPa to 𝜓 = 12.4 mm−1 at the matrix cracking saturation stress 𝜎 sat = 311 MPa, and the interface debonding ratio increases from 𝜂 = 0.7% to 75.4%; at T = 873 K, the matrix cracking density increases from

3.4 Time-Dependent Matrix Multiple Cracking Evolution of C/SiC Composites

Figure 3.10 (a) The experimental and theoretical matrix cracking density versus applied stress curves and (b) the fiber/matrix interface debonding ratio versus applied stress curves of unidirectional SiC/SiC composite.

(a)

(b)

𝜓 = 0.45 mm−1 at the first matrix cracking stress 𝜎 mc = 206 MPa to 𝜓 = 11.2 mm−1 at the matrix cracking saturation stress 𝜎 sat = 288 MPa, and the interface debonding ratio increases from 𝜂 = 0.7% to 72.3%; at T = 973 K, the matrix cracking density increases from 𝜓 = 0.4 mm−1 at the first matrix cracking stress 𝜎 mc = 188 MPa to 𝜓 = 10.2 mm−1 at the matrix cracking saturation stress 𝜎 sat = 263 MPa, and the interface debonding ratio increases from 𝜂 = 0.8% to 69.5%, and at T = 1073 K, the matrix cracking density increases from 𝜓 = 0.35 mm−1 at the first matrix cracking stress 𝜎 mc = 169 MPa to 𝜓 = 9.4 mm−1 at the matrix cracking saturation stress 𝜎 sat = 236 MPa, and the interface debonding ratio increases from 𝜂 = 0.8% to 66.9%.

3.4 Time-Dependent Matrix Multiple Cracking Evolution of C/SiC Composites In this section, time-dependent matrix multiple cracking of C/SiC composite is investigated considering interface oxidation. Time-dependent interface oxidation

101

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3 Matrix Multiple Cracking Evolution of Fiber-Reinforced CMCs at Elevated Temperature

length and temperature-dependent interface shear stress in the oxidation and debonding region, interface debonding energy, fiber and matrix modulus, and matrix fracture energy are considered in the analysis for the microstress field, interface debonding criterion, and CMSE model. Relationships between time, temperature, applied stress, fiber/matrix interface debonding, and matrix fracture are established. Effects of interface shear stress, interface frictional coefficient, interface debonding energy, and matrix fracture energy on matrix cracking density and interface oxidation ratio are discussed for different temperatures and oxidation durations. Experimental matrix cracking density and interface oxidation ratio for unidirectional C/SiC composite at different testing temperatures and oxidation durations are predicted.

3.4.1

Theoretical Models

Because of a mismatch between axial thermal expansion coefficients of the fiber and matrix, unavoidable micro-cracks were created in the matrix when the composite was cooled down from a high fabricated temperature to ambient temperature. These processing-induced micro-cracks, which were present on the surface of the material, served as avenues for the ingress of environmental atmosphere into the composite at elevated temperature. Time-dependent interface oxidation length 𝜁 is [49] ]} { [ 𝜑 (T)t 𝜁(t, T) = 𝜑1 (T) 1 − exp − 2 (3.30) b where b is the delay factor considering the deceleration of reduced oxygen activity, t denotes the oxidation time, and 𝜑1 (T) and 𝜑2 (T) are temperature-dependent parameters described using Arrhenius law [49]. ) ( 8231 (3.31) 𝜑1 (T) = 7.021 × 10−3 × exp T ) ( 17 090 (3.32) 𝜑2 (T) = 227.1 × exp − T Temperature-dependent interface shear stress in the oxidation and debonding region are [43] |𝛼rf (T) − 𝛼rm (T)|(Tm − T) (3.33) A |𝛼 (T) − 𝛼rm (T)|(Tm − T) (3.34) 𝜏f (T) = 𝜏s + 𝜇 rf A where 𝜏 0 and 𝜏 s denote the initial interface shear stress, 𝜇 denotes the interface frictional coefficient, 𝛼 rf and 𝛼 rm denote the fiber and matrix radial thermal expansion coefficient, respectively, T m and T denote the processing temperature and testing temperature, and A is a constant dependent on the elastic properties of the matrix and fibers. 𝜏i (T) = 𝜏0 + 𝜇

3.4.1.1 Time-Dependent Stress Analysis

Temperature-dependent fiber axial stress 𝜎 f (x, T), matrix axial stress 𝜎 m (x, T), and the interface shear stress 𝜏 i (x, T) can be described using the following equations.

3.4 Time-Dependent Matrix Multiple Cracking Evolution of C/SiC Composites

⎧ 𝜎 − 2𝜏f (T) x, x ∈ [0, 𝜁(T)] rf ⎪ Vf ⎪𝜎 2𝜏 (T) 2𝜏 (T) x ∈ [𝜁(T), ld (T)] ⎪ V − fr 𝜁(T) − ri [x − 𝜁(T)], f f f { [ ] 𝜎f (x, T) = ⎨ l (T) Vm 2𝜏f (T) x ∈ ld (T), c 2 ⎪𝜎fo (T) + V 𝜎mo (T) − r 𝜁(T) f } f[ ] ⎪ 2𝜏i (T) x−l (T) , ⎪ − r [ld (T) − 𝜁(T)] exp −𝜌 rd f f ⎩

(3.35)

⎧ Vf 𝜏f (T) x ∈ [0, 𝜁(T)] ⎪2 Vm rf x, Vf 𝜏i (T) ⎪ Vf 𝜏f (T) x ∈ [𝜁(T), ld (T)] ⎪2 V r 𝜁(T) + 2 V r [x − 𝜁(T)], [ ] (3.36) 𝜎m (x, T) = ⎨ m f { Vf 𝜏f (T)m f lc (T) Vf 𝜏i (T) (T) + 2 𝜁(T) + 2 x ∈ l (T), 𝜎 d ⎪ mo Vm rf Vm 2 [ rf ] } ⎪ x−ld (T) (T) − 𝜁(T)] − 𝜎 (T) exp −𝜌 , [l mo ⎪ d rf ⎩ ⎧ 𝜏f (T), x ∈ [0, 𝜁(T)] ⎪ x ld (T)]] 𝜏i (x, T) = ⎨ 𝜏i (T), [ ] [ ] ∈ [𝜁(T), [ ⎪ 𝜌 Vm 𝜎mo − 2𝜏i (T) ld (T) exp −𝜌 x−ld (T) , x ∈ ld (T), lc (T) rf rf 2 ⎩ 2 Vf

(3.37)

where Ef (T) 𝜎 + Ef (T)[𝛼lc (T) − 𝛼lf (T)]ΔT Ec (T) E (T) 𝜎 + Em (T)[𝛼lc (T) − 𝛼lm (T)]ΔT 𝜎mo (T) = m Ec (T) 𝜎fo (T) =

(3.38) (3.39)

3.4.1.2 Time-Dependent Interface Debonding

Temperature-dependent interface debonding criterion can be described using the following equation [44]. Γd (T) =

ld 𝜕v(T) F 𝜕wf (T) 1 − 𝜏i (x, T) dx ∫ 4𝜋rf 𝜕ld 2 0 𝜕ld

(3.40)

where F(= 𝜋rf2 𝜎∕Vf ) denotes the fiber load at the matrix cracking plane, wf (T) denotes the temperature-dependent fiber axial displacement at the matrix cracking plane, and v(T) denotes the temperature-dependent relative displacement between the fiber and the matrix. The temperature-dependent axial displacements of the fiber and matrix, i.e. wf (x, T) and wm (x, T), can be described using the following equations. wf (x, T) = =

lc (T) 2

∫x

𝜎f (x, T) dx Ef (T)

𝜏 (T) 𝜎 [l (T) − x] − f [2𝜁(T)ld (T) − 𝜁 2 (T) − x2 ] Vf Ef (T) d rf Ef (T) ( ) 𝜎 (T) lc (T) 𝜏 (T) [ld (T) − 𝜁(T)]2 + fo − ld (T) − i rf Ef (T) Ef (T) 2 { } Vm 2𝜏f (T) 2𝜏i (T) rf 𝜎 (T) − 𝜁(T) − [ld (T) − 𝜁(T)] + 𝜌Ef (T) Vf mo rf rf )] [ ( l (T)∕2 − ld (T) (3.41) × 1 − exp −𝜌 c rf

103

104

3 Matrix Multiple Cracking Evolution of Fiber-Reinforced CMCs at Elevated Temperature lc (T) 2

𝜎m (x, T) dx ∫x Em (T) Vf 𝜏f (T) [2𝜁(T)ld (T) − 𝜁 2 (T) − x2 ] = rf Vm Em (T) [ ] 𝜎mo (T) lc (T) Vf 𝜏i (T) 2 [l (T) − 𝜁(T)] + − ld (T) + rf Vm Em (T) d Em (T) 2 { } V 𝜏 (T) V 𝜏 (T) rf 𝜎mo (T) − 2 f f 𝜁(T) − 2 f i [ld (T) − 𝜁(T)] − 𝜌Em (T) rf Vm rf Vm )] [ ( lc (T)∕2 − ld (T) (3.42) × 1 − exp −𝜌 rf

wm (x, T) =

Temperature-dependent relative displacement between the fiber and the matrix, i.e. v(x, T), can be described using the following equation. v(x, T) = |wf (x, T) − wm (x, T)| =

Ec (T)𝜏i (T) 𝜎 [ld (T) − x] − [l (T) − 𝜁(T)]2 Vf Ef (T) rf Vm Em (T)Ef (T) d Ec (T)𝜏f (T) − [2𝜁(T)ld (T) − 𝜁 2 (T) − x2 ] rf Vm Em (T)Ef (T) { } rf Ec (T) 𝜏 (T) 𝜏 (T) 𝜎mo (T) − 2 f 𝜁(T) − 2 i [ld (T) − 𝜁(T)] + 𝜌Vm Em (T)Ef (T) rf rf )] [ ( lc (T)∕2 − ld (T) (3.43) × 1 − exp −𝜌 rf

Substituting wf (x = 0, T) and v(x, T) into Eq. (3.40), it leads to the following equation. Ec (T)𝜏i2 (T)

[ld (T) − 𝜁(T)]2 +

Ec (T)𝜏i2 (T)

[l (T) − 𝜁(T)] rf Vm Em (T)Ef (T) 𝜌Vm Em (T)Ef (T) d 2E (T)𝜏f (T)𝜏i (T) 𝜏 (T)𝜎 [l (T) − 𝜁(T)] + c [l (T) − 𝜁(T)]𝜁(T) − i Vf Ef (T) d rf Vm Em (T)Ef (T) d Ec (T)𝜏f2 (T) E (T)𝜏f (T)𝜏i (T) r 𝜏 (T)𝜎 + 𝜁 2 (T) + c 𝜁(T) − f i 2𝜌Vf Ef (T) rf Vm Em (T)Ef (T) 𝜌Vm Em (T)Ef (T) r V E (T)𝜎 2 𝜏 (T)𝜎 − Γd (T) = 0 𝜁(T) + f 2m m − f Vf Ef (T) 4V Ef (T)Ec (T)

(3.44)

f

Solving Eq. (3.44), the temperature-dependent interface debonding length ld (T) can be described using the following equation. ] ( ) [ r Vm Em (T)𝜎 𝜏 (T) 1 ld (T) = 1 − f 𝜁(T) + f − 𝜏i (T) 2 Vf Ec (T)𝜏i (T) 𝜌 √ √( )2 √ r r V E (T)E (T) −√ f Γd (T) + f m m 2 f (3.45) 2𝜌 Ec (T)𝜏i (T)

3.4 Time-Dependent Matrix Multiple Cracking Evolution of C/SiC Composites

3.4.1.3 Time-Dependent Matrix Multiple Cracking

Temperature-dependent matrix strain energy can be described using the following equation. Um (T) =

1 2Em (T) ∫Am ∫0

lc (T)

2 𝜎m (x, T) dx dAm

(3.46)

where Am is the cross-sectional area of matrix in the unit cell, Em (T) denotes the temperature-dependent matrix modulus, lc (T) denotes the temperature-dependent matrix crack spacing, and 𝜎 m (T) denotes the temperature-dependent matrix axial stress. Substituting temperature-dependent matrix axial stresses in Eq. (3.36) into Eq. (3.46), the matrix strain energy for the interface partially debonding can be described using the following equation. { ( )2 ) ( Am Vf 𝜏f (T) 2 4 Vf 𝜏f (T) 𝜁 3 (T) + 4 [ld (T) − 𝜁(T)]𝜁 2 (T) Um (T) = Em 3 Vm rf (T) Vm rf ) ( Vf 2 𝜏f (T)𝜏i (T)𝜁(T)[ld (T) − 𝜁(T)]2 +4 rf Vm ]2 [ [ ] lc (T) 4 Vf 𝜏i (T) 2 − ld (T) + [ld (T) − 𝜁(T)]3 + 𝜎mo 3 Vm rf 2 [ ] 2rf 𝜎mo (T) Vf 𝜏f (T) Vf 𝜏i (T) 2 𝜁(T) + 2 [l (T) − 𝜁(T)] − 𝜎mo (T) + 𝜌 V r Vm rf d )] [ ( m f l (T)∕2 − ld (T) × 1 − exp −𝜌 c rf [ ]2 Vf 𝜏f (T) Vf 𝜏i (T) rf 2 𝜁(T) + 2 (l (T) − 𝜁(T)) − 𝜎mo (T) + 2𝜌 Vm rf Vm rf d )]} [ ( l (T)∕2 − ld (T) (3.47) × 1 − exp −2𝜌 c rf where the temperature-dependent interface oxidation length 𝜁(T) can be determined by Eq. (3.30). The temperature-dependent fiber/matrix interface debonded length can be determined by Eq. (3.45). The matrix strain energy for the interface completely debonding can be described using the following equation. { [ ]2 ] [ Am Vf 𝜏f (T) 2 4 Vf 𝜏f (T) 3 Um (T) = 𝜁 (T) + 4 [ld (T) − 𝜁(T)]𝜁 2 (T) Em 3 Vm rf Vm rf ) ( Vf 2 𝜏f (T)𝜏i (T)𝜁(T)[ld (T) − 𝜁(T)]2 +4 rf Vm } [ ]2 4 Vf 𝜏i (T) 3 (3.48) + [ld (T) − 𝜁(T)] 3 Vm rf The CMSE U mcr at the critical stress 𝜎 cr can be described using the following equation.

105

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3 Matrix Multiple Cracking Evolution of Fiber-Reinforced CMCs at Elevated Temperature

𝜎 (T) 1 kA l mocr 2 m 0 Em (T) 2

Umcr (T) =

(3.49)

where k (k ∈ [0,1]) is the critical matrix strain energy parameter, l0 is the initial matrix crack spacing, and 𝜎 mocr (T) can be described using the following equation. 𝜎mocr (T) =

Em (T) 𝜎 (T) + Em (T)[𝛼lc (T) − 𝛼lm (T)]ΔT Ec (T) cr

(3.50)

where 𝜎 cr (T) denotes the temperature-dependent critical stress corresponding to composite’s proportional limit stress defined by the ACK model [38]. ]1 [ 2 6Vf Ef (T)Ec2 (T)𝜏i (T)Γm (T) 3 − Ec (T)[𝛼lc (T) − 𝛼lm (T)]ΔT (3.51) 𝜎cr (T) = 2 rf Vm Em (T) where Γm (T) denotes the temperature-dependent matrix fracture energy. The matrix multi-cracking evolution can be determined using the following equation. Um (𝜎 > 𝜎cr , T) = Ucrm (𝜎cr , T) The interface oxidation ratio is defined as 𝜁 𝜔= ld

3.4.2

(3.52)

(3.53)

Results and Discussion

The ceramic composite system of C/SiC is used for the case study and its material properties are given by V f = 35%, r f = 3.5 μm, Γm = 25 J/m2 (at room temperature), and Γd = 0.1 J/m2 (at room temperature). Effects of interface shear stress, interface frictional coefficient, interface debonding energy, and matrix fracture energy on time-dependent matrix cracking density and interface oxidation ratio are discussed. 3.4.2.1 Time-Dependent Matrix Multiple Cracking of C/SiC Composite for Different Interface Shear Stress

Figure 3.11 shows the effect of interface shear stress (i.e. 𝜏 0 = 30 and 40 MPa) on temperature-dependent matrix multiple cracking and interface oxidation of C/SiC composite at T = 873, 973, and 1073 K for the oxidation duration t = 1 and 3 hours. When 𝜏 0 = 30 MPa at T = 873 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.19 mm−1 at 𝜎 mc = 117 MPa to 𝜓 = 3.49 mm−1 at 𝜎 sat = 235 MPa, and the interface oxidation ratio decreases from 𝜔 = 9% to 𝜔 = 3.4%, and at the oxidation duration t = 3 hours, the matrix cracking density increases from 𝜓 = 0.17 mm−1 at 𝜎 mc = 117 MPa to 𝜓 = 3.4 mm−1 at 𝜎 sat = 235 MPa, and the interface oxidation ratio decreases from 𝜔 = 25.3% to 𝜔 = 9.9%. At T = 973 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.17 mm−1 at 𝜎 mc = 137 MPa to 𝜓 = 4.2 mm−1 at 𝜎 sat = 275 MPa, and the interface oxidation ratio decreases from 𝜔 = 23% to 𝜔 = 9.9%, and at the oxidation duration t = 3 hours, the matrix cracking density increases from 𝜓 = 0.14 mm−1 at 𝜎 mc = 137 MPa to

3.4 Time-Dependent Matrix Multiple Cracking Evolution of C/SiC Composites

Figure 3.11 (a) The matrix cracking density versus applied stress curves for different oxidation temperatures and times when 𝜏 0 = 30 MPa; (b) the interface oxidation ratio versus applied stress curves for different oxidation temperatures and times when 𝜏 0 = 30 MPa; (c) the matrix cracking density versus applied stress curves for different oxidation temperatures and times when 𝜏 0 = 40 MPa; and (d) the interface oxidation ratio versus applied stress curves for different oxidation temperatures and times when 𝜏 0 = 40 MPa.

(a)

(b)

𝜓 = 3.9 mm−1 at 𝜎 sat = 275 MPa, and the interface oxidation ratio decreases from 𝜔 = 55.9% to 𝜔 = 26.9%. At T = 1073 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.16 mm−1 at 𝜎 mc = 153 MPa to 𝜓 = 4.9 mm−1 at 𝜎 sat = 306 MPa, and the interface oxidation ratio decreases from 𝜔 = 46% to 𝜔 = 22.9%, and at the oxidation duration t = 3 hours, the matrix cracking density increases from 𝜓 = 0.12 mm−1 at 𝜎 mc = 153 MPa to 𝜓 = 4.3 mm−1 at 𝜎 sat = 306 MPa, and the interface oxidation ratio decreases from 𝜔 = 90.7% to 𝜔 = 54.5%. When 𝜏 0 = 40 MPa at T = 873 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.21 mm−1 at 𝜎 mc = 169 MPa to 𝜓 = 5.13 mm−1 at 𝜎 sat = 338 MPa, and the interface oxidation ratio decreases from 𝜔 = 9.7% to 𝜔 = 4.2%, and at the oxidation duration t = 3 hours, the matrix cracking density increases from 𝜓 = 0.18 mm−1 at 𝜎 mc = 169 MPa to 𝜓 = 4.9 mm−1 at 𝜎 sat = 338 MPa, and the interface oxidation ratio decreases from 𝜔 = 25.7% to 𝜔 = 11.9%. At T = 973 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.2 mm−1 at 𝜎 mc = 180 MPa to 𝜓 = 5.8 mm−1 at 𝜎 sat = 360 MPa, and the interface oxidation ratio decreases from 𝜔 = 25.2% to 𝜔 = 12%, and at oxidation duration t = 3 hours,

107

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3 Matrix Multiple Cracking Evolution of Fiber-Reinforced CMCs at Elevated Temperature

Figure 3.11

(Continued)

(c)

(d)

the matrix cracking density increases from 𝜓 = 0.15 mm−1 at 𝜎 mc = 180 MPa to 𝜓 = 5.2 mm−1 at 𝜎 sat = 360 MPa, and the interface oxidation ratio decreases from 𝜔 = 55.5% to 30.7%. At T = 1073 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.18 mm−1 at 𝜎 mc = 188 MPa to 𝜓 = 6.4 mm−1 at 𝜎 sat = 377 MPa, and the interface oxidation ratio decreases from 𝜔 = 48.5% to 𝜔 = 26.5%, and at oxidation duration t = 3 hours, the matrix cracking density increases from 𝜓 = 0.16 mm−1 at 𝜎 mc = 188 MPa to 𝜓 = 6.2 mm−1 at 𝜎 sat = 377 MPa, and the interface oxidation ratio decreases from 𝜔 = 85.3% to 𝜔 = 57.4%. 3.4.2.2 Time-Dependent Matrix Multiple Cracking of C/SiC Composite for Different Interface Frictional Coefficients

Figure 3.12 shows the effect of interface frictional coefficient (i.e. 𝜇 = 0.03 and 0.06) on the temperature-dependent matrix multiple cracking and interface oxidation of C/SiC composite at T = 873, 973, and 1073 K for the oxidation duration t = 1 and 3 hours.

3.4 Time-Dependent Matrix Multiple Cracking Evolution of C/SiC Composites

Figure 3.12 (a) The matrix cracking density versus applied stress curves for different oxidation temperatures and times when 𝜇 = 0.03; (b) the interface oxidation ratio versus applied stress curves for different oxidation temperatures and times when 𝜇 = 0.03; (c) the matrix cracking density versus applied stress curves for different oxidation temperatures and times when 𝜇 = 0.06; and (d) the interface oxidation ratio versus applied stress curves for different oxidation temperatures and times when 𝜇 = 0.06.

(a)

(b)

When 𝜇 = 0.03 at T = 873 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.2 mm−1 at 𝜎 mc = 159 MPa to 𝜓 = 4.8 mm−1 at 𝜎 sat = 320 MPa, and the interface oxidation ratio decreases from 𝜔 = 9.5% to 𝜔 = 4%, and at oxidation duration t = 3 hours, the matrix cracking density increases from 𝜓 = 0.18 mm−1 at 𝜎 mc = 159 MPa to 𝜓 = 4.6 mm−1 at 𝜎 sat = 320 MPa, and the interface oxidation ratio decreases from 𝜔 = 25.3% to 𝜔 = 11.4%. At T = 973 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.19 mm−1 at 𝜎 mc = 166 MPa to 𝜓 = 5.2 mm−1 at 𝜎 sat = 334 MPa, and the interface oxidation ratio decreases from 𝜔 = 24.3% to 𝜔 = 11.3%, and at the oxidation duration t = 3 hours, the matrix cracking density increases from 𝜓 = 0.15 mm−1 at 𝜎 mc = 167 MPa to 𝜓 = 4.8 mm−1 at 𝜎 sat = 334 MPa, and the interface oxidation ratio decreases from 𝜔 = 55.2% to 𝜔 = 29.4%. At T = 1073 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.17 mm−1 at 𝜎 mc = 172 MPa to 𝜓 = 5.7 mm−1 at 𝜎 sat = 345 MPa, and the interface oxidation ratio decreases from 𝜔 = 47.2% to 𝜔 = 24.8%, and at the oxidation duration t = 3 hours, the matrix cracking density increases from 𝜓 = 0.13 mm−1 at

109

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3 Matrix Multiple Cracking Evolution of Fiber-Reinforced CMCs at Elevated Temperature

Figure 3.12

(Continued)

(c)

(d)

𝜎 mc = 172 MPa to 𝜓 = 5.2 mm−1 at 𝜎 sat = 345 MPa, and the interface oxidation ratio decreases from 𝜔 = 86.9% to 𝜔 = 55.9%. When 𝜇 = 0.06 at T = 873 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.2 mm−1 at 𝜎 mc = 88 MPa to 𝜓 = 2.8 mm−1 at 𝜎 sat = 176 MPa, and the interface oxidation ratio decreases from 𝜔 = 10.4% to 3%, and at the oxidation duration t = 3 hours, the matrix cracking density increases from 𝜓 = 0.21 mm−1 at 𝜎 mc = 88 MPa to 𝜓 = 2.7 mm−1 at 𝜎 sat = 176 MPa, and the interface oxidation ratio decreases from 𝜔 = 30.7% to 𝜔 = 9.4%. At T = 973 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.17 mm−1 at 𝜎 mc = 119 MPa to 𝜓 = 3.6 mm−1 at 𝜎 sat = 238 MPa, and the interface oxidation ratio decreases from 𝜔 = 22.8% to 𝜔 = 9%, and at the oxidation duration t = 3 hours, the matrix cracking density increases from 𝜓 = 0.15 mm−1 at 𝜎 mc = 119 MPa to 𝜓 = 3.4 mm−1 at 𝜎 sat = 238 MPa, and the interface oxidation ratio decreases from 𝜔 = 59% to 𝜔 = 25.8%. At T = 1073 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.15 mm−1 at 𝜎 mc = 142 MPa to 𝜓 = 4.5 mm−1 at 𝜎 sat = 284 MPa, and the interface oxidation ratio decreases from 𝜔 = 45.7% to 𝜔 = 21.8%, and at the oxidation duration t = 3 hours,

3.4 Time-Dependent Matrix Multiple Cracking Evolution of C/SiC Composites

the matrix cracking density increases from 𝜓 = 0.11 mm−1 at 𝜎 mc = 142 MPa to 𝜓 = 4 mm−1 at 𝜎 sat = 284 MPa, and the interface oxidation ratio decreases from 𝜔 = 94.6% to 𝜔 = 53.9%. 3.4.2.3 Time-Dependent Matrix Multiple Cracking of C/SiC Composite for Different Interface Debonding Energies

Figure 3.13 shows the effect of fiber/matrix interface debonding energy (i.e. Γd = 0.3 and 0.5 J/m2 ) on temperature-dependent matrix multiple cracking and interface oxidation of C/SiC composite at T = 873, 973, and 1073 K for the oxidation duration t = 1 and 3 hours. When Γd = 0.3 J/m2 at T = 873 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.36 mm−1 at 𝜎 mc = 118 MPa to 𝜓 = 4.2 mm−1 at 𝜎 sat = 235 MPa, and the interface oxidation ratio decreases from 𝜔 = 18.8% to 𝜔 = 4.2%, and at the oxidation duration t = 3 hours, the matrix cracking density increases from 𝜓 = 0.32 mm−1 at 𝜎 mc = 117 MPa to 𝜓 = 4.1 mm−1 at 𝜎 sat = 235 MPa, and the interface oxidation ratio decreases from 𝜔 = 48.8% to 𝜔 = 12.2%. At T = 973 K Figure 3.13 (a) The matrix cracking density versus applied stress curves for different oxidation temperatures and times when Γd = 0.3 J/m2 ; (b) the interface oxidation ratio versus applied stress curves for different oxidation temperatures and times when Γd = 0.3 J/m2 ; (c) the matrix cracking density versus applied stress curves for different oxidation temperatures and times when Γd = 0.5 J/m2 ; and (d) the interface oxidation ratio versus applied stress curves for different oxidation temperatures and times when Γd = 0.5 J/m2 .

(a)

(b)

111

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3 Matrix Multiple Cracking Evolution of Fiber-Reinforced CMCs at Elevated Temperature

Figure 3.13

(Continued)

(c)

(d)

and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.24 mm−1 at 𝜎 mc = 137 MPa to 𝜓 = 4.7 mm−1 at 𝜎 sat = 275 MPa, and the interface oxidation ratio decreases from 𝜔 = 34.1% to 𝜔 = 11.5%, and at oxidation duration t = 3 hours, the matrix cracking density increases from 𝜓 = 0.19 mm−1 at 𝜎 mc = 137 MPa to 𝜓 = 4.3 mm−1 at 𝜎 sat = 275 MPa, and the interface oxidation ratio decreases from 𝜔 = 75.9% to 𝜔 = 30.9%. At T = 1073 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.19 mm−1 at 𝜎 mc = 153 MPa to 𝜓 = 5.3 mm−1 at 𝜎 sat = 306 MPa, and the interface oxidation ratio decreases from 𝜔 = 57.6% to 𝜔 = 25.4%, and at the oxidation duration t = 3 hours, the matrix cracking density increases from 𝜓 = 0.13 mm−1 at 𝜎 mc = 153 MPa to 𝜓 = 4.5 mm−1 at 𝜎 sat = 306 MPa, and the interface oxidation ratio decreases from 𝜔 = 1 to 𝜔 = 59%. When Γd = 0.5 J/m2 at T = 873 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 1 mm−1 at 𝜎 mc = 118 MPa to 𝜓 = 5.4 mm−1 at 𝜎 sat = 235 MPa, and the interface oxidation ratio decreases from 𝜔 = 72.9% to 𝜔 = 5%, and at the oxidation duration t = 3 hours, the matrix cracking density increases from 𝜓 = 0.73 mm−1 at 𝜎 mc = 117 MPa to 𝜓 = 5 mm−1 at 𝜎 sat = 235 MPa, and the interface

3.4 Time-Dependent Matrix Multiple Cracking Evolution of C/SiC Composites

oxidation ratio decreases from 𝜔 = 1 to 𝜔 = 14.5%. At T = 973 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.35 mm−1 at 𝜎 mc = 137 MPa to 𝜓 = 5.3 mm−1 at 𝜎 sat = 275 MPa, and the interface oxidation ratio decreases from 𝜔 = 51% to 𝜔 = 12.9%, and at the oxidation duration t = 3 hours, the matrix cracking density increases from 𝜓 = 0.24 mm−1 at 𝜎 mc = 137 MPa to 𝜓 = 4.7 mm−1 at 𝜎 sat = 275 MPa, and the interface oxidation ratio decreases from 𝜔 = 1% to 34.3%. At T = 1073 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.22 mm−1 at 𝜎 mc = 153 MPa to 𝜓 = 5.6 mm−1 at 𝜎 sat = 306 MPa, and the interface oxidation ratio decreases from 𝜔 = 69.6% to 27.5%, and at the oxidation duration t = 3 hours, the matrix cracking density increases from 𝜓 = 0.14 mm−1 at 𝜎 mc = 153 MPa to 𝜓 = 4.6 mm−1 at 𝜎 sat = 306 MPa, and the interface oxidation ratio decreases from 𝜔 = 1% to 𝜔 = 62.8%. 3.4.2.4 Time-Dependent Matrix Multiple Cracking of C/SiC Composite for Different Matrix Fracture Energies

Figure 3.14 shows the effect of matrix fracture energy (i.e. Γm = 15 and 30 J/m2 ) on temperature-dependent matrix multiple cracking and interface oxidation of C/SiC composite at T = 873, 973, and 1073 K for the oxidation duration t = 1 and 3 hours. When Γm = 15 J/m2 at T = 873 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.33 mm−1 at 𝜎 mc = 85 MPa to 𝜓 = 4.4 mm−1 at 𝜎 sat = 171 MPa, and the interface oxidation ratio decreases from 𝜔 = 16.6% to 5.1%, and at the oxidation duration t = 3 hours, the matrix cracking density increases from 𝜓 = 0.29 mm−1 at 𝜎 mc = 85 MPa to 𝜓 = 4.2 mm−1 at 𝜎 sat = 171 MPa, and the interface oxidation ratio decreases from 𝜔 = 43.9% to 𝜔 = 14.8%. At T = 973 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.24 mm−1 at 𝜎 mc = 104 MPa to 𝜓 = 5 mm−1 at 𝜎 sat = 208 MPa, and the interface oxidation ratio decreases from 𝜔 = 33.9% to 13.6%, and at the oxidation duration t = 3 hours, the matrix cracking density increases from 𝜓 = 0.19 mm−1 at 𝜎 mc = 104 MPa to 𝜓 = 4.5 mm−1 at 𝜎 sat = 208 MPa, and the interface oxidation ratio decreases from 𝜔 = 75.7% to 36%. At T = 1073 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.2 mm−1 at 𝜎 mc = 119 MPa to 𝜓 = 5.6 mm−1 at 𝜎 sat = 239 MPa, and the interface oxidation ratio decreases from 𝜔 = 59.1% to 𝜔 = 29.3%, and at the oxidation duration t = 3 hours, the matrix cracking density increases from 𝜓 = 0.14 mm−1 at 𝜎 mc = 119 MPa to 𝜓 = 4.8 mm−1 at 𝜎 sat = 239 MPa, and the interface oxidation ratio decreases from 𝜔 = 1% to 𝜔 = 66%. When Γm = 30 J/m2 at T = 873 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.16 mm−1 at 𝜎 mc = 130 MPa to 𝜓 = 3.2 mm−1 at 𝜎 sat = 261 MPa, and the interface oxidation ratio decreases from 𝜔 = 7.6% to 𝜔 = 2.9%, and at oxidation duration t = 3 hours, the matrix cracking density increases from 𝜓 = 0.15 mm−1 at 𝜎 mc = 130 MPa to 𝜓 = 3.1 mm−1 at 𝜎 sat = 261 MPa, and the interface oxidation ratio decreases from 𝜔 = 21.6% to 𝜔 = 8.7%. At T = 973 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.15 mm−1 at 𝜎 mc = 150 MPa to 𝜓 = 3.9 mm−1 at 𝜎 sat = 301 MPa, and the interface oxidation ratio decreases from 𝜔 = 20.4% to 𝜔 = 8.9%, and at oxidation duration t = 3 hours, the matrix

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3 Matrix Multiple Cracking Evolution of Fiber-Reinforced CMCs at Elevated Temperature

Figure 3.14 (a) The matrix cracking density versus applied stress curves for different oxidation temperatures and times when Γm = 15 J/m2 ; (b) the interface oxidation ratio versus applied stress curves for different oxidation temperatures and times when Γm = 15 J/m2 ; (c) the matrix cracking density versus applied stress curves for different oxidation temperatures and times when Γm = 30 J/m2 ; and (d) the interface oxidation ratio versus applied stress curves for different oxidation temperatures and times when Γm = 30 J/m2 .

(a)

(b)

cracking density increases from 𝜓 = 0.13 mm−1 at 𝜎 mc = 150 MPa to 𝜓 = 3.7 mm−1 at 𝜎 sat = 301 MPa, and the interface oxidation ratio decreases from 𝜔 = 50.7% to 𝜔 = 24.5%. At T = 1073 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.15 mm−1 at 𝜎 mc = 166 MPa to 𝜓 = 4.6 mm−1 at 𝜎 sat = 333 MPa, and the interface oxidation ratio decreases from 𝜔 = 42.3% to 𝜔 = 21%, and at the oxidation duration t = 3 hours, the matrix cracking density increases from 𝜓 = 0.11 mm−1 at 𝜎 mc = 166 MPa to 𝜓 = 4.2 mm−1 at 𝜎 sat = 333 MPa, and the interface oxidation ratio decreases from 𝜔 = 85.7% to 𝜔 = 50.9%.

3.4.3

Experimental Comparisons

Figure 3.15 shows the experimental and theoretical matrix cracking density and interface debonding of unidirectional C/SiC composite at room temperature, T = 773 and 873 K for the oxidation duration t = 1 and 3 hours. At room temperature, matrix cracking starts from 𝜎 mc = 100 MPa and approaches to saturation at 𝜎 sat = 220 MPa; the matrix cracking density increases from

3.4 Time-Dependent Matrix Multiple Cracking Evolution of C/SiC Composites

Figure 3.14

(Continued)

(c)

(d)

𝜓 = 4.2 mm−1 to 𝜓 = 9.4 mm−1 . Experimental matrix cracking density of C/SiC composite is obtained through the optical microscope observation on the surface of the specimen. The matrix cracking on the surface of the specimen first occurs at low stress level in the matrix rich region, and these surface cracks do not propagate through the thickness of the whole specimen. However, the model in the present analysis is suitable for predicting the evolution of matrix cracking, which propagates through the thickness of the specimen. The predicted matrix cracking stress is higher than the experimental data, and the saturation of the matrix cracking density is lower than the experimental data. At T = 773 K, matrix cracking density increases from 𝜓 = 1.3 mm−1 at 𝜎 mc = 140 MPa to 𝜓 = 8.7 mm−1 at 𝜎 sat = 281 MPa, and at oxidation duration t = 1 hour, the matrix cracking density increases from 𝜓 = 0.84 mm−1 at 𝜎 mc = 140 MPa to 𝜓 = 7.8 mm−1 at 𝜎 sat = 281 MPa, and the interface oxidation ratio decreases from 𝜔 = 13.9% to 2.2%, and at oxidation duration t = 3 hours, the matrix cracking density increases from 𝜓 = 0.74 mm−1 at 𝜎 mc = 140 MPa to 𝜓 = 7.5 mm−1

115

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3 Matrix Multiple Cracking Evolution of Fiber-Reinforced CMCs at Elevated Temperature

Figure 3.15 (a) The experimental and theoretical matrix cracking density versus applied stress curves for different oxidation temperatures and times and (b) the interface oxidation ratio versus applied stress curves for different oxidation temperatures and times of unidirectional C/SiC composite.

(a)

(b)

at 𝜎 sat = 281 MPa, and the interface oxidation ratio decreases from 𝜔 = 35% to 𝜔 = 6.5%. At T = 873 K, the matrix cracking density increases from 𝜓 = 1.1 mm−1 at 𝜎 mc = 182 MPa to 𝜓 = 9.4 mm−1 at 𝜎 sat = 365 MPa, and at the oxidation duration t = 1 hour, the matrix cracking density increases from 𝜓 = 0.38 mm−1 at 𝜎 mc = 182 MPa to 𝜓 = 6.8 mm−1 at 𝜎 sat = 365 MPa, and the interface oxidation ratio decreases from 𝜔 = 19.2% to 𝜔 = 6.1%, and at the oxidation duration t = 3 hours, the matrix cracking density increases from 𝜓 = 0.31 mm−1 at 𝜎 mc = 182 MPa to 𝜔 = 6.4 mm−1 at 𝜎 sat = 365 MPa, and the interface oxidation ratio decreases from 𝜔 = 45.3% to 𝜔 = 16.9%.

3.5 Time-Dependent Matrix Multiple Cracking Evolution of SiC/SiC Composites In this section, time-dependent matrix multiple cracking of SiC/SiC composite is investigated considering the interface oxidation. Effects of fiber volume, interface

3.5 Time-Dependent Matrix Multiple Cracking Evolution of SiC/SiC Composites

shear stress, interface frictional coefficient, interface debonding energy, and matrix fracture energy on matrix cracking density and interface oxidation ratio of SiC/SiC composite are discussed for different temperatures and oxidation durations. Experimental matrix cracking density and interface oxidation ratio for unidirectional and mini SiC/SiC composites at different testing temperatures and oxidation durations are predicted.

3.5.1

Results and Discussion

The ceramic composite system of SiC/SiC is used for the case study, and its material properties are given by V f = 30%, r f = 7.5 μm, Ef = 230 GPa, Γm = 15 J/m2 , Γd = 0.4 J/m2 , 𝜏 0 = 10 MPa, 𝛼 rf = 2.9 × 10−6 K−1 , and 𝛼 lf = 3.9 × 10−6 K−1 . Effects of fiber volume, interface shear stress, interface frictional coefficient, interface debonding energy, and matrix fracture energy on time-dependent matrix cracking density and interface oxidation ratio are discussed. 3.5.1.1 Time-Dependent Matrix Multiple Cracking of SiC/SiC Composite for Different Fiber Volumes

Figure 3.16 shows the effect of fiber volume (i.e. V f = 25%, 30%, and 35%) on time-dependent matrix cracking density and interface oxidation ratio of SiC/SiC composite at T = 873, 973, and 1073 K for the oxidation duration t = 1 and 3 hours. When V f = 25% at T = 873 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 1.2 mm−1 at 𝜎 mc = 110 MPa to 𝜓 = 11.9 mm−1 at 𝜎 sat = 120 MPa, and the interface oxidation ratio decreases from 𝜔 = 6% to 𝜔 = 3.4%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.67 mm−1 at 𝜎 mc = 110 MPa to 𝜓 = 10.7 mm−1 at 𝜎 sat = 123 MPa, and the interface oxidation ratio decreases from 𝜔 = 16.6% to 𝜔 = 9.8%. At T = 973 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.24 mm−1 at 𝜎 mc = 104 MPa to 𝜓 = 9.2 mm−1 at 𝜎 sat = 133 MPa, and the interface oxidation ratio decreases from 𝜔 = 16.7% to 𝜔 = 9.5%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.17 mm−1 at 𝜎 mc = 104 MPa to 𝜓 = 7.4 mm−1 at 𝜎 sat = 140 MPa, and the interface oxidation ratio decreases from 𝜔 = 40.5% to 𝜔 = 25.1%. At T = 1073 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.15 mm−1 at 𝜎 mc = 96 MPa to 𝜓 = 7 mm−1 at 𝜎 sat = 144 MPa, and the interface oxidation ratio decreases from 𝜔 = 36.7% to 𝜔 = 21.1%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.09 mm−1 at 𝜎 mc = 96.3 MPa to 𝜓 = 5.0 mm−1 at 𝜎 sat = 144 MPa, and the interface oxidation ratio decreases from 𝜔 = 72.8% to 𝜔 = 48.9%. When V f = 30% at T = 873 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.47 mm−1 at 𝜎 mc = 146 MPa to 𝜓 = 10.36 mm−1 at 𝜎 sat = 167 MPa, and the interface oxidation ratio decreases from 𝜔 = 5.6% to 𝜔 = 3.2%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.35 mm−1 at 𝜎 mc = 146 MPa to 𝜓 = 3.4 mm−1 at 𝜎 sat = 170 MPa, and the interface oxidation ratio decreases from 𝜔 = 15.6% to 𝜔 = 9.3%. At T = 973 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.21 mm−1 at 𝜎 mc = 134 MPa to 𝜓 = 8.46 mm−1 at 𝜎 sat = 178 MPa,

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3 Matrix Multiple Cracking Evolution of Fiber-Reinforced CMCs at Elevated Temperature

(a)

(b)

(c)

Figure 3.16 (a) The matrix cracking density versus applied stress curves for different oxidation temperatures and times when V f = 25%; (b) the interface oxidation ratio versus applied stress curves for different oxidation temperatures and times when V f = 25%; (c) the matrix cracking density versus applied stress curves for different oxidation temperatures and times when V f = 30%; (d) the interface oxidation ratio versus applied stress curves for different oxidation temperatures and times when V f = 30%; (e) the matrix cracking density versus applied stress curves for different oxidation temperatures and times when V f = 35%; and (f) the interface oxidation ratio versus applied stress curves for different oxidation temperatures and times when V f = 35%.

3.5 Time-Dependent Matrix Multiple Cracking Evolution of SiC/SiC Composites

Figure 3.16

(Continued)

(d)

(e)

(f)

119

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3 Matrix Multiple Cracking Evolution of Fiber-Reinforced CMCs at Elevated Temperature

and the interface oxidation ratio decreases from 𝜔 = 15.9% to 𝜔 = 9.1%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.15 mm−1 at 𝜎 mc = 134 MPa to 𝜓 = 6.9 mm−1 at 𝜎 sat = 186 MPa, and the interface oxidation ratio decreases from 𝜔 = 38.9% to 𝜔 = 24.3%. At T = 1073 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.14 mm−1 at 𝜎 mc = 122 MPa to 𝜓 = 6.7 mm−1 at 𝜎 sat = 183 MPa, and the interface oxidation ratio decreases from 𝜔 = 35.7% to 𝜔 = 20.7%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.09 mm−1 at 𝜎 mc = 122 MPa to 𝜓 = 4.8 mm−1 at 𝜎 sat = 183 MPa, and the interface oxidation ratio decreases from 𝜔 = 71.5% to 𝜔 = 48.2%. When V f = 35% at T = 873 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.35 mm−1 at 𝜎 mc = 180 MPa to 𝜓 = 9.75 mm−1 at 𝜎 sat = 215 MPa, and the interface oxidation ratio decreases from 𝜔 = 5.5% to 𝜔 = 3.2%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.28 mm−1 at 𝜎 mc = 180 MPa to 𝜓 = 8.9 mm−1 at 𝜎 sat = 219 MPa, and the interface oxidation ratio decreases from 𝜔 = 15.4% to 9.2%. At T = 973 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.2 mm−1 at 𝜎 mc = 165 MPa to 𝜓 = 8.2 mm−1 at 𝜎 sat = 223 MPa, and the interface oxidation ratio decreases from 𝜔 = 15.9% to 𝜔 = 9.2%, and at t = 3 hours, the matrix multi-cracking density increases from 𝜓 = 0.15 mm−1 at 𝜎 mc = 165 MPa to 𝜓 = 6.7 mm−1 at 𝜎 sat = 234 MPa, and the interface oxidation ratio decreases from 𝜔 = 38.8% to 𝜔 = 24.3%. At T = 1073 K and t = 1 hour, the matrix multi-cracking density increases from 𝜓 = 0.14 mm−1 at 𝜎 mc = 148 MPa to 𝜓 = 6.6 mm−1 at 𝜎 sat = 222 MPa, and the interface oxidation ratio decreases from 𝜔 = 36% to 𝜔 = 21%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.09 mm−1 at 𝜎 mc = 148 MPa to 𝜓 = 4.8 mm−1 at 𝜎 sat = 222 MPa, and the interface oxidation ratio decreases from 𝜔 = 71.9% to 48.7%. 3.5.1.2 Time-Dependent Matrix Multiple Cracking of SiC/SiC Composite for Different Interface Shear Stress

Figure 3.17 shows the effect of interface shear stress (i.e. 𝜏 0 = 15, 20, and 25 MPa) on time-dependent matrix cracking density and interface oxidation ratio of SiC/SiC composite at T = 873, 973, and 1073 K for t = 1 and 3 hours. When 𝜏 0 = 15 MPa at T = 873 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.34 mm−1 at 𝜎 mc = 191 MPa to 𝜓 = 10.33 mm−1 at 𝜎 sat = 234 MPa, and the interface oxidation ratio decreases from 𝜔 = 6% to 𝜔 = 3.5%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.27 mm−1 at 𝜎 mc = 191 MPa to 𝜓 = 9.3 mm−1 at 𝜎 sat = 240 MPa, and the interface oxidation ratio decreases from 𝜔 = 16.6% to 𝜔 = 10.2%. At T = 973 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.21 mm−1 at 𝜎 mc = 176 MPa to 𝜓 = 8.7 mm−1 at 𝜎 sat = 245 MPa, and the interface oxidation ratio decreases from 𝜔 = 17.1% to 𝜔 = 10.1%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.15 mm−1 at 𝜎 mc = 176 MPa to 𝜓 = 7 mm−1 at 𝜎 sat = 258 MPa, and the interface oxidation ratio decreases from 𝜔 = 40.6% to 𝜔 = 26.3%. At T = 1073 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.15 mm−1 at 𝜎 mc = 159 MPa to 𝜓 = 7 mm−1 at 𝜎 sat = 239 MPa, and the interface oxidation ratio decreases from 𝜔 = 37.5% to 𝜔 = 22.8%, and at t = 3 hours, the matrix cracking

3.5 Time-Dependent Matrix Multiple Cracking Evolution of SiC/SiC Composites

Figure 3.17 (a) The matrix cracking density versus applied stress curves for different oxidation temperatures and times when 𝜏 0 = 15 MPa; (b) the interface oxidation ratio versus applied stress curves for different oxidation temperatures and times when 𝜏 0 = 15 MPa; (c) the matrix cracking density versus applied stress curves for different oxidation temperatures and times when 𝜏 0 = 20 MPa; and (d) the interface oxidation ratio versus applied stress curves for different oxidation temperatures and times when 𝜏 0 = 20 MPa; (e) the matrix cracking density versus applied stress curves for different oxidation temperatures and times when 𝜏 0 = 25 MPa; and (f) the interface oxidation ratio versus applied stress curves for different oxidation temperatures and times when 𝜏 0 = 25 MPa.

(a)

(b)

(c)

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3 Matrix Multiple Cracking Evolution of Fiber-Reinforced CMCs at Elevated Temperature

Figure 3.17

(d)

(e)

(f)

(Continued)

3.5 Time-Dependent Matrix Multiple Cracking Evolution of SiC/SiC Composites

density increases from 𝜓 = 0.09 mm−1 at 𝜎 mc = 159 MPa to 𝜓 = 4.8 mm−1 at 𝜎 sat = 239 MPa, and the interface oxidation ratio decreases from 𝜔 = 71.7% to 𝜔 = 50.9%. When 𝜏 0 = 20 MPa at T = 873 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.34 mm−1 at 𝜎 mc = 201 MPa to 𝜓 = 10.89 mm−1 at 𝜎 sat = 251 MPa, and the interface oxidation ratio decreases from 𝜔 = 6.5% to 𝜔 = 3.9%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.27 mm−1 at 𝜎 mc = 201 MPa to 𝜓 = 9.7 mm−1 at 𝜎 sat = 260 MPa, and the interface oxidation ratio decreases from 𝜔 = 17.7% to 𝜔 = 11%. At T = 973 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.22 mm−1 at 𝜎 mc = 186 MPa to 𝜓 = 9.26 mm−1 at 𝜎 sat = 265 MPa, and the interface oxidation ratio decreases from 𝜔 = 18.2% to 𝜔 = 11%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.15 mm−1 at 𝜎 mc = 186 MPa to 𝜓 = 7.3 mm−1 at 𝜎 sat = 280 MPa, and the interface oxidation ratio decreases from 𝜔 = 42.3% to 𝜔 = 28.1%. At T = 1073 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.15 mm−1 at 𝜎 mc = 169 MPa to 𝜓 = 7.4 mm−1 at 𝜎 sat = 254 MPa, and the interface oxidation ratio decreases from 𝜔 = 39% to 𝜔 = 24.5%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.09 mm−1 at 𝜎 mc = 169 MPa to 𝜓 = 4.9 mm−1 at 𝜎 sat = 254 MPa, and the interface oxidation ratio decreases from 𝜔 = 72.1% to 𝜔 = 52.9%. When 𝜏 0 = 25 MPa at T = 873 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.35 mm−1 at 𝜎 mc = 211 MPa to 𝜓 = 11.4 mm−1 at 𝜎 sat = 270 MPa, and the interface oxidation ratio decreases from 𝜔 = 7% to 𝜔 = 4.2%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.28 mm−1 at 𝜎 mc = 211 MPa to 𝜓 = 10.2 mm−1 at 𝜎 sat = 277 MPa, and the interface oxidation ratio decreases from 𝜔 = 18.8% to 𝜔 = 11.8%. At T = 973 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.23 mm−1 at 𝜎 mc = 195 MPa to 𝜓 = 9.7 mm−1 at 𝜎 sat = 282 MPa, and the interface oxidation ratio decreases from 𝜔 = 19.3% to 𝜔 = 11.8%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.15 mm−1 at 𝜎 mc = 195 MPa to 𝜓 = 7.5 mm−1 at 𝜎 sat = 293 MPa, and the interface oxidation ratio decreases from 𝜔 = 43.8% to 𝜔 = 29.7%. At T = 1073 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.16 mm−1 at 𝜎 mc = 178 MPa to 𝜓 = 7.7 mm−1 at 𝜎 sat = 268 MPa, and the interface oxidation ratio decreases from 𝜔 = 40.4% to 𝜔 = 26.1%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.09 mm−1 at 𝜎 mc = 178 MPa to 𝜓 = 5 mm−1 at 𝜎 sat = 268 MPa, and the interface oxidation ratio decreases from 𝜔 = 72.6% to 𝜔 = 54.6%. Figure 3.18 shows the effect of interface shear stress (i.e. 𝜏 f = 1, 3, and 5 MPa) on time-dependent matrix cracking density and interface oxidation ratio of SiC/SiC composite at T = 873, 973, and 1073 K for t = 1 and 3 hours. When 𝜏 f = 1 MPa at T = 873 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.34 mm−1 at 𝜎 mc = 180 MPa to 𝜓 = 9.6 mm−1 at 𝜎 sat = 214 MPa, and the interface oxidation ratio decreases from 𝜔 = 5.5% to 𝜔 = 3.2%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.27 mm−1 at 𝜎 mc = 180 MPa to 𝜓 = 8.6 mm−1 at 𝜎 sat = 220 MPa, and the interface oxidation

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3 Matrix Multiple Cracking Evolution of Fiber-Reinforced CMCs at Elevated Temperature

(a)

(b)

(c)

Figure 3.18 (a) The matrix cracking density versus applied stress curves for different oxidation temperatures and times when 𝜏 f = 1 MPa; (b) the interface oxidation ratio versus applied stress curves for different oxidation temperatures and times when 𝜏 f = 1 MPa; (c) the matrix cracking density versus applied stress curves for different oxidation temperatures and times when 𝜏 f = 3 MPa; (d) the interface oxidation ratio versus applied stress curves for different oxidation temperatures and times when 𝜏 f = 3 MPa; (e) the matrix cracking density versus applied stress curves for different oxidation temperatures and times when 𝜏 f = 5 MPa; and (f) the interface oxidation ratio versus applied stress curves for different oxidation temperatures and times when 𝜏 f = 5 MPa.

3.5 Time-Dependent Matrix Multiple Cracking Evolution of SiC/SiC Composites

Figure 3.18

(Continued)

(d)

(e)

(f)

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3 Matrix Multiple Cracking Evolution of Fiber-Reinforced CMCs at Elevated Temperature

ratio decreases from 𝜔 = 15% to 𝜔 = 9.1%. At T = 973 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.19 mm−1 at 𝜎 mc = 165 MPa to 𝜓 = 7.9 mm−1 at 𝜎 sat = 225 MPa, and the interface oxidation ratio decreases from 𝜔 = 15.4% to 𝜔 = 9%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.13 mm−1 at 𝜎 mc = 165 MPa to 𝜓 = 6.3 mm−1 at 𝜎 sat = 237 MPa, and the interface oxidation ratio decreases from 𝜔 = 36.2% to 𝜔 = 23.3%. At T = 1073 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.13 mm−1 at 𝜎 mc = 148 MPa to 𝜓 = 6.3 mm−1 at 𝜎 sat = 222 MPa, and the interface oxidation ratio decreases from 𝜔 = 33.5% to 𝜔 = 20%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.07 mm−1 at 𝜎 mc = 148 MPa to 𝜓 = 4.2 mm−1 at 𝜎 sat = 222 MPa, and the interface oxidation ratio decreases from 𝜔 = 62.7% to 𝜔 = 44.3%. When 𝜏 f = 3 MPa at T = 873 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.35 mm−1 at 𝜎 mc = 180 MPa to 𝜓 = 9.7 mm−1 at 𝜎 sat = 214 MPa, and the interface oxidation ratio decreases from 𝜔 = 5.5% to 𝜔 = 3.2%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.28 mm−1 at 𝜎 mc = 180 MPa to 𝜓 = 8.7 mm−1 at 𝜎 sat = 220 MPa, and the interface oxidation ratio decreases from 𝜔 = 15.2% to 𝜔 = 9.2%. At T = 973 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.2 mm−1 at 𝜎 mc = 165 MPa to 𝜓 = 8.0 mm−1 at 𝜎 sat = 224 MPa, and the interface oxidation ratio decreases from 𝜔 = 15.6% to 𝜔 = 9.1%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.14 mm−1 at 𝜎 mc = 165 MPa to 𝜓 = 6.5 mm−1 at 𝜎 sat = 236 MPa, and the interface oxidation ratio decreases from 𝜔 = 37.5% to 𝜔 = 23.8%. At T = 1073 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.14 mm−1 at 𝜎 mc = 148 MPa to 𝜓 = 6.4 mm−1 at 𝜎 sat = 222 MPa, and the interface oxidation ratio decreases from 𝜔 = 34.7% to 𝜔 = 20.5%, and at t = 3 hours, the matrix multi-cracking density increases from 𝜓 = 0.08 mm−1 at 𝜎 mc = 148 MPa to 𝜓 = 4.5 mm−1 at 𝜎 sat = 222 MPa, and the interface oxidation ratio decreases from 𝜔 = 67% to 𝜔 = 46.4%. When 𝜏 f = 5 MPa at T = 873 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.35 mm−1 at 𝜎 mc = 180 MPa to 𝜓 = 9.7 mm−1 at 𝜎 sat = 214 MPa, and the interface oxidation ratio decreases from 𝜔 = 5.5% to 𝜔 = 3.2%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.29 mm−1 at 𝜎 mc = 180 MPa to 𝜓 = 8.9 mm−1 at 𝜎 sat = 220 MPa, and the interface oxidation ratio decreases from 𝜔 = 15.4% to 𝜔 = 9.3%. At T = 973 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.2 mm−1 at 𝜎 mc = 165 MPa to 𝜓 = 8.2 mm−1 at 𝜎 sat = 224 MPa, and the interface oxidation ratio decreases from 𝜔 = 15.9% to 𝜔 = 9.2%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.15 mm−1 at 𝜎 mc = 165 MPa to 𝜓 = 6.7 mm−1 at 𝜎 sat = 234 MPa, and the interface oxidation ratio decreases from 𝜔 = 38.8% to 𝜔 = 24.3%. At T = 1073 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.15 mm−1 at 𝜎 mc = 148 MPa to 𝜓 = 6.6 mm−1 at 𝜎 sat = 222 MPa, and the interface oxidation ratio decreases from 𝜔 = 36% to 𝜔 = 21%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.09 mm−1 at 𝜎 mc = 148 MPa to 𝜓 = 4.8 mm−1 at 𝜎 sat = 222 MPa, and the interface oxidation ratio decreases from 𝜔 = 72% to 𝜔 = 48.7%.

3.5 Time-Dependent Matrix Multiple Cracking Evolution of SiC/SiC Composites

3.5.1.3 Time-Dependent Matrix Multiple Cracking of SiC/SiC Composite for Different Interface Frictional Coefficients

Figure 3.19 shows the effect of fiber/matrix interface frictional coefficient (i.e. 𝜇 = 0.05, 0.1, and 0.15) on time-dependent matrix cracking density and interface oxidation ratio of SiC/SiC composite at T = 873, 973, and 1073 K for t = 1 and 3 hours. When 𝜇 = 0.05 at T = 873 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 1.2 mm−1 at 𝜎 mc = 128 MPa to 𝜓 = 9.9 mm−1 at 𝜎 sat = 136 MPa, and the interface oxidation ratio decreases from 𝜔 = 4.9% to 𝜔 = 2.7%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.71 mm−1 at 𝜎 mc = 128 MPa to 𝜓 = 9.2 mm−1 at 𝜎 sat = 138 MPa, and the interface oxidation ratio decreases from 𝜔 = 13.8% to 𝜔 = 7.9%. At T = 973 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.21 mm−1 at 𝜎 mc = 120 MPa to 𝜓 = 7.9 mm−1 at 𝜎 sat = 151 MPa, and the interface oxidation ratio decreases from 𝜔 = 14.5% to 𝜔 = 7.9%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.16 mm−1 at 𝜎 mc = 120 MPa to 𝜓 = 6.7 mm−1 at 𝜎 sat = 156 MPa, and the interface oxidation ratio decreases from 𝜔 = 37.1% to 𝜔 = 21.7%. At T = 1073 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.14 mm−1 at 𝜎 mc = 110 MPa to 𝜓 = 6.3 mm−1 at 𝜎 sat = 162 MPa, and the interface oxidation ratio decreases from 𝜔 = 34.8% to 𝜔 = 18.8%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.09 mm−1 at 𝜎 mc = 110 MPa to 𝜓 = 4.8 mm−1 at 𝜎 sat = 166 MPa, and the interface oxidation ratio decreases from 𝜔 = 73.7% to 𝜔 = 46.1%. When 𝜇 = 0.1 at T = 873 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.47 mm−1 at 𝜎 mc = 146 MPa to 𝜓 = 10.3 mm−1 at 𝜎 sat = 167 MPa, and the interface oxidation ratio decreases from 𝜔 = 5.6% to 𝜔 = 3.2%, and at t = 3 hour, the matrix cracking density increases from 𝜓 = 0.35 mm−1 at 𝜎 mc = 146 MPa to 𝜓 = 9.4 mm−1 at 𝜎 sat = 170 MPa, and the interface oxidation ratio decreases from 𝜔 = 15.6% to 𝜔 = 9.3%. At T = 973 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.21 mm−1 at 𝜎 mc = 134 MPa to 𝜓 = 8.4 mm−1 at 𝜎 sat = 178 MPa, and the interface oxidation ratio decreases from 𝜔 = 15.9% to 𝜔 = 9.1%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.15 mm−1 at 𝜎 mc = 134 MPa to 𝜓 = 6.9 mm−1 at 𝜎 sat = 186 MPa, and the interface oxidation ratio decreases from 𝜔 = 38.9% to 𝜔 = 24.3%. At T = 1073 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.14 mm−1 at 𝜎 mc = 122 MPa to 𝜓 = 6.7 mm−1 at 𝜎 sat = 183 MPa, and the interface oxidation ratio decreases from 𝜔 = 35.7% to 𝜔 = 20.7%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.09 mm−1 at 𝜎 mc = 122 MPa to 𝜓 = 4.8 mm−1 at 𝜎 sat = 183 MPa, and the interface oxidation ratio decreases from 𝜔 = 71.5% to 𝜔 = 48.2%. When 𝜇 = 0.15 at T = 873 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.4 mm−1 at 𝜎 mc = 160 MPa to 𝜓 = 10.9 mm−1 at 𝜎 sat = 193 MPa, and the interface oxidation ratio decreases from 𝜔 = 6.3% to 𝜔 = 3.7%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.31 mm−1 at 𝜎 mc = 160 MPa to 𝜓 = 9.8 mm−1 at 𝜎 sat = 198 MPa, and the interface oxidation ratio decreases from 𝜔 = 17.1% to 𝜔 = 10.5%. At T = 973 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.22 mm−1 at 𝜎 mc = 147 MPa to 𝜓 = 8.9 mm−1 at 𝜎 sat = 201 MPa,

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3 Matrix Multiple Cracking Evolution of Fiber-Reinforced CMCs at Elevated Temperature

(a)

(b)

(c)

Figure 3.19 (a) The matrix cracking density versus applied stress curves for different oxidation temperatures and times when 𝜇 = 0.05; (b) the interface oxidation ratio versus applied stress curves for different oxidation temperatures and times when 𝜇 = 0.05; (c) the matrix cracking density versus applied stress curves for different oxidation temperatures and times when 𝜇 = 0.1; (d) the interface oxidation ratio versus applied stress curves for different oxidation temperatures and times when 𝜇 = 0.1; (e) the matrix cracking density versus applied stress curves for different oxidation temperatures and times when 𝜇 = 0.15; and (f) the interface oxidation ratio versus applied stress curves for different oxidation temperatures and times when 𝜇 = 0.15.

3.5 Time-Dependent Matrix Multiple Cracking Evolution of SiC/SiC Composites

Figure 3.19

(Continued)

(d)

(e)

(f)

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3 Matrix Multiple Cracking Evolution of Fiber-Reinforced CMCs at Elevated Temperature

and the interface oxidation ratio decreases from 𝜔 = 17.2% to 𝜔 = 10.2%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.15 mm−1 at 𝜎 mc = 147 MPa to 𝜓 = 7.2 mm−1 at 𝜎 sat = 212 MPa, and the interface oxidation ratio decreases from 𝜔 = 40.7% to 𝜔 = 26.5%. At T = 1073 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.15 mm−1 at 𝜎 mc = 132 MPa to 𝜓 = 7 mm−1 at 𝜎 sat = 198 MPa, and the interface oxidation ratio decreases from 𝜔 = 36.9% to 𝜔 = 22.4%, and at t = 3 hours, the matrix multi-cracking density increases from 𝜓 = 0.09 mm−1 at 𝜎 mc = 132 MPa to 𝜓 = 4.8 mm−1 at 𝜎 sat = 198 MPa, and the interface oxidation ratio decreases from 𝜔 = 71% to 𝜔 = 50.1%. 3.5.1.4 Time-Dependent Matrix Multiple Cracking of SiC/SiC Composite for Different Interface Debonding Energies

Figure 3.20 shows the effect of interface debonding energy (i.e. Γd = 0.3, 0.5, and 0.7 J/m2 ) on time-dependent matrix cracking density and interface oxidation ratio of SiC/SiC composite at T = 873, 973, and 1073 K for t = 1 and 3 hours. When Γd = 0.3 J/m2 at T = 873 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.7 mm−1 at 𝜎 mc = 146 MPa to 𝜓 = 10.3 mm−1 at 𝜎 sat = 160 MPa, and the interface oxidation ratio decreases from 𝜔 = 5.2% to 𝜔 = 3.1%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.49 mm−1 at 𝜎 mc = 146 MPa to 𝜓 = 9.4 mm−1 at 𝜎 sat = 164 MPa, and the interface oxidation ratio decreases from 𝜔 = 14.7% to 𝜔 = 9.0%. At T = 973 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.24 mm−1 at 𝜎 mc = 134 MPa to 𝜓 = 8.4 mm−1 at 𝜎 sat = 171 MPa, and the interface oxidation ratio decreases from 𝜔 = 14.7% to 𝜔 = 8.7%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.16 mm−1 at 𝜎 mc = 134 MPa to 𝜓 = 6.9 mm−1 at 𝜎 sat = 179 MPa, and the interface oxidation ratio decreases from 𝜔 = 36.5% to 𝜔 = 23.3%. At T = 1073 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.15 mm−1 at 𝜎 mc = 122 MPa to 𝜓 = 6.7 mm−1 at 𝜎 sat = 183 MPa, and the interface oxidation ratio decreases from 𝜔 = 32.6% to 𝜔 = 19.6%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.09 mm−1 at 𝜎 mc = 122 MPa to 𝜓 = 4.8 mm−1 at 𝜎 sat = 183 MPa, and the interface oxidation ratio decreases from 𝜔 = 67.3% to 𝜔 = 46.3%. When Γd = 0.5 J/m2 at T = 873 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.3 mm−1 at 𝜎 mc = 146 MPa to 𝜓 = 10.3 mm−1 at 𝜎 sat = 173 MPa, and the interface oxidation ratio decreases from 𝜔 = 5.9% to 𝜔 = 3.3%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.29 mm−1 at 𝜎 mc = 146 MPa to 𝜓 = 9.4 mm−1 at 𝜎 sat = 176 MPa, and the interface oxidation ratio decreases from 𝜔 = 16.5% to 𝜔 = 9.6%. At T = 973 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.2 mm−1 at 𝜎 mc = 134 MPa to 𝜓 = 8.4 mm−1 at 𝜎 sat = 184 MPa, and the interface oxidation ratio decreases from 𝜔 = 17.1% to 𝜔 = 9.5%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.15 mm−1 at 𝜎 mc = 134 MPa to 𝜓 = 6.9 mm−1 at 𝜎 sat = 192 MPa, and the interface oxidation ratio decreases from 𝜔 = 41.2% to 𝜔 = 25.2%. At T = 1073 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.15 mm−1 at 𝜎 mc = 122 MPa to 𝜓 = 6.7 mm−1 at 𝜎 sat = 183 MPa, and the interface oxidation ratio

3.5 Time-Dependent Matrix Multiple Cracking Evolution of SiC/SiC Composites

Figure 3.20 (a) The matrix cracking density versus applied stress curves for different oxidation temperatures and times when Γd = 0.3 J/m2 ; (b) the interface oxidation ratio versus applied stress curves for different oxidation temperatures and times when Γd = 0.3 J/m2 ; (c) the matrix cracking density versus applied stress curves for different oxidation temperatures and times when Γd = 0.5 J/m2 ; (d) the interface oxidation ratio versus applied stress curves for different oxidation temperatures and times when Γd = 0.5 J/m2 ; (e) the matrix cracking density versus applied stress curves for different oxidation temperatures and times when Γd = 0.7 J/m2 ; and (f) the interface oxidation ratio versus applied stress curves for different oxidation temperatures and times when Γd = 0.7 J/m2 .

(a)

(b)

(c)

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3 Matrix Multiple Cracking Evolution of Fiber-Reinforced CMCs at Elevated Temperature

Figure 3.20

(d)

(e)

(f)

(Continued)

3.5 Time-Dependent Matrix Multiple Cracking Evolution of SiC/SiC Composites

decreases from 𝜔 = 38.9% to 𝜔 = 21.7%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.09 mm−1 at 𝜎 mc = 122 MPa to 𝜓 = 4.8 mm−1 at 𝜎 sat = 183 MPa, and the interface oxidation ratio decreases from 𝜔 = 75.6% to 𝜔 = 50%. When Γd = 0.7 J/m2 at T = 873 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.3 mm−1 at 𝜎 mc = 146 MPa to 𝜓 = 10.3 mm−1 at 𝜎 sat = 183 MPa, and the interface oxidation ratio decreases from 𝜔 = 6.7% to 𝜔 = 3.6%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.24 mm−1 at 𝜎 mc = 146 MPa to 𝜓 = 9.4 mm−1 at 𝜎 sat = 186 MPa, and the interface oxidation ratio decreases from 𝜔 = 18.2% to 𝜔 = 10.2%. At T = 973 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.2 mm−1 at 𝜎 mc = 134 MPa to 𝜓 = 8.4 mm−1 at 𝜎 sat = 202 MPa, and the interface oxidation ratio decreases from 𝜔 = 19.7% to 𝜔 = 10.3%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.15 mm−1 at 𝜎 mc = 134 MPa to 𝜓 = 6.9 mm−1 at 𝜎 sat = 202 MPa, and the interface oxidation ratio decreases from 𝜔 = 46% to 𝜔 = 27%. At T = 1073 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.16 mm−1 at 𝜎 mc = 122 MPa to 𝜓 = 6.8 mm−1 at 𝜎 sat = 183 MPa, and the interface oxidation ratio decreases from 𝜔 = 46% to 𝜔 = 23.8%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.09 mm−1 at 𝜎 mc = 122 MPa to 𝜓 = 4.9 mm−1 at 𝜎 sat = 183 MPa, and the interface oxidation ratio decreases from 𝜔 = 84.1% to 𝜔 = 53.6%. 3.5.1.5 Time-Dependent Matrix Cracking Stress of SiC/SiC Composite for Different Matrix Fracture Energies

Figure 3.21 shows the effect of matrix fracture energy (i.e. Γm = 20, 25, and 30 J/m2 ) on time-dependent matrix cracking density and interface oxidation ratio of SiC/SiC composite at T = 873, 973, and 1073 K for t = 1 and 3 hours. When Γm = 20 J/m2 at T = 873 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.3 mm−1 at 𝜎 mc = 162 MPa to 𝜓 = 8.4 mm−1 at 𝜎 sat = 195 MPa, and the interface oxidation ratio decreases from 𝜔 = 4.8% to 𝜔 = 2.8%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.24 mm−1 at 𝜎 mc = 162 MPa to 𝜓 = 7.8 mm−1 at 𝜎 sat = 199 MPa, and the interface oxidation ratio decreases from 𝜔 = 13.6% to 𝜔 = 8.2%. At T = 973 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.18 mm−1 at 𝜎 mc = 149 MPa to 𝜓 = 7.1 mm−1 at 𝜎 sat = 204 MPa, and the interface oxidation ratio decreases from 𝜔 = 13.7% to 𝜔 = 8%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.13 mm−1 at 𝜎 mc = 149 MPa to 𝜓 = 6 mm−1 at 𝜎 sat = 212 MPa, and the interface oxidation ratio decreases from 𝜔 = 34.4% to 𝜔 = 21.6%. At T = 1073 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.13 mm−1 at 𝜎 mc = 135 MPa to 𝜓 = 5.8 mm−1 at 𝜎 sat = 203 MPa, and the interface oxidation ratio decreases from 𝜔 = 30.9% to 𝜔 = 18.2%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.08 mm−1 at 𝜎 mc = 135 MPa to 𝜓 = 4.3 mm−1 at 𝜎 sat = 203 MPa, and the interface oxidation ratio decreases from 𝜔 = 64.7% to 𝜔 = 43.6%. When Γm = 25 J/m2 at T = 873 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.23 mm−1 at 𝜎 mc = 175 MPa to 𝜓 = 7.3 mm−1 at 𝜎 sat = 218 MPa, and the interface oxidation ratio decreases from 𝜔 = 4.3% to 𝜔 = 2.6%, and

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(a)

(b)

(c)

Figure 3.21 (a) The matrix cracking density versus applied stress curves for different oxidation temperatures and times when Γm = 20 J/m2 ; (b) the interface oxidation ratio versus applied stress curves for different oxidation temperatures and times when Γm = 20 J/m2 ; (c) the matrix cracking density versus applied stress curves for different oxidation temperatures and times when Γm = 25 J/m2 ; (d) the interface oxidation ratio versus applied stress curves for different oxidation temperatures and times when Γm = 25 J/m2 ; (e) the matrix cracking density versus applied stress curves for different oxidation temperatures and times when Γm = 30 J/m2 ; and (f) the interface oxidation ratio versus applied stress curves for different oxidation temperatures and times when Γm = 30 J/m2 .

3.5 Time-Dependent Matrix Multiple Cracking Evolution of SiC/SiC Composites

Figure 3.21

(Continued)

(d)

(e)

(f)

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3 Matrix Multiple Cracking Evolution of Fiber-Reinforced CMCs at Elevated Temperature

at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.2 mm−1 at 𝜎 mc = 175 MPa to 𝜓 = 6.8 mm−1 at 𝜎 sat = 222 MPa, and the interface oxidation ratio decreases from 𝜔 = 12.2% to 𝜔 = 7.5%. At T = 973 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.15 mm−1 at 𝜎 mc = 162 MPa to 𝜓 = 6.2 mm−1 at 𝜎 sat = 243 MPa, and the interface oxidation ratio decreases from 𝜔 = 12.3% to 𝜔 = 7.3%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.12 mm−1 at 𝜎 mc = 162 MPa to 𝜓 = 5.4 mm−1 at 𝜎 sat = 234 MPa, and the interface oxidation ratio decreases from 𝜔 = 31.3% to 𝜔 = 19.8%. At T = 1073 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.12 mm−1 at 𝜎 mc = 146 MPa to 𝜓 = 5.2 mm−1 at 𝜎 sat = 220 MPa, and the interface oxidation ratio decreases from 𝜔 = 27.7% to 𝜔 = 16.5%, and at t = 3 hours, the matrix multi-cracking density increases from 𝜓 = 0.08 mm−1 at 𝜎 mc = 146 MPa to 𝜓 = 4.0 mm−1 at 𝜎 sat = 220 MPa, and the interface oxidation ratio decreases from 𝜔 = 60% to 𝜔 = 40.4%. When Γm = 30 J/m2 at T = 873 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.2 mm−1 at 𝜎 mc = 187 MPa to 𝜓 = 6.5 mm−1 at 𝜎 sat = 239 MPa, and the interface oxidation ratio decreases from 𝜔 = 4% to 𝜔 = 2.4%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.17 mm−1 at 𝜎 mc = 167 MPa to 𝜓 = 6.1 mm−1 at 𝜎 sat = 242 MPa, and the interface oxidation ratio decreases from 𝜔 = 11.3% to 𝜔 = 6.9%. At T = 973 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.14 mm−1 at 𝜎 mc = 172 MPa to 𝜓 = 5.6 mm−1 at 𝜎 sat = 243 MPa, and the interface oxidation ratio decreases from 𝜔 = 11.2% to 𝜔 = 6.7%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.11 mm−1 at 𝜎 mc = 172 MPa to 𝜓 = 4.9 mm−1 at 𝜎 sat = 253 MPa, and the interface oxidation ratio decreases from 𝜔 = 29.1% to 𝜔 = 18.4%. At T = 1073 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.11 mm−1 at 𝜎 mc = 156 MPa to 𝜓 = 4.8 mm−1 at 𝜎 sat = 234 MPa, and the interface oxidation ratio decreases from 𝜔 = 25.4% to 𝜔 = 15.3%, and at t = 3 hours, the matrix cracking density increases from 𝜓 = 0.07 mm−1 at 𝜎 mc = 156 MPa to 𝜓 = 3.7 mm−1 at 𝜎 sat = 234 MPa, and the interface oxidation ratio decreases from 𝜔 = 56.3% to 𝜔 = 37.9%.

3.5.2

Experimental Comparisons

Figures 3.22 and 3.23 show the experimental and theoretical matrix cracking density and fiber/matrix interface oxidation ratio of unidirectional SiC/SiC [50] and mini SiC/SiC [51] composites at room temperature, T = 773, 873, 973, and 1073 K for t = 1 and 3 hours.

3.5.2.1 Unidirectional SiC/SiC Composite

Beyerle et al. [50] investigated the damage evolution of matrix multiple cracking in unidirectional SiC/SiC composite. At room temperature, the matrix multiple cracking evolution starts at 𝜎 mc = 240 MPa and approaches to saturation at 𝜎 sat = 320 MPa, and the matrix cracking density increases from 𝜓 = 1.1 mm−1 to 𝜓 = 13 mm−1 .

3.5 Time-Dependent Matrix Multiple Cracking Evolution of SiC/SiC Composites

Figure 3.22 (a) The experimental and theoretical matrix cracking density versus applied stress curves for different oxidation temperatures and times; and (b) the interface oxidation ratio versus applied stress curves for different oxidation temperatures and times of unidirectional SiC/SiC composite.

(a)

(b)

At T = 773 K, the matrix cracking density increases from 𝜓 = 0.5 mm−1 at 𝜎 mc = 222 MPa to 𝜓 = 12.4 mm−1 at 𝜎 sat = 311 MPa; at T = 773 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.36 mm−1 at 𝜎 mc = 222 MPa to 𝜓 = 11.9 mm−1 at 𝜎 sat = 272 MPa, and the interface oxidation ratio decreases from 𝜔 = 2% at 𝜎 mc = 222 MPa to 𝜔 = 1.1% at 𝜎 sat = 272 MPa, and at T = 773 K and t = 3 hours, the matrix cracking density increases from 𝜓 = 0.34 mm−1 at 𝜎 mc = 222 MPa to 𝜓 = 11.5 mm−1 at 𝜎 sat = 273 MPa, and the interface oxidation ratio decreases from 𝜔 = 5.8% at 𝜎 mc = 222 MPa to 𝜔 = 3.4% at 𝜎 sat = 273 MPa. At T = 873 K, the matrix cracking density increases from 𝜓 = 0.45 mm−1 at 𝜎 mc = 206 MPa to 𝜓 = 11.2 mm−1 at 𝜎 sat = 288 MPa; at T = 873 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.25 mm−1 at 𝜎 mc = 206 MPa to 𝜓 = 10.3 mm−1 at 𝜎 sat = 288 MPa, and the interface oxidation ratio decreases from 𝜔 = 8.3% at 𝜎 mc = 206 MPa to 𝜔 = 4.6% at 𝜎 sat = 288 MPa, and at T = 873 K and t = 3 hours, the matrix cracking density increases from 𝜓 = 0.21 mm−1 at 𝜎 mc = 206 MPa to 𝜓 = 9.4 mm−1 at 𝜎 sat = 288 MPa, and the interface oxidation ratio decreases from 𝜔 = 22.3% at 𝜎 mc = 206 MPa to 𝜔 = 13% at 𝜎 sat = 288 MPa.

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Figure 3.23 (a) The experimental and theoretical matrix cracking density versus the applied stress curves for different oxidation temperatures and times and (b) the interface oxidation ratio versus applied stress curves for different oxidation temperatures and times of mini SiC/SiC composite.

(a)

(b)

At T = 973 K, the matrix cracking density increases from 𝜓 = 0.4 mm−1 at 𝜎 mc = 188 MPa to 𝜓 = 10.2 mm−1 at 𝜎 sat = 263 MPa; at T = 973 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.21 mm−1 at 𝜎 mc = 188 MPa to 𝜓 = 8.8 mm−1 at 𝜎 sat = 263 MPa, and the interface oxidation ratio decreases from 𝜔 = 26.3% at 𝜎 mc = 188 MPa to 𝜔 = 13.7% at 𝜎 sat = 263 MPa, and at T = 973 K and t = 3 hours, the matrix cracking density increases from 𝜓 = 0.15 mm−1 at 𝜎 mc = 188 MPa to 𝜓 = 7.1 mm−1 at 𝜎 sat = 263 MPa, and the interface oxidation ratio decreases from 𝜔 = 57.2% at 𝜎 mc = 188 MPa to 𝜔 = 34.5% at 𝜎 sat = 263 MPa. At T = 1073 K, the matrix cracking density increases from 𝜓 = 0.35 mm−1 at 𝜎 mc = 169 MPa to 𝜓 = 9.4 mm−1 at 𝜎 sat = 236 MPa; at T = 1073 K and t = 1 hour, the matrix cracking density increases from 𝜓 = 0.2 mm−1 at 𝜎 mc = 169 MPa to 𝜓 = 7.7 mm−1 at 𝜎 sat = 236 MPa, and the interface oxidation ratio decreases from 𝜔 = 65.7% at 𝜎 mc = 169 MPa to 𝜔 = 33.2% at 𝜎 sat = 236 MPa, and at T = 1073 K and t = 3 hours, the matrix cracking density increases from 𝜓 = 0.11 mm−1 at 𝜎 mc = 169 MPa to 𝜓 = 5.2 mm−1 at 𝜎 sat = 236 MPa, and the interface oxidation ratio decreases from 𝜔 = 1 at 𝜎 mc = 169 MPa to 𝜔 = 68.1% at 𝜎 sat = 236 MPa.

3.6 Conclusion

3.5.2.2 SiC/SiC Minicomposite

Zhang et al. [51] investigated the damage evolution of matrix multiple cracking in mini-SiC/SiC composite. At room temperature, the matrix cracking evolution starts from the applied stress 𝜎 mc = 135 MPa and approaches to saturation at 𝜎 sat = 250 MPa; the matrix cracking density increases from 𝜓 = 0.4 mm−1 to 𝜓 = 2.4 mm−1 . At T = 973 K and t = 3 hours, the matrix cracking density increases from 𝜓 = 0.06 mm−1 at 𝜎 mc = 112 MPa to 𝜓 = 2 mm−1 at 𝜎 sat = 140 MPa, and the interface oxidation ratio decreases from 𝜔 = 9.8% to 𝜔 = 4.6%. At T = 1073 K and t = 3 hours, the matrix cracking density increases from 𝜓 = 0.04 mm−1 at 𝜎 mc = 98.4 MPa to 𝜓 = 1.9 mm−1 at 𝜎 sat = 136 MPa, and the interface oxidation ratio decreases from 𝜔 = 23.7% to 𝜔 = 11.6%. At T = 1173 K and t = 3 hours, the matrix multi-cracking density increases from 𝜓 = 0.03 mm−1 at 𝜎 mc = 83.4 MPa to 𝜓 = 1.7 mm−1 at 𝜎 sat = 133 MPa, and the interface oxidation ratio decreases from 𝜔 = 46.2% to 𝜔 = 24.5%.

3.6 Conclusion In this chapter, temperature-dependent matrix multiple cracking evolution of fiber-reinforced CMCs was investigated using the CMSE criterion. Temperature-dependent fiber/matrix interface shear stress, Young’s modulus of the fibers and the matrix, the matrix fracture energy, and the fiber/matrix interface debonding energy were considered in the microstress field analysis, fiber/matrix interface debonding criterion, and matrix multiple cracking evolution model. Effects of fiber volume, fiber/matrix interface shear stress, fiber/matrix interface debonding energy, matrix fracture energy, temperature, and duration on matrix multiple cracking evolution and fiber/matrix interface debonding were discussed. Experimental matrix multiple cracking evolution and fiber/matrix interface debonding of unidirectional C/SiC composite at elevated temperatures were predicted. ●

●

●

●

With increasing fiber/matrix interface shear stress, the first matrix cracking stress, matrix cracking saturation stress, and saturation matrix cracking density increased, the matrix cracking evolved with higher applied stress, and the fiber/matrix interface debonding length decreased. With increasing fiber/matrix interface debonding energy, the matrix saturation cracking stress decreased, and the saturation matrix cracking density increased, and the rate of matrix cracking development increased because of the decrease of fiber/matrix interface debonding length. With increasing matrix fracture energy, the first matrix cracking stress and matrix saturation cracking stress increased, the saturation matrix cracking density decreased, and the fiber/matrix interface debonded length decreased. With increasing oxidation duration at elevated temperature, the saturation matrix cracking density decreased because of the decrease of the interface shear stress in the oxidation region.

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28 Li, L.B. (2015). Modeling for matrix multicracking evolution of cross-ply ceramic-matrix composites using energy balance approach. Appl. Compos. Mater. 22: 733–755. https://doi.org/10.1007/s10443-014-9433-0. 29 Solti, J.P., Mall, S., and Robertson, D.D. (1997). Modeling of matrix failure in ceramic matrix composites. J. Compos. Tech. Res. 19: 29–40. 30 Li, L.B. (2015). Effect of fiber Poisson contraction on matrix multicracking evolution of fiber-reinforced ceramic-matrix composites. Appl. Compos. Mater. 22: 583–598. https://doi.org/10.1007/s10443-014-9426-z. 31 Curtin, W.A. (1993). Multiple matrix cracking in brittle matrix composites. Acta Metall. Mater. 41: 1369–1377. https://doi.org/10.1016/0956-7151(93)90246-O. 32 Curtin, W.A., Ahn, B.K., and Takeda, N. (1998). Modeling brittle and toughness stress-strain behavior in unidirectional ceramic matrix composites. Acta Mater. 46: 3409–3420. https://doi.org/10.1016/S1359-6454(98)00041-X. 33 Smith, C.E., Morscher, G.N., and Xia, Z.H. (2008). Monitoring damage accumulation in ceramic matrix composites using electrical resistivity. Scr. Mater. 59: 463–466. https://doi.org/10.1016/j.scriptamat.2008.04.033. 34 Morscher, G.N. and Gordon, N.A. (2017). Acoustic emission and electrical resistance in SiC-based laminate ceramic composites tested under tensile loading. J. Eur. Ceram. Soc. 37: 3861–3872. https://doi.org/10.1016/j.jeurceramsoc.2017.05 .003. 35 Simon, C., Rebillat, F., Herb, V., and Camus, G. (2017). Monitoring damage evolution of SiCf /[Si-B-C]m composites using electrical resistivity: crack density-based electromechanical modeling. Acta Mater. 124: 579–587. https://doi .org/10.1016/j.actamat.2016.11.036. 36 Racle, E., Godin, N., Reynaud, P., and Fantozzi, G. (2017). Fatigue lifetime of ceramic matrix composites at intermediate temperature by acoustic emission. Materials 10: 658. https://doi.org/10.3390/ma10060658. 37 Guo, S. and Kagawa, Y. (2000). Temperature dependence of in situ conTM stituent properties of polymer-infiltration-pyrolysis-processed Nicalon SiC fiber-reinforced SiC matrix composite. J. Mater. Res. 15: 951–960. https://doi.org/ 10.1557/JMR.2000.0136. 38 Li, L.B. (2020). Modeling stress-dependent matrix multiple fracture of fiber-reinforced ceramic-matrix composites considering fiber oxidation and fracture. Compos. Interfaces https://doi.org/10.1080/09276440.2020.1775016. 39 Li, L.B. (2020). Effect of temperature on matrix multicracking evolution of C/SiC fiber reinforced ceramic-matrix composites. High Temp. Mater. Processes (London) 39: 189–199. https://doi.org/10.1515/htmp-2020-0044. 40 Li, L.B. (2019). Modeling matrix multi-fracture in SiC/SiC ceramic-matrix composites at elevated temperatures. J. Aust. Ceram. Soc. 55: 1115–1126. https://doi .org/10.1007/s41779-019-00326-6. 41 Li, L.B. (2020). Time-dependent matrix fracture of carbon fiber-reinforced silicon carbide ceramic-matrix composites considering interface oxidation. Compos. Interfaces 27: 551–567. https://doi.org/10.1080/09276440.2019.1667194. 42 Li, L.B. (2019). Time-dependent matrix multi-fracture of SiC/SiC ceramic-matrix composites considering interface oxidation. Ceram.-Silik. 63: 131–148.

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43 Reynaud, P., Douby, D., and Fantozzi, G. (1998). Effects of temperature and of oxidation on the interfacial shear stress between fibers and matrix in ceramic-matrix composites. Acta Mater. 46: 2461–2469. https://doi.org/10.1016/ S1359-6454(98)80029-3. 44 Gao, Y.C., Mai, Y., and Cotterell, B. (1988). Fracture of fiber-reinforced materials. J. Appl. Math. Phys. 39: 550–572. https://doi.org/10.1007/BF00948962. 45 Sauder, C., Lamon, J., and Pailler, R. (2004). The tensile behavior of carbon fibers at high temperatures up to 2400∘ C. Carbon 42: 715–725. https://doi.org/10 .1016/j.carbon.2003.11.020. 46 Snead, L.L., Nozawa, T., Katoh, Y. et al. (2007). Handbook of SiC properties for fuel performance modeling. J. Nucl. Mater. 371: 329–377. https://doi.org/10.1016/ j.jnucmat.2007.05.016. 47 Pradere, C. and Sauder, C. (2008). Transverse and longitudinal coefficient of thermal expansion of carbon fibers at high temperatures (300–2500 K). Carbon 46: 1874–1884. https://doi.org/10.1016/j.carbon.2008.07.035. 48 Wang, R.Z., Li, W.G., Li, D.Y., and Fang, D.N. (2015). A new temperature dependent fracture strength model for the ZrB2 -SiC composites. J. Eur. Ceram. Soc. 35: 2957–2962. https://doi.org/10.1016/j.jeurceramsoc.2015.03.025. 49 Casas, L. and Martinez-Esnaola, J.M. (2003). Modelling the effect of oxidation on the creep behavior of fiber-reinforced ceramic matrix composites. Acta Mater. 51: 3745–3757. https://doi.org/10.1016/S1359-6454(03)00189-7. 50 Beyerle, D.S., Spearing, S.M., Zok, F.W., and Evans, A.G. (1992). Damage and failure in unidirectional ceramic matrix composites. J. Am. Ceram. Soc. 75: 2719–2725. https://doi.org/10.1111/j.1151-2916.1993.tb03828.x. 51 Zhang, S., Gao, X., Chen, J. et al. (2016). Strength model of the matrix element in SiC/SiC composites. Mater. Des. 101: 66–71. https://doi.org/10.1016/j.matdes .2016.03.166.

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4 Time-Dependent Tensile Behavior of Ceramic-Matrix Composites 4.1 Introduction Ceramics possess extremely high resistance to heat, chemicals, abrasion, and wear and typified by chemical inertness, low density, and high strength and hardness. However, in a monolithic form, even these advanced ceramics have their limitations, i.e. low damage tolerance or low fracture toughness. Once a crack begins to develop during loading or from a manufacturing defect, the ceramic structures fail catastrophically in fracture. Ceramic-matrix composite (CMC) incorporated with the reinforcing fiber in the ceramic-matrix has been found to drastically improve the fracture toughness over that of monolithic ceramics. The outstanding thermal and mechanical properties, increased fracture toughness, and damage tolerance ability make CMCs ideal candidates for use as structural or other components in high-temperature applications [1–5]. The mechanical properties reflect the damage resistance of composites under different loading conditions, which is an important index for the safety design of components [6–9]. It is directly related to the safety and reliability of components in service and the predictability of damage [10–14]. The tensile stress–strain behavior reflects the strength of the composite material to resist the damage of external tensile loading. The tensile properties (i.e. proportional limit stress, matrix crack spacing, tensile strength, and fracture strain) can be obtained from the tensile stress–strain curves and can be used for component design [15–17]. Jia et al. [18] investigated the relationship between the interphase and tensile strength of SiC fiber monofilament. The tensile strength of the SiC fiber monofilament decreases with the increasing coating layers. The SiC fibers with single boron nitride (BN) coating have the high monofilament strength retention of about 70%, and 42.1% with two BN coatings, and 32.3% with four BN coatings. The minicomposite comprises one single fiber tow, interphase, and matrix and can be used to optimize the fiber–matrix interfacial zone and to generate micromechanical data necessary for modeling the mechanical behavior [19]. Almansour [20], Sauder et al. [21], and Yang et al. [22] performed investigations on the tensile behavior of SiC/SiC minicomposites with different fiber types and interface properties. Shi et al. [23] performed an investigation on the variability in tensile behavior of SiC/SiC minicomposite. The tensile strength of the SiC/SiC minicomposite satisfied the Weibull distribution. He et al. [24] performed High Temperature Mechanical Behavior of Ceramic-Matrix Composites, First Edition. Longbiao Li. © 2021 WILEY-VCH GmbH. Published 2021 by WILEY-VCH GmbH.

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an investigation on the tensile behavior of SiC/SiC minicomposites with different interphase thicknesses. The tensile strength and fiber pullout length increase with the interphase thickness. Chateau et al. [25] investigated the damage evolution and final fracture in SiC/SiC minicomposite using the in situ X-ray microtomography under tensile loading. Zeng et al. [26] performed experimental and theoretical investigations on the tensile damage evolution of unidirectional C/SiC composite at room temperature. Ma et al. [27], Wang et al. [28], Liang and Jiao [29], and Hu et al. [30] performed investigations on the tensile damage and fracture of 2.5D and 3D CMCs. The nonlinearity appears in the tensile curves both along the warp and the weft direction. Under tensile loading, the matrix cracking first occurs because of the local stress concentration of the pores inside of the composite, and the transverse cracks and longitudinal interlaminar cracks result the final brittle fracture of the composite. The acoustic emission technique is used to monitor the damage evolution of 3D-needled C/SiC composite [31]. The damage signal contained three main frequencies of 240, 370, and 455 kHz, corresponding to the damage mechanisms of the interface damage, matrix damage, and fiber fracture, respectively. Wang et al. [32] compared the tensile behavior of C/SiC composites with different fiber preforms. The minicomposite has the largest strength, modulus, and strain energy density to failure in contrast to the lowest values of 2D composite and the intermediate properties of 3D composite. The tensile behavior of CMCs is affected by the temperature [33–35]. For the unidirectional C/SiC composite at 1300 ∘ C, the composite tensile strength was 𝜎 UTS = 374 MPa and the composite tensile modulus was Ec = 134 GPa; and at 1450 ∘ C, the composite tensile strength was 𝜎 UTS = 338 MPa and the composite tensile modulus was Ec = 116 GPa. For the 2D SiC/SiC composite, the fracture strain at 1200 ∘ C is higher than that at room temperature because of the interface oxidation. For the 3D C/SiC composite, when the temperature increases from room temperature to 1500 ∘ C, the composite elastic modulus and the strain for saturation matrix cracking remained unchanged; the first matrix cracking stress, matrix cracking saturation stress, and fracture stress all increased first with temperature to the peak value at the temperature range of 1100–1300 ∘ C and then decreased with temperature. Luo and Qiao [36] investigated the effect of loading rate on tensile behavior of 3D C/SiC composite at room temperature, 1100, and 1500 ∘ C. At room temperature, the fracture stress increased with loading rate; at 1500 ∘ C, the fracture stress decreased with loading rate; and at 1100 ∘ C, the fracture stress remained without changing with loading rate. At elevated oxidizing temperature, the applied stress opens existing cracks and allows for easier ingress of oxygen to the fibers [37, 38]. Under thermal and mechanical load cycling in oxidative environment, the strain is damage dependent and a combination of physical mechanism in the form of matrix micro-cracking and fiber debonding and chemical mechanism of fiber oxidation. Li et al. [39, 40] and Li [41] developed a micromechanical approach to predict the tensile behavior of CMCs with different fiber preforms considering multiple damage mechanisms. Li [42] predicted the time-dependent proportional limit stress of C/SiC composites with different fiber volumes, interface properties, and matrix damage. Li [43] analyzed matrix multi-cracking of fiber-reinforced CMCs considering the interface oxidation

4.1 Introduction

and compared the matrix cracking evolution of C/SiC composite with/without the interface oxidation. Martinez-Fernandez and Morscher [44] investigated the tensile properties of single tow Hi-NicalonTM SiC/PyC/SiC composite at room temperature, 700, 950, and 1200 ∘ C. The elevated temperature stress rupture behavior was dependent on the precrack stress, and the stress rupture life increases with the decreasing precrack stress. Forio et al. [45] investigated the lifetime of SiC multifilament tows under static fatigue in air at a temperature range of 600–700 ∘ C. A slow-crack-growth mechanism is considered in the analysis of delayed failure of SiC/SiC minicomposite under low stress state. Morscher and Cawley [46] investigated the time-dependent strength degradation of woven SiC/BN/SiC composite at intermediate temperatures. The strength degradation is dependent on the kinetics for fusion of fibers to one another, the number of matrix cracks, and the applied stress state. Larochelle and Morscher [47] investigated the tensile stress rupture behavior of the woven Sylramic-iBN/BN/SiC composite at 550 and 750 ∘ C in humid environment. The stress rupture strengths decreased with respect to time with the rate of decrease related to the temperature and the amount of moisture content. Pailler and Lamon [48] developed a micromechanics-based model of fatigue/oxidation for CMCs considering thermally induced residual stresses and kinetics of interphase degradation or crack healing. Santhosh et al. [49, 50] investigated the time-dependent deformation and damage of 2D SiC/SiC composite under multiaxial stress and dwell fatigue at 1204 ∘ C. Morscher et al. [51] investigated the damage evolution and failure mechanisms of 2D Sylramic-iBN SiC/SiC composite under tensile creep and fatigue loading at 1204 ∘ C in air condition. The damage development was the growth of matrix cracks and increasing number of matrix cracks with stress and time. Four dominant failure criterions are present in the literature for modeling matrix crack evolution of CMCs: maximum stress theories, energy balance approach, critical matrix strain energy criterion, and statistical failure approach. The maximum stress criterion assumes that a new matrix crack will form whenever the matrix stress exceeds the ultimate strength of the matrix, which is assumed to be single-valued and a known material property [52]. The energy balance failure criteria involve calculation of the energy balance relationship before and after the formation of a single dominant crack as originally proposed by Aveston et al. [53]. The progression of matrix cracking as determined by the energy criterion is dependent on matrix strain energy release rate. The energy criterion is represented by Zok and Spearing [54] and Zhu and Weitsman [55]. The concept of a critical matrix strain energy criterion presupposes the existence of an ultimate or critical strain energy limit beyond which the matrix fails. Beyond this, as more energy is placed into the composite, the matrix, unable to support the additional load, continues to fail. As more energy is placed into the system, the matrix fails such that all the additional energy is transferred to the fibers. Failure may consist of the formation of matrix cracks, the propagation of existing cracks, or interface debonding [56]. Statistical failure approach assumes that matrix cracking is governed by statistical relations, which relate the size and spatial distribution of matrix flaws to their relative propagation stress [57].

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In this chapter, time-dependent tensile behavior of fiber-reinforced CMCs is investigated using the micromechanical approach. Damages in the matrix, interface, and fibers were taken into consideration for the tensile damage and fracture analysis. Experimental time-dependent tensile damage and fracture of SiC/SiC and C/SiC composites are predicted.

4.2 Theoretical Analysis When no damage occurs, the composite strain is 𝜀c = 𝜎∕Ec

(4.1)

When damages of matrix cracking, interface debonding, and fiber fracture occur, the composite strain is 𝜀c =

2 𝜎 (x) dx − (𝛼c − 𝛼f )ΔT Ef lc ∫lc ∕2 f

(4.2)

where Ef is the fiber Young’s modulus, lc is the matrix crack spacing, 𝛼 f and 𝛼 c denote the fiber and composite thermal expansion coefficient, respectively, and ΔT denotes the temperature difference between test temperature and fabricated temperature. )m ]}−1 { [ ( 𝜎 − (𝜎mc − 𝜎th ) (4.3) lc (𝜎, t) = Λ𝛿R (t) 1 − exp − (𝜎R − 𝜎th ) − (𝜎mc − 𝜎th ) where Λ denotes the final nominal crack space, 𝜎 R is the matrix cracking characteristic stress, and m is the matrix Weibull modulus. ) ( r V E 𝜏f 𝜁(t) + f m m 𝜎R (4.4) 𝛿R (t) = 1 − 𝜏i 2𝜏i Vf Ec where 𝜏 i and 𝜏 f are the interface shear stress in the debonding and oxidation region, respectively, V f is the fiber volume, and Em and Ec are Young’s modulus of the matrix and the composite, respectively. When matrix cracking and interface debonding occur, the time-dependent composite strain is ]2 2𝜏f 2𝜏i [ 4𝜏 l (t) 2𝜎 ld (t) + 𝜁 2 (t) − f d 𝜁 (t) − ld (t) − 𝜁 (t) 𝜀c (t) = Vf Ef lc (t) rf Ef lc (t) rf Ef lc (t) rf Ef lc (t) ( { ) } ] lc (t) 2𝜎fo 2rf 2𝜏 2𝜏 [ 𝜎 + − f 𝜁 (t) − i ld (t) − 𝜁 (t) − 𝜎fo − ld (t) + 2 rf rf Ef lc (t) 𝜌Ef lc (t) Vf [ ( )] l (t)∕2 − ld (t) × 1 − exp −𝜌 c (4.5) − (𝛼c − 𝛼f )ΔT rf where 𝜎 fo is the fiber stress in the interface bonding region, 𝜁 is the interface oxidation length, and ld is the interface debonding length. { [ ]} 𝜙t (4.6) 𝜁(t) = 𝜙1 1 − exp − 2 b √ ) ( ) √ ( √( r )2 r V E E r f Vm E m 𝜎 1 𝜏f f 𝜁(t) + −√ f ld (t) = 1 − − + f m m Γd (4.7) 𝜏i 2 Vf Ec 𝜏i 𝜌 2𝜌 Ec 𝜏i2

4.3 Results and Discussion

where 𝜑1 , 𝜑2 , and b are interface oxidation model parameters, 𝜌 is the shear-lag model parameter, and Γd is the interface debonding energy. When fiber failure occurs, the time-dependent composite strain is ]2 4𝜏 l (t) 2𝜏f 2𝜏i [ Φ(t) 2ld (t) + 𝜁 2 (t) − f d 𝜁 (t) − l (t) − 𝜁 (t) 𝜀c (t) = Ef lc (t) rf Ef lc (t) rf Ef lc (t) rf Ef lc (t) d ( { ) } ] lc (t) 2𝜎fo 2𝜏 2rf 2𝜏 [ Φ(t) − f 𝜁 (t) − i ld (t) − 𝜁 (t) − 𝜎fo + − ld (t) + 2 rf rf Ef lc (t) 𝜌Ef lc (t) [ ( )] lc (t)∕2 − ld (t) × 1 − exp −𝜌 (4.8) − (𝛼c − 𝛼f )ΔT rf where Φ is the intact fiber stress.

4.3 Results and Discussion Effects of fiber volume, fiber radius, matrix Weibull modulus, matrix cracking characteristic strength, matrix cracking saturation spacing, interface shear stress, interface debonded energy, fiber strength, fiber Weibull modulus, and oxidation time (the oxidation tine is an exposure time before applying tensile stress) on the time-dependent tensile stress–strain curves, interface debonding, and fiber failure of fiber-reinforced unidirectional Hi-Nicalon SiC/SiC are analyzed.

4.3.1 Time-Dependent Tensile Behavior of SiC/SiC Composite for Different Fiber Volumes Figure 4.1 shows the time-dependent tensile stress–strain curves and interface debonding for different fiber volumes (i.e. V f = 35%, 40%, and 45%) when the oxidation duration is t = 3000 seconds. When the fiber volume increases from V f = 35% to 45%, the composite tensile strength and fracture strain increase, and the composite strain at the damage stages of the matrix cracking and interface debonding decreases. Lissart and Lamon [58] found that the tensile strength and failure strain increase with the fiber volume. At high fiber volume, the stress carried by the fiber at the matrix crack plane decreases, leading to the decrease of the interface debonding length. When the fiber volume is V f = 35%, the composite ultimate tensile strength is 𝜎 UTS = 640 MPa, and the corresponding failure strain is 𝜀f = 0.38%; the interface debonding ratio increases to 𝜂 = 56%, and the interface oxidation ratio decreases to 𝜔 = 9.7%. When the fiber volume is V f = 45%, the composite ultimate tensile strength is 𝜎 UTS = 823 MPa and the failure strain is 𝜀f = 0.43%; the interface debonding ratio increases to 𝜂 = 62.4%, and the interface oxidation ratio decreases to 𝜔 = 11.3%.

4.3.2 Time-Dependent Tensile Behavior of SiC/SiC Composite for Different Fiber Radii Figure 4.2 shows the time-dependent tensile stress–strain curves and interface debonding for different fiber radii (i.e. r f = 5, 6, and 7 μm) when the oxidation

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(a)

(b)

(c)

Figure 4.1 Effect of fiber volume on (a) the time-dependent tensile stress–strain curves; (b) the time-dependent interface debonding ratio versus the applied stress curves; and (c) the time-dependent interface oxidation ratio versus the applied stress curves of unidirectional Hi-NicalonTM SiC/SiC minicomposite.

4.3 Results and Discussion

(a)

(b)

(c)

Figure 4.2 Effect of fiber radius on (a) the time-dependent tensile stress–strain curves; (b) the time-dependent interface debonding ratio versus the applied stress curves; and (c) the time-dependent interface oxidation ratio versus the applied stress curves of unidirectional Hi-NicalonTM SiC/SiC minicomposite.

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4 Time-Dependent Tensile Behavior of Ceramic-Matrix Composites

duration is t = 3000 seconds. When the fiber radius increases from r f = 5 to 7 μm, the composite failure strain increases, the interface debonding length increases, and the interface oxidation ratio decreases. When the fiber radius increases, the interface debonding length increases, leading to the decrease of the interface oxidation ratio and the increase of the composite failure strain. When the fiber radius is r f = 5 μm, the composite ultimate failure strain is 𝜀f = 0.4%; the interface debonding length increases to 𝜂 = 49.3%; and the interface oxidation ratio decreases to 𝜔 = 14.3%. When the fiber radius is r f = 7 μm, the composite ultimate failure strain is 𝜀f = 0.44%; the interface debonding length increases to 𝜂 = 68.1%; and the interface oxidation ratio decreases to 𝜔 = 10.3%.

4.3.3 Time-Dependent Tensile Behavior of SiC/SiC Composite for Different Matrix Weibull Moduli Figure 4.3 shows the time-dependent tensile stress–strain curves and interface debonding for different matrix Weibull moduli (i.e. m = 3, 4, and 5) when the oxidation duration is t = 3000 seconds. When the matrix Weibull modulus increases from m = 3 to 5, the composite failure strain increases, the interface debonding length decreases first and then increases, and the interface oxidation ratio remains unchanged. Lissart and Lamon [58] found that the composite failure strain decreases with matrix Weibull modulus. At high matrix Weibull modulus, the matrix strength distribution becomes much more concentrated, and the matrix crack spacing increases, and the interface debonding length decreases at low stress. When m = 3, the composite failure strain is 𝜀f = 0.418% with interface debonding ratio 𝜂 = 57.9%, and the interface oxidation ratio 𝜔 = 11%. When m = 6, the composite failure strain is 𝜀f = 0.429% with the interface debonding ratio 𝜂 = 62.7% and the interface oxidation ratio 𝜔 = 11%.

4.3.4 Time-Dependent Tensile Behavior of SiC/SiC Composite for Different Matrix Cracking Characteristic Strengths Figure 4.4 shows the time-dependent tensile stress–strain curves and interface debonding for different matrix cracking characteristic strengths (i.e. 𝜎 R = 300, 400, and 500 MPa) when the oxidation duration is t = 3000 seconds. When the matrix cracking characteristic strength increases from 𝜎 R = 300 to 500 MPa, the composite strain decreases at the damage stage of the matrix cracking and interface debonding, and interface debonding length decreases at the stage of matrix cracking, and the interface oxidation ratio remains unchanged. At higher matrix cracking characteristic strength, the matrix cracking occurs at the higher applied stress level, the matrix crack spacing increases at low stress level, and the interface debonding length decreases at the damage stage of the matrix cracking.

4.3 Results and Discussion

(a)

(b)

(c)

Figure 4.3 Effect of matrix Weibull modulus on (a) the time-dependent tensile stress–strain curves; (b) the time-dependent interface debonding ratio versus the applied stress curves; and (c) the time-dependent interface oxidation ratio versus the applied stress curves of unidirectional Hi-NicalonTM SiC/SiC minicomposite.

153

154

4 Time-Dependent Tensile Behavior of Ceramic-Matrix Composites

(a)

(b)

(c)

Figure 4.4 Effect of the matrix cracking characteristic strength on (a) the time-dependent tensile stress–strain curves; (b) the time-dependent interface debonding ratio versus the applied stress curves; and (c) the time-dependent interface oxidation ratio versus the applied stress curves of unidirectional Hi-NicalonTM SiC/SiC minicomposite.

4.3 Results and Discussion

4.3.5 Time-Dependent Tensile Behavior of SiC/SiC Composite for Different Matrix Cracking Saturation Spacings Figure 4.5 shows the time-dependent tensile stress–strain curves and interface debonding for different matrix cracking saturation spacings (i.e. lsat = 200, 300, and 400 μm) when the oxidation duration is t = 3000 seconds. When the saturation matrix crack spacing increases from lsat = 200 to 400 μm, the composite failure strain decreases, the interface debonding length decreases, and the interface oxidation ratio decreases when the interface complete debonding. At high saturation matrix crack spacing, the matrix crack spacing increases, and the interface debonding length, interface oxidation ratio, and composite failure strain decrease. When lsat = 200 μm, the composite failure strain is 𝜀f = 0.53% with the interface debonding ratio 𝜂 = 1, and the interface oxidation ratio 𝜔 = 12.3%. When lsat = 400 μm, the composite failure strain is 𝜀f = 0.41% with the interface debonding ratio 𝜂 = 55.4% and the interface oxidation ratio 𝜔 = 11%.

4.3.6 Time-Dependent Tensile Behavior of SiC/SiC Composite for Different Interface Shear Stress Figure 4.6 shows the time-dependent tensile stress–strain curves and interface debonding for different interface shear stress (i.e. 𝜏 i = 10, 20, and 30 MPa) when the oxidation duration is t = 3000 seconds. When the interface shear stress increases from 𝜏 i = 10 to 30 MPa, the composite failure strain decreases, the interface debonding length decreases, and the interface oxidation ratio increases. Lissart and Lamon [58] found that the composite tensile failure strain decreases with the interface shear stress. At high interface shear stress, the stress transfer between the fiber and the matrix increases, the interface debonding length decreases, the interface oxidation ratio increases, and the composite failure strain and fiber pullout length decrease. When 𝜏 i = 10 MPa, the composite failure strain is εf = 0.56% with the interface debonding ratio 𝜂 = 1 and the interface oxidation ratio 𝜔 = 7%. When 𝜏 i = 30 MPa, the composite failure strain is εf = 0.43%, with the interface debonding ratio 𝜂 = 63.4% and the interface oxidation ratio 𝜔 = 11%.

4.3.7 Time-Dependent Tensile Behavior of SiC/SiC Composite for Different Interface Debonding Energies Figure 4.7 shows the time-dependent tensile stress–strain curves and interface debonding for different interface debonding energy (i.e. Γd = 0.1, 0.3, and 0.5 J/m2 ) when the oxidation duration is t = 3000 seconds. When the interface debonding energy increases from Γd = 0.1 to 0.5 J/m2 , the composite tensile failure strain decreases, the interface debonding length decreases, and the interface oxidation ratio increases. When the interface debonding energy increases, the energy needed for the interface debonding propagation increases, leading to the decrease of the interface debonding length, the increase of the interface oxidation ratio, and the decrease of the composite failure strain.

155

156

4 Time-Dependent Tensile Behavior of Ceramic-Matrix Composites

(a)

(b)

(c)

Figure 4.5 Effect of matrix cracking saturation spacing on (a) the time-dependent tensile stress–strain curves; (b) the time-dependent interface debonding ratio versus the applied stress curves; and (c) the time-dependent interface oxidation ratio versus the applied stress curves of unidirectional Hi-NicalonTM SiC/SiC minicomposite.

4.3 Results and Discussion

(a)

(b)

(c)

Figure 4.6 The effect of the interface shear stress on (a) the time-dependent tensile stress–strain curves; (b) the time-dependent interface debonding ratio versus the applied stress curves; and (c) the time-dependent interface oxidation ratio versus the applied stress curves of unidirectional Hi-NicalonTM SiC/SiC minicomposite.

157

158

4 Time-Dependent Tensile Behavior of Ceramic-Matrix Composites

(a)

(b)

(c)

Figure 4.7 Effect of interface debonding energy on (a) the time-dependent tensile stress–strain curves; (b) the time-dependent interface debonding ratio versus the applied stress curves; and (c) the time-dependent interface oxidation ratio versus the applied stress curves of unidirectional Hi-NicalonTM SiC/SiC minicomposite.

4.3 Results and Discussion

When Γd = 0.1 J/m2 , the composite failure strain is εf = 0.43% with the interface debonding ratio 𝜂 = 63%, and the interface oxidation ratio 𝜔 = 11%. When Γd = 0.5 J/m2 , the composite failure strain is εf = 0.42%, with the interface debonding ratio 𝜂 = 54% and the interface oxidation ratio 𝜔 = 13%.

4.3.8 Time-Dependent Tensile Behavior of SiC/SiC Composite for Different Fiber Strengths Figure 4.8 shows the time-dependent tensile stress–strain curves and fiber failure for different fiber strengths (i.e. 𝜎 o = 2.0, 2.5, and 3 GPa) when the oxidation duration is t = 3000 seconds. When the fiber strength increases from 𝜎 o = 2 to 3 GPa, the

(a)

(b)

Figure 4.8 Effect of fiber strength on (a) the time-dependent tensile stress–strain curves and (b) the time-dependent broken fiber fraction versus the applied stress curves of unidirectional Hi-NicalonTM SiC/SiC minicomposite.

159

160

4 Time-Dependent Tensile Behavior of Ceramic-Matrix Composites

composite tensile strength and failure strain increase, and the fiber failure probability at low stress decreases. Lissart and Lamon [58] found that the composite tensile strength and failure strain increase with fiber strength. When 𝜎 o = 2 GPa, the composite ultimate tensile strength is 𝜎 UTS = 612 MPa with the failure strain of εf = 0.28%, and the fiber failure probability is q = 13.5%; when 𝜎 o = 2.5 GPa, the composite ultimate tensile strength is 𝜎 UTS = 765 MPa with a failure strain of εf = 0.4% and the fiber failure probability is q = 13.5%, and when 𝜎 o = 3 GPa, the composite ultimate tensile strength is 𝜎 UTS = 805 MPa with a failure strain of εf = 0.43%, and the fiber failure probability is q = 13.5%.

4.3.9 Time-Dependent Tensile Behavior of SiC/SiC Composite for Different Fiber Weibull Moduli Figure 4.9 shows the time-dependent tensile stress–strain curves and fiber failure for different fiber Weibull moduli (i.e. mf = 3, 4, and 5) when the oxidation duration is t = 3000 seconds. When the fiber Weibull modulus increases from mf = 3 to 5, the composite tensile strength and failure strain increase, and the fiber failure probability at low stress decreases. Lissart and Lamon [58] found that the composite tensile strain and failure strain increase with fiber Weibull modulus. At high fiber Weibull modulus, the fiber strength distribution becomes much more concentrated, leading to the higher composite tensile strength and failure strain. When mf = 3, the composite ultimate tensile strength is 𝜎 UTS = 671 MPa with a failure strain of εf = 0.36%, and the fiber failure probability is q = 18.2%; when mf = 4, the composite ultimate tensile strength is 𝜎 UTS = 706 MPa with a failure strain of εf = 0.37%, and the fiber failure probability is q = 15.5%, and when mf = 5, the composite ultimate tensile strength is 𝜎 UTS = 734 MPa with a failure strain of εf = 0.38%, and the fiber failure probability is q = 13%.

4.3.10 Time-Dependent Tensile Behavior of SiC/SiC Composite for Different Oxidation Durations Figure 4.10 shows the time-dependent tensile stress–strain curves, interface debonding, and fiber failure for different oxidation durations (i.e. t = 1, 2, and 3 hours) at an elevated temperature of T = 800 ∘ C. The degradation of the fiber strength and the increase of the interface oxidation length both increase with oxidation time. When the oxidation time increases, the composite tensile strength and failure strain decrease, the interface debonding length increases, the interface oxidation ratio increases, and the fiber failure probability at low stress increases. Li [42, 43] investigated the time-dependent matrix cracking evolution and proportional limit stress of fiber-reinforced CMCs. It was found that the interface debonding length and the interface oxidation ratio increase with the oxidation time. When t = 1 hour, the composite ultimate tensile strength is 𝜎 UTS = 734 MPa with the failure strain of 𝜀f = 0.38%, the interface debonding ratio 𝜂 = 57%, the interface oxidation ratio 𝜔 = 14.4%, and the fiber failure probability q = 13%. When t = 3 hours, the composite ultimate tensile strength is 𝜎 UTS = 584 MPa with the failure strain of

4.4 Experimental Comparisons

(a)

(b)

Figure 4.9 Effect of fiber Weibull modulus on (a) the time-dependent tensile stress–strain curves and (b) the time-dependent broken fiber fraction versus the applied stress curves of unidirectional Hi-NicalonTM SiC/SiC minicomposite.

εf = 0.27%, the interface debonding ratio 𝜂 = 34.9%, the interface oxidation ratio 𝜔 = 41.8%, and the fiber failure probability q = 12.6%.

4.4 Experimental Comparisons 4.4.1

Time-Dependent Tensile Behavior of SiC/SiC Composite

Almansour [20] and Sauder et al. [21] investigated the tensile behavior of unidirectional SiC/SiC minicomposites with different fiber types and interface properties. Material properties of four different unidirectional SiC/SiC minicomposites are listed in Table 4.1. The tensile stress–strain curves, matrix cracking

161

162

4 Time-Dependent Tensile Behavior of Ceramic-Matrix Composites

(a)

(b)

(c)

Figure 4.10 Effect of the oxidation time on (a) the time-dependent tensile stress–strain curves; (b) the time-dependent interface debonding ratio versus the applied stress curves; (c) the time-dependent interface oxidation ratio versus the applied stress curves; and (d) the time-dependent broken fiber fraction versus the applied stress curves of unidirectional Hi-NicalonTM SiC/SiC minicomposite.

4.4 Experimental Comparisons

(d)

Figure 4.10 Table 4.1

(Continued) Material properties of SiC/SiC minicomposites. Hi-NicalonTM SiC/SiC [8]

Hi-NicalonTM type S SiC/SiC [9]

TyrannoTM SA3 SiC/SiC [9]

r f (μm)

7

6.5

3.5

5.5

V f (%)

25.8

46

43

27.5

Items

TyrannoTM ZMI SiC/SiC [9]

Ef (GPa)

350

372

387

170

Em (GPa) 𝛼 (10−6 /∘ C)

400

400

400

350

3.5

3.5

4

4

𝛼 m (10−6 /∘ C)

4.6

4.6

4.6

4.6

𝜏 i (MPa)

40

9

100

30

𝜏 f (MPa)

1

1

1

1

f

2

Γd (J/m ) 𝜎 UTS (MPa) mf

3

0.1

0.1

3

644

940

1116

498

5

5

5

5

evolution, interface debonding, and fiber failure for different oxidation times at T = 800 ∘ C are predicted. Figure 4.11 shows the experimental and predicted tensile damage and fracture of unidirectional Hi-Nicalon SiC/SiC minicomposite. When the composite is without oxidation, the composite ultimate tensile strength is 𝜎 UTS = 642 MPa with the failure strain 𝜀f = 0.57%; the matrix cracking density increases from 𝜓 = 0.06 mm−1 at 𝜎 = 260 MPa to 𝜓 = 1.77 mm−1 at 𝜎 = 566 MPa. When the oxidation duration is t = 1 hour, the composite ultimate tensile strength is 𝜎 UTS = 451 MPa with the failure strain 𝜀f = 0.34%, the interface debonding ratio increases to 𝜂 = 22%, the interface

163

164

4 Time-Dependent Tensile Behavior of Ceramic-Matrix Composites

(a)

(b) Figure 4.11 (a) Experimental and predicted tensile stress–strain curves for the oxidation duration t = 0, 1, 2, and 3 hours; (b) experimental and predicted matrix cracking density versus the applied stress curves; (c) time-dependent interface debonding ratio versus the applied stress curves when t = 1, 2, and 3 hours; (d) the time-dependent interface oxidation ratio versus the applied stress curves when t = 1, 2, and 3 hours; and (e) the time-dependent broken fiber fraction versus the applied stress curves when t = 1, 2, and 3 hours of unidirectional Hi-NicalonTM SiC/SiC minicomposite.

4.4 Experimental Comparisons

(c)

(d)

(e)

Figure 4.11

(Continued)

165

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4 Time-Dependent Tensile Behavior of Ceramic-Matrix Composites

oxidation ratio decreases to 𝜔 = 18%, and the fiber failure probability increases to q = 13.1%. When the oxidation duration is t = 2 hours, the composite ultimate tensile strength is 𝜎 UTS = 379 MPa with the failure strain 𝜀f = 0.24%, the interface debonding ratio increases to 𝜂 = 11%, the interface oxidation ratio decreases to 𝜔 = 39%, and the fiber failure probability increases to q = 12.6%. When the oxidation duration is t = 3 hours, the composite ultimate tensile strength is 𝜎 UTS = 342 MPa with the failure strain 𝜀f = 0.2%, the interface debonding ratio increases to 𝜂 = 7%, the interface oxidation ratio decreases to 𝜔 = 55.4%, and the fiber failure probability increases to q = 12%. Figure 4.12 shows experimental and predicted tensile damage and fracture of unidirectional Hi-Nicalon type S SiC/SiC minicomposite. When the composite is without oxidation, the composite ultimate tensile strength is 𝜎 UTS = 940 MPa with the failure strain 𝜀f = 0.62%, and the matrix cracking density increases from 𝜓 = 0.08 mm−1 at 𝜎 = 375 MPa to 𝜓 = 2.85 mm−1 at 𝜎 = 937 MPa. When the oxidation duration is t = 1 hour, the composite ultimate tensile strength is 𝜎 UTS = 804 MPa with the failure strain 𝜀f = 0.54%, the interface debonding ratio increases to 𝜂 = 1, the interface oxidation ratio decreases to 𝜔 = 8.4%, and the fiber failure probability increases to q = 12.9%. When the oxidation duration is t = 2 hours, the composite ultimate tensile strength is 𝜎 UTS = 676 MPa with the failure strain 𝜀f = 0.45%, the interface debonding ratio increases to 𝜂 = 1, the interface oxidation ratio decreases to 𝜔 = 14.8%, and the fiber failure probability increases to q = 12.9%. When the oxidation duration is t = 3 hours, the composite tensile strength is 𝜎 UTS = 611 MPa with the failure strain 𝜀f = 0.4%, the interface debonding ratio increases to 𝜂 = 1, the interface oxidation ratio decreases to 𝜔 = 17.1%, and the fiber failure probability increases to q = 13.1%. Figure 4.13 shows experimental and predicted tensile damage and fracture of unidirectional TyrannoTM SA3 SiC/SiC minicomposite. When the composite is without oxidation, the composite ultimate tensile strength is 𝜎 UTS = 1106 MPa with the failure strain 𝜀f = 0.63%, and the matrix cracking density increases from 𝜓 = 1.6 mm−1 at 𝜎 = 593 MPa to 𝜓 = 23 mm−1 at 𝜎 = 1031 MPa. When the oxidation duration is t = 1 hour, the composite ultimate tensile strength is 𝜎 UTS = 752 MPa with the failure strain 𝜀f = 0.36%, the interface debonding ratio increases to 𝜂 = 39%, the interface oxidation ratio decreases to 𝜔 = 53%, and the fiber failure probability increases to q = 13.5%. When the oxidation duration is t = 2 hours, the composite ultimate tensile strength is 𝜎 UTS = 632 MPa with the failure strain 𝜀f = 0.26%, the interface debonding ratio increases to 𝜂 = 22.4%, the interface oxidation ratio decreases to 𝜔 = 74.7%, and the fiber failure probability increases to q = 13%. When the oxidation duration is t = 3 hours, the composite ultimate tensile strength is 𝜎 UTS = 571 MPa with the failure strain 𝜀f = 0.23%, the interface debonding ratio increases to 𝜂 = 16.8%, the interface oxidation ratio decreases to 𝜔 = 83.8%, and the fiber failure probability increases to q = 12.9%. Figure 4.14 shows experimental and predicted tensile damage and fracture of unidirectional Tyranno ZMI SiC/SiC minicomposite. When the composite is without oxidation, the composite ultimate tensile strength is 𝜎 UTS = 497 MPa with the failure strain 𝜀f = 0.5%, and the matrix cracking density increases from 𝜓 = 0.02 mm−1

4.4 Experimental Comparisons

(a)

(b) Figure 4.12 (a) Experimental and predicted tensile stress–strain curves for the oxidation duration t = 0, 1, 2, and 3 hours; (b) experimental and predicted matrix cracking density versus the applied stress curves; (c) the time-dependent interface debonding ratio versus the applied stress curves when t = 1, 2, and 3 hours; (d) the time-dependent interface oxidation ratio versus the applied stress curves when t = 1, 2, and 3 hours; and (e) the time-dependent broken fiber fraction versus the applied stress curves when t = 1, 2, and 3 hours of unidirectional Hi-NicalonTM type S SiC/SiC minicomposite.

167

168

4 Time-Dependent Tensile Behavior of Ceramic-Matrix Composites

(c)

(d)

(e)

Figure 4.12

(Continued)

4.4 Experimental Comparisons

(a)

(b)

Figure 4.13 (a) Experimental and predicted tensile stress–strain curves for the oxidation duration t = 0, 1, 2, and 3 hours; (b) experimental and predicted matrix cracking density versus the applied stress curves; (c) the time-dependent interface debonding ratio versus the applied stress curves when t = 1, 2, and 3 hours; (d) the time-dependent interface oxidation ratio versus the applied stress curves when t = 1, 2, and 3 hours; and (e) the time-dependent broken fiber fraction versus the applied stress curves when t = 1, 2, and 3 hours of unidirectional TyrannoTM SA3 SiC/SiC minicomposite.

169

170

4 Time-Dependent Tensile Behavior of Ceramic-Matrix Composites

(c)

(d)

(e)

Figure 4.13

(Continued)

4.4 Experimental Comparisons

(a)

(b)

Figure 4.14 (a) Experimental and predicted tensile stress–strain curves for the oxidation duration t = 0, 1, 2, and 3 hours; (b) experimental and predicted matrix cracking density versus the applied stress curves; (c) the time-dependent interface debonding ratio versus the applied stress curves when t = 1, 2, and 3 hours; (d) the time-dependent interface oxidation ratio versus the applied stress curves when t = 1, 2, and 3 hours; and (e) the time-dependent broken fiber fraction versus the applied stress curves when t = 1, 2, and 3 hours of unidirectional TyrannoTM ZMI SiC/SiC minicomposite.

171

172

4 Time-Dependent Tensile Behavior of Ceramic-Matrix Composites

(c)

(d)

(e)

Figure 4.14

(Continued)

4.4 Experimental Comparisons

at 𝜎 = 158 MPa to 𝜓 = 1.5 mm−1 at 𝜎 = 388 MPa. When the oxidation duration is t = 1 hour, the composite ultimate tensile strength is 𝜎 UTS = 481 MPa with the failure strain 𝜀f = 0.52%, the interface debonding ratio increases to 𝜂 = 29%, the interface oxidation ratio decreases to 𝜔 = 15%, and the fiber failure probability increases to q = 13.8%. When the oxidation duration is t = 2 hours, the composite ultimate tensile strength is 𝜎 UTS = 404 MPa with the failure strain 𝜀f = 0.43%, the interface debonding ratio increases to 𝜂 = 27%, the interface oxidation ratio decreases to 𝜔 = 33%, and the fiber failure probability increases to q = 12.7%. When the oxidation duration is t = 3 hours, the composite ultimate tensile strength is 𝜎 UTS = 365 MPa with the failure strain 𝜀f = 0.41%, the interface debonding ratio increases to 𝜂 = 28%, the interface oxidation ratio decreases to 𝜔 = 47.7%, and the fiber failure probability increases to q = 12.6%.

4.4.2

Time-Dependent Tensile Behavior of C/SiC Composite

Wang et al. [32] investigated the tensile behavior of 1D, 2D, and 3D C/SiC composites at room temperature. Zhang et al. [59] investigated the tensile behavior of 2.5D C/SiC composite after exposure at elevated temperature. The material properties of 1D, 2D, 2.5D, and 3D C/SiC composites are listed in Table 4.2. Figure 4.15 shows experimental and predicted tensile stress–strain curves, interface debonding and oxidation ratio, and broken fiber fraction of 1D C/SiC composite without and with oxidation at T = 800 ∘ C and t = 10, 20, and 30 hours. With Table 4.2

Material properties of C/SiC composites.

Items

𝜆 r f (μm)

1D C/SiC

2D C/SiC

2.5D C/SiC

3D C/SiC

1

0.5

0.75

0.93

3.5

3.5

3.5

3.5

V f (%)

30

35

40

40

Ef (GPa)

230

230

230

230

𝛼 f (10−6 /K)

0

0

0.5

0

𝛼 m (10−6 /K)

4.6

4.6

4.6

4.6

3

5

6

5

𝜎 R (MPa)

100

40

80

80

lsat (μm)

120

300

80

80

𝜏 i (MPa)

10

11

5

9

m

𝜏 f (MPa)

1

1

1

1

Γd (J/m2 )

0.1

0.3

0.1

0.1

𝜎 UTS (MPa)

333

149

226

206

𝜀f (%)

0.59

0.34

0.56

0.37

5

5

5

5

mf

173

174

4 Time-Dependent Tensile Behavior of Ceramic-Matrix Composites

(a)

(b)

(c)

Figure 4.15 (a) Experimental and predicted tensile stress–strain curves; (b) the interface debonding ratio versus the applied stress curves; (c) the interface oxidation ratio versus the applied stress curves; and (d) the broken fiber fraction versus applied stress curves of unidirectional C/SiC composite.

4.4 Experimental Comparisons

(d)

Figure 4.15

(Continued)

increasing oxidation duration, both the composite tensile strength and failure strain decrease; the interface debonding ratio and interface oxidation ratio increase, and the broken fiber fraction increases at low stress level. Without oxidation, the composite tensile strength is 𝜎 UTS = 333 MPa with the failure strain 𝜀f = 0.68%; when oxidation duration is t = 10 hours, the composite tensile strength is 𝜎 UTS = 314 MPa with the failure strain 𝜀f = 0.67%, the interface debonding ratio increases to 𝜂 = 1, the interface oxidation ratio decreases to 𝜔 = 24.8%, and the broken fiber fraction increases to q = 12%; when the oxidation duration is t = 20 hours, the composite tensile strength is 𝜎 UTS = 264 MPa with the failure strain 𝜀f = 0.61%, the interface debonding ratio increases to 𝜂 = 1, the interface oxidation ratio decreases to 𝜔 = 49%, and the broken fiber fraction increases to q = 12%; when the oxidation duration is t = 30 hours, the composite tensile strength is 𝜎 UTS = 239 MPa with the failure strain 𝜀f = 0.59%, the interface debonding ratio increases to 𝜂 = 1, the interface oxidation ratio decreases to 𝜔 = 74.4%, and the broken fiber fraction increases to q = 12.8%. Figure 4.16 shows experimental and predicted tensile stress–strain curves, interface debonding and oxidation ratio, and broken fiber fraction of 2D C/SiC composite without and with oxidation at T = 800 ∘ C and t = 10, 20, and 30 hours. With increasing oxidation time, both the composite tensile strength and failure strain decrease; the interface debonding ratio and interface oxidation ratio increase; and the broken fiber fraction increases at low stress level. Without oxidation, the composite tensile strength is 𝜎 UTS = 148 MPa with the failure strain 𝜀f = 0.35%; when the pre-exposure time is t = 10 hours, the composite tensile strength is 𝜎 UTS = 148 MPa with the failure strain 𝜀f = 0.37%, the interface debonding ratio increases to 𝜂 = 38.4%, the interface oxidation ratio decreases to 𝜔 = 8.6%, and the broken fiber fraction increases to q = 11.7%; when the oxidation duration is t = 20 hours, the composite tensile strength is 𝜎 UTS = 130 MPa with the failure strain 𝜀f = 0.33%, the interface debonding ratio increases to 𝜂 = 35.4%, the interface oxidation ratio decreases to 𝜔 = 18.6%, and the broken fiber fraction increases to q = 12.2%; when the oxidation duration is

175

176

4 Time-Dependent Tensile Behavior of Ceramic-Matrix Composites

(a)

(b)

(c)

Figure 4.16 (a) Experimental and predicted tensile stress–strain curves; (b) the interface debonding ratio versus the applied stress curves; (c) the interface oxidation ratio versus the applied stress curves; and (d) the broken fiber fraction versus applied stress curves of 2D C/SiC composite.

4.4 Experimental Comparisons

(d)

Figure 4.16

(Continued)

t = 30 hours, the composite tensile strength is 𝜎 UTS = 117 MPa with the failure strain 𝜀f = 0.3%, the interface debonding ratio increases to 𝜂 = 34%, the interface oxidation ratio decreases to 𝜔 = 29%, and the broken fiber fraction increases to q = 10.9%. Figure 4.17 shows experimental and predicted tensile stress–strain curves, interface debonding and oxidation ratio, and broken fiber fraction of 2.5D C/SiC composite without and with oxidation at T = 900 ∘ C and t = 10 hours. Without oxidation, the composite tensile strength is 𝜎 UTS = 225 MPa with the failure strain 𝜀f = 0.54%; when the oxidation duration is t = 10 hours, the composite tensile strength is 𝜎 UTS = 191 MPa with the failure strain 𝜀f = 0.48%, the interface debonding ratio increases to 𝜂 = 1, the interface oxidation ratio decreases to 𝜔 = 25%, and the broken fiber fraction increases to q = 23.5%. Figure 4.18 shows experimental and predicted tensile stress–strain curves, interface debonding and oxidation ratio, and broken fiber fraction of 3D C/SiC composite without and with oxidation at T = 800 ∘ C and t = 10, 20, and 30 hours. With increasing oxidation duration, both the composite tensile strength and failure strain decrease; the interface debonding ratio and interface oxidation ratio increase; and the broken fiber fraction increases at low stress level. Without oxidation, the composite tensile strength is 𝜎 UTS = 203 MPa with the failure strain 𝜀f = 0.38%; when the oxidation duration is t = 10 hours, the composite tensile strength is 𝜎 UTS = 192 MPa with the failure strain 𝜀f = 0.38%, the interface debonding ratio increases to 𝜂 = 83%, the interface oxidation ratio decreases to 𝜔 = 15%, and the broken fiber fraction increases to q = 13.1%; when oxidation duration is t = 20 hours, the composite tensile strength is 𝜎 UTS = 161 MPa with the failure strain 𝜀f = 0.33%, the interface debonding ratio increases to 𝜂 = 79%, the interface oxidation ratio decreases to 𝜔 = 31%, and the broken fiber fraction increases to q = 11.7%; when the oxidation duration is t = 30 hours, the composite tensile strength is 𝜎 UTS = 145 MPa with the failure strain 𝜀f = 0.32%, the interface debonding ratio increases to 𝜂 = 83%, the interface oxidation ratio decreases to 𝜔 = 44%, and the broken fiber fraction increases to q = 10.8%.

177

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4 Time-Dependent Tensile Behavior of Ceramic-Matrix Composites

(a)

(b)

(c)

Figure 4.17 (a) Experimental and predicted tensile stress–strain curves; (b) the interface debonding ratio versus the applied stress curves; (c) the interface oxidation ratio versus the applied stress curves; and (d) the broken fiber fraction versus applied stress curves of 2.5D C/SiC composite.

4.5 Conclusion

(d)

Figure 4.17

(Continued)

4.5 Conclusion In this chapter, time-dependent tensile damage and fracture of fiber-reinforced CMCs is investigated considering the interface and fiber oxidation. Time-dependent damage mechanisms of matrix cracking, interface debonding, fiber failure, and interface and fiber oxidation are considered in the analysis of the tensile stress–strain curve. Experimental time-dependent tensile stress–strain curves, matrix cracking, interface debonding, and fiber failure of different SiC/SiC and C/SiC composites are predicted for different oxidation durations. 1) When the fiber volume increases, the composite ultimate tensile strength and failure strain increase, and the composite strain at the damage stages of matrix cracking, interface debonding, and fiber failure decrease; when the fiber radius increases, the composite failure strain increases, the interface debonding ratio increases, and the interface oxidation ratio decreases. 2) When the matrix Weibull modulus increases, the composite ultimate failure strain increases, and the interface debonding ratio decreases first and then increases; when the matrix cracking characteristic strength increases, the composite strain at the damage stages of matrix cracking and interface debonding decreases, and the interface debonding ratio at the stage of matrix cracking decreases; when the saturation matrix crack spacing increases, the composite failure strain decreases and the interface debonding ratio decreases, and the interface oxidation ratio decreases during the interface complete debonding. 3) When the interface shear stress and interface debonding energy increase, the composite failure strain decreases, the interface debonding ratio decreases, and the interface oxidation ratio increases. 4) When the fiber strength and fiber Weibull modulus increase, the composite tensile strength and failure strain increase, and the fiber failure probability at low stress decreases.

179

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4 Time-Dependent Tensile Behavior of Ceramic-Matrix Composites

(a)

(b)

(c)

Figure 4.18 (a) Experimental and predicted tensile stress–strain curves; (b) the interface debonding ratio versus the applied stress curves; (c) the interface oxidation ratio versus the applied stress curves; and (d) the broken fiber fraction versus applied stress curves of 3D C/SiC composite.

References

(d)

Figure 4.18

(Continued)

5) When the oxidation time increases, the composite tensile strength and failure strain decrease, the interface debonding ratio increases, the interface oxidation ratio increases, and the fiber failure probability at low stress increases.

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36 Luo, G. and Qiao, S. (2003). Influence of loading rates on 3D-C/SiC tensile properties at different temperature. Mater. Eng. 10: 9–10. 37 Halbig, M.C., Brewer, D.N., and Eckel, A.J. (2000). Degradation of continuous fiber ceramic matrix composites under constant-load conditions. NASA/TM-2000-209681. 38 Mei, H., Cheng, L., Zhang, L., and Xu, Y. (2007). Modeling the effects of thermal and mechanical load cycling on a C/SiC composite in oxygen/argon mixtures. Carbon 45: 2195–2204. https://doi.org/10.1016/j.carbon.2007.06.051. 39 Li, L.B., Song, Y., and Sun, Y. (2014). Modeling the tensile behavior of unidirectional C/SiC ceramic-matrix composites. Mech. Compos. Mater. 49: 659–672. https://doi.org/10.1007/s11029-013-9382-y. 40 Li, L.B., Song, Y., and Sun, Y. (2015). Modeling the tensile behavior of cross-ply C/SiC ceramic-matrix composites. Mech. Compos. Mater. 51: 359–376. https://doi .org/10.1007/s11029-015-9507-6. 41 Li, L.B. (2018). Modeling the monotonic and cyclic tensile stress–strain behavior of 2D and 2.5D woven C/SiC ceramic-matrix composites. Mech. Compos. Mater. 54: 165–178. https://doi.org/10.1007/s11029-018-9729-5. 42 Li, L. (2019). Time-dependent proportional limit stress of carbon fiber-reinforced silicon carbide-matrix composites considering interface oxidation. J. Ceram. Soc. Jpn. 127: 279–287. https://doi.org/10.2109/jcersj2.18176. 43 Li, L.B. (2018). Effect of interface oxidation on matrix multi-cracking evolution of fiber-reinforced ceramic-matrix composites at elevated temperatures. J. Ceram. Sci. Technol. 9: 397–410. 44 Martinez-Fernandez, J. and Morscher, G.N. (2000). Room and elevated temperature tensile properties of single tow Hi-Nicalon, carbon interphase, CVI SiC matrix minicomposites. J. Eur. Ceram. Soc. 20: 2627–2636. https://doi.org/10 .1016/S0955-2219(00)00138-2. 45 Forio, P., Lavaire, F., and Lamon, J. (2004). Delayed failure at intermediate tem∘ ∘ peratures (600 C–700 C) in air in silicon carbide multifilament tows. J. Am. Ceram. Soc. 87: 888–893. https://doi.org/10.1111/j.1551-2916.2004.00888.x. 46 Morscher, G.N. and Cawley, J.D. (2002). Intermediate temperature strength degradation in SiC/SiC composites. J. Eur. Ceram. Soc. 22: 2777–2787. https://doi .org/10.1016/S0955-2219(02)00144-9. 47 Larochelle, K.J. and Morscher, G.N. (2006). Tensile stress rupture behavior of a woven ceramic matrix composite in humid environments at intermediate temperature – Part I. Appl. Compos. Mater. 13: 147–172. https://doi.org/10.1007/ s10443-006-9009-8. 48 Pailler, F. and Lamon, J. (2005). Micromechanics based model of fatigue/oxidation for ceramic matrix composites. Compos. Sci. Technol. 65: 369–374. https://doi.org/10.1016/j.compscitech.2004.09.029. 49 Santhosh, U., Ahmad, J., Ojard, G. et al. (2016). Deformation and damage modeling of ceramic matrix composites under multiaxial stresses. Compos. Part B: Eng. 90: 97–106. https://doi.org/10.1016/j.compositesb.2015.12.017.

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50 Santhosh, U., Ahmad, J., Kalarikkal, S. et al. (2018). Time-dependent deformation and damage modeling of a SiC/SiC composite. J. Aerosp. Eng. 31: 04018086. https://doi.org/10.1061/(ASCE)AS.1943-5525.0000921. 51 Morscher, G.N., Ojard, G., Miller, R. et al. (2008). Tensile creep and fatigue of Sylramic-iBN melt-infiltrated SiC matrix composites: retained properties, damage development, and failure mechanisms. Compos. Sci. Technol. 68: 3305–3313. https://doi.org/10.1016/j.compscitech.2008.08.028. 52 Daniel, I.M., Anastassopoulos, G., and Lee, J.W. (1993). The behavior of ceramic matrix fiber composites under longitudinal loading. Compos. Sci. Technol. 46: 105–113. https://doi.org/10.1016/0266-3538(93)90166-E. 53 Aveston, J., Cooper, G.A., and Kelly, A. (1971). Single and multiple fracture. In: Properties of Fiber Composites: Conference on Proceedings (ed. A. Kelly), 15–26. London, England: National Physical Laboratory, IPC. 54 Zok, F.W. and Spearing, S.M. (1992). Matrix crack spacing in brittle matrix composites. Acta Metall. Mater. 40: 2033–2043. https://doi.org/10.1016/09567151(92)90189-L. 55 Zhu, H. and Weitsman, Y. (1994). The progression of failure mechanisms in unidirectional reinforced ceramic composites. J. Mech. Phys. Solids 42: 1601–1632. https://doi.org/10.1016/0022-5096(94)90089-2. 56 Solti, J.P., Mall, S., and Robertson, D.D. (1997). Modeling of matrix failure in ceramic matrix composites. J. Compos. Tech. Res. 19: 29–40. https://doi.org/10 .1520/CTR10011J. 57 Curtin, W.A. (1993). Multiple matrix cracking in brittle matrix composites. Acta Metall. Mater. 41: 1369–1377. https://doi.org/10.1016/0956-7151(93)90246-O. 58 Lissart, N. and Lamon, J. (1997). Damage and failure in ceramic matrix minicomposites: experimental study and model. Acta Mater. 45: 1025–1044. https:// doi.org/10.1016/S1359-6454(96)00224-8. 59 Zhang, C., Zhao, M., Liu, Y. et al. (2016). Tensile strength degradation of a 2.5D-C/SiC composite under thermal cycles in air. J. Euro. Ceram. Soc. 36: 3011–3019.

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5 Fatigue Behavior of Ceramic-Matrix Composites at Elevated Temperature 5.1 Introduction At present, superalloy is still the main material of high-temperature structure in aeroengines (i.e. combustion chamber and turbine). After more than 40 years of development, the temperature resistance of metal materials represented by a single-crystal alloy has been greatly improved, but the difference between them and the combustion temperature of aeroengines is still large, and the gap is gradually increasing in the new generation of aeroengines. In order to improve the temperature resistance, most designers adopt the active cooling of “thermal barrier coating + film cooling.” However, the introduction of cooling air affects the combustion efficiency, and the higher the combustion temperature, the greater the effect. Therefore, the improvement of temperature resistance is the key technology to develop next-generation aeroengines. Ceramic materials with high temperature resistance, good mechanical properties, and low density have long been considered as ideal materials for high-temperature structures in aeroengines. However, because of the low toughness of ceramics, once damaged, it will cause catastrophic consequences for aeroengine, which limits its application. In order to improve the toughness of ceramic materials, the fiber-reinforced ceramic-matrix composites (CMCs) have been developed. SiCf /SiC composite possesses low density and long lifetime at high temperature up to thousands of hours, which is an ideal material for the hot section components of commercial aeroengine [1–4]. Under long-term applications at elevated temperature, CMCs are subjected to mechanical or thermal cyclic fatigue loading [5–7]. Understanding the fatigue damage mechanisms of CMCs at elevated temperature is necessary for hot section component designers [8–10]. To reduce the failure risk of CMC hot section components in aeroengines during operation, it is necessary to investigate the cyclic-dependent fatigue damage evolution in high-temperature environment and to develop related damage models, prediction methods, and computation tools [11–13]. Oxidation is the key factor to limit the application of CMCs on hot section load-carrying components of aeroengines. Combining carbides deposited by chemical vapor infiltration (CVI) process with specific sequences, a new generation of SiC/SiC composite with self-healing matrix has been developed to improve the oxidation resistance [14, 15]. The self-sealing matrix forms a glass with oxygen High Temperature Mechanical Behavior of Ceramic-Matrix Composites, First Edition. Longbiao Li. © 2021 WILEY-VCH GmbH. Published 2021 by WILEY-VCH GmbH.

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at high temperature and consequently prevent oxygen diffusion inside the material. At low temperature of 650–1000 ∘ C in dry and wet oxygen atmosphere, the self-healing 2.5D NicalonTM NL202 SiC/[Si-B-C] with a pyrolytic carbon (PyC) interphase exhibits a better oxidation resistance compared to SiC/SiC with PyC because of the presence of boron compounds [16]. The fatigue lifetime duration in air atmosphere at intermediate and high temperature is considerably reduced beyond the elastic yield point. For the Nicalon SiC/[Si-B-C] composite, the elastic yield point is about 𝜎 = 80 MPa. The lifetime duration was about t = 10–20 hours at T = 873 K and less than t = 1 hour at T = 1123 K under 𝜎 max = 120 MPa. For the self-healing Hi-NicalonTM SiC/SiC composite, a duration of t = 1000 hours without failure is reached at 𝜎 max = 170 MPa, and a duration higher than t = 100 hours at 𝜎 max = 200 MPa at T = 873 K [17]. For the self-healing Hi-Nicalon SiC/[SiC-B4 C] composite, at 1200 ∘ C, there was little influence on fatigue performance at f = 1.0 Hz but noticeably degraded fatigue lifetime at f = 0.1 Hz with the presence of steam [18, 19]. Increase in temperature from T = 1200 to 1300 ∘ C slightly degrades fatigue performance in air atmosphere but not in steam atmosphere [20]. The crack growth in the SiC fiber controls the fatigue lifetime of self-healing Hi-Nicalon SiC/[Si-B-C] at T = 873 K, and the fiber creep controls the fatigue lifetime of self-healing SiC/[Si-B-C] at T = 1200 ∘ C [21]. The typical cyclic fatigue behavior of a self-healing Hi-Nicalon SiC/[Si-B-C] composite involves an initial decrease of effective modulus to a minimum value, followed by a stiffening, and the time-to-the minimum modulus is in inverse proportion to the loading frequency [22]. The initial cracks within the longitudinal tows caused by interphase oxidation contributes to the initial decrease of modulus. The glass produced by the oxidation of self-healing matrix may contribute to the stiffening of the composite either by sealing the cracks or by bonding the fiber to the matrix [23]. The damage evolution of self-healing Hi-Nicalon SiC/[Si-B-C] composite at elevated temperature can be monitored using acoustic emission (AE) [24, 25]. The relationship between interface oxidation and AE energy under static fatigue loading at elevated temperature has been developed [26]. However, at high temperature above 1000 ∘ C, AE cannot be applied for cyclic fatigue damage monitoring. The complex fatigue damage mechanisms of self-healing CMCs affect damage evolution and lifetime. Hysteresis loops related with cyclic-dependent fatigue damage mechanisms [27–29]. The damage parameters derived from hysteresis loops have already been applied for analyzing fatigue damage and fracture of different non-oxide CMCs at elevated temperatures [30–33]. However, the cyclic-dependent damage evolution and accumulation of self-healing CMCs are much different from previous analysis results especially at elevated temperatures. In this chapter, cyclic-dependent damage development in self-healing 2.5D woven Hi-Nicalon SiC/[Si-B-C] and 2D woven Hi-Nicalon SiC/[SiC-B4 C] composites at T = 600 and 1200 ∘ C are investigated. Cyclic-dependent damage parameters of internal friction, dissipated energy, Kachanov’s damage parameter, and broken fiber fraction are obtained to analyze damage development in self-healing CMCs. Relationships between cyclic-dependent damage parameters and multiple fatigue damage mechanisms are established. Experimental fatigue damage evolution of

5.2 Theoretical Analysis

self-healing Hi-Nicalon SiC/[Si-B-C] and Hi-Nicalon SiC/[SiC-B4 C] composites are predicted. Effects of fatigue peak stress, testing environment, and loading frequencies on the evolution of internal damage and final fracture are analyzed.

5.2 Theoretical Analysis When peak stress is higher than the first matrix cracking stress, under cyclic fatigue loading, multiple fatigue damage mechanisms of matrix cracking, interface debonding, wear and oxidation, and fiber fracture occur [5–8, 34]. Hysteresis loops appear and evolve with cycle number upon unloading and reloading because of internal multiple damages in CMCs [27–37]. A unit cell is extracted from the damaged CMCs, as shown in Figure 5.1. The total length of the unit cell is half of a matrix crack spacing lc /2, and the interface debonding length between the space of matrix cracking is ld . Upon unloading, the debonding zone is divided into counter-slip zone with length ly and slip zone with length ld − ly , as shown in Figure 5.1a, and upon reloading, the debonding zone is divided into new slip zone with length lz , counter-slip region with length ly − lz , and slip region with length ld − ly , as shown in Figure 5.1b. Based on the interface debonding and slip state between the space of matrix cracking, the type of hysteresis loops can be divided into four cases, as shown in Table 5.1. For cases 1 and 2 in Table 5.3, the unloading and reloading composite hysteresis strain is a function of cyclic-dependent unloading intact fiber stress ΦU (N), reloading intact fiber stress ΦR (N), interface shear stress 𝜏 i (N), interface debonding length ld (N), and interface slip length ly (N) and lz (N). Cyclic-dependent unloading composite hysteresis strain 𝜀U (N) and reloading composite hysteresis strain 𝜀R (N) are 𝜀U (N) =

(a)

Figure 5.1

2 ΦU (N) + Σ(N) 𝜏 (N) ly +4 i Ef Ef rf lc )( ) ( 𝜏i (N) 2ly (N) − ld (N) 2ly (N) + ld (N) − lc (N) − − (𝛼c − 𝛼f )ΔT Ef rf lc

(b)

Unit cell of damaged CMCs upon (a) unloading and (b) reloading.

(5.1)

189

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5 Fatigue Behavior of Ceramic-Matrix Composites at Elevated Temperature

Table 5.1

Interface debonding and slip state in CMCs.

Case

Interface debonding condition

Interface counter slip condition

Interface new slip condition

Case 1

ld (𝜎 max ) < lc /2

ly (𝜎 min ) = ld (𝜎 max )

lz (𝜎 max ) = ld (𝜎 max )

Case 2

ld (𝜎 max ) < lc /2

ly (𝜎 min ) < ld (𝜎 max )

lz (𝜎 max ) < ld (𝜎 max )

Case 3

ld (𝜎 max ) = lc /2

ly (𝜎 min ) < lc /2

lz (𝜎 max ) < lc /2

Case 4

ld (𝜎 max ) = lc /2

ly (𝜎 min ) = lc /2

lz (𝜎 max ) = lc /2

)2 ( ΦR (N) + Σ(N) 𝜏i (N) l2z 𝜏i (N) ly (N) − 2lz (N) 𝜀R (N) = −4 +4 Ef E f r f lc Ef r f lc )( ) ( (N) − 2l (N) + 2l (N) l (N) + 2l l 𝜏i (N) d y z d y (N) − 2lz (N) − lc (N) +2 Ef r f lc − (𝛼c − 𝛼f )ΔT

(5.2)

where Σ is the additional stress in intact fibers resulting from gross slip of adjacent fractured fibers, r f is the fiber radius, Ef is the fiber Young’s modulus, 𝛼 f and 𝛼 c denote the fiber and composite thermal expansion coefficient, respectively, and ΔT denotes the temperature difference between tested and fabricated temperature. For cases 3 and 4 in Table 5.1, cyclic-dependent unloading composite hysteresis strain 𝜀U (N) and reloading composite hysteresis strain 𝜀R (N) are ( )2 2 ΦU (N) + Σ(N) 𝜏i (N) ly (N) 𝜏i (N) 2ly (N) − lc (N)∕2 𝜀U (N) = +4 −2 − (𝛼c − 𝛼f )ΔT Ef Ef rf lc Ef rf lc (5.3) 2

𝜀R (N) =

ΦR (N) + Σ(N) 𝜏 (N) l2z 𝜏 (N) (ly − 2lz ) −4 i +4 i Ef Ef rf lc Ef rf lc ( )2 𝜏 (N) lc ∕2 − 2ly + 2lz −2 i − (𝛼c − 𝛼f )ΔT Ef rf lc

(5.4)

Cyclic-dependent internal damage parameter is defined by ΔW/W e , where W e is the maximum elastic energy stored during a cycle [30]. 𝜎max [ ] 𝜀U (N) − 𝜀R (N) d𝜎 (5.5) ΔW(N) = ∫𝜎min Substituting Eqs. (5.1)–(5.4) into Eq. (5.5), the damage parameter ΔW(N) can be obtained, which is a function of cyclic-dependent unloading intact fiber stress ΦU (N), reloading intact fiber stress ΦR (N), interface shear stress 𝜏 i (N), and interface debonding and slip length ld (N), ly (N), and lz (N). It should be noted that the cyclic-dependent unloading intact fiber stress (ΦU (N)) and reloading intact fiber stress (ΦR (N)) consider fiber failure and broken fiber fraction. The mean elastic modulus E is the mean slope of hysteresis loop. This modulus is usually normalized by Young’s modulus E0 of an undamaged composite, leading to the plotting of E/E0 . A Kachanov’s damage parameter D is

5.3 Experimental Comparisons

D=1−

E E0

(5.6)

The Kachanov’s damage parameter D is another way to describe evolution of composite mean elastic modulus E under cyclic fatigue but contains the same information than the normalized modulus E/E0 .

5.3 Experimental Comparisons Monotonic tensile and cyclic-dependent damage evolution of self-healing 2.5D woven Hi-Nicalon SiC/[Si-B-C] and 2D woven Hi-Nicalon SiC/[SiC-B4 C] composites are analyzed for different temperatures. The monotonic tensile curves exhibit obvious nonlinear at elevated temperatures. For 2.5D woven Hi-Nicalon SiC/[Si-B-C] at 600 and 1200 ∘ C, the tensile curves can be divided into three main zones; however, for 2D woven Hi-Nicalon SiC/[SiC-B4 C] at 1200 ∘ C, the tensile curve can only be divided into two main zones. Cyclic-dependent damage parameters of internal friction ΔW(N)/W e (N), dissipated energy ΔW(N), interface shear stress 𝜏 i (N), Kachanov’s damage parameter D(N), and broken fiber fraction q(N) versus applied cycle number are analyzed for different temperatures, peak stresses, and loading frequencies. The interface shear stress decreases with applied cycle number; and the Kachanov’s damage parameter and broken fiber fraction increase with applied cycle number. However, the evolution of internal frictional and dissipated energy with increasing applied cycles is much more complex, which depends on the peak stress, temperature, and testing environment. The internal damage evolution of 2.5D woven Hi-Nicalon SiC/[Si-B-C] and 2D woven Hi-Nicalon SiC/[SiC-B4 C] subjected to cyclic fatigue loading are obtained.

5.3.1 2.5D Woven Hi-NicalonTM SiC/[Si-B-C] at 600 ∘ C in Air Atmosphere Figure 5.2 shows the monotonic tensile curve of 2.5D woven self-healing Hi-Nicalon SiC/[Si-B-C] composite at T = 600 ∘ C in air atmosphere. The self-healing Hi-Nicalon SiC/[Si-B-C] composite fractures at 𝜎 UTS = 341 MPa with the failure strain 𝜀f = 0.64%. The tensile curve exhibits obvious nonlinear and can be divided into three zones, including (i) the linear elastic zone with an elastic modulus Ec = 195 ± 20 GPa; (ii) the nonlinear zone due to multiple matrix cracking; and (iii) the second linear zone after saturation of matrix cracking up to final fracture, with an elastic modulus of Ec = 23 ± 1 GPa, which is half of theoretical value Ef V fl (43 GPa) when the load is supported only by the longitudinal fiber. Experimental cyclic-dependent internal friction parameter ΔW/W e versus cycle number curves of 2.5D woven self-healing Hi-Nicalon SiC/[Si-B-C] composite under 𝜎 min = −50/𝜎 max = 300 MPa and 𝜎 min = 0/𝜎 max = 200 MPa at T = 600 ∘ C in air atmosphere are shown in Figure 5.3. Cyclic-dependent internal friction parameter ΔW/W e decreases first, followed by a short stabilization, and increases again before

191

192

5 Fatigue Behavior of Ceramic-Matrix Composites at Elevated Temperature

Figure 5.2 Tensile curve of 2.5D woven self-healing Hi-NicalonTM SiC/[Si-B-C] composite at 600 ∘ C in air atmosphere.

Figure 5.3 Experimental ΔW/W e versus cycle number curves of 2.5D woven self-healing Hi-NicalonTM SiC/[Si-B-C] composite at 600 ∘ C in air atmosphere.

reaching a plateau, and finally a sharp increase when the composite approaches failure. During initial stage of cyclic fatigue loading, matrix cracking and interface debonding occur when the fatigue peak stress is higher than the first matrix cracking stress. Under repeated unloading and reloading, the sliding between the fiber and the matrix leads to the interface wear and oxidation, which decreases the interface shear stress [5, 8, 27, 29]. The initial decrease of internal friction parameter ΔW/W e is mainly attributed to matrix cracking, cyclic-dependent interface debonding, and interface wear. However, with increasing applied cycle number, the interface wear and oxidation decrease the interface shear stress to a constant value [30], leading to the stabilization of cyclic-dependent interface debonding and slip length and also

5.3 Experimental Comparisons

the internal friction damage parameter ΔW/W e . The interface wear and oxidation also decrease the fiber strength, leading to the gradual fracture of fiber, and the sudden increase of internal friction at the end of the test correspond to fibers broken [30, 33]. Experimental and predicted cyclic-dependent internal friction parameter ΔW(N)/W e (N) and broken fiber fraction q(N) versus the interface shear stress curves, and the cyclic-dependent Kachanov’s damage parameter D(N) and the interface shear stress 𝜏 i (N) versus cycle number curves of 2.5D woven self-healing Hi-Nicalon SiC/[Si-B-C] at 600 ∘ C in air atmosphere are shown in Figure 5.4 and Table 5.2. Under 𝜎 max = 200 and 300 MPa, the internal damage parameter ΔW(N)/W e (N) first decreases with the interface shear stress mainly because of the interface wear and oxidation and then increases with the interface shear stress mainly because of the fiber broken, corresponding to the interface slip case 4 in Table 5.1, as shown in Figure 5.4a. Under 𝜎 max = 300 MPa, the broken fiber fraction q(N) at higher interface shear stress is much higher than that under 𝜎 max = 200 MPa, mainly because of higher peak stress, as shown in Figure 5.4b, and the Kachanov’s damage parameter D(N) is also higher than that under 𝜎 max = 200 MPa, as shown in Figure 5.4c. The interface shear stress under 𝜎 max = 300 MPa is also higher than that under 𝜎 max = 200 MPa, mainly because of the scatter of interface shear stress or compressive stress of 𝜎 min = −50 MPa acting on the composite, as shown in Figure 5.4d. Under 𝜎 max = 200 MPa, the cyclic-dependent damage parameter ΔW/W e decreases first, i.e. from ΔW/W e = 0.054 at 𝜏 i = 15.7 MPa to ΔW/W e = 0.034 at 𝜏 i = 8.0 MPa, and then increases from ΔW/W e = 0.034 at 𝜏 i = 8.0 MPa to ΔW/W e = 0.062 at 𝜏 i = 6.0 MPa. The cyclic-dependent broken fiber fraction increases from q = 0.004 at 𝜏 i = 15.7 MPa to q = 0.24 at 𝜏 i = 6.0 MPa. The cyclic-dependent Kachanov’s damage parameter D increases from D = 0 at N = 1 to D = 0.21 at N = 89 459. The cyclic-dependent interface shear stress 𝜏 i decreases from 𝜏 i = 15.7 MPa at N = 1 to 𝜏 i = 6.0 MPa at N = 33 788. Under 𝜎 max = 300 MPa, the cyclic-dependent damage parameter ΔW/W e decreases first, i.e. from ΔW/W e = 0.067 at 𝜏 i = 19.5 MPa to ΔW/W e = 0.058 at 𝜏 i = 14.7 MPa and then increases from ΔW/W e = 0.058 at 𝜏 i = 14.7 MPa to ΔW/W e = 0.091 at 𝜏 i = 10.5 MPa. The cyclic-dependent broken fiber fraction increases from q = 0.029 at 𝜏 i = 19.5 MPa to q = 0.347 at 𝜏 i = 10.5 MPa. The cyclic-dependent Kachanov’s damage parameter D increases from D = 0 at N = 1 to D = 0.265 at N = 23 666. The interface shear stress decreases from 𝜏 i = 19.5 MPa at N = 1 to 𝜏 i = 10.5 MPa at N = 19 812.

5.3.2 2.5D Woven Hi-NicalonTM SiC/[Si-B-C] at 1200 ∘ C in Air Atmosphere Monotonic tensile curve of 2.5D woven self-healing Hi-Nicalon SiC/[Si-B-C] composite at T = 1200 ∘ C in air atmosphere is shown in Figure 5.5. The composite tensile fractured at 𝜎 UTS = 354 MPa with 𝜀f = 0.699%. The tensile curve can also be divided into three zones, including (i) the initial linear elastic zone, (ii) the non-linear zone,

193

194

5 Fatigue Behavior of Ceramic-Matrix Composites at Elevated Temperature

(a)

(b)

(c)

Figure 5.4 (a) Experimental and predicted internal friction parameter ΔW/W e versus the interface shear stress curves; (b) the broken fiber fraction q versus the interface shear stress curves; (c) the experimental and predicted Kachanov’s damage parameter D versus cycle number curves; and (d) the interface shear stress 𝜏 i (N) versus cycle number curve of 2.5D woven self-healing Hi-NicalonTM SiC/[Si-B-C] composite at 600 ∘ C in air atmosphere.

5.3 Experimental Comparisons

Figure 5.4

(Continued)

(d)

Table 5.2 Cyclic-dependent damage evolution of 2.5D woven self-healing Hi-NicalonTM SiC/[Si-B-C] composite at 600 ∘ C in air atmosphere. Cycle number

𝚫W /W e

𝝉 i (MPa)

q (%)

0/200 MPa at 600 ∘ C in air atmosphere 1

0.054

15.7

0.4

1 055

0.034

8.0

4.5

2 993

0.037

7.1

7.7

5 041

0.042

6.5

12.3

10 343

0.045

6.3

15.1

33 788

0.062

6

24.0

−50/300 MPa at 600 ∘ C in air atmosphere 1

0.067

19.5

2.9

10

0.058

14.7

7.0

100

0.059

13.0

10.9

500

0.067

11.5

18.4

1 000

0.071

11.2

21.1

2 080

0.074

11.0

23.5

4 410

0.075

10.9

24.9

10 406

0.079

10.8

26.7

19 812

0.091

10.5

34.7

195

196

5 Fatigue Behavior of Ceramic-Matrix Composites at Elevated Temperature

Figure 5.5 Tensile curve of 2.5D woven self-healing Hi-NicalonTM SiC/[Si-B-C] composite at 1200 ∘ C in air atmosphere.

Figure 5.6 Experimental internal friction parameter ΔW/W e versus cycle number curves of 2.5D woven self-healing Hi-NicalonTM SiC/[Si-B-C] composite at 1200 ∘ C in air atmosphere.

and (iii) the second linear region with fiber broken. The average fracture strength and failure strain of 2.5D woven Hi-Nicalon SiC/[Si-B-C] composite is slightly lower at T = 1200 ∘ C, i.e. 𝜎 UTS = 320 MPa and 𝜖 f = 0.62%, against 𝜎 UTS = 332 MPa and 𝜖 f = 0.658% at T = 600 ∘ C. Experimental cyclic-dependent internal friction parameter ΔW/W e versus cycle number curves of 2.5D woven self-healing Hi-Nicalon SiC/[Si-B-C] at T = 1200 ∘ C in air atmosphere are shown in Figure 5.6. The cyclic-dependent internal friction parameter ΔW/W e decreases continuously, and finally a sharp increase when the composite approaches fail. The internal friction decreases as the interface wear reduces the interface shear stress. The sudden increase of internal friction at the end of the test corresponds to the fiber broken.

5.3 Experimental Comparisons

Figure 5.7 (a) Experimental and predicted internal friction parameter ΔW/W e versus the interface shear stress curves; (b) the broken fiber fraction versus the interface shear stress curves; (c) the experimental and predicted Kachanov’s damage parameter versus cycle number curves; and (d) the interface shear stress versus cycle number curves of 2.5D woven self-healing Hi-NicalonTM SiC/[Si-B-C] composite at 1200 ∘ C in air atmosphere.

(a)

(b)

(c)

197

198

5 Fatigue Behavior of Ceramic-Matrix Composites at Elevated Temperature

Figure 5.7

(Continued)

(d)

Experimental and predicted cyclic-dependent internal friction parameter ΔW(N)/W e (N) and the broken fiber fraction q(N) versus the interface shear stress curves, and the cyclic-dependent Kachanov’s damage parameter D(N) and the interface shear stress 𝜏 i (N) versus cycle number curves of 2.5D woven self-healing Hi-Nicalon SiC/[Si-B-C] composite at 1200 ∘ C in air atmosphere are shown in Figure 5.7 and Table 5.3. Under 𝜎 max = 170 and 200 MPa, the internal damage parameter ΔW/W e decreases with the interface shear stress, corresponding to the interface slip case 4 in Table 5.1. Under 𝜎 max = 200 MPa, the broken fiber fraction is higher than that under 𝜎 max = 170 MPa at the same interface shear stress; and the Kachanov’s damage parameter D is also higher than that under 𝜎 max = 170 MPa at the same cycle number. However, the value of the interface shear stress under 𝜎 max = 200 MPa is close to that under 𝜎 max = 170 MPa. Under 𝜎 max = 170 MPa, the cyclic-dependent internal friction parameter ΔW/W e increases to the peak value first and then decreases, i.e. from ΔW/W e = 0.2 at 𝜏 i = 150 MPa to ΔW/W e = 0.42 at 𝜏 i = 39.5 MPa, and then to ΔW/W e = 0.04 at 𝜏 i = 1.25 MPa. The broken fiber fraction increases from q = 0.0007 at 𝜏 i = 150 MPa to q = 0.12 at 𝜏 i = 1.2 MPa. The Kachanov’s damage parameter increases from D = 0 at N = 1 to D = 0.068 at N = 13 202. The interface shear stress decreases from 𝜏 i = 5.5 MPa at N = 1 to 𝜏 i = 1.7 MPa at N = 32 334. Under 𝜎 max = 200 MPa, the cyclic-dependent internal friction parameter ΔW/W e increases to the peak value first and then decreases, i.e. from ΔW/W e = 0.161 at 𝜏 i = 150 MPa to ΔW/W e = 0.42 at 𝜏 i = 26.4 MPa, and then to ΔW/W e = 0.05 at 𝜏 i = 1.37 MPa. The broken fiber fraction increases from q = 0.0008 at 𝜏 i = 150 MPa to q = 0.14 at 𝜏 i = 1.3 MPa. The Kachanov’s damage parameter increases from D = 0 at N = 1 to D = 0.2 at N = 60 530. The interface shear stress decreases from 𝜏 i = 5.1 MPa at N = 1 to 𝜏 i = 2 MPa at N = 66 794.

5.3 Experimental Comparisons

Table 5.3 Cyclic-dependent damage evolution of 2.5D woven self-healing Hi-NicalonTM SiC/[Si-B-C] composite at 1200 ∘ C in air atmosphere. Cycle number

𝚫W /W e

𝝉 i (MPa)

q (%)

0/170 MPa at 1200 ∘ C in air atmosphere 1

0.097

5.5

2.0

10

0.076

4.7

2.4

517

0.068

4.3

2.6

1 281

0.062

3.6

3.2

13 700

0.047

2.5

4.9

32 334

0.044

1.7

7.8

−50/200 MPa at 1200 ∘ C in air atmosphere 1

0.133

5.1

2.7

5

0.096

3.6

4.0

20

0.090

3.4

4.3

100

0.087

3.3

4.4

450

0.085

3.2

4.6

790

0.082

3.1

4.8

8 910

0.073

2.6

5.9

66 794

0.061

2.0

8.1

5.3.3 2D Woven Self-Healing Hi-NicalonTM SiC/[SiC-B4 C] at 1200 ∘ C in Air and in Steam Atmospheres Monotonic tensile curve of 2D woven self-healing Hi-Nicalon SiC/[SiC-B4 C] composite at T = 1200 ∘ C in air atmosphere is shown in Figure 5.8. The composite tensile fractured at 𝜎 UTS = 345 MPa with 𝜀f = 0.79%. Experimental cyclic-dependent dissipated energy (ΔW(N)) versus cycle number curves of 2D woven self-healing Hi-Nicalon SiC/[SiC-B4 C] composite under 𝜎 max = 140 MPa at T = 1200 ∘ C in air atmosphere and in steam atmosphere with f = 0.1 and 1 Hz are shown in Figure 5.9. Experimental and predicted cyclic-dependent dissipated energy ΔW(N) and broken fiber fraction versus the interface shear stress curves, and the cyclic-dependent interface shear stress 𝜏 i (N) and dissipated energy ΔW(N) versus cycle number curves of 2D woven self-healing Hi-Nicalon SiC/[SiC-B4 C] composite under 𝜎 max = 140 MPa and f = 0.1 and 1 Hz at T = 1200 ∘ C in air and in steam atmospheres are shown in Figure 5.10 and Table 5.4. The damage parameter ΔW increases with the interface shear stress, corresponding to the interface slip case 2 in Table 5.1, as shown in Figure 5.10a. The broken fiber fraction increases with decreasing interface shear stress, as shown in Figure 5.10b. The interface shear stress degrades with cycle number, and the degradation rate of the interface shear stress under f = 0.1 Hz is higher than that under f = 1 Hz. When f = 1 Hz, the degradation rate of the

199

200

5 Fatigue Behavior of Ceramic-Matrix Composites at Elevated Temperature

Figure 5.8 1200 ∘ C.

Tensile curve of 2D woven self-healing Hi-NicalonTM SiC/[SiC-B4 C] composite at

Figure 5.9 Experimental cyclic-dependent dissipated energy (ΔW) versus cycle number curves of 2D woven self-healing Hi-NicalonTM SiC/[SiC-B4 C] composite under 𝜎 max = 140 MPa at 1200 ∘ C in air and in steam atmospheres.

interface shear stress is less affected in air or in steam condition; however, when f = 0.1 Hz, the degradation rate of the interface shear stress in steam is higher than that in air condition. At T = 1200 ∘ C in air condition, under 𝜎 max = 140 MPa and f = 1 Hz, the experimental cyclic-dependent dissipated energy ΔW increases from ΔW = 8.5 kPa at 𝜏 i = 34 MPa to ΔW = 32.4 kPa at 𝜏 i = 9.2 MPa; the broken fiber fraction increases from P = 0.001 at 𝜏 i = 150 MPa to P = 0.167 at 𝜏 i = 1.57 MPa; the experimental interface shear stress decreases from 𝜏 i = 34 MPa at N = 1000 to 𝜏 i = 9.2 MPa at N = 60 000,

5.3 Experimental Comparisons

Figure 5.10 (a) Experimental and predicted dissipated energy ΔW versus the interface shear stress curves; (b) the broken fiber fraction versus the interface shear stress curves; (c) the experimental and predicted interface shear stress versus cycle number curves; and (d) the experimental and predicted dissipated energy versus cycle number curves of 2D woven self-healing Hi-NicalonTM SiC/[SiC-B4 C] composite at 1200 ∘ C in air atmosphere.

(a)

(b)

(c)

201

202

5 Fatigue Behavior of Ceramic-Matrix Composites at Elevated Temperature

Figure 5.10

(d)

Table 5.4 Cyclic-dependent damage evolution of 2D woven self-healing Hi-NicalonTM SiC/[SiC-B4 C] composite under 𝜎 max = 140 MPa at 1200 ∘ C in air and in steam atmospheres. Cycle number

𝚫W (kPa)

𝝉 i (MPa)

q (%)

𝜎 max = 140 MPa and f = 1.0 Hz at 1200 ∘ C in air atmosphere 1 000

8.5

34

0.48

10 000

10.1

28

0.58

30 000

15.6

18

0.9

60 000

32.4

9.2

1.8

𝜎 max = 140 MPa and f = 0.1 Hz at 1200 ∘ C in air atmosphere 1 000

12.7

23

0.7

10 000

18.8

15.7

1

30 000

34.2

8.7

2

𝜎 max = 140 MPa and f = 1.0 Hz at 1200 ∘ C in steam atmosphere 1 000

8.5

34

0.48

10 000

10.8

27.1

0.6

30 000

18.1

16.3

1

𝜎 max = 140 MPa and f = 0.1 Hz at 1200 ∘ C in steam atmosphere 1 000

14.8

20

10 000

17.5

16.5

10 000

27

11

0.8 1 1.5

(Continued)

5.3 Experimental Comparisons

and the experimental dissipated energy increases from ΔW = 8.4 kPa at N = 1000 to ΔW = 32.4 kPa at N = 60 000. Under 𝜎 max = 140 MPa and f = 0.1 Hz, the experimental cyclic-dependent dissipated energy increases from ΔW = 12.7 kPa at 𝜏 i = 23 MPa to ΔW = 34.2 kPa at 𝜏 i = 8.7 MPa; the broken fiber fraction increases from q = 0.001 at 𝜏 i = 150 MPa to q = 0.167 at 𝜏 i = 1.57 MPa; the experimental interface shear stress decreases from 𝜏 i = 23 MPa at N = 1000 to 𝜏 i = 8.7 MPa at N = 30 000; and the experimental dissipated energy increases from ΔW = 12.7 kPa at N = 1000 to ΔW = 34.2 kPa at N = 30 000. At T = 1200 ∘ C in steam condition, under 𝜎 max = 140 MPa and f = 1 Hz, the experimental cyclic-dependent dissipated energy increases from ΔW = 8.5 kPa at 𝜏 i = 34 MPa to ΔW = 18.1 kPa at 𝜏 i = 16.3 MPa; the broken fiber fraction increases from P = 0.001 at 𝜏 i = 150 MPa to P = 0.167 at 𝜏 i = 1.57 MPa; the experimental interface shear stress decreases from 𝜏 i = 34 MPa at N = 1000 to 𝜏 i = 16.3 MPa at N = 30 000; and the experimental dissipated energy increases from ΔW = 8.5 kPa at N = 1000 to ΔW = 18.1 kPa at N = 30 000. At T = 1200 ∘ C in steam condition, under 𝜎 max = 140 MPa and f = 0.1 Hz, the experimental cyclic-dependent dissipated energy increases with decreasing interface shear stress, i.e. from ΔW = 14.8 kPa at 𝜏 i = 20 MPa to ΔW = 27 kPa at 𝜏 i = 11 MPa; the broken fiber fraction increases from q = 0.001 at 𝜏 i = 150 MPa to q = 0.167 at 𝜏 i = 1.57 MPa; the experimental interface shear stress decreases from 𝜏 i = 20 MPa at N = 100 to 𝜏 i = 11 MPa at N = 10 000; and the experimental dissipated energy increases from ΔW = 14.8 kPa at N = 100 to ΔW = 27 kPa at N = 10 000.

5.3.4

Discussion

The evolution curves of cyclic-dependent interface shear stress 𝜏 i (N) and broken fiber fraction versus cycle number of 2.5D woven self-healing Hi-Nicalon SiC/[Si-B-C] composite under different peak stresses at T = 600 and 1200 ∘ C in air atmosphere are shown in Figure 5.11. For 2.5D SiC/[Si-B-C] composite, the temperature is a govern parameter for the fatigue damage process. When the temperature increases from T = 600 to 1200 ∘ C, the interface shear stress decreases a lot, and the interface shear stress degradation rate and broken fiber fraction increase with peak stress. At T = 600 ∘ C and 𝜎 max = 200 MPa, the interface shear stress decreases from 𝜏 i = 15.7 MPa at N = 1 to 𝜏 i = 6.0 MPa at N = 33 788, and the broken fiber fraction increases from q = 0.004 to q = 0.24, and under 𝜎 max = 300 MPa, the interface shear stress decreases from 𝜏 i = 19.5 MPa at N = 1 to 𝜏 i = 10.5 MPa at N = 19 812, and the broken fiber fraction increases from q = 0.029 to q = 0.347. However, at T = 1200 ∘ C and under 𝜎 max = 170 MPa, the interface shear stress decreases from 𝜏 i = 5.5 MPa at N = 1 to 𝜏 i = 1.7 MPa at N = 32 334, and the broken fiber fraction increases from q = 0.0007 to q = 0.12, and under 𝜎 max = 200 MPa, the interface shear stress decreases from 𝜏 i = 5.1 MPa at N = 1 to 𝜏 i = 2 MPa at N = 66 794, and the broken fiber fraction increases from q = 0.0008 to q = 0.14.

203

204

5 Fatigue Behavior of Ceramic-Matrix Composites at Elevated Temperature

(a)

(b)

Figure 5.11 (a) Experimental and predicted interface shear stress versus cycle number curves and (b) the broken fiber fraction versus cycle number curves of 2.5D woven self-healing Hi-NicalonTM SiC/[Si-B-C] composite at 600–1200 ∘ C in air and in steam atmospheres.

The evolution curves of cyclic-dependent interface shear stress 𝜏 i (N) and the broken fiber fraction versus cycle number of 2D woven self-healing Hi-Nicalon SiC/[SiC-B4 C] composite under 𝜎 max = 140 MPa at T = 1200 ∘ C in air and in steam atmospheres with f = 0.1 and 1 Hz are shown in Figure 5.12. For the 2D SiC/[SiC-B4 C] composite, the loading frequency is the govern parameter for the fatigue process. Under 𝜎 max = 140 MPa, at the same cycle number, the interface shear stress decreases with decreasing loading frequency, and the interface shear stress in steam atmosphere is lower than that in air atmosphere, and the broken fiber fraction increases with decreasing loading frequency, and the broken fiber fraction in steam atmosphere is higher than that in air atmosphere.

5.3 Experimental Comparisons

(a)

(b)

Figure 5.12 (a) Experimental and predicted interface shear stress versus cycle number curves and (b) the broken fiber fraction versus cycle number curves of 2D woven self-healing Hi-NicalonTM SiC/[SiC-B4 C] composite under 𝜎 max = 140 MPa at 1200 ∘ C in air and in steam atmospheres.

At T = 1200 ∘ C in air atmosphere and f = 1 Hz, the interface shear stress decreases from 𝜏 i = 34 MPa at N = 1000 to 𝜏 i = 9.2 MPa at N = 60 000, and the broken fiber fraction increases from q = 0.0048 to q = 0.018; when f = 0.1 Hz, the interface shear stress decreases from 𝜏 i = 23 MPa at N = 1000 to 𝜏 i = 8.7 MPa at N = 30 000, and the broken fiber fraction increases from q = 0.007 to q = 0.02. At T = 1200 ∘ C in steam atmosphere and f = 1 Hz, the interface shear stress decreases from 𝜏 i = 34 MPa at N = 1000 to 𝜏 i = 16.3 MPa at N = 30 000, and the broken fiber fraction increases from q = 0.0048 to q = 0.01; when f = 0.1 Hz, the interface shear stress decreases from 𝜏 i = 20 MPa at N = 100 to 𝜏 i = 11 MPa at N = 10 000, and the broken fiber fraction increases from q = 0.008 to q = 0.015.

205

206

5 Fatigue Behavior of Ceramic-Matrix Composites at Elevated Temperature

5.4 Conclusion In this chapter, cyclic-dependent damage evolution of self-healing 2.5D woven Hi-Nicalon SiC/[Si-B-C] and 2D woven Hi-Nicalon SiC/[SiC-B4 C] composites under different peak stresses and loading frequencies at T = 600 and 1200 ∘ C are investigated. Damage parameters of internal friction ΔW/W e , dissipated energy ΔW, Kachanov’s damage parameter D, broken fiber fraction, and interface shear stress are used to describe fatigue damage evolution. ●

●

For 2.5D woven self-healing Hi-Nicalon SiC/[Si-B-C] composite, temperature is a govern parameter for the fatigue process. At T = 600 ∘ C in air atmosphere, ΔW/W e first decreases and then increases with cycle number, and at T = 1200 ∘ C in air atmosphere, ΔW/W e decreases with cycle number. The degradation rate of the interface shear stress and broken fiber faction increases with peak stress. For 2D woven self-healing Hi-Nicalon SiC/[SiC-B4 C] composite at T = 1200 ∘ C, loading frequency is a govern parameter for the fatigue process. ΔW increases with cycle number; under 𝜎 max = 140 MPa, at the same applied cycle number, the interface shear stress decreases with the loading frequency, and the interface shear stress in steam atmosphere is lower than that in air atmosphere, and the broken fiber fraction increases with decreasing loading frequency, and the broken fiber fraction in steam atmosphere is higher than that in air atmosphere.

References 1 Naslain, R. (2004). Design, preparation and properties of non-oxide CMCs for application in engines and nuclear reactors: an overview. Compos. Sci. Technol. 64: 155–170. https://doi.org/10.1016/S0266-3538(03)00230-6. 2 Bendnarcyk, B.A., Mital, S.K., Pineda, E.J., and Arnold, S.M. (2015). Multiscale modeling of ceramic matrix composites. In: The 56th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Kissimmee, FL, USA (5–9 January 2015). Reston, USA: American Institute of Aeronautics and Astronautics. https://doi.org/10.2514/6.2015-1191. 3 Padture, N.P. (2016). Advanced structural ceramics in aerospace propulsion. Nat. Mater. 15: 804. https://doi.org/10.1038/nmat4687. 4 Steibel, J., Blank, J., and Dix, B. (2017). Ceramic-matrix composites at GE Aviation. In: Advanced Ceramic Matrix Composites: Science and Technology of Materials, Design, Applications, Performance and Integration, La Fonda on the Plaza, Santa Fe, NM, USA (5–9 November 2017). New York, USA: Engineering Conferences International. 5 Zhu, S.J., Mizuno, M., Kagawa, Y., and Mutoh, Y. (1999). Monotonic tension, fatigue and creep behavior of SiC-fiber-reinforced SiC-matrix composites: a review. Compos. Sci. Technol. 59: 833–851. https://doi.org/10.1016/S02663538(99)00014-7.

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20 Ruggles-Wrenn, M.B. and Lee, M.D. (2016). Fatigue behavior of an advanced SiC/SiC composite with a self-healing matrix at 1300 ∘ C in air and in steam. Mater. Sci. Eng., A 677: 438–445. https://doi.org/10.1016/j.msea.2016.09.076. 21 Reynaud, P., Rouby, D., and Fantozzi, G. (2005). Cyclic fatigue behaviour at high temperature of a self-healing ceramic matrix composite. Ann. Chim. Sci. Mat. 30: 649–658. 22 Carrere, P. and Lamon, J. (1999). Fatigue behavior at high temperature in air of a 2D woven SiC/SiBC with a self healing matrix. Key Eng. Mater. 164–165: 321–324. https://doi.org/10.4028/http://www.scientific.net/KEM.164-165.321. 23 Forio, P. and Lamon, J. (2001). Fatigue behavior at high temperatures in air of a 2D SiC/Si-B-C composite with a self-healing multilayered matrix. In: Advances in Ceramic Matrix Composites VII (eds. N.P. Bansal, J.P. Singh and H.-T. Lin), 127–141. The American Ceramic Society. https://doi.org/10.1002/9781118380925 .ch10. 24 Simon, C., Rebillat, F., and Camus, G. (2017). Electrical resistivity monitoring of a SiC/[Si-B-C] composite under oxidizing environments. Acta Mater. 132: 586–597. https://doi.org/10.1016/j.actamat.2017.04.070. 25 Simon, C., Rebillat, F., Herb, V., and Camus, G. (2017). Monitoring damage evolution of SiCf /[Si-B-C]m composites using electrical resistivity: crack density-based electromechanical modeling. Acta Mater. 124: 579–587. https://doi .org/10.1016/j.actamat.2016.11.036. 26 Moevus, M., Reynaud, P., R’Mili, M. et al. (2006). Static fatigue of a 2.5D SiC/[Si-B-C] composite at intermediate temperature under air. Adv. Sci. Technol. 50: 141–146. https://doi.org/10.4028/www.scientific.net/AST.50.141. 27 Reynaud, P. (1996). Cyclic fatigue of ceramic-matrix composites at ambient and elevated temperatures. Compos. Sci. Technol. 56: 809–814. https://doi.org/10 .1016/0266-3538(96)00025-5. 28 Dalmaz, A., Reynaud, P., Rouby, D., and Fantozzi, A.F. (1998). Mechanical behavior and damage development during cyclic fatigue at high-temperature of a 2.5D carbon/sic composite. Compos. Sci. Technol. 58: 693–699. https://doi.org/10 .1016/S0266-3538(97)00150-4. 29 Fantozzi, G. and Reynaud, P. (2009). Mechanical hysteresis in ceramic matrix composites. Mater. Sci. Eng., A 521–522: 18–23. https://doi.org/10.1016/j.msea .2008.09.128. 30 Li, L.B. (2015). A hysteresis dissipated energy-based damage parameter for life prediction of carbon fiber-reinforced ceramic-matrix composites under fatigue loading. Compos. Part B 82: 108–128. https://doi.org/10.1016/j.compositesb.2015 .08.026. 31 Li, L.B. (2018). Damage monitoring and life prediction of carbon fiber-reinforced ceramic-matrix composites at room and elevated temperatures using hysteresis dissipated energy-based damage parameter. Compos. Interfaces 25: 335–356. https://doi.org/10.1080/09276440.2018.1439621.

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32 Li, L.B., Reynaud, P., and Fantozzi, G. (2019). Mechanical hysteresis and damage evolution in C/SiC composites under fatigue loading at room and elevated temperatures. Int. J. Appl. Ceram. Technol. 16: 2214–2228. https://doi.org/10.1111/ijac .13300. 33 Li, L.B. (2019). Failure analysis of long-fiber-reinforced ceramic-matrix composites subjected to in-phase thermomechanical and isothermal cyclic loading. Eng. Fail. Anal. 104: 856–872. https://doi.org/10.1016/j.engfailanal.2019.06.082. 34 Li, L.B. (2019). Time-dependent damage and fracture of fiber-reinforced ceramic-matrix composites at elevated temperatures. Compos. Interfaces 26 (11): 963–988. https://doi.org/10.1080/09276440.2019.1569397. 35 Penas, O. (2002). Etude de composites SiC/SiBC à matrice multiséquencée en fatigue cyclique à hautes températures sous air. PhD thesis. INSA de Lyon. 36 Budiansky, B., Hutchinson, J.W., and Evans, A.G. (1986). Matrix fracture in fiber-reinforced ceramics. J. Mech. Phys. Solids 34: 167–189. https://doi.org/10 .1016/0022-5096(86)90035-9. 37 Evans, A.G. (1997). Design and life prediction issues for high-temperature engineering ceramics and their composites. Acta Mater. 45: 23–40. https://doi.org/10 .1016/S1359-6454(96)00143-7.

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6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature 6.1 Introduction The improvement of thrust–weight ratio and the reduction of fuel consumption have become the trend of aeroengine development. Two most effective ways to increase the thrust–weight ratio of aeroengine are to reduce the component weight and to increase the turbine inlet temperature. In general, aeroengines had high thrust–weight ratios of approximately 12–15, the inlet temperature at turbine approaches 1800 ∘ C, which exceeds the limit of the service temperature of superalloy materials. To satisfy the design requirements of aeroengines, especially those with high thrust–weight ratios and low fuel consumptions, it is necessary to develop a new type of lightweight material for hot-section components, which can withstand higher temperature, possess higher strength, and longer life in order to replace the traditional superalloy materials [1–3]. In general, ceramic-matrix composites (CMCs) exhibit high strength and modulus and corrosion resistance especially at elevated temperature. Compared with the traditional superalloy materials, CMCs can withstand higher operating temperature, simplify or even remove cooling structure, optimize aeroengine structure design, and improve the service life of aeroengine hot-section components [4–8]. Silicon carbide fiber-reinforced silicon carbide CMCs (SiC/SiC) and oxide fiber-reinforced oxide CMCs (Oxide/Oxide) have already been used in aeroengines. SiC/SiC composite consists of the SiC fiber with a diameter of 12 μm, the interface layer with a thickness of 0.2–0.5 μm, and the SiC matrix. SiC/SiC composite has high oxidation resistance, lightweight (i.e. density 2.1–2.8 g/cm3 ), high-temperature (1200–1400 ∘ C) gas lifetime up to thousands of hours, and is the most ideal material for the hot end structure of aeroengines. In 1992, with the project support from the U.S. Department of Energy, General Electric (GE) company developed the prepreg infiltration process, realized the rapid and low-cost preparation of high-performance SiC/SiC composite, and carried out the verification process of different levels, including material level, component level, and engine level. In 2010, GE aviation carried out turbine rotor blade test with F414 engine as the verification platform to build a ceramic aeroengine. In 2016, SiC/SiC composite material was first used in the turbine outer ring of Leading Edge Aviation Propulsion (LEAP) aeroengine and has been produced in batches. Then, this material was applied in the combustion chamber, guide vane, turbine High Temperature Mechanical Behavior of Ceramic-Matrix Composites, First Edition. Longbiao Li. © 2021 WILEY-VCH GmbH. Published 2021 by WILEY-VCH GmbH.

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6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

outer ring, and other structures of the new GE9x commercial aeroengine. The fuel consumption rate was 10% lower than that of GE90-115B [9]. Because of the long service lifetime requirements of CMCs components in aeroengines, it is necessary to understand the complex damage mechanisms and develop methods or tools to predict the damage evolution of CMCs [10–16]. At the intermediate temperature, SiC/SiC composite lifetime is decreased because of interface oxidation and lowered fiber content following exposure to stress rupture loading [11]. At elevated temperature range of 700–1200 ∘ C, the SiC/SiC minicomposite with the pyrolytic carbon (PyC) interphase exhibited severe damages of oxidation embrittlement, and the SiC/SiC minicomposite with the boron nitride (BN) interphase showed only mild degradation subjected to the stress rupture loading [17]. Martinez-Fernandez and Morscher [18] investigated the monotonic tensile, stress rupture under constant load, and low-cycle fatigue of single tow Hi-NicalonTM , PyC interphase, chemical vapor infiltration (CVI) SiC matrix minicomposites at room temperature, 700, 950, and 1200 ∘ C in air atmosphere. The stress rupture behavior at elevated temperature depended on the precracking stress level. However, for the 2D woven melt-infiltrated (MI) Hi-Nicalon SiC/SiC composite, the stress rupture properties were obviously worse than SiC/SiC minicomposite properties under similar testing conditions because of complex fiber preform and damage evolution mechanisms [19]. Verrilli et al. [20] investigated the lifetime of C/SiC composite at elevated temperatures of 600 and 1200 ∘ C subjected to the stress rupture loading in different environments. Stress rupture lives in air and in steam containing environments were similar at a low stress level of 69 MPa at an elevated temperature of 1200 ∘ C. The fiber oxidation rate correlated with the composite stress rupture lifetime in various environments. In the theoretical research area, Marshall et al. [21] and Zok and Spearing [22] applied the fracture mechanics approach to explore nonsteady first matrix cracking stress and multiple matrix cracking in fiber-reinforced CMCs. The energy balance relationship before and after the matrix cracking is established considering the mutual inference factors of the stress field between the adjacent matrix cracks. Curtin [23] investigated multiple matrix cracking in CMCs in the presence of matrix internal flaws. Evans [24] reported a method to predict design and life problems in fiber-reinforced CMCs. In addition, a connection between the macromechanical behavior and constituent properties of CMCs was established based on these predictions. McNulty and Zok [25] investigated the low-cycle fatigue damage mechanism and reported predictive damage models to describe the low-cycle CMC fatigue life. The degradation of the interface properties and fiber strength controls the fatigue life of CMCs. Lara-Curzio [26] established a micromechanical model for fiber-reinforced CMC reliability and time-to-failure estimation, particularly following the application of stresses greater than the first matrix cracking stress. The relationship between internal damage mechanisms and lifetime was established. In addition, the stress and temperature influences on the fiber-reinforced CMCs were investigated. Halverson and Curtin [27] developed a micromechanically based model for composite strength, and stress rupture lifetime of oxide/oxide fiber-reinforced CMCs considering the degradation of the fiber, matrix damage, and

6.2 Stress Rupture of Ceramic-Matrix Composites

fiber pullout. Casas and Martinez-Esnaola [28] produced a fiber-reinforced CMC micromechanical creep–oxidation model, which was used to characterize oxidation at the CMC interface and the matrix, fiber creep, and fiber degradation with respect to time. Pailler and Lamon [29] developed a micromechanics-based model for the thermomechanical behavior of minicomposites based on multi-matrix cracking and fiber failure, which was derived from a fracture statistics-based model. Dassios et al. [30] analyzed the micromechanical behavior and micromechanics of crack growth resistance and bridging laws. The contributions of intact and pulled-out fibers on the bridging strain were discussed. Baranger [31] developed a reduced constitutive law to characterize the complex material behavior and applied the constitutive law to the mechanical modeling of SiC/SiC composites. Li [32–34] developed micromechanical damage models and constitutive relationship of cross-ply CMCs subjected to the dwell fatigue loading at elevated temperature. Li et al. [35, 36] and Li [37] developed a micromechanical constitutive relationship to predict the damage and fracture of different fiber-reinforced CMCs subjected to tensile loading considering multiple damage mechanisms. Nonetheless, the above research did not consider fiber-reinforced CMC time-dependent deformation, damage, and fracture following the application of stress rupture loading at intermediate environmental temperatures. In this chapter, time-dependent deformation, damage, and fracture of fiber-reinforced CMCs that were exposed to stress rupture loading at intermediate environmental temperatures are investigated. The composite microstress field and tensile constitutive relationship of the damaged CMCs were examined to characterize their time-dependent damage mechanisms. Relationships between stress rupture lifetime, peak stress level, time-dependent composite deformation, and evolution of internal damages are established. Effects of composite material properties, composite damage state, and environmental temperature on stress rupture lifetime, time-dependent composite deformation, and evolution of the internal damages of SiC/SiC are analyzed. Experimental stress rupture lifetime, time-dependent composite deformation, and composite internal damage evolution of SiC/SiC composite subjected to the stress rupture loading are evaluated.

6.2 Stress Rupture of Ceramic-Matrix Composites Under Constant Stress at Intermediate Temperature In this section, time-dependent deformation, damage, and fracture tendencies of fiber-reinforced CMCs that were exposed to stress rupture loading at intermediate environmental temperatures are discussed. Relationships between stress rupture lifetime, constant peak stress level, time-dependent composite deformation, and evolution of internal damages are established. Effects of composite material properties, composite damage state, and environmental temperature on stress rupture lifetime, time-dependent composite deformation, and evolution of the internal damages of SiC/SiC are analyzed.

213

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6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

6.2.1

Theoretical Models

Under the stress rupture loading at intermediate temperature, multiple timedependent damage mechanisms occur and are affected by the environmental temperature. The time-dependent composite damage mechanisms are characterized by combining microstress field analysis and different damage models. The time-dependent composite strain is determined considering time-dependent composite damage mechanisms, damage modes, and microstress field analysis. Two damage parameters of debonding and oxidation fraction at the interface are developed to analyze the internal time-dependent damage evolution. When multiple damage modes existed inside of the fiber-reinforced CMCs, the time-dependent composite strain 𝜀c (t) is given by 𝜀c (t) =

2 𝜎 (x, t) dx − (𝛼c − 𝛼f )ΔT Ef lc ∫lc ∕2 f

(6.1)

where Ef is the fiber Young’s modulus, lc is the matrix crack spacing, 𝜎 f (x, t) is the fiber axial stress, 𝛼 c and 𝛼 f are the composite and fiber thermal expansion coefficients, and ΔT is the temperature difference between testing and fabricated temperature. The time-dependent fiber axial stress is ⎧Φ(t) − 2𝜏f x, x ∈ [0, 𝜁(t)] rf ⎪ 2𝜏f 2𝜏i ⎪ x ∈ [𝜁(t), ld ] 𝜎f (x, t) = ⎨Φ(t) − rf 𝜁(t) − rf (x − 𝜁(t)), [ ] ( ) [ ] ⎪ ⎪𝜎fo + Φ(t) − 𝜎fo − 2𝜏f 𝜁(t) − 2𝜏i (ld − 𝜁(t)) exp −𝜌 x−ld , x ∈ ld , lc rf rf rf 2 ⎩ (6.2) where Φ is the stress carried by the intact fiber, 𝜏 f and 𝜏 i are the interface shear stress in the oxidation and debonding region, r f is the fiber radius, 𝜌 is the shear-lag model parameter, 𝜎 fo is the fiber axial stress in the interface bonding region, ld is the interface debonding length, and ζ is the interface oxidation length. The interface debonding length can be determined using the fracture mechanics approach, as shown in Eq. (6.3). The intact fiber stress is determined by combining Eqs. (6.4)–(6.7). ( ) ) ( r Vm Em Φ(t) 1 𝜏 − ld (t) = 1 − f 𝜁(t) + f 𝜏i 2 E c 𝜏i 𝜌 √ √( )2 ) ( 2 √ r rf2 Vf Vm Ef Em Φ (t) rV EE 𝜎 f √ + f m f2 m Γd (6.3) 1− − − 2𝜌 Vf Φ(t) 4Ec2 𝜏i2 E c 𝜏i 2𝜏 𝜎 = Φ(1 − q(Φ)) + f ⟨L⟩q(Φ) (6.4) Vf rf [ ( ) ] mf +1 Φ q(Φ) = 1 − exp − (6.5) 𝜎fc ( 𝜎fc =

m l0 𝜎0 f (t)𝜏i

rf

)

1 mf +1

,

1 ⎛ m ⎜ 𝜎0 (t)rf l0 f 𝛿fc = ⎜ 𝜏i ⎜ ⎝

mf

⎞ mf +1 ⎟ ⎟ ⎟ ⎠

(6.6)

6.2 Stress Rupture of Ceramic-Matrix Composites

( )4 ⎧ 1 KIC ⎪ 𝜎0 , t ≤ k Y 𝜎0 ( )4 𝜎0 (t) = ⎨ (6.7) KIC K , t > k1 Y 𝜎IC ⎪ √ 4 0 ⎩ Y kt where mf is the Weibull modulus of the fiber, 𝜎 0 is the fiber strength, K IC is the fracture toughness, Y is the geometric parameter, and k is the parabolic rate constant. Substituting the time-dependent fiber axial stress in Eq. (6.2) into Eq. (6.1), the time-dependent composite strain 𝜀c (t) is 𝜏 l

𝜏 l (t)

𝜏 l (t)

Φ(t) ⎧ E 𝜂(t) + Ef rd 𝜂(t)𝜔2 (t) − 2 Ef dr 𝜂(t)𝜔(t) − Ei dr 𝜂(t)(1 − 𝜔(t))2 f f f f f f ⎪ f { } ⎪ + 𝜎fo (1 − 𝜂(t)) + 2 1 rf Φ(t) − 𝜏 lc 𝜂(t)𝜔(t) − 𝜏 lc 𝜂(t)(1 − 𝜔(t)) − 𝜎 f i fo 𝜌Ef lc rf rf ⎪ Ef 𝜀c (t) = ⎨ [ ( )] 𝜌 lc ⎪ × 1 − exp − (1 − 𝜂(t)) − (𝛼c − 𝛼f )ΔT, 𝜂(t) < 1 2 rf ⎪ ⎪Φ(t) 𝜂(t) + 𝜏f ld 𝜂(t)𝜔2 (t) − 2 𝜏f ld (t) 𝜂(t)𝜔(t) − 𝜏i ld (t) 𝜂(t)(1 − 𝜔(t))2 , 𝜂(t) = 1 ⎩E E r E r E r f

f

f

f

f

f

f

(6.8) where 𝜂(t) is the time-dependent debonding fraction at the interface and 𝜔(t) is the time-dependent oxidation fraction at the interface. l (t) 𝜁(t) (6.9) 𝜂(t) = 2 d , 𝜔(t) = lc ld (t)

6.2.2

Results and Discussion

Under the stress rupture loading at intermediate temperature, the time-dependent deformation, damage, and fracture evolution are affected by the composite material properties, composite damage state, and intermediate environmental temperature. In the present analysis, effects of the composite material properties, constant peak stress level, composite damage state, and environmental temperature on the evolution of the time-dependent deformation and fraction of SiC/SiC composite subjected to the stress rupture loading are analyzed. Material properties of SiC/SiC composite are listed in Table 6.1. 6.2.2.1 Stress Rupture of SiC/SiC Composite for Different Fiber Volumes

Figure 6.1 and Table 6.2 show the effect of the fiber volume (i.e. V f = 20%, 30%, and 40%) on the stress rupture lifetime, time-dependent composite deformation, debonding fraction at the interface, and oxidation fraction at the interface. In addition, Figure 6.1 and Table 6.2 present the broken fiber fraction of the SiC/SiC composite at a constant peak stress of 𝜎 max = 200 MPa at T = 800 ∘ C. When the fiber volume increases, the stress rupture lifetime increases, and the final composite fracture strain is the highest at the fiber volume of V f = 30%; at V f = 40%, the failure strain is more comparable to the conditions of V f = 20%, which indicates that the composite failure strain depends on the fiber volume, and the failure strain is low for low or high fiber volume, as shown in Figure 6.1a; the time-dependent debonding fraction at the interface increases to the complete debonding (i.e. 𝜂 = 1) at the fiber volume of V f = 30% and 40%, and the time for the

215

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6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

Table 6.1 Material properties of SiC/SiC composite. Vf

0.4

Ef (GPa)

270

Em (GPa)

400

r f (μm)

7

m

3

𝛼 f (10 /∘ C) 𝛼 (10−6 /∘ C) −6

3.5 4.6

m

ΔT (∘ C)

−1000

Γd (J/m2 )

0.1

𝜏 i (MPa)

50

𝜏 f (MPa)

1

𝜎 0 (GPa)

2.59

l0 (mm)

25

mf

5

Table 6.2 Evolution of the time-dependent stress rupture of SiC/SiC composite under constant peak stress level of 𝜎 max = 200 MPa for different fiber volumes. 𝜺c (%)

𝜼

𝝎

q

0

0.1

0.25

0

4.8 × 10−5

93.5

0.19

0.54

0.45

0.16

0

0.072

0.136

0

4.0 × 10−6

335

0.234

1.0

0.872

0.05

385

0.244

1.0

1.0

0.06

752

0.274

1.0

1.0

0.25

0

0.065

0.08

0

4.0 × 10−6

361

0.173

1.0

0.938

0.009

t (ks)

V f = 20%

V f = 30%

V f = 40%

385

0.175

1.0

1.0

0.01

3015

0.205

1.0

1.0

0.27

6.2 Stress Rupture of Ceramic-Matrix Composites

(a)

(b)

Figure 6.1 Effect of the fiber volume (i.e. V f = 20%, 30%, and 40%) on (a) the evolution of the time-dependent stress rupture curves; (b) the evolution of the time-dependent debonding fraction at the interface versus time curves; (c) the evolution of the time-dependent oxidation fraction at the interface versus time curves; and (d) the evolution of the time-dependent broken fraction of the fiber versus time curves of SiC/SiC composite under constant peak stress level of 𝜎 max = 200 MPa.

complete debonding at the interface for the fiber volume of V f = 30% is less than that at V f = 40%, as shown in Figure 6.1b; the time-dependent oxidation fraction at the interface increases to the complete interface oxidation (i.e. 𝜔 = 1.0) at the fiber volume of V f = 30% and 40%, and the time for the complete interface oxidation is the same at the fiber volume of V f = 30% and 40%, as shown in Figure 6.1c, and the time-dependent failure probability of the fiber at the final composite fracture increases with the fiber volume, as shown in Figure 6.1d. At the fiber volume of V f = 30%, there is a large increase in the composite fracture strain because of the

217

218

6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

(c)

(d)

Figure 6.1

(Continued)

decrease of the debonding fraction at the interface and the increase of the oxidation fraction at the interface, whereas at the fiber volume of V f = 40%, the composite fracture strain is more comparable to the conditions of the fiber volume of V f = 20% because of the low broken fiber fraction before complete debonding and oxidation of the interface. 6.2.2.2 Stress Rupture of SiC/SiC Composite for Different Peak Stress Levels

Figure 6.2 and Table 6.3 show the effect of the constant peak stress level 𝜎 max = 150, 250, and 350 MPa on the stress rupture lifetime, time-dependent composite deformation, debonding fraction at the interface, and oxidation fraction at the interface. In addition, Figure 6.2 and Table 6.3 present the broken fiber fraction of the SiC/SiC composite at T = 800 ∘ C.

6.2 Stress Rupture of Ceramic-Matrix Composites

(a)

(b)

Figure 6.2 Effect of the constant peak stress level (i.e. 𝜎 max = 150, 250, and 300 MPa) on (a) the evolution of the time-dependent stress rupture curves; (b) the evolution of the time-dependent debonding fraction at the interface versus time curves; (c) the evolution of the time-dependent oxidation fraction at the interface versus time curves; and (d) the evolution of the time-dependent broken fraction of the fiber versus time curves of SiC/SiC composite.

The time-dependent damage evolution and composite final fracture are affected by the peak stress level. When the peak stress level increases, the stress rupture lifetime decreases, the final composite fracture strain is the highest at the peak stress level of 𝜎 max = 250 MPa, under high peak stress of 𝜎 max = 350 MPa, the composite failure strain is low because of the composite fracture with partial debonding at the interface, as shown in Figure 6.2a; the composite fracture occurs at the condition of partial debonding at the interface corresponding to the high constant peak stresses of 𝜎 max = 250 and 300 MPa, and the composite fracture

219

220

6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

(c)

(d)

Figure 6.2

(Continued)

occurs at the condition of complete debonding and oxidation at the interface corresponding to the low constant peak stress level of 𝜎 max = 150 MPa, as shown in Figure 6.2b; the composite fracture occurs at the condition of the partial oxidation at the interface corresponding to the high constant peak stresses of 𝜎 max = 250 and 300 MPa, and the composite fracture occurs at the condition of the complete oxidation at the interface corresponding to the low constant peak stress level of 𝜎 max = 150 MPa, as shown in Figure 6.2c; the failure probability of the fiber at composite final fracture increases with the decrease of the peak stress, as shown in Figure 6.2d. At the high constant peak stresses of 𝜎 max = 250 and 350 MPa, the tensile composite strain increases to the final fracture, and at low constant peak stress level of 𝜎 max = 150 MPa, the tensile composite strain evolution with time can be divided into two stages, i.e. (i) Stage I, the condition of the partial debonding and oxidation at the interface (i.e. 𝜂 < 1 and 𝜔 < 1) and (ii) Stage II, the

6.2 Stress Rupture of Ceramic-Matrix Composites

Table 6.3 Evolution of the time-dependent stress rupture of SiC/SiC composite for different constant peak stress levels. 𝜺c (%)

𝜼

𝝎

q

0

0.046

0

0

1 × 10−6

688

0.167

1.0

0.968

0.02

710

0.168

1.0

1.0

0.021

3015

0.196

1.0

1.0

0.27

0

0.105

0.25

0

1.6 × 10−5

203

0.293

1.0

0.718

0.13

227

0.31

1.0

0.8

0.16

0

0.18

0.43

0

1.2 × 10−4

32.7

0.238

0.63

0.21

0.08

t (ks)

𝜎 max = 150 MPa

𝜎 max = 250 MPa

𝜎 max = 300 MPa

condition of the complete debonding and oxidation at the interface (i.e. 𝜂 = 1 and 𝜔 = 1). 6.2.2.3 Stress Rupture of SiC/SiC Composite for Different Saturation Spaces Between Matrix Cracking

Figure 6.3 and Table 6.4 show the effect of the saturation space between the matrix cracking (lsat = 150, 250, and 350 μm) on the stress rupture lifetime, time-dependent composite deformation, debonding fraction at the interface, and oxidation fraction at the interface. In addition, Figure 6.3 and Table 6.4 present the broken fiber fraction of the SiC/SiC composite at a constant peak stress level of 𝜎 max = 200 MPa at T = 800 ∘ C. When the saturation spacing between the matrix cracks increases, the time-dependent composite strain decreases, which indicates that the composite failure strain depends on the damage in matrix, as shown in Figure 6.3a; the time for the condition of the complete debonding at the interface (i.e. 𝜂 = 1) increases, as shown in Figure 6.3b; the time for the condition of the complete oxidation at the interface (i.e. 𝜔 = 1) increases, as shown in Figure 6.3c; and the failure probability of the fiber remains the same, as shown in Figure 6.3d. 6.2.2.4 Stress Rupture of SiC/SiC Composite for Different Interface Shear Stress

Figure 6.4 and Table 6.5 show the effect of the interface shear stress in the slip region 𝜏 i = 20, 30, and 40 MPa on the stress rupture lifetime, time-dependent composite deformation, debonding fraction at the interface, and oxidation fraction

221

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6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

Table 6.4 Evolution of the time-dependent stress rupture of SiC/SiC composite for different spacings between saturation matrix cracking. 𝜺c (%)

𝜼

𝝎

q

0

0.08

0.22

0

4 × 10−6

180

0.228

1.0

0.782

0.02

230

0.244

1.0

1.0

0.03

752

0.278

1.0

1.0

0.25

0

0.07

0.13

0

4 × 10−6

335

0.234

1.0

0.872

0.05

384

0.244

1.0

1.0

0.06

752

0.274

1.0

1.0

0.25

0

0.067

0.09

0

4 × 10−6

490

0.238

1.0

0.909

0.09

540

0.246

1.0

1.0

0.109

752

0.27

1.0

1.0

0.25

t (ks)

lsat = 150 𝜇m

lsat = 250 𝜇m

lsat = 350 𝜇m

Table 6.5 Evolution of the time-dependent stress rupture of SiC/SiC composite under constant peak stress level of 𝜎 max = 200 MPa for different interface shear stress in slip region. 𝜺c (%)

𝜼

𝝎

q

0

0.085

0.25

0

1.1 × 10−5

354

0.233

0.99

0.99

0.23

0

0.075

0.16

0

7 × 10−6

442

0.241

1.0

0.82

0.167

491

0.262

1.0

0.91

0.23

0

0.07

0.12

0

5 × 10−6

474

0.239

1.0

0.879

0.119

540

0.253

1.0

1.0

0.155

628

0.269

1.0

1.0

0.25

t (ks)

𝜏 i = 20 MPa

𝜏 i = 30 MPa

𝜏 i = 40 MPa

6.2 Stress Rupture of Ceramic-Matrix Composites

(a)

(b)

Figure 6.3 Effect of the space between saturation matrix cracking (i.e. lsat = 150, 250, and 350 μm) on (a) the evolution of the time-dependent stress rupture curves; (b) the evolution of the time-dependent debonding fraction at the interface versus time curves; (c) the evolution of the time-dependent oxidation fraction at the interface versus time curves; and (d) the evolution of the time-dependent broken fraction of the fiber versus time curves of SiC/SiC composite under constant peak stress level of 𝜎 max = 200 MPa.

at the interface. In addition, Figure 6.4 and Table 6.5 present the broken fiber fraction of the SiC/SiC composite at a constant peak stress level of 𝜎 max = 200 MPa at T = 800 ∘ C. When the interface shear stress in the slip region increases, the stress rupture lifetime increases, and the composite final fracture strain increases, under low interface shear stress (i.e. 𝜏 i = 20 MPa), the strain curve has a slight curvature at higher times because of a large amount of fiber failure without interface complete debonding; however, under high interface shear stress (i.e. 𝜏 i = 30

223

224

6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

(c)

(d)

Figure 6.3

(Continued)

or 40 MPa), the strain curve shows linearity till final fracture because of composite final fracture with complete interface debonding, as shown in Figure 6.4a; the time-dependent debonding fraction at the interface increases to the condition of the complete debonding at high values of interface shear stress, i.e. 𝜏 i = 30, and 40 MPa, as shown in Figure 6.4b; and the time-dependent oxidation fraction at the interface increases to the complete oxidation at high values of interface shear stress, i.e. 𝜏 i = 30 and 40 MPa, as shown in Figure 6.4c, and the time-dependent failure probability of the fiber at the condition of composite final fracture increases with the interface shear stress, as shown in Figure 6.4d. At a low interface shear stress of 𝜏 i = 20 MPa, the tensile fracture occurs at the condition of partial debonding and oxidation at the interface (i.e. 𝜂 < 1 and 𝜔 < 1), and the debonding fraction of the interface increases fast near the condition of

6.2 Stress Rupture of Ceramic-Matrix Composites

(a)

(b)

Figure 6.4 Effect of the interface shear stress in slip region (i.e. 𝜏 i = 20, 30, and 40 MPa) on (a) the evolution of the time-dependent stress rupture curves; (b) the evolution of the time-dependent debonding fraction at the interface versus time curves; (c) the evolution of the time-dependent oxidation fraction at the interface versus time curves; and (d) the evolution of the time-dependent broken fraction of the fiber versus time curves of SiC/SiC composite under constant peak stress level of 𝜎 max = 200 MPa.

composite final fracture, leading to the decrease of the oxidation fraction at the interface approaching final fracture; at the interface shear stress of 𝜏 i = 30 MPa, the tensile composite fracture occurs at the condition of complete debonding and partial oxidation at the interface (i.e. 𝜂 = 1 and 𝜔 < 1), and at the interface shear stress of 𝜏 i = 40 MPa, the tensile composite fracture occurs at the condition of complete debonding and oxidation at the interface (i.e. 𝜂 = 1 and 𝜔 = 1). Figure 6.5 and Table 6.6 show the effect of the interface shear stress in the oxidation region (i.e. 𝜏 f = 1, 3, and 5 MPa) on the stress rupture lifetime, time-dependent

225

226

6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

(c)

(d)

Figure 6.4

(Continued)

composite deformation, debonding fraction at the interface, and oxidation fraction at the interface. In addition, Figure 6.5 and Table 6.6 present the broken fiber fraction of the SiC/SiC composite at a constant peak stress level of 𝜎 max = 200 MPa at T = 800 ∘ C. When the interface shear stress in the oxidation region increases, the time-dependent composite strain decreases because of the decrease of the interface debonding fraction, as shown in Figure 6.5a; the time for the condition of complete debonding at the interface (i.e. 𝜂 = 1) increases, as shown in Figure 6.5b; the time for the condition of the complete oxidation at the interface (i.e. 𝜔 = 1) remains the same, as shown in Figure 6.5c, and the failure probability of the fiber remains the same; in the present analysis, the fiber failure in the oxidation region is not considered, as shown in Figure 6.5d. At the interface shear stress of 𝜏 f = 5 MPa, the time for the condition of complete debonding at the interface is the highest

6.2 Stress Rupture of Ceramic-Matrix Composites

(a)

(b)

Figure 6.5 The effect of the interface shear stress in oxidation region (i.e. 𝜏 f = 1, 3, and 5 MPa) on (a) the evolution of the time-dependent stress rupture curves; (b) the evolution of the time-dependent debonding fraction at the interface versus time curves; (c) the evolution of the time-dependent oxidation fraction at the interface versus time curves; and (d) the time-dependent broken fraction of the fiber versus time curves of SiC/SiC composite under constant peak stress level of 𝜎 max = 200 MPa.

compared with that of low interface shear stress of 𝜏 f = 1 and 3 MPa, and the time for the condition of complete oxidation at the interface is the same for 𝜏 f = 1, 3, and 5 MPa. 6.2.2.5 Stress Rupture of SiC/SiC Composite for Different Fiber Weibull Modulus

Figure 6.6 and Table 6.7 show the effect of the Weibull modulus of the fiber mf = 4, 5, and 6 on the stress rupture lifetime, time-dependent composite deformation, debonding fraction at the interface, and oxidation fraction at the interface. In

227

228

6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

(c)

(d)

Figure 6.5

(Continued)

addition, Figure 6.6 and Table 6.7 present the broken fiber fraction of the SiC/SiC composite at a constant peak stress level of 𝜎 max = 200 MPa at T = 800 ∘ C. When the Weibull modulus of the fiber increases, the stress rupture lifetime decreases, and the composite fracture strain also decreases, under high fiber Weibull modulus, most of the fiber fractures at low stress level with complete interface debonding, leading to the decrease of the composite failure strain, as shown in Figure 6.6a; the composite fracture occurs at the condition of partial debonding at the interface (i.e. 𝜂 < 1) for the high Weibull modulus of the fiber, i.e. mf = 6, and at the condition of complete debonding at the interface (i.e. 𝜂 = 1) for the low Weibull modulus of the fiber, i.e. mf = 4 and 5, as shown in Figure 6.6b; the composite fracture occurs at the condition of partial oxidation at the interface (i.e. 𝜔 < 1) for the high Weibull modulus of the fiber, i.e. mf = 6, and the condition of complete oxidation at the interface (i.e. 𝜔 = 1) for low Weibull modulus of the fiber,

6.2 Stress Rupture of Ceramic-Matrix Composites

Table 6.6 Evolution of the time-dependent stress rupture of SiC/SiC composite under constant peak stress level of 𝜎 max = 200 MPa for different interface shear stress in oxidation region. 𝜺c (%)

𝜼

𝝎

q

0

0.067

0.09

0

4 × 10−6

490

0.238

1.0

0.909

0.092

t (ks)

𝜏 f = 1 MPa

540

0.246

1.0

1.0

0.109

752

0.27

1.0

1.0

0.25

0

0.067

0.09

0

4 × 10−6

511

0.214

1.0

0.946

0.099

540

0.217

1.0

1.0

0.109

752

0.24

1.0

1.0

0.25

0

0.067

0.09

0

4 × 10−6

533

0.187

1.0

0.987

0.106

540

0.188

1.0

1.0

0.109

752

0.21

1.0

1.0

0.25

𝜏 f = 3 MPa

𝜏 f = 5 MPa

i.e. mf = 4 and 5, as shown in Figure 6.6c, and the failure probability of the fiber at final composite fracture increases with decreasing Weibull modulus of the fiber, as shown in Figure 6.6d. 6.2.2.6 Stress Rupture of SiC/SiC Composite for Different Environmental Temperatures

Figure 6.7 and Table 6.8 show the effect of the environmental temperature T = 600, 700, and 800 ∘ C on the stress rupture lifetime, time-dependent composite deformation, debonding fraction at the interface, and oxidation fraction at the interface. In addition, Figure 6.7 and Table 6.8 present the broken fiber fraction of the SiC/SiC composite at a constant peak stress level of 𝜎 max = 200 MPa. When the environmental temperature increases, the stress rupture lifetime decreases, and the final composite fracture strain increases, as shown in Figure 6.7a and Table 6.8; the time-dependent debonding fraction at the interface and the oxidation fraction at the interface increase, as shown in Figure 6.7b,c and Table 6.8, and the time-dependent broken fraction of the fiber remains the same, as shown in Figure 6.7d and Table 6.8. The fracture occurs at the condition of partial debonding and oxidation at the interface (i.e. 𝜂 < 1 and 𝜔 < 1), and the rate for interface debonding, interface oxidation, and fiber failure probability all increase.

229

230

6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

(a)

(b)

Figure 6.6 Effect of the Weibull modulus of the fiber (i.e. mf = 4, 5, and 6) on (a) the evolution of the time-dependent stress rupture curves; (b) the evolution of the time-dependent debonding fraction at the interface versus time curves; (c) the evolution of the time-dependent oxidation fraction at the interface versus time curves; and (d) the evolution of the time-dependent broken fraction of the fiber versus time curves of SiC/SiC composite under constant peak stress level of 𝜎 max = 200 MPa.

6.2.3

Experimental Comparisons

Lara-Curzio [11] reported the SiC/SiC composite stress rupture behavior at T = 950 ∘ C. The experimental data and predicted results of the stress rupture lifetime, time-dependent composite deformation, debonding fraction at the interface, oxidation fraction at the interface, and the broken fraction of the fiber curves under 𝜎 max = 80, 100, and 120 MPa are shown in Figures 6.8–6.10. At a constant peak stress level of 𝜎 max = 80 MPa, the time-dependent composite strain increased from 𝜀c = 0.085% at t = 0 seconds to 𝜀c = 0.27% at t = 83 600 seconds;

6.2 Stress Rupture of Ceramic-Matrix Composites

(c)

(d)

Figure 6.6

(Continued)

the time-dependent debonding fraction at the interface increases from partial debonding 𝜂 = 0.06 at t = 0 seconds under the peak stress level 𝜎 max = 80 MPa to partial debonding 𝜂 = 0.55 at t = 83 600 seconds because of the oxidation propagation at the interface and increasing broken fiber probability; the time-dependent oxidation fraction at the interface increases from 𝜔 = 0 at t = 0 seconds to partial oxidation 𝜔 = 0.84 at t = 83 600 seconds because of the oxidation propagation at the interface; finally, a higher time-dependent fiber broken fraction of q = 0.29 was observed at t = 83 600 seconds as compared to that at t = 0 seconds (q = 0). At a constant peak stress level of 𝜎 max = 100 MPa, the time-dependent composite strain increases from 𝜀c = 0.1% at t = 0 seconds to 𝜀c = 0.2% at t = 33 700 seconds; the time-dependent debonding fraction at the interface increases from partial

231

232

6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

Table 6.7 Evolution of the time-dependent stress rupture of SiC/SiC composite under constant peak stress level of 𝜎 max = 200 MPa for different Weibull moduli of the fiber. 𝜺c (%)

𝜼

𝝎

q

0

0.067

0.09

0

1.6 × 10−5

494

0.232

1.0

0.915

0.044

540

0.238

1.0

1.0

0.048

1855

0.28

1.0

1.0

0.32

0

0.067

0.09

0

4 × 10−6

490

0.238

1.0

0.909

0.092

t (ks)

mf = 4

mf = 5

540

0.246

1.0

1.0

0.109

752

0.27

1.0

1.0

0.25

0

0.067

0.09

0

1 × 10−6

426

0.24

0.91

0.86

0.23

mf = 6

Table 6.8 Evolution of the time-dependent stress rupture of SiC/SiC composite under constant peak stress level of 𝜎 max = 200 MPa for different environmental temperatures. 𝜺c (%)

𝜼

𝝎

q

0

0.067

0.09

0

1 × 10−6

2745

0.23

0.9

0.85

0.23

0

0.067

0.09

0

1 × 10−6

983

0.23

0.91

0.86

0.23

0

0.067

0.09

0

1 × 10−6

426

0.24

0.91

0.86

0.23

t (ks)

T = 600 ∘ C

T = 700 ∘ C

T = 800 ∘ C

debonding 𝜂 = 0.07 at t = 0 seconds under the peak stress level 𝜎 max = 100 MPa to the partial debonding at the interface 𝜂 = 0.27 at t = 33 700 seconds because of the oxidation propagation at the interface and increasing broken fiber probability; the time-dependent oxidation fraction at the interface increases from 𝜔 = 0 at t = 0 seconds to partial oxidation at the interface 𝜔 = 0.61 at t = 33 700 seconds

6.2 Stress Rupture of Ceramic-Matrix Composites

(a)

(b)

Figure 6.7 Effect of the environmental temperature (i.e. T = 600, 700, and 800 ∘ C) on (a) the evolution of the time-dependent stress rupture curves; (b) the evolution of the time-dependent debonding fraction at the interface versus time curves; (c) the evolution of the time-dependent oxidation fraction at the interface versus time curves; and (d) the evolution of the time-dependent broken fraction of the fiber versus time curves of SiC/SiC composite under constant peak stress level of 𝜎 max = 200 MPa.

because of the oxidation propagation at the interface; lastly, a higher time-dependent broken fiber fraction of q = 0.22 at was observed at t = 33 700 seconds as compared to that at t = 0 seconds (q = 1 × 10−6 ). At a constant peak stress level of 𝜎 max = 120 MPa, the time-dependent composite strain increases from 𝜀c = 0.16% at t = 0 seconds to 𝜀c = 0.27% at t = 10 500 seconds; the time-dependent debonding fraction at the interface increases from partial debonding 𝜂 = 0.19 at t = 0 seconds under the peak stress level 𝜎 max = 120 MPa to the partial debonding 𝜂 = 0.42 at t = 10 500 seconds because of the oxidation propagation at the interface and increasing broken fiber probability; the time-dependent

233

234

6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

(c)

(d)

Figure 6.7

(Continued)

oxidation fraction at the interface increases from 𝜔 = 0 at t = 0 seconds to partial oxidation 𝜔 = 0.4 at t = 10 500 seconds because of the oxidation propagation at the interface; lastly, a higher time-dependent broken fiber fraction of q = 0.17 was observed at t = 10 500 seconds as compared to that at t = 0 seconds (q = 2 × 10−6 ).

6.3 Stress Rupture of Ceramic-Matrix Composites Under Stochastic Loading Stress and Time at Intermediate Temperature In this section, synergistic effects of stochastic loading stress and time interval on stress rupture damage evolution and lifetime of fiber-reinforced CMCs at

6.3 Stress Rupture of Ceramic-Matrix Composites

(a)

(b)

Figure 6.8 (a) Evolution of the time-dependent stress rupture curves of experimental data and theoretical results; (b) the evolution of the time-dependent debonding fraction at the interface versus time curves; (c) the evolution of the time-dependent oxidation fraction at the interface versus time curves; and (d) the evolution of the time-dependent broken fraction of the fiber versus time curves of SiC/SiC composite under constant peak stress level of 𝜎 max = 80 MPa.

intermediate temperatures are investigated. Relationships between the stochastic loading stress and stochastic time interval, internal damage evolution, macrostrain, broken fiber fraction, and stress rupture lifetime of fiber-reinforced CMCs are established. Effects of the stochastic loading stress and stochastic time interval, constituent properties, damage condition, and environment temperature on the stress rupture time, strain evolution, and internal damage of SiC/SiC composite under stress rupture loading are discussed. Experimental stress rupture behavior of SiC/SiC composite under constant and stochastic loading condition is predicted.

235

236

6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

(c)

(d)

Figure 6.8

6.3.1

(Continued)

Results and Discussion

Effects of stochastic stress and time interval, fiber volume, matrix crack spacing, fiber/matrix interface shear stress, and environmental temperature on the stress rupture damage evolution and lifetime of SiC/SiC composite are analyzed. 6.3.1.1 Stress Rupture of SiC/SiC Composite Under Stochastic Loading for Different Stochastic Stress Levels

Figure 6.11 and Table 6.9 shows the effect of the stochastic stress level (i.e. 𝜎 s = 200, 220, 240, 260, and 280 MPa) and time interval of Δt = 36 ks on the time-dependent strain, interface debonding ratio, interface oxidation ratio, broken fiber fraction, and stress rupture lifetime of SiC/SiC composite under constant stress 𝜎 = 180 MPa at T = 850 ∘ C in air atmosphere. When the stochastic stress level increases, the stress rupture lifetime decreases; the time for the fiber/matrix

6.3 Stress Rupture of Ceramic-Matrix Composites

(a)

(b)

Figure 6.9 (a) Evolution of the time-dependent stress rupture curves of experimental data and theoretical results; (b) the evolution of the time-dependent debonding fraction at the interface versus time curves; (c) the evolution of the time-dependent oxidation fraction at the interface versus time curves; and (d) the evolution of the time-dependent broken fraction of the fiber versus time curves of SiC/SiC composite under constant peak stress level of 𝜎 max = 100 MPa.

interface complete debonding and oxidation decreases; and the broken fiber fraction at the stage of stochastic loading increases. The stress rupture lifetime decreases from t = 1223.6 ks at 𝜎 s = 200 MPa to t = 1186 ks at 𝜎 s = 280 MPa. The time for the interface complete debonding decreases from t = 232.9 ks at 𝜎 s = 200 MPa to t = 143.3 ks at 𝜎 s = 280 MPa. The time for the interface complete oxidation decreases from t = 267.1 ks at 𝜎 s = 200 MPa to t = 179.9 ks at 𝜎 s = 280 MPa. When the stochastic stress level is 𝜎 s = 200 MPa, the strain increases from 𝜀c = 0.062–0.077% under 𝜎 = 180 MPa, and at t = 36 ks increases to 𝜀c = 0.095%

237

238

6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

(c)

(d)

Figure 6.9

(Continued)

under 𝜎 s = 200 MPa, and increases to 𝜀c = 0.115% at t = 72 ks under 𝜎 s = 200 MPa, and decreases to 𝜀c = 0.102% at t = 72 ks under 𝜎 = 180 MPa, and increases to 𝜀c = 0.213% at t = 1223.6 ks. The interface debonding ratio increases from 𝜂 = 0.125 to 0.23 under 𝜎 = 180 MPa, and at t = 36 ks increases to 𝜂 = 0.31 under 𝜎 s = 200 MPa, and increases to 𝜂 = 0.44 at t = 72 ks under 𝜎 s = 200 MPa, and remains to 𝜂 = 0.44 till t = 79 ks under 𝜎 = 180 MPa, and increases to 𝜂 = 1.0 at t = 232.9 ks. The interface oxidation ratio increases from 𝜔 = 0 to 0.47 under 𝜎 = 180 MPa, and at t = 36 ks decreases to 𝜔 = 0.43 under 𝜎 s = 200 MPa, and increases to 𝜔 = 0.61 at t = 72 ks under 𝜎 s = 200 MPa, and increases to 𝜔 = 1.0 at t = 267.1 ks under 𝜎 = 180 MPa. The broken fiber fraction increases from q = 2 × 10−6 to 1.5 × 10−3 under 𝜎 = 180 MPa, and at t = 36 ks increases to q = 2.9 × 10−3 under 𝜎 s = 200 MPa, and increases to q = 7 × 10−3 at t = 72 ks under 𝜎 s = 200 MPa, and increases to q = 0.285 at t = 1223.6 ks under 𝜎 = 180 MPa.

6.3 Stress Rupture of Ceramic-Matrix Composites

(a)

(b)

Figure 6.10 (a) Evolution of the time-dependent stress rupture curves of experimental data and theoretical results; (b) the evolution of the time-dependent debonding fraction at the interface versus time curves; (c) the evolution of the time-dependent oxidation fraction at the interface versus time curves; and (d) the evolution of the time-dependent broken fraction of the fiber versus time curves of SiC/SiC composite under constant peak stress level of 𝜎 max = 120 MPa.

When the stochastic stress level is 𝜎 s = 280 MPa, the strain increases from 𝜀c = 0.062% to 0.077% under 𝜎 = 180 MPa, and at t = 36 ks increases to 𝜀c = 0.178% under 𝜎 s = 280 MPa, and increases to 𝜀c = 0.225% at t = 72 ks under 𝜎 s = 280 MPa, and decreases to 𝜀c = 0.123% at t = 72 ks under 𝜎 = 180 MPa, and increases to 𝜀c = 0.21% at t = 1186 ks. The interface debonded ratio increases from 𝜂 = 0.12 to 0.23 under 𝜎 = 180 MPa, and at t = 36 ks increases to 𝜂 = 0.61 under 𝜎 s = 200 MPa, and increases to 𝜂 = 0.83 at t = 72 ks under 𝜎 s = 280 MPa, and remains to 𝜂 = 0.83 till t = 112.1 ks under 𝜎 = 180 MPa, and increases to 𝜂 = 1.0 at t = 143.3 ks. The interface oxidation ratio increases from 𝜔 = 0 to 0.47 under 𝜎 = 180 MPa, and at

239

240

6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

(c)

(d)

Figure 6.10

(Continued)

t = 36 ks decreases to 𝜔 = 0.32 under 𝜎 s = 280 MPa, and increases to 𝜔 = 0.48 at t = 72 ks under 𝜎 s = 280 MPa, and increases to 𝜔 = 1.0 at t = 179.9 ks under 𝜎 = 180 MPa. The broken fiber fraction increases from q = 2 × 10−6 to 1.5 × 10−3 under 𝜎 = 180 MPa, and at t = 36 ks increases to q = 0.023 under 𝜎 s = 280 MPa, and increases to q = 0.06 at t = 72 ks under 𝜎 s = 280 MPa, and increases to q = 0.285 at t = 1186 ks under 𝜎 = 180 MPa. 6.3.1.2 Stress Rupture of SiC/SiC Composite Under Stochastic Loading for Different Stochastic Loading Time Intervals

Figure 6.12 and Table 6.10 show the effect of the stochastic loading time (i.e. Δt = 18, 36, 54, 72, and 90 ks) and stochastic stress level 𝜎 s = 240 MPa on the time-dependent strain, interface debonding ratio, interface oxidation ratio, broken fiber fraction, and the stress rupture lifetime curves of SiC/SiC composite under

6.3 Stress Rupture of Ceramic-Matrix Composites

(a)

(b)

Figure 6.11 Effect of stochastic stress (i.e. 𝜎 s = 200, 220, 240, 260, and 280 MPa) and time (Δt = 36 ks) on (a) the strain versus the time curves; (b) the interface debonding ratio versus the time curves; (c) the interface oxidation ratio versus the time curves; (d) the broken fiber fraction versus the time curves; and (e) the stress rupture lifetime versus the stochastic stress curves of SiC/SiC composite under stress rupture loading 𝜎 = 180 MPa at T = 850 ∘ C in air atmosphere.

constant stress 𝜎 = 180 MPa at T = 850 ∘ C in air atmosphere. When stochastic loading time increases, the stress rupture time decreases; and the time for the interface complete debonding and oxidation remains the same. The stress rupture lifetime decreases from t = 1220.8 ks at Δt = 18 ks to t = 1204.2 ks at Δt = 90 ks. When the stochastic loading time is Δt = 18 ks, the strain increases from 𝜀c = 0.062% to 0.077% under 𝜎 = 180 MPa, and at t = 36 ks increases to 𝜀c = 0.134% under 𝜎 s = 240 MPa, and increases to 𝜀c = 0.15% at t = 54 ks under 𝜎 s = 240 MPa, and decreases to 𝜀c = 0.106% at t = 54 ks under 𝜎 = 180 MPa, and increases to 𝜀c = 0.214% at t = 1220.8 ks. The interface debonded ratio increases from 𝜂 = 0.12

241

242

6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

(c)

(d)

(e)

Figure 6.11

(Continued)

6.3 Stress Rupture of Ceramic-Matrix Composites

Table 6.9 The strain, interface debonding and oxidation ratio, and broken fiber fraction for different stochastic stress levels. 𝝈 (MPa) 180

180

200

200

180

180

180

t (ks)

0

36

36

72

72

79

232.9 267.1 1223.6

𝜀c (%)

0.062

0.077

0.095

0.115

0.102

0.104

0.176 0.182 0.213

𝜂

0.125

0.23

0.31

0.44

0.44

0.44

1.0

1.0

1.0

𝜔

0

0.47

0.43

0.61

0.61

0.67

0.87

1.0

1.0

q

2 × 10−6 1.5 × 10−3 2.9 × 10−3 7.0 × 10−3 7.0 × 10−3 7.4 × 10−3 0.019 0.023 0.285

180

180

𝝈 (MPa) 180

180

220

220

180

180

180

180

t (ks)

0

36

36

72

72

86.2

193.9

229.2 1222

𝜀c (%)

0.062

0.077

0.114

0.14

0.11

0.117

0.176

0.183 0.214

𝜂

0.12

0.23

0.39

0.54

0.54

0.54

1.0

1.0

1.0

𝜔

0

0.47

0.40

0.58

0.58

0.69

0.85

1.0

1.0

q

2 × 10−6 1.5 × 10−3 5.2 × 10−3 0.0125 0.0125 0.0135 0.0219 0.025 0.285

180

𝝈 (MPa) 180

180

240

240

180

180

180

180

t (ks)

0

36

36

72

72

93.8

169

204.9 1218.3

𝜀c (%)

0.062

0.077

0.134

0.167

0.117

0.13

0.176

0.183 0.213

𝜂

0.12

0.23

0.47

0.644

0.644

0.644 1.0

1.0

1.0

𝜔

0

0.47

0.37

0.547

0.547

0.712 0.826

1.0

1.0

q

2 × 10−6 1.5 × 10−3 8.8 × 10−3 0.0216 0.0216 0.023 0.0289 0.031 0.285

180

𝝈 (MPa)

180

180

260

260

180

180

180

180

180

t (ks)

0

36

36

72

72

102.2

153.2

189.5

1209.3

𝜀c (%)

0.062

0.077

0.156

0.195

0.122

0.142

0.176

0.184

0.212

𝜂

0.12

0.23

0.54

0.74

0.74

0.74

1.0

1.0

1.0

𝜔

0

0.47

0.35

0.51

0.51

0.73

0.81

1.0

1.0

q

2 × 10−6

1.5 × 10−3

0.014

0.036

0.036

0.038

0.042

0.045

0.285

𝝈 (MPa)

180

180

280

280

180

180

180

180

180

t (ks)

0

36

36

72

72

112.1

143.4

179.9

1186

𝜀c (%)

0.062

0.077

0.178

0.225

0.123

0.154

0.176

0.184

0.21

𝜂

0.12

0.23

0.61

0.83

0.83

0.83

1.0

1.0

1.0

𝜔

0

0.47

0.32

0.48

0.48

0.75

0.798

1.0

1.0

q

2 × 10−6

1.5 × 10−3

0.023

0.06

0.06

0.063

0.065

0.068

0.285

243

244

6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

(a)

(b)

Figure 6.12 Effect of stochastic stress (i.e. 𝜎 s = 240 MPa) and time interval (i.e. Δt = 18, 36, 54, 72, and 90 ks) on (a) the strain versus the time curves; (b) the interface debonding ratio versus the time curves; (c) the interface oxidation ratio versus the time curves; (d) the broken fiber fraction versus the time curves; and (e) the stress rupture lifetime versus the stochastic time interval curve of SiC/SiC composite under stress rupture constant stress 𝜎 = 180 MPa at T = 850 ∘ C in air atmosphere.

to 0.23 under 𝜎 = 180 MPa, and at t = 36 ks increases to 𝜂 = 0.47 under 𝜎 s = 240 MPa, and increases to 𝜂 = 0.557 at t = 54 ks under 𝜎 s = 240 MPa, and remains to 𝜂 = 0.557 till t = 75.3 ks under 𝜎 = 180 MPa, and increases to 𝜂 = 1.0 at t = 169 ks. The interface oxidation ratio increases from 𝜔 = 0 to 0.47 under 𝜎 = 180 MPa, and at t = 36 ks decreases to 𝜔 = 0.37 under 𝜎 s = 240 MPa, and increases to 𝜔 = 0.475 at t = 54 ks under 𝜎 s = 240 MPa, and increases to 𝜔 = 1.0 at t = 204.9 ks under 𝜎 = 180 MPa. The broken fiber fraction increases from q = 2 × 10−6 to 1.5 × 10−3 under 𝜎 = 180 MPa, and at t = 36 ks increases to q = 8.8 × 10−3 under 𝜎 s = 240 MPa, and increases to q = 0.014 at t = 54 ks under 𝜎 s = 240 MPa, and increases to q = 0.285 at t = 1220.8 ks under 𝜎 = 180 MPa.

6.3 Stress Rupture of Ceramic-Matrix Composites

(c)

(d)

(e)

Figure 6.12

(Continued)

245

246

6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

Table 6.10 The strain, interface debonding and oxidation ratio, and broken fiber fraction for different stochastic times. 𝝈 (MPa) 180

180

240

240

180

180

180

180

t (ks)

0

36

36

54

54

75.3

169

204.9 1220.8

𝜀c (%)

0.062

0.077

0.134

0.15

0.106

0.118

0.176 0.183 0.214

𝜂

0.12

0.23

0.47

0.557

0.557

0.557

1.0

1.0

1.0

𝜔

0

0.47

0.37

0.475

0.475

0.662

0.826 1.0

1.0

q

2 × 10−6 1.5 × 10−3 8.8 × 10−3 0.0148 0.0148 0.0162 0.023 0.026 0.285

180

𝝈 (MPa) 180

180

240

240

180

180

180

180

t (ks)

0

36

36

72

72

93.8

169

204.9 1218.3

𝜀c (%)

0.062

0.077

0.134

0.167

0.117

0.13

0.176

0.183 0.213

𝜂

0.12

0.23

0.47

0.644

0.644

0.644 1.0

1.0

1.0

𝜔

0

0.47

0.37

0.547

0.547

0.712 0.826

1.0

1.0

q

2 × 10−6 1.5 × 10−3 8.8 × 10−3 0.0216 0.0216 0.023 0.0289 0.031 0.285

𝝈 (MPa)

180

180

240

240

180

180

180

180

180

t (ks)

0

36

36

90

90

112.4

169

204.9

1214.9

𝜀c (%)

0.062

0.077

0.134

0.183 0.129 0.142

𝜂

0.12

0.23

0.47

0.732 0.732 0.732

1.0

1.0

1.0

𝜔

0

0.47

0.37

0.602 0.602 0.751

0.826 1.0

1.0

q

2 × 10−6

1.5 × 10−3

8.8 × 10−3

0.029 0.029 0.031

0.035 0.038

0.285

𝝈 (MPa)

180

180

240

240

180

180

180

180

180

t (ks)

0

36

36

108

108

131

169

204.9

1210.3

𝜀c (%)

0.062

0.077

0.134

0.199 0.14

𝜂

0.12

0.23

0.47

0.821 0.821 0.821 1.0

𝜔

0

0.47

0.37

0.644 0.644 0.78

q

2 × 10−6

1.5 × 10−3

8.8 × 10−3

0.037 0.037 0.039 0.042 0.045

0.285

𝝈 (MPa)

180

180

240

240

180

180

180

180

180

t (ks)

0

36

36

126

126

149.8

169

204.9

1204.2

𝜀c (%)

0.062

0.077

0.134

0.216 0.151 0.165

0.176 0.183

0.21

𝜂

0.12

0.23

0.47

0.909 0.909 0.909

1.0

1.0

1.0

𝜔

0

0.47

0.37

0.678 0.678 0.805

0.826 1.0

1.0

q

2 × 10−6

1.5 × 10−3

8.8 × 10−3

0.046 0.046 0.048

0.049 0.053

0.285

0.176 0.183

0.153 0.176 0.183

180

0.212

0.211

1.0

1.0

0.826 1.0

1.0

6.3 Stress Rupture of Ceramic-Matrix Composites

When the stochastic loading time is Δt = 90 ks, the strain increases from 𝜀c = 0.062% to 0.077% under 𝜎 = 180 MPa, and at t = 36 ks increases to 𝜀c = 0.134% under 𝜎 s = 240 MPa, and increases to 𝜀c = 0.216% at t = 126 ks under 𝜎 s = 240 MPa, and decreases to 𝜀c = 0.151% at t = 126 ks under 𝜎 = 180 MPa, and increases to 𝜀c = 0.21% at t = 1204.2 ks. The interface debonding ratio increases from 𝜂 = 0.12 to 0.23 under 𝜎 = 180 MPa, and at t = 36 ks increases to 𝜂 = 0.47 under 𝜎 s = 240 MPa, and increases to 𝜂 = 0.909 at t = 126 ks under 𝜎 s = 240 MPa, and remains to 𝜂 = 0.557 till t = 149.8 ks under 𝜎 = 180 MPa, and increases to 𝜂 = 1.0 at t = 169 ks. The interface oxidation ratio increases from 𝜔 = 0 to 0.47 under 𝜎 = 180 MPa, and at t = 36 ks decreases to 𝜔 = 0.37 under 𝜎 s = 240 MPa, and increases to 𝜔 = 0.678 at t = 126 ks under 𝜎 s = 240 MPa, and increases to 𝜔 = 1.0 at t = 204.9 ks under 𝜎 = 180 MPa. The broken fiber fraction increases from q = 2 × 10−6 to 1.5 × 10−3 under 𝜎 = 180 MPa, and at t = 36 ks increases to q = 8.8 × 10−3 under 𝜎 s = 240 MPa, and increases to q = 0.046 at t = 126 ks under 𝜎 s = 240 MPa, and increases to q = 0.285 at t = 1204.2 ks under 𝜎 = 180 MPa. 6.3.1.3 Stress Rupture of SiC/SiC Composite Under Stochastic Loading for Different Fiber Volumes

Figure 6.13 and Table 6.11 show the effect of the fiber volume (i.e. V f = 20%, 25%, 30%, 35%, and 40%) on the time-dependent strain, interface debonding ratio, interface oxidation ratio, broken fiber fraction, and stress rupture lifetime curves of SiC/SiC composite under constant stress 𝜎 = 160 MPa, stochastic stress level 𝜎 s = 180 MPa, and time interval Δt = 36 ks at T = 850 ∘ C in air atmosphere. When the fiber volume increases, the stress rupture time increases; the time for the interface complete debonding increases; the time for the interface complete oxidation remains the same; and the broken fiber fraction at the stage of stochastic loading decreases. The stress rupture lifetime increases from t = 133.1 ks at V f = 20% to t = 4090.2 ks at V f = 40%. The time for the interface complete debonding decreases from t = 266.7 ks at V f = 25% to t = 311.3 ks at V f = 40%. When the fiber volume is V f = 20%, the strain increases from 𝜀c = 0.074% to 0.096% under 𝜎 = 160 MPa, and at t = 36 ks increases to 𝜀c = 0.135% under 𝜎 s = 180 MPa, and increases to 𝜀c = 0.179% at t = 72 ks under 𝜎 s = 180 MPa, and decreases to 𝜀c = 0.143% at t = 72 ks under 𝜎 = 160 MPa, and increases to 𝜀c = 0.205% at t = 133.1 ks. The interface debonding ratio increases from 𝜂 = 0.206 to 0.294 under 𝜎 = 160 MPa, and at t = 36 ks increases to 𝜂 = 0.431 under 𝜎 s = 180 MPa, and increases to 𝜂 = 0.584 at t = 72 ks under 𝜎 s = 180 MPa, and remains to 𝜂 = 0.584 till t = 97.4 ks under 𝜎 = 160 MPa, and increases to 𝜂 = 0.722 at t = 133.1 ks. The interface oxidation ratio increases from 𝜔 = 0 to 0.292 under 𝜎 = 160 MPa, and at t = 36 ks decreases to 𝜔 = 0.257 under 𝜎 s = 180 MPa, and increases to 𝜔 = 0.38 at t = 72 ks under 𝜎 s = 180 MPa, and increases to 𝜔 = 0.567 at t = 133.1 ks under 𝜎 = 160 MPa. The broken fiber fraction increases from q = 2.5 × 10−5 to 0.023 under 𝜎 = 160 MPa, and at t = 36 ks increases to q = 0.05 under 𝜎 s = 180 MPa, and increases to q = 0.162 at t = 72 ks under 𝜎 s = 180 MPa, and increases to q = 0.285 at t = 133.1 ks under 𝜎 = 160 MPa.

247

248

6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

(a)

(b)

Figure 6.13 Effect of fiber volume (i.e. V f = 20%, 25%, 30%, 35%, and 40%) on (a) the strain versus the time curves; (b) the interface debonding ratio versus the time curves; (c) the interface oxidation ratio versus the time curves; (d) the broken fiber fraction versus the time curves; and (e) the stress rupture lifetime versus the fiber volume curve of SiC/SiC composite under stress rupture constant stress of 𝜎 = 160 MPa and stochastic stress of 𝜎 s = 180 MPa and Δt = 36 ks at T = 850 ∘ C in air atmosphere.

When the fiber volume is V f = 40%, the strain increases from 𝜀c = 0.05% to 0.059% under 𝜎 = 160 MPa, and at t = 36 ks increases to 𝜀c = 0.071% under 𝜎 s = 180 MPa, and increases to 𝜀c = 0.083% at t = 72 ks under 𝜎 s = 180 MPa, and decreases to 𝜀c = 0.073% at t = 72 ks under 𝜎 = 160 MPa, and increases to 𝜀c = 0.161% at t = 4090.2 ks. The interface debonding ratio increases from 𝜂 = 0.06 to 0.144 under 𝜎 = 160 MPa, and at t = 36 ks increases to 𝜂 = 0.202 under 𝜎 s = 180 MPa, and increases to 𝜂 = 0.31 at t = 72 ks under 𝜎 s = 180 MPa, and remains to 𝜂 = 0.31 till t = 77.6 ks under 𝜎 = 160 MPa, and increases to 𝜂 = 1.0 at t = 311.3 ks. The interface oxidation ratio increases from 𝜔 = 0 to 0.597 under 𝜎 = 160 MPa, and at t = 36 ks

6.3 Stress Rupture of Ceramic-Matrix Composites

(c)

(d)

(e)

Figure 6.13

(Continued)

249

250

6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

Table 6.11 The strain, interface debonding and oxidation ratio, and broken fiber fraction for different fiber volumes. 𝝈 (MPa)

160

160

180

180

160

160

160

t (ks)

0

36

36

72

72

97.4

133.1

𝜀c (%)

0.0746

0.0967

0.135

0.179

0.143

0.166

0.205

𝜂

0.206

0.294

0.431

0.584

0.584

0.584

0.722

𝜔

0

0.292

0.257

0.38

0.38

0.513

0.567

0.023

0.05

0.162

0.162

0.199

0.285

V f = 20%

2.5 × 10

q

−5

𝝈 (MPa)

160

160

180

180

160

160

160

160

160

t (ks)

0

36

36

72

72

83.8

266.7

326.9

426.8

V f = 25% 𝜀c (%)

0.0618

0.0779

0.102

0.127

0.109

0.115

0.223

0.24

0.263

𝜂

0.147

0.232

0.332

0.445

0.445

0.445

1.0

1.0

1.0

𝜔

0

0.371

0.334

0.5

0.5

0.581

0.817

1.0

1.0

q

7 × 10−6

5.8 × 10−3

0.012

0.029

0.029

0.032

0.103

0.14

0.285

𝝈 (MPa) 160

160

180

180

160

160

160

t (ks)

36

36

72

72

80.7

288.7 326.9 1027.3

0.068

0.086

0.105

0.092

0.095

0.182

0

160

160

V f = 30% 𝜀c (%)

0.055

𝜂

0.108

0.192

0.274

0.382

0.382

0.382

1.0

1.0

1.0

𝜔

0

0.447

0.406

0.581

0.581

0.651

0.884

1.0

1.0

q

2 × 10−6 1.9 × 10−3 3.9 × 10−3 9.4 × 10−3 9.4 × 10−3 0.010

0.032

0.037

0.285

160

160

0.188

0.219

𝝈 (MPa) 160

160

180

180

160

160

160

t (ks)

36

36

72

72

78.9

301.9 326.9 2154.3

0

V f = 35% 𝜀c (%)

0.052

0.062

0.077

0.092

0.081

0.083

0.155 0.157 0.186

𝜂

0.081

0.164

0.233

0.34

0.34

0.34

1.0

1.0

1.0

𝜔

0

0.523

0.477

0.65

0.65

0.71

0.924 1.0

1.0

q

1 × 10−6 7.6 × 10−4 1.5 × 10−3 3.7 × 10−3 3.7 × 10−3 3.9 × 10−3 0.013 0.0142 0.285 (Continued)

6.3 Stress Rupture of Ceramic-Matrix Composites

Table 6.11

(Continued)

𝝈 (MPa)160

160

180

180

160

160

160

160

160

t (ks)

36

36

72

72

77.6

311.3

326.9

4090.2

𝜀c (%) 0.0506 0.059

0.071

0.083

0.073

0.075

0.134

0.135

0.161

𝜂

0.06

0.144

0.202

0.31

0.31

0.31

1.0

1.0

1.0

𝜔

0

0.597

0.548

0.716

0.716

0.772

0.953

1.0

1.0

q

0

3.4 × 10−4 6.9 × 10−4 1.6 × 10−3 1.6 × 10−3 1.7 × 10−3 5.9 × 10−3 6.2 × 10−3 0.285

0

V f = 40%

decreases to 𝜔 = 0.548 under 𝜎 s = 180 MPa, and increases to 𝜔 = 0.716 at t = 72 ks under 𝜎 s = 180 MPa, and increases to 𝜔 = 1.0 at t = 326.9 ks under 𝜎 = 160 MPa. The broken fiber fraction increases from q = 0 to 3.4 × 10−4 under 𝜎 = 160 MPa, and at t = 36 ks increases to q = 6.9 × 10−4 under 𝜎 s = 180 MPa, and increases to q = 1.6 × 10−3 at t = 72 ks under 𝜎 s = 180 MPa, and increases to q = 0.285 at t = 4090.2 ks under 𝜎 = 160 MPa. 6.3.1.4 Stress Rupture of SiC/SiC Composite Under Stochastic Loading for Different Matrix Crack Spacings

Figure 6.14 and Table 6.12 show the effect of the saturation matrix crack spacing (i.e. lsat = 100, 150, 200, 250, and 300 μm) on the strain, interface debonding ratio, interface oxidation ratio, and broken fiber fraction versus time curves of SiC/SiC composite under constant stress 𝜎 = 180 MPa, stochastic stress level 𝜎 s = 200 MPa, and stochastic time Δt = 36 ks at T = 850 ∘ C in air atmosphere. When the saturation matrix crack spacing increases, the stress rupture time remains the same; the time for the interface complete debonding and oxidation increases; and the broken fiber fraction remains the same. The time for the interface complete debonding increases from t = 60.7 ks at lsat = 100 μm to t = 288.4 ks at lsat = 300 μm. The time for the interface complete oxidation increases from t = 106.3 ks at lsat = 100 μm to t = 321.1 ks at lsat = 300 μm. When the saturation matrix crack spacing is lsat = 100 μm, the strain increases from 𝜀c = 0.08% to 0.118% under 𝜎 = 180 MPa, and at t = 36 ks increases to 𝜀c = 0.152% under 𝜎 s = 200 MPa, and increases to 𝜀c = 0.196% at t = 72 ks under 𝜎 s = 200 MPa, and decreases to 𝜀c = 0.174% at t = 72 ks under 𝜎 = 180 MPa, and increases to 𝜀c = 0.219% at t = 1223.6 ks. The interface debonding ratio increases from 𝜂 = 0.313 to 0.582 under 𝜎 = 180 MPa, and at t = 36 ks increases to 𝜂 = 0.774 under 𝜎 s = 200 MPa, and increases to 𝜂 = 1.0 at t = 60.7 ks under 𝜎 s = 200 MPa. The interface oxidation ratio increases from 𝜔 = 0 to 0.477 under 𝜎 = 180 MPa, and at t = 36 ks decreases to 𝜔 = 0.438 under 𝜎 s = 200 MPa, and increases to 𝜔 = 0.678 at t = 72 ks under 𝜎 s = 200 MPa, and increases to 𝜔 = 1.0 at t = 106.3 ks under 𝜎 = 180 MPa. The broken fiber fraction increases from q = 2 × 10−6 to 1.5 × 10−3 under 𝜎 = 180 MPa, and at t = 36 ks increases to q = 2.9 × 10−3 under 𝜎 s = 200 MPa,

251

252

6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

(a)

(b)

Figure 6.14 Effect of saturation matrix crack spacing (i.e. lsat = 100, 150, 200, 250, and 300 μm) on (a) the strain versus the time curves; (b) the interface debonding ratio versus the time curves; (c) the interface oxidation ratio versus the time curves; and (d) the broken fiber fraction versus the time curves of SiC/SiC composite under stress rupture constant stress 𝜎 = 180 MPa and stochastic stress 𝜎 s = 200 MPa and Δt = 36 ks at T = 850 ∘ C in air atmosphere.

and increases to q = 7 × 10−3 at t = 72 ks under 𝜎 s = 200 MPa, and increases to q = 0.285 at t = 1223.6 ks under 𝜎 = 180 MPa. When the saturation matrix crack spacing is lsat = 300 μm, the strain increases from 𝜀c = 0.061% to 0.073% under 𝜎 = 180 MPa, and at t = 36 ks increases to 𝜀c = 0.088% under 𝜎 s = 200 MPa, and increases to 𝜀c = 0.105% at t = 72 ks under 𝜎 s = 200 MPa, and decreases to 𝜀c = 0.093% at t = 72 ks under 𝜎 = 180 MPa, and increases to 𝜀c = 0.211% at t = 1223.6 ks. The interface debonded ratio increases from 𝜂 = 0.104 to 0.194 under 𝜎 = 180 MPa, and at t = 36 ks increases to 𝜂 = 0.258 under 𝜎 s = 200 MPa, and increases to 𝜂 = 0.368 at t = 72 ks under 𝜎 s = 200 MPa,

6.3 Stress Rupture of Ceramic-Matrix Composites

(c)

(d)

Figure 6.14

(Continued)

and remains to 𝜂 = 0.368 till t = 79 ks under 𝜎 = 180 MPa, and increases to 𝜂 = 1.0 at t = 288.4 ks. The interface oxidation ratio increases from 𝜔 = 0 to 0.477 under 𝜎 = 180 MPa, and at t = 36 ks decreases to 𝜔 = 0.438 under 𝜎 s = 200 MPa, and increases to 𝜔 = 0.614 at t = 72 ks under 𝜎 s = 200 MPa, and increases to 𝜔 = 1.0 at t = 321.1 ks under 𝜎 = 180 MPa. The broken fiber fraction increases from q = 2 × 10−6 to 1.5 × 10−3 under 𝜎 = 180 MPa, and at t = 36 ks increases to q = 2.9 × 10−3 under 𝜎 s = 200 MPa, and increases to q = 7 × 10−3 at t = 72 ks under 𝜎 s = 200 MPa, and increases to q = 0.285 at t = 1223.6 ks under 𝜎 = 180 MPa. 6.3.1.5 Stress Rupture of SiC/SiC Composite Under Stochastic Loading for Different Interface Shear Stress

Figure 6.15 and Table 6.13 show the effect of the interface shear stress in the slip region (i.e. 𝜏 i = 10, 20, 30, 40, and 50 MPa) on the strain, interface debonding ratio, interface oxidation ratio, broken fiber fraction, and stress rupture lifetime

253

254

6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

Table 6.12 The strain, interface debonding and oxidation ratio, and broken fiber fraction for different saturation matrix crack spacings. 𝝈 (MPa) 180

180

200

200

200

180

180

180

t (ks)

36

36

60.7

72

72

106.3

1223.6

0

lsat = 100 𝜇m 𝜀c (%)

0.08

0.118

0.152

0.185

0.196

0.174

0.186

0.219

𝜂

0.313

0.582

0.774

1.0

1.0

1.0

1.0

1.0

𝜔

0

0.477

0.438

0.572

0.678

0.678

1.0

q

2 × 10

−6

1.5 × 10

−3

2.9 × 10

−3

5.6 × 10

−3

7 × 10

−3

7 × 10

−3

1.0

9.3 × 10

−3

0.285

𝝈 (MPa) 180

180

200

200

180

180

180

t (ks)

36

36

72

72

79

122.6 159.7 1223.6

0

180

180

lsat = 150 𝜇m 𝜀c (%)

0.07

0.095

0.12

0.154

0.136

0.141

0.175 0.185 0.219

𝜂

0.208

0.388

0.516

0.736

0.736

0.736

1.0

1.0

1.0

𝜔

0

0.477

0.438

0.614

0.614

0.674

0.769 1.0

1.0

q

2 × 10−6 1.5 × 10−3 2.9 × 10−3 7 × 10−3 7 × 10−3 7.4 × 10−3 0.0105 0.0134 0.285

𝝈 (MPa) 180

180

200

200

180

180

180

t (ks)

36

36

72

72

79

177.7 213.3 1223.6

0.084

0.104

0.129

0.115

0.118

0.176 0.183 0.215

0

180

180

lsat = 200 𝜇m 𝜀c (%)

0.065

𝜂

0.156

0.291

0.387

0.552

0.552

0.552

1.0

1.0

1.0

𝜔

0

0.477

0.438

0.614

0.614

0.674

0.834 1.0

1.0

q

2 × 10−6 1.5 × 10−3 2.9 × 10−3 7 × 10−3 7 × 10−3 7.4 × 10−3 0.015 0.018 0.285

𝝈 (MPa) 180

180

200

200

180

180

180

t (ks)

36

36

72

72

79

232.9 267.1 1223.6

0

180

180

lsat = 250 𝜇m 𝜀c (%)

0.062

0.077

0.095

0.115

0.102

0.104

0.176 0.182 0.213

𝜂

0.125

0.23

0.31

0.44

0.44

0.44

1.0

1.0

1.0

𝜔

0

0.47

0.43

0.61

0.61

0.67

0.87

1.0

1.0

q

2 × 10−6 1.5 × 10−3 2.9 × 10−3 7 × 10−3 7 × 10−3 7.4 × 10−3 0.019 0.023 0.285 (Continued)

6.3 Stress Rupture of Ceramic-Matrix Composites

Table 6.12

(Continued)

𝝈 (MPa) 180

180

200

200

180

180

180

t (ks)

36

36

72

72

79

288.4 321.1 1223.6

0.073

0.088

0.105

0.093

0.095

0.176 0.18

0

180

180

lsat = 300 𝜇m 𝜀c (%)

0.061

𝜂

0.104

0.194

0.258

0.368

0.368

0.368

1.0

1.0

1.0

𝜔

0

0.477

0.438

0.614

0.614

0.674

0.899 1.0

1.0

q

2 × 10−6 1.5 × 10−3 2.9 × 10−3 7 × 10−3 7 × 10−3 7.4 × 10−3 0.025 0.028 0.285

0.211

curves of SiC/SiC composite under constant stress 𝜎 = 180 MPa, stochastic stress 𝜎 s = 200 MPa, and stochastic time interval of Δt = 36 ks at T = 850 ∘ C in air atmosphere. When the interface shear stress in the slip region increases, the stress rupture time increases; the time for the interface complete debonding increases; the time for the interface complete oxidation remains the same; and the broken fiber fraction at the stage of stochastic loading decreases. The stress rupture lifetime increases from t = 507.2 ks at 𝜏 i = 10 MPa to t = 1841.8 ks at 𝜏 i = 50 MPa. The time for the interface complete debonding decreases from t = 150.9 ks at 𝜏 i = 10 MPa to t = 247.9 ks at 𝜏 i = 50 MPa. When the interface shear stress in the slip region is 𝜏 i = 10 MPa, the strain increases from 𝜀c = 0.086% to 0.1% under 𝜎 = 180 MPa, and at t = 36 ks increases to 𝜀c = 0.13% under 𝜎 s = 200 MPa, and increases to 𝜀c = 0.15% at t = 72 ks under 𝜎 s = 200 MPa, and decreases to 𝜀c = 0.13% at t = 72 ks under 𝜎 = 180 MPa, and increases to 𝜀c = 0.212% at t = 507.2 ks. The interface debonding ratio increases from 𝜂 = 0.387 to 0.489 under 𝜎 = 180 MPa, and at t = 36 ks increases to 𝜂 = 0.674 under 𝜎 s = 200 MPa, and increases to 𝜂 = 0.805 at t = 72 ks under 𝜎 s = 200 MPa, and remains to 𝜂 = 0.805 till t = 95.4 ks under 𝜎 = 180 MPa, and increases to 𝜂 = 1.0 at t = 150.9 ks. The interface oxidation ratio increases from 𝜔 = 0 to 0.227 under 𝜎 = 180 MPa, and at t = 36 ks decreases to 𝜔 = 0.201 under 𝜎 s = 200 MPa, and increases to 𝜔 = 0.336 at t = 72 ks under 𝜎 s = 200 MPa, and increases to 𝜔 = 1.0 at t = 267.1 ks under 𝜎 = 180 MPa. The broken fiber fraction increases from q = 5 × 10−6 to 4.6 × 10−3 under 𝜎 = 180 MPa, and at t = 36 ks increases to q = 8.8 × 10−3 under 𝜎 s = 200 MPa, and increases to q = 0.022 at t = 72 ks under 𝜎 s = 200 MPa, and increases to q = 0.285 at t = 507.2 ks under 𝜎 = 180 MPa. When the interface shear stress in the slip region is 𝜏 i = 50 MPa, the strain increases from 𝜀c = 0.058% to 0.073% under 𝜎 = 180 MPa, and at t = 36 ks increases to 𝜀c = 0.088% under 𝜎 s = 200 MPa, and increases to 𝜀c = 0.108% at t = 72 ks under 𝜎 s = 200 MPa, and decreases to 𝜀c = 0.096% at t = 72 ks under 𝜎 = 180 MPa, and increases to 𝜀c = 0.213% at t = 1841.8 ks. The interface debonding ratio increases from 𝜂 = 0.072 to 0.181 under 𝜎 = 180 MPa, and at t = 36 ks increases to 𝜂 = 0.237 under 𝜎 s = 200 MPa, and increases to 𝜂 = 0.37 at t = 72 ks under 𝜎 s = 200 MPa,

255

256

6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

(a)

(b)

Figure 6.15 Effect of the interface shear stress in the slip region (i.e. 𝜏 i = 10, 20, 30, 40, and 50 MPa) on (a) the strain versus the time curves; (b) the interface debonding ratio versus the time curves; (c) the interface oxidation ratio versus the time curves; (d) the broken fiber fraction versus the time curves; and (e) the stress rupture lifetime versus the interface shear stress curves of SiC/SiC composite under stress rupture constant stress 𝜎 = 180 MPa and stochastic stress 𝜎 s = 200 MPa and Δt = 36 ks at T = 850 ∘ C in air atmosphere.

and remains to 𝜂 = 0.37 till t = 76.1 ks under 𝜎 = 180 MPa, and increases to 𝜂 = 1.0 at t = 247.9 ks. The interface oxidation ratio increases from 𝜔 = 0 to 0.611 under 𝜎 = 180 MPa, and at t = 36 ks decreases to 𝜔 = 0.573 under 𝜎 s = 200 MPa, and increases to 𝜔 = 0.733 at t = 72 ks under 𝜎 s = 200 MPa, and increases to 𝜔 = 1.0 at t = 267.1 ks under 𝜎 = 180 MPa. The broken fiber fraction increases from q = 1 × 10−6 to 9.2 × 10−4 under 𝜎 = 180 MPa, and at t = 36 ks increases to q = 1.7 × 10−3 under 𝜎 s = 200 MPa, and increases to q = 4.1 × 10−3 at t = 72 ks under 𝜎 s = 200 MPa, and increases to q = 0.285 at t = 1841.8 ks under 𝜎 = 180 MPa.

6.3 Stress Rupture of Ceramic-Matrix Composites

(c)

(d)

(e)

Figure 6.15

(Continued)

257

258

6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

Table 6.13 The strain, interface debonding and oxidation ratio, and broken fiber fraction for different interface shear stress in the slip region. 𝝈 (MPa)

180

180

200

200

180

180

180

180

180

t (ks)

0

36

36

72

72

95.4

150.9

267.1

507.2

0.1

0.13

0.15

0.13

0.139

0.165

0.186

0.212

𝜏 i = 10 MPa 𝜀c (%)

0.086

𝜂

0.387

0.489

0.674

0.805

0.805

0.805

1.0

1.0

1.0

𝜔

0

0.227

0.201

0.336

0.336

0.446

0.567

1.0

1.0

q

5 × 10−6

4.6 × 10−3

8.8 × 10−3

0.022

0.022

0.026

0.04

0.077

0.285

𝝈 (MPa) 180

180

200

200

180

180

180

180

180

t (ks)

36

36

72

72

82.8

213.5

267.1

884.3

0

𝜏 i = 20 MPa 𝜀c (%)

0.068

0.083

0.103

0.123 0.109 0.113

0.174

0.183

0.213

𝜂

0.19

0.297

0.374

0.531 0.531 0.531

1.0

1.0

1.0

𝜔

0

0.4

0.339

0.51

0.51

0.587

0.8

1.0

1.0

q

3 × 10−6 2.3 × 10−3 4.3 × 10−3 0.01

0.01

0.0116 0.0276 0.0355 0.285

𝝈 (MPa) 180

180

200

200

180

180

180

t (ks)

36

36

72

72

79

232.9 267.1 1223.6

0.104

0.176 0.182 0.213

0

180

180

𝜏 i = 30 MPa 𝜀c (%)

0.062

0.077

0.095

0.115

0.102

𝜂

0.125

0.23

0.31

0.44

0.44

0.44

1.0

1.0

1.0

𝜔

0

0.47

0.43

0.61

0.61

0.67

0.87

1.0

1.0

q

2 × 10−6 1.5 × 10−3 2.9 × 10−3 7 × 10−3 7 × 10−3 7.4 × 10−3 0.019 0.023 0.285

𝝈 (MPa) 180

180

200

200

180

180

180

t (ks)

36

36

72

72

77.1

242.3 267.1

0

180

180 1540.5

𝜏 i = 40 MPa 𝜀c (%)

0.0598 0.074

0.09

0.11

0.098

0.1

0.177 0.181 0.213

𝜂

0.092

0.201

0.264

0.396

0.396

0.396

1.0

1.0

1.0

𝜔

0

0.553

0.514

0.683

0.683

0.732

0.908 1.0

1.0

q

1 × 10−6 1.1 × 10−3 2 × 10−3 5.2 × 10−3 5.2 × 10−3 5.4 × 10−3 0.015 0.017 0.285 (Continued)

6.3 Stress Rupture of Ceramic-Matrix Composites

Table 6.13

(Continued)

𝝈 (MPa) 180

180

200

200

180

180

180

t (ks)

36

36

72

72

76.1

247.9 267.1 1841.8

0.073

0.088

0.108

0.096

0.098

0.178 0.181 0.213

0

180

180

𝜏 i = 50 MPa 𝜀c (%)

0.058

𝜂

0.072

0.181

0.237

0.37

0.37

0.37

1.0

1.0

1.0

𝜔

0

0.611

0.573

0.733

0.733

0.774

0.928 1.0

1.0

q

1 × 10−6 9.2 × 10−4 1.7 × 10−3 4.1 × 10−3 4.1 × 10−3 4.3 × 10−3 0.012 0.013 0.285

Figure 6.16 and Table 6.14 show the effect of the interface shear stress in the oxidation region (i.e. 𝜏 f = 1, 2, 3, 4, and 5 MPa) on the strain, interface debonding ratio, interface oxidation ratio, and broken fiber fraction versus time curves of SiC/SiC composite under constant stress 𝜎 = 180 MPa, stochastic stress 𝜎 s = 200 MPa, and stochastic time interval Δt = 36 ks at T = 850 ∘ C in air atmosphere. When the interface shear stress in the oxidation region increases, the stress rupture time remains the same; the time for the interface complete debonding increases; the time for the interface complete oxidation decreases; and the broken fiber fraction remains the same. The time for the interface complete debonding increases from t = 232.9 ks at 𝜏 f = 1 MPa to t = 267.2 ks at 𝜏 f = 5 MPa. The time for the interface complete oxidation decreases from t = 267.1 ks at 𝜏 f = 1 MPa to t = 250.5 ks at 𝜏 f = 5 MPa. When the interface shear stress in the oxidation region is 𝜏 f = 1 MPa, the strain increases from 𝜀c = 0.062% to 0.077% under 𝜎 = 180 MPa, and at t = 36 ks increases to 𝜀c = 0.095% under 𝜎 s = 200 MPa, and increases to 𝜀c = 0.115% at t = 72 ks under 𝜎 s = 200 MPa, and decreases to 𝜀c = 0.102% at t = 72 ks under 𝜎 = 180 MPa, and increases to 𝜀c = 0.213% at t = 1223.6 ks. The interface debonding ratio increases from 𝜂 = 0.125 to 0.23 under 𝜎 = 180 MPa, and at t = 36 ks increases to 𝜂 = 0.31 under 𝜎 s = 200 MPa, and increases to 𝜂 = 0.44 at t = 72 ks under 𝜎 s = 200 MPa, and remains to 𝜂 = 0.44 till t = 79 ks under 𝜎 = 180 MPa, and increases to 𝜂 = 1.0 at t = 232.9 ks. The interface oxidation ratio increases from 𝜔 = 0 to 0.47 under 𝜎 = 180 MPa, and at t = 36 ks decreases to 𝜔 = 0.43 under 𝜎 s = 200 MPa, and increases to 𝜔 = 0.61 at t = 72 ks under 𝜎 s = 200 MPa, and increases to 𝜔 = 1.0 at t = 267.1 ks under 𝜎 = 180 MPa. The broken fiber fraction increases from q = 2 × 10−6 to 1.5 × 10−3 under 𝜎 = 180 MPa, and at t = 36 ks increases to q = 2.9 × 10−3 under 𝜎 s = 200 MPa, and increases to q = 7 × 10−3 at t = 72 ks under 𝜎 s = 200 MPa, and increases to q = 0.285 at t = 1223.6 ks under 𝜎 = 180 MPa. When the interface shear stress in the oxidation region is 𝜏 f = 5 MPa, the strain increases from 𝜀c = 0.062% to 0.075% under 𝜎 = 180 MPa, and at t = 36 ks increases to 𝜀c = 0.091% under 𝜎 s = 200 MPa, and increases to 𝜀c = 0.107% at t = 72 ks under 𝜎 s = 200 MPa, and decreases to 𝜀c = 0.094% at t = 72 ks under 𝜎 = 180 MPa, and increases to 𝜀c = 0.171% at t = 1223.6 ks. The interface debonding ratio increases from 𝜂 = 0.125 to 0.218 under 𝜎 = 180 MPa, and at t = 36 ks increases to 𝜂 = 0.291

259

260

6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

(a)

(b)

Figure 6.16 Effect of interface shear stress in the oxidation region (i.e. 𝜏 f = 1, 2, 3, 4, and 5 MPa) on (a) the strain versus the time curves; (b) the interface debonding ratio versus the time curves; (c) the interface oxidation ratio versus the time curves; and (d) the broken fiber fraction versus the time curves of SiC/SiC composite under stress rupture loading of 𝜎 = 180 MPa and stochastic loading stress of 𝜎 s = 200 MPa and Δt = 36 ks at T = 850 ∘ C in air atmosphere.

under 𝜎 s = 200 MPa, and increases to 𝜂 = 0.405 at t = 72 ks under 𝜎 s = 200 MPa, and remains to 𝜂 = 0.405 till t = 80.1 ks under 𝜎 = 180 MPa, and increases to 𝜂 = 1.0 at t = 267.1 ks. The interface oxidation ratio increases from 𝜔 = 0 to 0.509 under 𝜎 = 180 MPa, and at t = 36 ks decreases to 𝜔 = 0.466 under 𝜎 s = 200 MPa, and increases to 𝜔 = 0.669 at t = 72 ks under 𝜎 s = 200 MPa, and increases to 𝜔 = 1.0 at t = 250.5 ks under 𝜎 = 180 MPa. The broken fiber fraction increases from q = 2 × 10−6 to 1.5 × 10−3 under 𝜎 = 180 MPa, and at t = 36 ks increases to q = 2.9 × 10−3 under 𝜎 s = 200 MPa, and increases to q = 7 × 10−3 at t = 72 ks under 𝜎 s = 200 MPa, and increases to q = 0.285 at t = 1223.6 ks under 𝜎 = 180 MPa.

6.3 Stress Rupture of Ceramic-Matrix Composites

(c)

(d)

Figure 6.16

(Continued)

6.3.1.6 Stress Rupture of SiC/SiC Composite Under Stochastic Loading for Different Environmental Temperatures

Figure 6.17 and Table 6.15 show the effect of the environmental temperature (i.e. T = 750, 800, 850, 900, and 950 ∘ C) on the strain, interface debonding ratio, interface oxidation ratio, and broken fiber fraction versus time curves of SiC/SiC composite under constant stress of 𝜎 = 180 MPa, stochastic stress of 𝜎 s = 200 MPa, and stochastic time interval of Δt = 36 ks in air atmosphere. When the environmental temperature increases, the stress rupture time decreases; the time for the interface complete debonding and oxidation decreases; and the broken fiber fraction at the stage of stochastic loading increases. The stress rupture lifetime decreases from t = 2614 ks at T = 750 ∘ C to t = 648 ks at T = 950 ∘ C. The time for the interface complete debonding decreases from t = 501.5 ks at T = 750 ∘ C to t = 123.1 ks at T = 950 ∘ C. The time for the interface complete oxidation decreases from t = 574.8 ks at T = 750 ∘ C to t = 141.3 ks at T = 950 ∘ C.

261

262

6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

Table 6.14 The strain, interface debonding and oxidation ratio, and broken fiber fraction for different interface shear stress in the oxidation region. 𝝈 (MPa)

180

180

200

200

180

180

180

t (ks)

0

36

36

72

72

79

232.9 267.1 1223.6

0.104

0.176 0.182 0.213

180

180

𝜏 f = 1 MPa 𝜀c (%)

0.062

0.077

0.095

0.115

0.102

𝜂

0.125

0.23

0.31

0.44

0.44

0.44

1.0

1.0

1.0

𝜔

0

0.47

0.43

0.61

0.61

0.67

0.87

1.0

1.0

q

2 × 10−6 1.5 × 10−3 2.9 × 10−3 7 × 10−3 7 × 10−3 7.4 × 10−3 0.019 0.023 0.285

𝝈 (MPa)

180

180

200

200

180

180

180

t (ks)

0

36

36

72

72

79.2

241.3 267.1 1223.6

180

180

𝜏 f = 2 MPa 𝜀c (%)

0.062

0.077

0.094

0.113

0.1

0.102

0.168 0.171 0.202

𝜂

0.125

0.229

0.305

0.432

0.432

0.432

1.0

1.0

1.0

𝜔

0

0.485

0.445

0.627

0.627

0.69

0.904 1.0

1.0

q

2 × 10−6 1.5 × 10−3 2.9 × 10−3 7 × 10−3 7 × 10−3 7.47 × 10−3 0.02

0.023 0.285

𝝈 (MPa)

180

180

200

200

180

180

180

180

t (ks)

0

36

36

72

72

79.5

250.3 267.1 1223.6

0.076

0.093

0.11

0.098

0.1

0.159 0.161 0.192

180

𝜏 f = 3 MPa 𝜀c (%)

0.062

𝜂

0.125

0.225

0.3

0.423

0.423

0.423

1.0

1.0

1.0

𝜔

0

0.493

0.45

0.64

0.64

0.707

0.937 1.0

1.0

q

2 × 10−6 1.5 × 10−3 2.9 × 10−3 7 × 10−3 7 × 10−3 7.49 × 10−3 0.021 0.023 0.285

𝝈 (MPa)

180

180

200

200

180

180

180

t (ks)

0

36

36

72

72

79.8

259.9 267.1 1223.6

180

180

𝜏 f = 4 MPa 𝜀c (%)

0.062

0.075

0.092

0.108

0.096

0.098

0.15

0.151 0.182

𝜂

0.125

0.221

0.296

0.414

0.414

0.414

1.0

1.0

1.0

𝜔

0

0.501

0.458

0.654

0.654

0.725

0.973 1.0

1.0

q

2 × 10−6 1.5 × 10−3 2.9 × 10−3 7 × 10−3 7 × 10−3 7.51 × 10−3 0.022 0.023 0.285 (Continued)

6.3 Stress Rupture of Ceramic-Matrix Composites

Table 6.14

(Continued)

𝝈 (MPa)

180

180

200

200

180

180

180

t (ks)

0

36

36

72

72

80.1

250.5 267.1 1223.6

0.075

0.091

0.107

0.094

0.096

0.138 0.14 0.171

180

180

𝜏 f = 5 MPa 𝜀c (%)

0.062

𝜂

0.125

0.218

0.291

0.405

0.405

0.405

0.938 1.0

1.0

𝜔

0

0.509

0.466

0.669

0.669

0.744

1.0

1.0

q

2 × 10

−6

1.5 × 10

−3

2.9 × 10

−3

7 × 10

−3

7 × 10

−3

7.53 × 10

−3

1.0

0.0214 0.023 0.285

When the environmental temperature is T = 750 ∘ C, the strain increases from 𝜀c = 0.062% to 0.069% under 𝜎 = 180 MPa, and at t = 36 ks increases to 𝜀c = 0.084% under 𝜎 s = 200 MPa, and increases to 𝜀c = 0.093% at t = 72 ks under 𝜎 s = 200 MPa, and decreases to 𝜀c = 0.083% at t = 72 ks under 𝜎 = 180 MPa, and increases to 𝜀c = 0.213% at t = 2614 ks. The interface debonding ratio increases from 𝜂 = 0.125 to 0.175 under 𝜎 = 180 MPa, and at t = 36 ks increases to 𝜂 = 0.238 under 𝜎 s = 200 MPa, and increases to 𝜂 = 0.299 at t = 72 ks under 𝜎 s = 200 MPa, and remains to 𝜂 = 0.299 till t = 86.8 ks under 𝜎 = 180 MPa, and increases to 𝜂 = 1.0 at t = 501.5 ks. The interface oxidation ratio increases from 𝜔 = 0 to 0.294 under 𝜎 = 180 MPa, and at t = 36 ks decreases to 𝜔 = 0.263 under 𝜎 s = 200 MPa, and increases to 𝜔 = 0.419 at t = 72 ks under 𝜎 s = 200 MPa, and increases to 𝜔 = 1.0 at t = 574.8 ks under 𝜎 = 180 MPa. The broken fiber fraction increases from q = 2 × 10−6 to 5.9 × 10−4 under 𝜎 = 180 MPa, and at t = 36 ks increases to q = 1.1 × 10−3 under 𝜎 s = 200 MPa, and increases to q = 2.6 × 10−3 at t = 72 ks under 𝜎 s = 200 MPa, and increases to q = 0.285 at t = 2614 ks under 𝜎 = 180 MPa. When the environmental temperature is T = 950 ∘ C, the strain increases from 𝜀c = 0.062% to 0.09% under 𝜎 = 180 MPa, and at t = 36 ks increases to 𝜀c = 0.113% under 𝜎 s = 200 MPa, and increases to 𝜀c = 0.15% at t = 72 ks under 𝜎 s = 200 MPa, and decreases to 𝜀c = 0.133% at t = 72 ks under 𝜎 = 180 MPa, and increases to 𝜀c = 0.212% at t = 648 ks. The interface debonding ratio increases from 𝜂 = 0.125 to 0.33 under 𝜎 = 180 MPa, and at t = 36 ks increases to 𝜂 = 0.428 under 𝜎 s = 200 MPa, and increases to 𝜂 = 0.678 at t = 72 ks under 𝜎 s = 200 MPa, and remains to 𝜂 = 0.678 till t = 75.8 ks under 𝜎 = 180 MPa, and increases to 𝜂 = 1.0 at t = 123.1 ks. The interface oxidation ratio increases from 𝜔 = 0 to 0.64 under 𝜎 = 180 MPa, and at t = 36 ks decreases to 𝜔 = 0.602 under 𝜎 s = 200 MPa, and increases to 𝜔 = 0.758 at t = 72 ks under 𝜎 s = 200 MPa, and increases to 𝜔 = 1.0 at t = 141.3 ks under 𝜎 = 180 MPa. The broken fiber fraction increases from q = 2 × 10−6 to 3.4 × 10−3 under 𝜎 = 180 MPa, and at t = 36 ks increases to q = 6.5 × 10−3 under 𝜎 s = 200 MPa, and increases to q = 0.016 at t = 72 ks under 𝜎 s = 200 MPa, and increases to q = 0.285 at t = 648 ks under 𝜎 = 180 MPa.

263

264

6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

(a)

(b)

Figure 6.17 Effect of environment temperature (i.e. T = 750, 800, 850, 900, and 950 ∘ C) on (a) the strain versus the time curves; (b) the interface debonding ratio versus the time curves; (c) the interface oxidation ratio versus the time curves; (d) the broken fiber fraction versus the time curves; and (e) the stress rupture lifetime versus the temperature curves of SiC/SiC composite under stress rupture constant stress of 𝜎 = 180 MPa and stochastic stress of 𝜎 s = 200 MPa and Δt = 36 ks in air atmosphere.

6.3.2

Experimental Comparisons

Lara-Curzio [11] investigated the stress rupture behavior of SiC/SiC composite at T = 950 ∘ C in air atmosphere. Experimental and predicted stress rupture curves, interface debonding and interface oxidation, broken fiber fraction versus time curves, and the stress versus lifetime curves of SiC/SiC composite under constant and stochastic stress with different time intervals at T = 950 ∘ C in air atmosphere are shown in Figures 6.18–6.20 and Tables 6.16–6.18.

6.3 Stress Rupture of Ceramic-Matrix Composites

(c)

(d)

(e)

Figure 6.17

(Continued)

265

266

6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

Table 6.15 The strain, interface debonding and oxidation ratio, and broken fiber fraction for different environment temperatures. 𝝈 (MPa)

180

180

200

200

180

180

180

t (ks)

0

36

36

72

72

86.8

501.5 574.8 2614

0.069

0.084

0.093

0.083

0.085

0.176 0.182 0.213

180

180

T = 750 ∘ C 𝜀c (%)

0.062

𝜂

0.125

0.175

0.238

0.299

0.299

0.299

1.0

1.0

1.0

𝜔

0

0.294

0.263

0.419

0.419

0.505

0.873 1.0

1.0

q

2 × 10−6 5.9 × 10−4 1.1 × 10−3 2.6 × 10−3 2.6 × 10−3 3 × 10−3 0.018 0.021 0.285

𝝈 (MPa)

180

180

200

200

180

180

180

t (ks)

0

36

36

72

72

81.9

335.6 384.7 1757.2

180

180

T = 800 ∘ C 𝜀c (%)

0.062

0.073

0.088

0.102

0.091

0.094

0.176 0.182 0.213

𝜂

0.125

0.199

0.269

0.36

0.36

0.36

1.0

1.0

1.0

𝜔

0

0.385

0.349

0.521

0.521

0.593

0.873 1.0

1.0

q

2 × 10−6 9.8 × 10−4 1.8 × 10−3 4.4 × 10−3 4.4 × 10−3 4.8 × 10−3 0.018 0.022 0.285

𝝈 (MPa)

180

180

200

200

180

180

180

t (ks)

0

36

36

72

72

79

232.9 267.1 1223.6

0.104

0.176 0.182 0.213

180

180

T = 850 ∘ C 𝜀c (%)

0.062

0.077

0.095

0.115

0.102

𝜂

0.125

0.23

0.31

0.44

0.44

0.44

1.0

1.0

1.0

𝜔

0

0.47

0.43

0.61

0.61

0.67

0.87

1.0

1.0

q

2 × 10−6 1.5 × 10−3 2.9 × 10−3 7 × 10−3 7 × 10−3 7.4 × 10−3 0.019 0.023 0.285

𝝈 (MPa)

180

180

200

200

180

180

180

180

180

t (ks)

0

36

36

72

72

77.1

166.9

191.5

878.6

0.116 0.118 0.176

0.182

0.213

1.0

1.0

T = 900 ∘ C 𝜀c (%)

0.062

0.083

0.103

0.13

𝜂

0.125

0.276

0.362

0.546 0.546 0.546 1.0

𝜔

0

0.563

0.524

0.693 0.693 0.742 0.873

1.0

1.0

q

2 × 10−6

2.3 × 10−3

4.4 × 10−3

0.01

0.024

0.285

0.01

0.011 0.021

(Continued)

6.3 Stress Rupture of Ceramic-Matrix Composites

Table 6.15

(Continued)

𝝈 (MPa)

180

180

200

200

180

180

180

t (ks)

0

36

36

72

72

75.8

123.1 141.3 648

𝜀c (%)

0.062

0.09

0.113

0.15

0.133 0.136

𝜂

0.125

0.33

0.428

𝜔

0

0.64

0.602

q

2 × 10−6 3.4 × 10−3 6.5 × 10−3 0.016 0.016 0.0163 0.024

180

180

T = 950 ∘ C

6.3.2.1

0.176

0.182

0.212

0.678 0.678 0.678

1.0

1.0

1.0

0.758 0.758 0.797

0.873

1.0

1.0

0.027

0.285

𝝈 = 80 MPa and 𝝈 s = 90 MPa with 𝚫t = 7.2, 10.8, and 14.4 ks

Experimental and predicted strain, interface debonding ratio, interface oxidation ratio, and broken fiber fraction versus time curves, and the stress versus lifetime curve of SiC/SiC composite under constant stress 𝜎 = 80 MPa, stochastic stress 𝜎 s = 90 MPa and Δt = 7.2, 10.8, and 14.4 ks at T = 950 ∘ C in air atmosphere are shown in Figure 6.18 and Table 6.16. Under constant stress 𝜎 = 80 MPa, the stress rupture lifetime is t = 83.6 ks with partial interface debonding 𝜂 = 0.557, partial interface oxidation 𝜔 = 0.846, and broken fiber fraction q = 0.291. Under stochastic stress 𝜎 s = 90 MPa occurs at t = 14.4 ks, the stress rupture lifetime decreases to t = 83.1 ks with partial interface debonding 𝜂 = 0.711, partial interface oxidation 𝜔 = 0.852, and the broken fiber fraction q = 0.283. Under 𝜎 s = 90 MPa and Δt = 7.2 ks, the stress rupture lifetime decreases to t = 82.6 ks with partial interface debonding 𝜂 = 0.705, partial interface oxidation 𝜔 = 0.854, and broken fiber fraction q = 0.284. Under 𝜎 s = 90 MPa and Δt = 10.8 ks, the stress rupture lifetime decreases to t = 82.2 ks with partial interface debonding 𝜂 = 0.7, partial interface oxidation of 𝜔 = 0.856, and broken fiber fraction q = 0.284. Under 𝜎 s = 90 MPa and Δt = 14.4 ks, the stress rupture lifetime decreases to t = 81.6 ks with partial interface debonding 𝜂 = 0.695, partial interface oxidation 𝜔 = 0.857, and broken fiber fraction q = 0.285. Compared with constant stress 𝜎 = 80 MPa, the stress rupture lifetime decreases with stochastic stress 𝜎 s = 90 MPa and the time interval. When the constant stress is 𝜎 = 80 MPa, the stress rupture lifetime is t = 83.6 ks; however, under constant stress 𝜎 = 80 MPa with the stochastic stress 𝜎 s = 90 MPa and stochastic time interval Δt = 14.4 ks, the stress rupture time decreases to t = 81.6 ks. 6.3.2.2

𝝈 = 100 MPa and 𝝈 s = 110 MPa with 𝚫t = 7.2 ks

Experimental and predicted strain, interface debonding ratio, interface oxidation ratio, and broken fiber fraction versus time curves, and the stress versus lifetime curve of SiC/SiC composite under constant stress 𝜎 = 100 MPa, stochastic stress 𝜎 s = 110 MPa, and Δt = 7.2 ks at T = 950 ∘ C in air atmosphere are shown in Figure 6.19 and Table 6.17.

267

268

6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

(a)

(b)

Figure 6.18 (a) The experimental and predicted strain versus time curves; (b) the interface debonding ratio versus time curves; (c) the interface oxidation ratio versus time curves; (d) the broken fiber fraction versus time curves; and (e) the stress versus lifetime curves of SiC/SiC composite under constant stress 𝜎 = 80 MPa, stochastic stress 𝜎 s = 90 MPa and Δt = 7.2, 10.8, and 14.4 ks at T = 950 ∘ C in air atmosphere.

Under 𝜎 = 100 MPa, the stress rupture lifetime is t = 33.7 ks with partial interface debonding 𝜂 = 0.279, partial interface oxidation 𝜔 = 0.617, and broken fiber fraction q = 0.228. Under 𝜎 s = 110 MPa occurs at t = 10.8 ks, the stress rupture lifetime decreases to t = 32.8 ks with partial interface debonding 𝜂 = 0.312, partial interface oxidation 𝜔 = 0.627, and broken fiber fraction q = 0.285. Compared with constant stress 𝜎 = 100 MPa, the stress rupture lifetime decreases with stochastic stress 𝜎 s = 110 MPa and the time interval. When the constant stress is 𝜎 = 100 MPa, the stress rupture lifetime is t = 33.7 ks; however, under constant stress 𝜎 = 100 MPa with the stochastic stress 𝜎 s = 110 MPa and stochastic time interval Δt = 7.2 ks, the stress rupture time decreases to t = 32.8 ks.

6.3 Stress Rupture of Ceramic-Matrix Composites

(c)

(d)

(e)

Figure 6.18

(Continued)

269

270

6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

Table 6.16 Experimental and theoretical strain, interface debonding and oxidation ratio, and broken fiber fraction of SiC/SiC composite under constant stress 𝜎 = 80 MPa, stochastic stress 𝜎 s = 90 MPa and Δt = 7.2, 10.8, and 14.4 ks at T = 950 ∘ C in air atmosphere.

𝜎 = 80 MPa (experimental data)

𝜎 = 80 MPa (theoretical values)

𝝈 (MPa)

80

80

80

80

80

t (ks)

8.5

21.6

42.9

55.5

78.4

𝜀c (%)

0.106

0.13

0.167

0.191

0.233

𝝈 (MPa)

80

80

80

80

80

80

t (ks)

8.5

21.6

42.9

55.5

78.4

83.6

𝜀c (%)

0.102

0.127

0.169

0.196

0.252

0.277

𝜂

0.112

0.186

0.307

0.379

0.516

0.557

𝜔

0.432

0.659

0.792

0.828

0.858

0.846

q

0.007

0.024

0.064

0.097

0.203

0.291

𝜎 = 80 MPa 𝜎 s = 90 MPa t = 14.4 ks (theoretical values)

𝜎 = 80 MPa 𝜎 s = 90 MPa t = 14.4 ks Δt = 7.2 ks (theoretical values)

𝜎 = 80 MPa 𝜎 s = 90 MPa t = 14.4 ks Δt = 10.8 ks (theoretical values)

𝝈 (MPa)

80

80

90

80

80

t (ks)

0

14.4

14.4

14.4

83.1

𝜀c (%)

0.085

0.113

0.145

0.126

0.331

𝜂

0.064

0.145

0.204

0.204

0.711

𝜔

0

0.563

0.517

0.517

0.852

q

0

0.014

0.03

0.03

0.283

𝝈 (MPa)

80

80

90

90

80

80

t (ks)

0

14.4

14.4

21.6

21.6

82.6

𝜀c (%)

0.085

0.113

0.145

0.167

0.144

0.327

𝜂

0.064

0.145

0.204

0.259

0.259

0.705

𝜔

0

0.563

0.517

0.612

0.612

0.854

q

0

0.014

0.03

0.054

0.054

0.284

𝝈 (MPa)

80

80

90

90

80

80

t (ks)

0

14.4

14.4

25.2

25.2

82.2

𝜀c (%)

0.085

0.113

0.145

0.178

0.154

0.324

𝜂

0.064

0.145

0.204

0.286

0.286

0.7

𝜔

0

0.563

0.517

0.645

0.645

0.856

q

0

0.014

0.03

0.068

0.068

0.284

(Continued)

6.3 Stress Rupture of Ceramic-Matrix Composites

Table 6.16

(Continued)

𝜎 = 80 MPa 𝜎 s = 90 MPa t = 14.4 ks Δt = 14.4 ks (theoretical values)

6.3.2.3

𝝈 (MPa)

80

80

90

90

80

80

t (ks)

0

14.4

14.4

28.8

28.8

81.6

𝜀c (%)

0.085

0.113

0.145

0.189

0.163

0.321

𝜂

0.064

0.145

0.204

0.314

0.314

0.695

𝜔

0

0.563

0.517

0.672

0.672

0.857

q

0

0.014

0.03

0.084

0.084

0.285

𝝈 = 120 MPa and 𝝈 s = 130 and 140 MPa with 𝚫t = 7.2 ks

Experimental and predicted strain, interface debonding ratio, interface oxidation ratio, broken fiber fraction versus time curves, and the stress versus lifetime curve of SiC/SiC composite under constant stress 𝜎 = 120 MPa, stochastic stress 𝜎 s = 130, 140 MPa, and Δt = 7.2 ks at T = 950 ∘ C in air atmosphere are shown in Figure 6.20 and Table 6.18. Under 𝜎 = 120 MPa, the stress rupture lifetime is t = 10.5 ks with partial interface debonding 𝜂 = 0.42, partial interface oxidation of 𝜔 = 0.406, and broken fiber fraction q = 0.17. Under 𝜎 s = 130 MPa occurs at t = 3.6 ks, the stress rupture lifetime decreases to t = 6.8 ks with partial interface debonding 𝜂 = 0.415, partial interface oxidation 𝜔 = 0.287, and broken fiber fraction q = 0.151. Under 𝜎 s = 140 MPa occurs at t = 3.6 ks, the stress rupture lifetime decreases to t = 4.4 ks with partial interface debonding 𝜂 = 0.407, partial interface oxidation 𝜔 = 0.199, and broken fiber fraction q = 0.128. Compared with constant stress 𝜎 = 120 MPa, the stress rupture lifetime decreases with stochastic stress 𝜎 s = 130 and 140 MPa and the time interval. When the constant stress is 𝜎 = 120 MPa, the stress rupture lifetime is t = 10.5 ks; however, under constant stress of 𝜎 = 120 MPa with the stochastic stress 𝜎 s = 140 MPa and stochastic time interval Δt = 7.2 ks, the stress rupture time decreases to t = 4.4 ks.

6.3.2.4 Discussion

Under stress rupture loading, damage evolution and lifetime are affected by stochastic stress level and stochastic time interval, and also the constituent properties of fiber volume and interface properties, damage state of matrix crack spacing, and environmental temperature. Under constant stress with stochastic loading, the stress rupture lifetime decreases with increase of stochastic stress level, as shown in Figures 6.11e and 6.20e. When the stochastic stress level increases, the broken fiber fraction increases at the high stochastic stress, leading to the decrease of the lifetime under stress rupture loading. The stress rupture lifetime decreases with increase of stochastic time interval, as shown in Figures 6.12e and 6.18e. When the stochastic time interval increases, the accumulation fiber broken fiber fraction at the high stochastic stress increases, leading to the decrease of the lifetime under stress rupture loading. The stress

271

272

6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

(a)

(b)

Figure 6.19 (a) Experimental and predicted strain versus time curves; (b) the interface debonding ratio versus time curves; (c) the interface oxidation ratio versus time curves; (d) the broken fiber fraction versus time curves; and (e) the stress versus lifetime curve of SiC/SiC composite under constant stress 𝜎 = 100 MPa, stochastic stress 𝜎 s = 110 MPa and Δt = 7.2 ks at T = 950 ∘ C in air atmosphere.

rupture lifetime increases with the fiber volume, as shown in Figure 6.13e, because of the decrease of the intact fiber stress and broken fiber fraction under stress rupture loading. The stress rupture lifetime increases with the interfacial shear stress at the slip region, as shown in Figure 6.15e, because of the higher stress transfer between the fiber and the matrix and between the intact and broken fiber. The stress rupture lifetime decreases with the environmental temperature, as shown in Figure 6.17e. When the temperature increases, the rate of the interface oxidation and fiber strength degradation increases, leading to the higher fiber broken under stress rupture loading.

6.3 Stress Rupture of Ceramic-Matrix Composites

(c)

(d)

(e)

Figure 6.19

(Continued)

273

274

6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

Table 6.17 Experimental and theoretical strain, interface debonding and oxidation ratio, and broken fiber fraction of SiC/SiC composite under constant stress 𝜎 = 100 MPa, stochastic stress 𝜎 s = 110 MPa and Δt = 7.2 ks at T = 950 ∘ C in air atmosphere.

𝜎 = 100 MPa (experimental data)

𝜎 = 100 MPa (theoretical values)

𝜎 = 100 MPa 𝜎 s = 110 MPa t = 10.8 ks Δt = 7.2 ks (theoretical values)

𝝈 (MPa)

100

100

100

100

100

t (ks)

8.3

13.8

21

25.3

33.3

𝜀c (%)

0.125

0.135

0.15

0.161

0.189

𝝈 (MPa)

100

100

100

100

100

100

t (ks)

8.3

13.8

21

25.3

33.3

33.7

𝜀c (%)

0.127

0.14

0.158

0.169

0.196

0.198

𝜂

0.122

0.152

0.192

0.216

0.273

0.279

𝜔

0.348

0.466

0.561

0.598

0.624

0.617

q

0.022

0.044

0.083

0.114

0.218

0.228

𝝈 (MPa)

100

100

110

110

100

100

t (ks)

0

10.8

10.8

18

18

32.8

𝜀c (%)

0.109

0.133

0.159

0.186

0.161

0.212

𝜂

0.078

0.135

0.174

0.229

0.229

0.312

𝜔

0

0.408

0.370

0.469

0.469

0.627

q

0

0.03

0.06

0.143

0.143

0.285

6.4 Stress Rupture of Ceramic-Matrix Composites Under Multiple Load Sequence at Intermediate Temperature In this section, effect of multiple loading sequence on time-dependent stress rupture of fiber-reinforced CMCs at intermediate temperatures in oxidative environment is investigated. Relationships between multiple loading sequence, composite strain evolution, time, matrix cracking, interface debonding and oxidation, and fibers fracture are established. Effects of fiber volume, matrix crack spacing, interface shear stress in the slip and oxidation region, and environment temperature on the stress/time-dependent strain, interface debonding and oxidation fraction, and fibers broken fraction of SiC/SiC composite are analyzed. Experimental stress rupture of SiC/SiC composite under single and multiple loading sequence at T = 950 ∘ C in air atmosphere is predicted.

6.4.1

Results and Discussion

Figure 6.21 shows multiple loading sequence for stress rupture of fiber-reinforced CMCs at elevated temperature. Effects of fiber volume, matrix crack spacing,

6.4 Stress Rupture of Ceramic-Matrix Composites Under Multiple Load Sequence

(a)

(b)

Figure 6.20 (a) Experimental and predicted strain versus time curves; (b) the interface debonding ratio versus time curves; (c) the interface oxidation ratio versus time curves; (d) the broken fiber fraction versus time curves; and (e) the stress and lifetime curve of SiC/SiC composite under constant stress 𝜎 = 120 MPa, stochastic stress 𝜎 s = 130 and 140 MPa and Δt = 7.2 ks at T = 950 ∘ C in air atmosphere.

interface shear stress, and environmental temperature on stress/time-dependent stress rupture of SiC/SiC composite are analyzed. 6.4.1.1 Stress Rupture of SiC/SiC Composite Under Multiple Loading Sequence for Different Fiber Volumes

Figure 6.22 and Table 6.19 show the effect of fiber volume (i.e. V f = 25%, 30%, and 35%) on stress rupture, fraction of interface debonding and oxidation, and fraction of fiber broken versus time curves of SiC/SiC composite under multiple loading sequence 𝜎 max1 = 100 MPa, 𝜎 max2 = 150 MPa, and 𝜎 max3 = 200 MPa at T = 850 ∘ C in air atmosphere. When fiber volume increases, the stress rupture lifetime increases;

275

276

6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

(c)

(d)

(e)

Figure 6.20

(Continued)

6.4 Stress Rupture of Ceramic-Matrix Composites Under Multiple Load Sequence

Table 6.18 The experimental and theoretical strain, interface debonding and oxidation length, and broken fiber fraction of SiC/SiC composite under constant stress 𝜎 = 120 MPa, stochastic stress 𝜎 s = 130, 140 MPa and Δt = 7.2 ks at T = 950 ∘ C in air atmosphere.

𝜎 = 120 MPa (experimental data)

𝜎 = 120 MPa (theoretical values)

𝜎 = 120 MPa 𝜎 s = 130 MPa t = 3.6 ks Δt = 7.2 ks (theoretical values)

𝜎 = 120 MPa 𝜎 s = 140 MPa t = 14.4 ks Δt = 7.2 ks (theoretical values)

𝝈 (MPa)

120

120

120

120

t (ks)

0.7

4.4

7.1

9.3

𝜀c (%)

0.17

0.2

0.23

0.268

𝝈 (MPa)

120

120

120

120

120

120

t (ks)

0

0.7

4.4

7.1

9.3

10.5

𝜀c (%)

0.168

0.174

0.207

0.234

0.26

0.277

𝜂

0.197

0.209

0.276

0.33

0.38

0.42

𝜔

0

0.005

0.259

0.35

0.397

0.406

q

0

0.003

0.04

0.08

0.13

0.17

𝝈 (MPa)

120

120

130

130

t (ks)

0

3.6

3.6

6.8

𝜀c (%)

0.168

0.2

0.23

0.281

𝜂

0.197

0.261

0.312

0.415

𝜔

0

0.224

0.202

0.287

q

0

0.03

0.05

0.151

𝝈 (MPa)

120

120

140

140

t (ks)

0

3.6

3.6

4.4

𝜀c (%)

0.168

0.2

0.272

0.288

𝜂

0.197

0.261

0.369

0.407

𝜔

0

0.224

0.18

0.199

q

0

0.03

0.09

0.128

the time-dependent strain decreases at the same time; the time-dependent fraction of interface debonding decreases under the same applied stress when the interface partial debonds, and the time for complete interface debonding (𝜂 = 1.0) increases; the time-dependent fraction of interface oxidation increases, and the time for complete interface oxidation (𝜔 = 1.0) remains the same for different fiber volumes, and the time-dependent fiber failure probability decreases. Under multiple loading sequence, the time-dependent strain and the fraction of interface debonding and broken fiber increase at higher applied stress; however, the time-dependent fraction of interface oxidation decreases at higher applied stress. At low fiber volume of V f = 25%, the composite fractures at short lifetime with partial interface debonding (𝜂 < 1.0) and partial interface oxidation (𝜔 < 1.0), and low fraction of fiber failure;

277

278

6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

Figure 6.21

Diagram of multiple loading sequence.

however, at high fiber volume of V f = 30% and 35%, the composite fractures at long lifetime with complete interface debonding (𝜂 = 1.0) and complete interface oxidation (𝜔 = 1.0), and high fraction of fiber failure. When V f = 25%, the strain increases from 𝜀c = 0.029% at t = 0 seconds to 𝜀c = 0.032% at t = 36 ks under the first stress level 𝜎 max1 = 100 MPa; when the stress level increases to 𝜎 max2 = 150 MPa at t = 36 ks, the strain increases to 𝜀c = 0.067%, and increases to 𝜀c = 0.08% at t = 72 ks, and when the stress level increases to 𝜎 max3 = 200 MPa at t = 72 ks, the strain increases to 𝜀c = 0.16%, and increases to 𝜀c = 0.236% at t = 129.3 ks. The interface debonding ratio increases from 𝜂 = 0.02 at t = 0 seconds to 𝜂 = 0.046 at t = 36 ks under the first stress level 𝜎 max1 = 100 MPa; when the stress level increases to 𝜎 max2 = 150 MPa at t = 36 ks, the interface debonding ratio increases to 𝜂 = 0.188, and increases to 𝜂 = 0.26 at t = 72 ks, and when the stress level increases to 𝜎 max3 = 200 MPa at t = 72 ks, the interface debonding ratio increases to 𝜂 = 0.59, and increases to 𝜂 = 0.85 at t = 129.3 ks. The interface oxidation ratio increases from 𝜔 = 0 at t = 0 seconds to 𝜔 = 0.54 at t = 36 ks under the first stress level 𝜎 max1 = 100 MPa; when the stress level increases to 𝜎 max2 = 150 MPa at t = 36 ks, the interface oxidation ratio decreases to 𝜔 = 0.39 and increases to 𝜔 = 0.56 at t = 72 ks, and when the stress level increases to 𝜎 max3 = 200 MPa at t = 72 ks, the interface oxidation ratio decreases to 𝜔 = 0.45 and increases to 𝜔 = 0.56 at t = 129.3 ks. The fiber failure probability increases from q = 0 at t = 0 seconds to q = 3.3 × 10−4 at t = 36 ks under the first stress level 𝜎 max1 = 100 MPa; when the stress level increases to 𝜎 max2 = 150 MPa at t = 36 ks, the fiber failure probability increases to q = 3.9 × 10−3 and increases to q = 9.4 × 10−3 at t = 72 ks, and when the stress level increases to 𝜎 max3 = 200 MPa at t = 72 ks, the fiber failure probability increases to q = 0.06 and increases to q = 0.17 at t = 129.3 ks. When V f = 35%, the strain increases from 𝜀c = 0.028% at t = 0 seconds to 𝜀c = 0.03% at t = 36 ks under the first stress level 𝜎 max1 = 100 MPa; when the stress level increases to 𝜎 max2 = 150 MPa at t = 36 ks, the strain increases to 𝜀c = 0.055% and increases to 𝜀c = 0.063% at t = 72 ks, and when the stress level increases to 𝜎 max3 = 200 MPa at t = 72 ks, the strain increases to 𝜀c = 0.115%, and increases to 𝜀c = 0.234% at t = 732.8 ks. The interface debonding ratio increases from 𝜂 = 0.01 at t = 0 seconds to 𝜂 = 0.034 at t = 36 ks under the first stress level 𝜎 max1 = 100 MPa; when the stress level increases to 𝜎 max2 = 150 MPa at t = 36 ks, the interface debonding ratio increases to

6.4 Stress Rupture of Ceramic-Matrix Composites Under Multiple Load Sequence

(a)

(b)

Figure 6.22 Effect of fiber volume (i.e. V f = 25%, 30%, and 35%) on (a) the composite strain versus time curves; (b) the fraction of the interface debonding versus time curves; (c) the fraction of the interface oxidation versus time curves; and (d) the broken fiber fraction versus time curves of SiC/SiC composite under multiple loading sequence 𝜎 max1 = 100 MPa, 𝜎 max2 = 150 MPa, and 𝜎 max3 = 200 MPa at T = 850 ∘ C in air atmosphere.

𝜂 = 0.134, and increases to 𝜂 = 0.20 at t = 72 ks, and when the stress level increases to 𝜎 max3 = 200 MPa at t = 72 ks, the interface debonding ratio increases to 𝜂 = 0.44 and increases to 𝜂 = 1.0 at t = 225 ks. The interface oxidation ratio increases from 𝜔 = 0 at t = 0 seconds to 𝜔 = 0.73 at t = 36 ks under the first stress level 𝜎 max1 = 100 MPa; when the stress level increases to 𝜎 max2 = 150 MPa at t = 36 ks, the interface oxidation ratio decreases to 𝜔 = 0.54 and increases to 𝜔 = 0.72 at t = 72 ks, and when the stress level increases to 𝜎 max3 = 200 MPa at t = 72 ks, the interface oxidation ratio decreases to 𝜔 = 0.61 and increases to 𝜔 = 1.0 at t = 267.1 ks. The fiber failure probability increases from q = 0 at t = 0 seconds to q = 4.5 × 10−5 at t = 36 ks under the first stress level 𝜎 max1 = 100 MPa; when the stress level increases to 𝜎 max2 = 150 MPa at t = 36 ks, the

279

280

6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

(c)

(d)

Figure 6.22

(Continued)

fiber failure probability increases to q = 5.1 × 10−4 and increases to q = 1.2 × 10−3 at t = 72 , and when the stress level increases to 𝜎 max3 = 200 MPa at t = 72 ks, the fiber failure probability increases to q = 0.007 and increases to q = 0.26 at t = 732.8 ks. 6.4.1.2 Stress Rupture of SiC/SiC Composite Under Multiple Loading Sequence for Different Matrix Crack Spacings

Figure 6.23 and Table 6.20 show the effect of saturation matrix crack spacing (i.e. lsat = 150, 200, and 350 μm) on stress rupture, fraction of interface debonding and oxidation, and fraction of broken fiber versus time curves of SiC/SiC composite under multiple loading sequence 𝜎 max1 = 100 MPa, 𝜎 max2 = 150 MPa, and 𝜎 max3 = 200 MPa at T = 850 ∘ C in air atmosphere. When saturation matrix crack spacing increases, the stress rupture lifetime remains the same; the time-dependent strain decreases at the same time; the time-dependent fraction of interface debonding decreases at

6.4 Stress Rupture of Ceramic-Matrix Composites Under Multiple Load Sequence

Table 6.19 Effect of fiber volume on stress rupture of SiC/SiC composite under multiple loading sequence 𝜎 max1 = 100 MPa, 𝜎 max2 = 150 MPa, and 𝜎 max3 = 200 MPa at T = 850 ∘ C in air atmosphere. 𝝈 max (MPa)

100

t (ks)

0

150 36

200

36

72

72

129.3

0.16

0.236

V f = 25% 𝜀 (%)

0.029

𝜂

0.02

0.046

0.188

0.26

0.59

0.85

𝜔

0

0.54

0.39

0.56

0.45

0.58

q

0

3.3 × 10−4

3.9 × 10−3

9.4 × 10−3

0.06

0.17

𝝈 max (MPa)

0.032

0.067

100

t (ks)

0

0.08

150 36

200

36

72

72

343

0.060 16

0.070 13

0.134 43

0.270 85

V f = 30% 𝜀 (%)

0.028 88

𝜂

0.014 94

0.039 29

0.156 94

0.228 45

0.500 04

1

𝜔

0

0.641 05

0.471 44

0.646 92

0.542 8

1

q

0

1.14 × 10−4

0.001 3

0.003 11

0.018 14

0.238 44

𝝈 max (MPa)

0.031 17

100

t (ks)

0

150 36

36

200 72

72

732.8

0.063 92

0.115 02

0.234 92

V f = 35% 𝜀 (%)

0.028 92

𝜂

0.010 06

0.034 41

0.134 67

0.206 07

0.441 59

1

𝜔

0

0.732 11

0.549 38

0.717 17

0.614 64

1

q

0

4.5 × 10−5

5.15 × 10−4

0.001 23

0.007

0.261 77

0.030 78

0.055 9

the same time, and the time for complete interface debonding (𝜂 = 1.0) increases; the time-dependent fraction of the interface oxidation remains the same when the interface partial debonds (𝜔 < 1.0), and the time for complete interface oxidation (𝜔 = 1.0) increases, and the time-dependent fraction of fiber broken remains the same at the same time. When lsat = 150 μm, the strain increases from 𝜀c = 0.029% at t = 0 seconds to 𝜀c = 0.033% at t = 36 ks under the first stress level 𝜎 max1 = 100 MPa; when the stress level increases to 𝜎 max2 = 150 MPa at t = 36 ks, the strain increases to 𝜀c = 0.072% and increases to 𝜀c = 0.089% at t = 72 ks, and when the stress level increases to 𝜎 max3 = 200 MPa at t = 72 ks, the strain increases to 𝜀c = 0.18% and increases to 𝜀c = 0.27% at t = 343 ks. The interface debonding ratio increases from 𝜂 = 0.025 at

281

282

6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

(a)

(b)

Figure 6.23 Effect of saturation matrix crack spacing (i.e. lsat = 150, 200, and 300 μm) on (a) the composite strain versus time curves; (b) the fraction of the interface debonding versus time curves; (c) the fraction of the interface oxidation versus time curves; and (d) the broken fiber fraction versus time curves of SiC/SiC composite under multiple loading sequence of 𝜎 max1 = 100 MPa, 𝜎 max2 = 150 MPa, and 𝜎 max3 = 200 MPa at T = 850 ∘ C in air atmosphere.

t = 0 seconds to 𝜂 = 0.065 at t = 36 ks under the first stress level 𝜎 max1 = 100 MPa; when the stress level increases to 𝜎 max2 = 150 MPa at t = 36 ks, the interface debonding ratio increases to 𝜂 = 0.26 and increases to 𝜂 = 0.38 at t = 72 ks, and when the stress level increases to 𝜎 max3 = 200 MPa at t = 72 ks, the interface debonding ratio increases to 𝜂 = 0.83 and increases to 𝜂 = 1.0 at t = 98.9 ks. The interface oxidation ratio increases from 𝜔 = 0 at t = 0 seconds to 𝜔 = 0.64 at t = 36 ks under the first stress level 𝜎 max1 = 100 MPa; when the stress level increases to 𝜎 max2 = 150 MPa at t = 36 ks, the interface oxidation ratio decreases to 𝜔 = 0.47 and increases to 𝜔 = 0.64 at t = 72 ks, and when the stress level increases to 𝜎 max3 = 200 MPa at t = 72 ks, the

6.4 Stress Rupture of Ceramic-Matrix Composites Under Multiple Load Sequence

(c)

(d)

Figure 6.23

(Continued)

interface oxidation ratio decreases to 𝜔 = 0.54 and increases to 𝜔 = 1.0 at t = 159.7 ks. The fiber failure probability increases from q = 0 at t = 0 seconds to q = 1.1 × 10−4 at t = 36 ks under the first stress level 𝜎 max1 = 100 MPa; when the stress level increases to 𝜎 max2 = 150 MPa at t = 36 ks, the fiber failure probability increases to q = 1.3 × 10−3 and increases to q = 3 × 10−3 at t = 72 ks, and when the stress level increases to 𝜎 max3 = 200 MPa at t = 72 ks, the fiber failure probability increases to q = 0.018 and increases to q = 0.23 at t = 343 ks. When lsat = 350 μm, the strain increases from 𝜀c = 0.028% at t = 0 seconds to 𝜀c = 0.03% at t = 36 ks under the first stress level 𝜎 max1 = 100 MPa; when the stress level increases to 𝜎 max2 = 150 MPa at t = 36 ks, the strain increases to 𝜀c = 0.057% and increases to 𝜀c = 0.065% at t = 72 ks, and when the stress level increases to 𝜎 max3 = 200 MPa at t = 72 ks, the strain increases to 𝜀c = 0.12% and increases to 𝜀c (t) = 0.26% at t = 343 ks. The interface debonding ratio increases from 𝜂 = 0.025

283

284

6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

Table 6.20 Effect of matrix crack spacing on stress rupture of SiC/SiC composite under multiple loading sequence of 𝜎 max1 = 100 MPa, 𝜎 max2 = 150 MPa, and 𝜎 max3 = 200 MPa at T = 850 ∘ C in air atmosphere. 𝝈 max (MPa)

100

t (ks)

0

150 36

200

36

72

72

343

0.072 56

0.089 22

0.187 12

0.275 04

lsat = 150 𝜇m 𝜀 (%)

0.029 66

𝜂

0.024 9

0.065 49

0.261 56

0.380 74

0.833 4

1

𝜔

0

0.641 05

0.471 44

0.646 92

0.542 8

1

q

0

1.14 × 10−4

0.001 3

0.003 11

0.018 14

0.234 16

𝝈 max (MPa)

0.033 49

100

t (ks)

0

150 36

200

36

72

72

343

0.064 81

0.077 31

0.154 19

0.272 94

lsat = 200 𝜇m 𝜀 (%)

0.029 17

𝜂

0.018 68

0.049 12

0.641 05

0.285 56

0.625 05

1

𝜔

0

0.196 17

0.471 44

0.646 92

0.542 8

1

q

0

1.14 × 10−4

0.001 3

0.003 11

0.018 14

0.234 16

𝝈 max (MPa)

0.032 04

100

t (ks)

0

150 36

200

36

72

72

343

0.057 06

0.065 39

0.121 26

0.268 76

lsat = 300 𝜇m 𝜀 (%)

0.028 68

𝜂

0.012 45

0.032 75

0.641 05

0.190 37

0.416 7

1

𝜔

0

0.130 78

0.471 44

0.646 92

0.542 8

1

q

0

1.14 × 10−4

0.001 3

0.003 11

0.018 14

0.234 16

0.030 6

at t = 0 seconds to 𝜂 = 0.032 at t = 36 ks under the first stress level 𝜎 max1 = 100 MPa; when the stress level increases to 𝜎 max2 = 150 MPa at t = 36 ks, the interface debonding ratio increases to 𝜂 = 0.13 and increases to 𝜂 = 0.19 at t = 72 ks, and when the stress level increases to 𝜎 max3 = 200 MPa at t = 72 ks, the interface debonding length increases to 𝜂 = 0.41 and increases to 𝜂 = 1.0 at t = 257.5 ks. The interface oxidation length increases from 𝜔 = 0 at t = 0 seconds to 𝜔 = 0.64 at t = 36 ks under the first stress level 𝜎 max1 = 100 MPa; when the stress level increases to 𝜎 max2 = 150 MPa at t = 36 ks, the interface oxidation ratio decreases to 𝜔 = 0.47 and increases to 𝜔 = 0.64 at t = 72 ks, and when the stress level increases to 𝜎 max3 = 200 MPa at t = 72 ks, the interface oxidation ratio decreases to 𝜔 = 0.54 and increases to 𝜔 = 1.0 at t = 321.1 ks.

6.4 Stress Rupture of Ceramic-Matrix Composites Under Multiple Load Sequence

The fiber failure probability increases from q = 0 at t = 0 seconds to q = 1.1 × 10−4 at t = 36 ks under the first stress level 𝜎 max1 = 100 MPa; when the stress level increases to 𝜎 max2 = 150 MPa at t = 36 ks, the fiber failure probability increases to q = 1.3 × 10−3 and increases to q = 3 × 10−3 at t = 72 ks, and when the stress level increases to 𝜎 max3 = 200 MPa at t = 72 ks, the fiber failure probability increases to q = 0.018 and increases to q = 0.23 at t = 343 ks. 6.4.1.3 Stress Rupture of SiC/SiC Composite Under Multiple Loading Sequence for Different Interface Shear Stress

Figure 6.24 and Table 6.21 show the effect of interface shear stress in the slip region (i.e. 𝜏 i = 20, 40, and 60 MPa) on stress rupture, fraction of the interface debonding and oxidation, and the fraction of fiber broken versus time curves of SiC/SiC composite under multiple loading sequence 𝜎 max1 = 100 MPa, 𝜎 max2 = 150 MPa, and 𝜎 max3 = 200 MPa at T = 850 ∘ C in air atmosphere. When interface shear stress in the slip region increases, the stress rupture lifetime increases; the time-dependent strain decreases at the same time; the time-dependent fraction of the interface debonding decreases when the interface partial debonds (𝜂 < 1.0); and the time for complete interface debonding (𝜂 = 1.0) increases; the time-dependent fraction of the interface oxidation increases when the interface partial debonds, and the time for complete interface oxidation (𝜔 = 1.0) remains the same; and the time-dependent fraction of fiber broken decreases. When 𝜏 i = 20 MPa, the strain increases from 𝜀c = 0.029% at t = 0 seconds to 𝜀c = 0.031% at t = 36 ks under the first stress level 𝜎 max1 = 100 MPa; when the stress level increases to 𝜎 max2 = 150 MPa at t = 36 ks, the strain increases to 𝜀c = 0.064% and increases to 𝜀c = 0.073% at t = 72 ks, and when the stress level increases to 𝜎 max3 = 200 MPa at t = 72 ks, the strain increases to 𝜀c = 0.148% and increases to 𝜀c = 0.268% at t = 246.4 ks. The interface debonding ratio increases from 𝜂 = 0.023 at t = 0 seconds to 𝜂 = 0.047 at t = 36 ks under the first stress level of 𝜎 max1 = 100 MPa; when the stress level increases to 𝜎 max2 = 150 MPa at t = 36 ks, the interface debonding ratio increases to 𝜂 = 0.2 and increases to 𝜂 = 0.27 at t = 72 ks, and when the stress level increases to 𝜎 max3 = 200 MPa at t = 72 ks, the interface debonding ratio increases to 𝜂 = 0.62 and increases to 𝜂 = 1.0 at t = 169.2 ks. The interface oxidation ratio increases from 𝜔 = 0 at t = 0 seconds to 𝜔 = 0.53 at t = 36 ks under the first stress level 𝜎 max1 = 100 MPa; when the stress level increases to 𝜎 max2 = 150 MPa at t = 36 ks, the interface oxidation ratio decreases to 𝜔 = 0.36 and increases to 𝜔 = 0.54 at t = 72 ks, and when the stress level increases to 𝜎 max3 = 200 MPa at t = 72 ks, the interface oxidation ratio decreases to 𝜔 = 0.43 and increases to 𝜔 = 0.92 at t = 246.4 ks. The fiber failure probability increases from q = 0 at t = 0 seconds to q = 1.7 × 10−4 at t = 36 ks under the first stress level 𝜎 max1 = 100 MPa; when the stress level increases to 𝜎 max2 = 150 MPa at t = 36 ks, the fiber failure probability increases to q = 1.9 × 10−3 and increases to q = 4.6 × 10−3 at t = 72 ks, and when the stress level increases to 𝜎 max3 = 200 MPa at t = 72 ks, the fiber failure probability increases to q = 0.028 and increases to q = 0.227 at t = 246.4 ks. When 𝜏 i = 60 MPa, the strain increases from 𝜀c = 0.028% at t = 0 seconds to 𝜀c = 0.03% at t = 36 ks under the first stress level 𝜎 max1 = 100 MPa; when the stress

285

286

6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

(a)

(b)

Figure 6.24 Effect of interface shear stress in the slip region (i.e. 𝜏 i = 20, 40, and 60 MPa) on (a) the composite strain versus time curves; (b) the fraction of the interface debonding versus time curves; (c) the fraction of the interface oxidation versus time curves; and (d) the fraction of the broken fiber versus time curves of SiC/SiC composite under multiple loading sequence of 𝜎 max1 = 100 MPa, 𝜎 max2 = 150 MPa, and 𝜎 max3 = 200 MPa at T = 850 ∘ C in air atmosphere.

level increases to 𝜎 max2 = 150 MPa at t = 36 ks, the strain increases to 𝜀c = 0.056% and increases to 𝜀c = 0.066% at t = 72 ks, and when the stress level increases to 𝜎 max3 = 200 MPa at t = 72 ks, the strain increases to 𝜀c = 0.12% and increases to 𝜀c = 0.27% at t = 604.8 ks. The interface debonding ratio increases from 𝜂 = 0.006 at t = 0 seconds to 𝜂 = 0.03 at t = 36 ks under the first stress level 𝜎 max1 = 100 MPa; when the stress level increases to 𝜎 max2 = 150 MPa at t = 36 ks, the interface debonding ratio increases to 𝜂 = 0.11 and increases to 𝜂 = 0.185 at t = 72 ks, and when the stress level increases to 𝜎 max3 = 200 MPa at t = 72 ks, the interface debonding ratio increases to 𝜂 = 0.38 and increases to 𝜂 = 1.0 at t = 239.4 ks. The interface oxidation

6.4 Stress Rupture of Ceramic-Matrix Composites Under Multiple Load Sequence

(c)

(d)

Figure 6.24

(Continued)

ratio increases from 𝜔 = 0 at t = 0 seconds to 𝜔 = 0.8 at t = 36 ks under the first stress level 𝜎 max1 = 100 MPa; when the stress level increases to 𝜎 max2 = 150 MPa at t = 36 ks, the interface oxidation ratio decreases to 𝜔 = 0.65 and increases to 𝜔 = 0.79 at t = 72 ks, and when the stress level increases to 𝜎 max3 = 200 MPa at t = 72 ks, the interface oxidation ratio decreases to 𝜔 = 0.71 and increases to 𝜔 = 1.0 at t = 267.1 ks. The fiber failure probability increases from q = 0 at t = 0 seconds to q = 5.7 × 10−5 at t = 36 ks under the first stress level 𝜎 max1 = 100 MPa; when the stress level increases to 𝜎 max2 = 150 MPa at t = 36 ks, the fiber failure probability increases to q = 6.5 × 10−4 and increases to q = 1.5 × 10−3 at t = 72 ks, and when the stress level increases to 𝜎 max3 = 200 MPa at t = 72 ks, the fiber failure probability increases to q = 0.008 and increases to q = 0.25 at t = 604.8 ks. Figure 6.25 and Table 6.22 show the effect of interface shear stress in oxidation region (i.e. 𝜏 f = 1, 3, and 5 MPa) on stress rupture, fraction of interface debonding

287

288

6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

Table 6.21 Effect of interface shear stress in slip region on stress rupture behavior of SiC/SiC composite under multiple loading sequence of 𝜎 max1 = 100 MPa, 𝜎 max2 = 150 MPa, and 𝜎 max3 = 200 MPa at T = 850 ∘ C in air atmosphere. 𝝈 max (MPa)

100

t (ks)

0

150 36

200

36

72

72

246.4

0.064 12

0.073 99

0.148 8

0.268 54

𝜏 i = 20 MPa 𝜀 (%)

0.029 43

𝜂

0.023 15

0.047 08

0.200 65

0.271 15

0.622 27

1

𝜔

0

0.535 03

0.368 73

0.545 04

0.436 18

0.923 23

q

0

1.7 × 10−4

0.001 96

0.004 68

0.027 9

0.227 62

𝝈 max (MPa)

0.031 69

100

t (ks)

0

150 36

36

200 72

72

432

0.068 25

0.127 42

0.270 96

𝜏 i = 40 MPa 𝜀 (%)

0.028 6

𝜂

0.010 83

0.035 39

0.135 07

0.207 12

0.439 91

1

𝜔

0

0.711 79

0.547 77

0.713 54

0.617

1

q

0

8.5 × 10−5

9.75 × 10−4

0.002 33

0.013 45

0.234 76

𝝈 max (MPa)

0.030 92

0.058 19

100

t (ks)

0

150 36

36

200 72

72

604.8

0.066 36

0.120 53

0.273 71

𝜏 i = 60 MPa 𝜀 (%)

0.028 33

𝜂

0.006 69

0.031 46

0.113 14

0.185 76

0.380 29

1

𝜔

0

0.800 68

0.653 92

0.795 6

0.713 72

1

q

0

5.7 × 10−5

6.5 × 10−4

0.001 55

0.008 86

0.250 24

0.030 66

0.056 22

and oxidation, and fiber failure probability versus time curves of SiC/SiC composite under multiple loading sequence 𝜎 max1 = 100 MPa, 𝜎 max2 = 150 MPa, and 𝜎 max3 = 200 MPa at T = 850 ∘ C in air atmosphere. When interface shear stress in oxidation region increases, the stress rupture lifetime remains the same; the time-dependent strain decreases at the same time; the time-dependent fraction of the interface debonding decreases when the interface partial debonds (𝜂 < 1.0), and the time for complete interface debonding (𝜂 = 1.0) increases; the time-dependent fraction of the interface oxidation increases when the interface partial debonds, and the time for complete interface oxidation (𝜔 = 1.0) remains the same, and the time-dependent fraction of fiber broken remains the same.

6.4 Stress Rupture of Ceramic-Matrix Composites Under Multiple Load Sequence

(a)

(b)

Figure 6.25 Effect of interface shear stress in the oxidation region (i.e. 𝜏 f = 1, 3, and 5 MPa) on (a) the composite strain versus time curves; (b) the fraction of the interface debonding versus time curves; (c) the fraction of the interface oxidation versus time curves; and (d) the fraction of the broken fiber versus time curves of SiC/SiC composite under multiple loading sequence of 𝜎 max1 = 100 MPa, 𝜎 max2 = 150 MPa, and 𝜎 max3 = 200 MPa at T = 850 ∘ C in air atmosphere.

When 𝜏 f = 1 MPa, the strain increases from 𝜀c = 0.028% at t = 0 seconds to 𝜀c = 0.031% at t = 36 ks under the first stress level 𝜎 max1 = 100 MPa; when the stress level increases to 𝜎 max2 = 150 MPa at t = 36 ks, the strain increases to 𝜀c = 0.06% and increases to 𝜀c = 0.07% at t = 72 ks, and when the stress level increases to 𝜎 max3 = 200 MPa at t = 72 ks, the strain increases to 𝜀c = 0.13% and increases to 𝜀c = 0.27% at t = 343 ks. The interface debonding ratio increases from 𝜂 = 0.015 at t = 0 seconds to 𝜂 = 0.04 at t = 36 ks under the first stress level 𝜎 max1 = 100 MPa; when the stress level increases to 𝜎 max2 = 150 MPa at t = 36 ks, the interface debonding ratio increases to 𝜂 = 0.156 and increases to 𝜂 = 0.228 at t = 72 ks, and when the

289

290

6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

(c)

(d)

Figure 6.25

(Continued)

stress level increases to 𝜎 max3 = 200 MPa at t = 72 ks, the interface debonding length increases to 𝜂 = 0.5 and increases to 𝜂 = 1.0 at t = 205.4 ks. The interface oxidation ratio increases from 𝜔 = 0 at t = 0 seconds to 𝜔 = 0.64 at t = 36 ks under the first stress level 𝜎 max1 = 100 MPa; when the stress level increases to 𝜎 max2 = 150 MPa at t = 36 ks, the interface oxidation ratio decreases to 𝜔 = 0.47 and increases to 𝜔 = 0.64 at t = 72 ks, and when the stress level increases to 𝜎 max3 = 200 MPa at t = 72 ks, the interface oxidation ratio decreases to 𝜔 = 0.54 and increases to 𝜔 = 1.0 at t = 267.1 ks. The fiber failure probability increases from q = 0 at t = 0 seconds to q = 1.1 × 10−4 at t = 36 ks under the first stress level 𝜎 max1 = 100 MPa; when the stress level increases to 𝜎 max2 = 150 MPa at t = 36 ks, the fiber failure probability increases to q = 1.3 × 10−3 and increases to q = 3 × 10−3 at t = 72 ks, and when the stress level increases to 𝜎 max3 = 200 MPa at t = 72 ks, the fiber failure probability increases to q = 0.018 and increases to q = 0.23 at t = 343 ks.

6.4 Stress Rupture of Ceramic-Matrix Composites Under Multiple Load Sequence

Table 6.22 Effect of interface shear stress in oxidation region on stress rupture behavior of SiC/SiC composite under multiple loading sequence of 𝜎 max1 = 100 MPa, 𝜎 max2 = 150 MPa, and 𝜎 max3 = 200 MPa at T = 850 ∘ C in air atmosphere. 𝝈 max (MPa)

100

t (ks)

0

150 36

200

36

72

72

343

0.060 16

0.070 15

0.134 43

0.270 85

𝜏 f = 1 MPa 𝜀 (%)

0.028 88

𝜂

0.014 94

0.039 29

0.156 94

0.228 45

0.500 04

1

𝜔

0

0.641 05

0.471 44

0.646 92

0.542 8

1

q

0

1.14 × 10−4

0.001 3

0.003 11

0.018 14

0.234 16

𝝈 max (MPa)

0.031 17

100

t (ks)

0

150 36

200

36

72

72

343

0.059 27

0.068 01

0.129 59

0.249 92

𝜏 f = 3 MPa 𝜀 (%)

0.028 88

𝜂

0.015 01

0.037 61

0.152

0.218 59

0.481 95

1

𝜔

0

0.669 67

0.486 74

0.676 08

0.563 18

1

q

0

1.14 × 10−4

0.001 3

0.003 11

0.018 14

0.234 16

𝝈 max (MPa)

0.030 95

100

t (ks)

0

150 36

200

36

72

72

0.058 4

0.065 98

0.124 96

343

𝜏 f = 5 MPa 𝜀 (%)

0.028 88

𝜂

0.015

0.035 94

0.147 07

0.208 74

0.463 85

1

𝜔

0

0.700 97

0.503 06

0.707 99

0.585 15

1

q

0

1.14 × 10−4

0.001 3

0.003 11

0.018 14

0.234 16

0.030 74

0.229

When 𝜏 f = 5 MPa, the strain increases from 𝜀c = 0.028% at t = 0 seconds to 𝜀c = 0.03% at t = 36 ks under the first stress level 𝜎 max1 = 100 MPa; when the stress level increases to 𝜎 max2 = 150 MPa at t = 36 ks, the strain increases to 𝜀c = 0.058% and increases to 𝜀c = 0.066% at t = 72 ks, and when the stress level increases to 𝜎 max3 = 200 MPa at t = 72 ks, the strain increases to 𝜀c = 0.125% and increases to 𝜀c = 0.23% at t = 343 ks. The interface debonding ratio increases from 𝜂 = 0.015 at t = 0 seconds to 𝜂 = 0.036 at t = 36 ks under the first stress level 𝜎 max1 = 100 MPa; when the stress level increases to 𝜎 max2 = 150 MPa at t = 36 ks, the interface debonding ratio increases to 𝜂 = 0.147 and increases to 𝜂 = 0.208 at t = 72 ks, and when the stress level increases to 𝜎 max3 = 200 MPa at t = 72 ks, the interface debonding ratio

291

292

6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

increases to 𝜂 = 0.46 and increases to 𝜂 = 1.0 at t = 236.3 ks. The interface oxidation ratio increases from 𝜔 = 0 at t = 0 seconds to 𝜔 = 0.7 at t = 36 ks under the first stress level 𝜎 max1 = 100 MPa; when the stress level increases to 𝜎 max2 = 150 MPa at t = 36 ks, the interface oxidation ratio decreases to 𝜔 = 0.5 and increases to 𝜔 = 0.7 at t = 72 ks, and when the stress level increases to 𝜎 max3 = 200 MPa at t = 72 ks, the interface oxidation ratio decreases to 𝜔 = 0.58 and increases to 𝜔 = 1.0 at t = 267.1 ks. The fiber failure probability increases from q = 0 at t = 0 seconds to q = 1.1 × 10−4 at t = 36 ks under the first stress level 𝜎 max1 = 100 MPa; when the stress level increases to 𝜎 max2 = 150 MPa at t = 36 ks, the fiber failure probability increases to q = 1.3 × 10−3 and increases to q = 3 × 10−3 at t = 72 ks, and when the stress level increases to 𝜎 max3 = 200 MPa at t = 72 ks, the fiber failure probability increases to q = 0.018 and increases to q = 0.23 at t = 343 ks. 6.4.1.4 Stress Rupture of SiC/SiC Composite Under Multiple Loading Sequence for Different Environment Temperatures

Figure 6.26 and Table 6.23 show the effect of environment temperature (i.e. T = 700, 800, and 900 ∘ C) on stress rupture, fraction of interface debonding and oxidation, and fraction of fiber broken versus time curves of SiC/SiC composite under multiple loading sequence 𝜎 max1 = 100 MPa, 𝜎 max2 = 150 MPa, and 𝜎 max3 = 200 MPa. When temperature increases, the stress rupture lifetime decreases; the time-dependent composite strain increases at the same time; the time-dependent fraction of the interface debonding increases when the interface partial debonds (𝜂 < 1.0), and the time for complete interface debonding (𝜂 = 1.0) decreases; the time-dependent fraction of the interface oxidation increases when the interface partial debonds, and the time for complete interface oxidation (𝜔 = 1.0) decreases, and the time-dependent fraction of fiber broken increases. When T = 700 ∘ C, the strain increases from 𝜀c = 0.028% at t = 0 seconds to 𝜀c = 0.029% at t = 36 ks under the first stress level 𝜎 max1 = 100 MPa; when the stress level increases to 𝜎 max2 = 150 MPa at t = 36 ks, the strain increases to 𝜀c = 0.053% and increases to 𝜀c = 0.056% at t = 72 ks, and when the stress level increases to 𝜎 max3 = 200 MPa at t = 72 ks, the strain increases to 𝜀c = 0.098% and increases to 𝜀c = 0.275% at t = 1155.4 ks. The interface debonding ratio increases from 𝜂 = 0.0149 at t = 0 seconds to 𝜂 = 0.022 at t = 36 ks under the first stress level 𝜎 max1 = 100 MPa; when the stress level increases to 𝜎 max2 = 150 MPa at t = 36 ks, the interface debonding ratio increases to 𝜂 = 0.106 and increases to 𝜂 = 0.127 at t = 72 ks, and when the stress level increases to 𝜎 max3 = 200 MPa at t = 72 ks, the interface debonding ratio increases to 𝜂 = 0.31 and increases to 𝜂 = 1.0 at t = 688.9 ks. The interface oxidation ratio increases from 𝜔 = 0 at t = 0 seconds to 𝜔 = 0.337 at t = 36 ks under the first stress level 𝜎 max1 = 100 MPa; when the stress level increases to 𝜎 max2 = 150 MPa at t = 36 ks, the interface oxidation ratio decreases to 𝜔 = 0.206 and increases to 𝜔 = 0.343 at t = 72 ks, and when the stress level increases to 𝜎 max3 = 200 MPa at t = 72 ks, the interface oxidation ratio decreases to 𝜔 = 0.258 and increases to 𝜔 = 1.0 at t = 895.5 ks. The fiber failure probability increases from q = 0 at t = 0 seconds to q = 2.5 × 10−5 at t = 36 ks under the first stress level 𝜎 max1 = 100 MPa; when the stress level increases to 𝜎 max2 = 150 MPa at t = 36 ks, the fiber failure probability

6.4 Stress Rupture of Ceramic-Matrix Composites Under Multiple Load Sequence

(a)

(b)

Figure 6.26 Effect of temperature (i.e. T = 700, 800, and 900 ∘ C) on (a) the composite strain versus time curves; (b) the fraction of the interface debonding versus time curves; (c) the fraction of the interface oxidation versus time curves; and (d) the broken fiber fraction versus time curves of SiC/SiC composite under multiple loading sequence of 𝜎 max1 = 100 MPa, 𝜎 max2 = 150 MPa, and 𝜎 max3 = 200 MPa.

increases to q = 2.9 × 10−4 and increases to q = 6.9 × 10−4 at t = 72 ks, and when the stress level increases to 𝜎 max3 = 200 MPa at t = 72 ks, the fiber failure probability increases to q = 3.9 × 10−3 and increases to q = 0.26 at t = 1155.4 ks. When T = 900 ∘ C, the strain increases from 𝜀c = 0.028% at t = 0 seconds to 𝜀c = 0.032% at t = 36 ks under the first stress level 𝜎 max1 = 100 MPa; when the stress level increases to 𝜎 max2 = 150 MPa at t = 36 ks, the strain increases to 𝜀c = 0.064% and increases to 𝜀c = 0.077% at t = 72 ks, and when the stress level increases to 𝜎 max3 = 200 MPa at t = 72 ks, the strain increases to 𝜀c = 0.154% and increases to 𝜀c (t) = 0.265% at t = 263.1 ks. The interface debonding ratio increases from 𝜂 = 0.0149 at t = 0 seconds to 𝜂 = 0.048 at t = 36 ks under the first stress level

293

294

6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

(c)

(d)

Figure 6.26

(Continued)

𝜎 max1 = 100 MPa; when the stress level increases to 𝜎 max2 = 150 MPa at t = 36 ks, the interface debonding ratio increases to 𝜂 = 0.185, and increases to 𝜂 = 0.285 at t = 72 ks, and when the stress level increases to 𝜎 max3 = 200 MPa at t = 72 ks, the interface debonding ratio increases to 𝜂 = 0.6 and increases to 𝜂 = 1.0 at t = 147.1 ks. The interface oxidation ratio increases from 𝜔 = 0 at t = 0 seconds to 𝜔 = 0.718 at t = 36 ks under the first stress level 𝜎 max1 = 100 MPa; when the stress level increases to 𝜎 max2 = 150 MPa at t = 36 ks, the interface oxidation ratio decreases to 𝜔 = 0.557 and increases to 𝜔 = 0.72 at t = 72 ks, and when the stress level increases to 𝜎 max3 = 200 MPa at t = 72 ks, the interface oxidation ratio decreases to 𝜔 = 0.62 and increases to 𝜔 = 1.0 at t = 191.5 ks. The fiber failure probability increases from q = 0 at t = 0 seconds to q = 1.7 × 10−4 at t = 36 ks under the first stress level 𝜎 max1 = 100 MPa; when the stress level increases to 𝜎 max2 = 150 MPa at t = 36 ks, the fiber failure probability increases to q = 1.9 × 10−3 and increases to q = 4.7 × 10−3

6.4 Stress Rupture of Ceramic-Matrix Composites Under Multiple Load Sequence

Table 6.23 Effect of temperature on stress rupture behavior of SiC/SiC composite under multiple loading sequence of 𝜎 max1 = 100 MPa, 𝜎 max2 = 150 MPa, and 𝜎 max3 = 200 MPa. 𝝈 max (MPa)

100

t (ks)

150

0

36

36

200 72

72

236.1

T = 700 ∘ C 𝜀 (%)

0.028 88

0.029 57

0.053 01

0.056 03

0.098 68

0.265

𝜂

0.014 94

0.022 17

0.106 54

0.127 8

0.311 86

1

𝜔

0

0.337 16

0.206 04

0.343 51

0.258 53

1

q

0

2.5 × 10−5

2.91 × 10−4

6.92 × 10−4

0.003 92

0.259 27

𝝈 max (MPa)

100

t (ks)

0

150 36

36

200 72

72

502.4

T = 800 ∘ C 𝜀 (%)

0.028 88

0.030 48

0.057 05

0.064 04

0.118 8

0.276 12

𝜂

0.014 94

0.031 81

0.134 92

0.184 52

0.417 76

1

𝜔

0

0.548 57

0.379 92

0.555 26

0.450 43

1

q

0

7.2 × 10−5

8.27 × 10−4

0.001 97

0.011 35

0.263 43

𝝈 max (MPa)

100

t (ks)

0

150 36

36

200 72

72

1155.4

T = 900 ∘ C 𝜀 (%)

0.028 88

0.032 07

0.064 17

0.077 97

0.154 75

0.275 35

𝜂

0.014 94

0.048 99

0.185 47

0.285 24

0.606 78

1

𝜔

0

0.718 89

0.557 71

0.723 53

0.624 67

1

q

0

1.72 × 10−4

0.001 97

0.004 72

0.028 14

0.199 87

at t = 72 ks, and when the stress level increases to 𝜎 max3 = 200 MPa at t = 72 ks, the fiber failure probability increases to q = 0.028 and increases to q = 0.199 at t = 236.1 ks.

6.4.2

Experimental Comparisons

Lara-Curzio [11] investigated the stress rupture of SiC/SiC composite at T = 950 ∘ C in air atmosphere. The experimental and predicted stress rupture curves, fraction of the interface debonding and oxidation, and fraction of broken fiber versus the time curves of SiC/SiC composite under single and multiple loading sequence at T = 950 ∘ C in air atmosphere are shown in Figures 6.27–6.29.

295

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6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

(a)

(b)

Figure 6.27 (a) Experimental and predicted composite strain versus time curves; (b) the fraction of the interface debonding versus time curves; (c) the fraction of the interface oxidation versus time curves; and (d) the broken fiber fraction versus time curves of SiC/SiC composite under single loading stress 𝜎 max = 80 MPa, and multiple loading stress 𝜎 max1 = 80 MPa and 𝜎 max2 = 90 MPa at T = 950 ∘ C in air atmosphere.

Under 𝜎 max = 80 MPa, experimental time-dependent composite strain increases from 𝜀c (t = 0.6 ks) = 0.091% to 𝜀c (t = 78.4 ks) = 0.23%; the predicted time-dependent composite strain increases from 𝜀c (t = 0 seconds) = 0.085% to 𝜀c (t = 83.6 ks) = 0.27%; the time-dependent fraction of interface debonding increases from 𝜂 (t = 0 seconds) = 0.06 to 𝜂 (t = 83.6 ks) = 0.55; the time-dependent fraction of the interface oxidation increases from 𝜔 (t = 0 seconds) = 0 to 𝜔 (t = 83.6 ks) = 0.84, and the time-dependent fraction of the broken fiber increases from q (t = 0 seconds) = 0 to q (t = 83.6 ks) = 0.29. Under 𝜎 max1 = 80 MPa and 𝜎 max2 = 90 MPa, the time-dependent composite strain increases from 𝜀c (t = 0 seconds) = 0.085% to 𝜀c (t = 7.2 ks) = 0.099% under

6.4 Stress Rupture of Ceramic-Matrix Composites Under Multiple Load Sequence

(c)

(d)

Figure 6.27

(Continued)

𝜎 max1 = 80 MPa; upon increasing stress level to 𝜎 max2 = 90 MPa at t = 7.2 ks, the time-dependent composite strain increases to 𝜀c (t = 7.2 ks) = 0.12% and then increases to 𝜀c (t = 39.8 ks) = 0.228%. The time-dependent fraction of the interface debonding increases from 𝜂 (t = 0 seconds) = 0.064 to 𝜂 (t = 7.2 ks) = 0.1 under 𝜎 max1 = 80 MPa; upon increasing stress level to 𝜎 max2 = 90 MPa at t = 7.2 ks, the time-dependent fraction of the interface debonding increases to 𝜂 (t = 7.2 ks) = 0.15 and increases to 𝜂 (t = 39.8 ks) = 0.4. The time-dependent fraction of the interface oxidation increases from 𝜔 (t = 0 seconds) = 0 to 𝜔 (t = 7.2 ks) = 0.38 under 𝜎 max1 = 80 MPa; upon increasing stress level to 𝜎 max2 = 90 MPa at t = 7.2 ks, the time-dependent fraction of the interface oxidation decreases to 𝜔 (t = 7.2 ks) = 0.35 and then increases to 𝜔 (t = 39.8 ks) = 0.72. The time-dependent fraction of the fiber broken increases from q (t = 0 seconds) = 0 to q (t = 7.2 ks) = 5.8 × 10−3 under 𝜎 max1 = 80 MPa; upon increasing stress level to 𝜎 max2 = 90 MPa at t = 7.2 ks, the

297

298

6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

(a)

(b)

Figure 6.28 (a) Experimental and predicted composite strain versus time curves; (b) the fraction of the interface debonding versus time curves; (c) the fraction of the interface oxidation versus time curves; and (d) the broken fiber fraction versus time curves of SiC/SiC composite under single loading stress 𝜎 max = 100 MPa, and multiple loading stress 𝜎 max1 = 100 MPa and 𝜎 max2 = 110 MPa at T = 950 ∘ C in air atmosphere.

fraction of the fiber broken increases to q = 1.2 × 10−2 and then increases to q (t = 39.8 ks) = 0.15. Under 𝜎 max = 100 MPa, the experimental time-dependent composite strain increases from 𝜀c (t = 1.1 ks) = 0.11% to 𝜀c (t = 33.3 ks) = 0.19%; the predicted time-dependent composite strain increases from 𝜀c (t = 0 seconds) = 0.1% to 𝜀c (t = 33.7 ks) = 0.2%; the time-dependent fraction of the interface debonding increases from 𝜂 (t = 0 seconds) = 0.07 to 𝜂 (t = 33.7 ks) = 0.27; the time-dependent fraction of the interface oxidation increases from 𝜔 (t = 0 seconds) = 0 to 𝜔 (t = 33.7 ks) = 0.61; and the time-dependent fraction of the broken fiber increases from q (t = 0 seconds) = 1 × 10−6 to q (t = 33.7 ks) = 0.22.

6.4 Stress Rupture of Ceramic-Matrix Composites Under Multiple Load Sequence

(c)

(d)

Figure 6.28

(Continued)

Under 𝜎 max1 = 100 MPa and 𝜎 max2 = 110 MPa, the time-dependent composite strain increases from 𝜀c (t = 0 seconds) = 0.1% to 𝜀c (t = 7.2 ks) = 0.12% under 𝜎 max1 = 100 MPa; upon increasing the stress level to 𝜎 max2 = 110 MPa at t = 7.2 ks, the time-dependent composite strain increases to 𝜀c (t = 7.2 ks) = 0.14% and increases to 𝜀c (t = 13.4 ks) = 0.168%. The time-dependent fraction of the interface debonding increases from 𝜂 (t = 0 seconds) = 0.07 to 𝜂 (t = 7.2 ks) = 0.116 under 𝜎 max1 = 100 MPa; upon increasing stress level to 𝜎 max2 = 110 MPa at t = 7.2 ks, the time-dependent fraction of the interface debonding increases to 𝜂 = 0.15 and then increases to 𝜂 (t = 13.4 ks) = 0.19. The time-dependent fraction of the interface oxidation increases from 𝜔 (t = 0 seconds) = 0 to 𝜔 (t = 7.2 ks) = 0.31 under 𝜎 max1 = 100 MPa; upon increasing stress level to 𝜎 max2 = 110 MPa at t = 7.2 ks, the time-dependent fraction of the interface oxidation decreases to 𝜔 (t = 7.2 ks) = 0.28 and increases to 𝜔 (t = 13.4 ks) = 0.41. The time-dependent fraction of the fiber

299

300

6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

(a)

(b)

Figure 6.29 (a) Experimental and predicted composite strain versus time curves; (b) the fraction of the interface debonding versus time curves; (c) the fraction of the interface oxidation versus time curves; and (d) the fraction of the broken fiber versus time curves of SiC/SiC composite under single loading stress 𝜎 max = 120 MPa, and multiple loading stress 𝜎 max1 = 120 MPa and 𝜎 max2 = 130 MPa at T = 950 ∘ C in air atmosphere.

broken increases from q (t = 0 seconds) = 0 to q (t = 7.2 ks) = 0.018 under 𝜎 max1 = 100 MPa; upon increasing stress level to 𝜎 max2 = 110 MPa at t = 7.2 ks, the time-dependent fraction of the fiber broken increases to q (t = 7.2 ks) = 0.034 and increases to q (t = 13.4 ks) = 0.084. Under 𝜎 max = 120 MPa, the experimental time-dependent composite strain increases from 𝜀c (t = 0.73 ks) = 0.17% to 𝜀c (t = 9.3 ks) = 0.26%; the predicted time-dependent composite strain increases from 𝜀c (t = 0 seconds) = 0.16% to 𝜀c (t = 10.5 ks) = 0.27%; the time-dependent fraction of the interface debonding increases from 𝜂 (t = 0 seconds) = 0.19 to 𝜂 (t = 10.5 ks) = 0.42; the time-dependent fraction of the interface oxidation increases from 𝜔 (t = 0 seconds) = 0 to 𝜔

6.4 Stress Rupture of Ceramic-Matrix Composites Under Multiple Load Sequence

(c)

(d)

Figure 6.29

(Continued)

(t = 10.5 ks) = 0.4; and the time-dependent fraction of the broken fiber increases from q (t = 0 seconds) = 2 × 10−6 to q (t = 10.5 ks) = 0.17. Under 𝜎 max1 = 120 MPa and 𝜎 max2 = 130 MPa, the time-dependent composite strain increases from 𝜀c (t = 0 seconds) = 0.16% to 𝜀c (t = 3.6 ks) = 0.19% under 𝜎 max1 = 120 MPa; upon increasing stress level to 𝜎 max2 = 130 MPa at t = 3.6 ks, the time-dependent composite strain increases to 𝜀c = 0.23% and increases to 𝜀c (t = 6.8 ks) = 0.28%. The time-dependent fraction of the interface debonding increases from 𝜂 (t = 0 seconds) = 0.19 to 𝜂 (t = 3.6 ks) = 0.26 under 𝜎 max1 = 120 MPa; upon increasing stress level to 𝜎 max2 = 130 MPa at t = 3.6 ks, the time-dependent fraction of the interface debonding increases to 𝜂 = 0.31 and then increases to 𝜂 (t = 6.8 ks) = 0.41. The time-dependent fraction of the interface oxidation increases from 𝜔 (t = 0 seconds) = 0 to 𝜔 (t = 3.6 ks) = 0.22 under 𝜎 max1 = 120 MPa; upon

301

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6 Stress Rupture of Ceramic-Matrix Composites at Elevated Temperature

increasing stress level to 𝜎 max2 = 130 MPa at t = 3.6 ks, the time-dependent fraction of the interface oxidation decreases to 𝜔 (t = 3.6 ks) = 0.2 and increases to 𝜔 (t = 6.8 ks) = 0.28. The time-dependent fraction of the fiber broken increases from q (t = 0 seconds) = 0 to q (t = 3.6 ks) = 0.03 under 𝜎 max1 = 120 MPa; upon increasing stress level to 𝜎 max2 = 130 MPa at t = 3.6 ks, the time-dependent fraction of the fiber broken increases to q = 0.05 and increases to q (t = 6.8 ks) = 0.15.

6.5 Conclusion In this chapter, stress rupture of fiber-reinforced CMCs at elevated environmental temperatures is investigated. Time-dependent damage mechanisms were considered in the analysis of composite’s microstress field and tensile constitutive relationship. Relationships between stress rupture lifetime, peak stress level, time-dependent composite deformation, and evolution of internal damages are established. Experimental stress rupture lifetime and internal damage evolution of SiC/SiC composite are evaluated. ●

●

When the stochastic stress level increases, the stress rupture time decreases; the time for the interface complete debonding and oxidation decreases; and the broken fiber fraction at the stage of stochastic loading increases. When the stochastic loading time interval increases, the stress rupture time decreases; and the time for the interface complete debonding and oxidation remains the same. When temperature increased, the stress rupture lifetime decreased; the time-dependent composite strain increased at the same time; the time-dependent fraction of the interface debonding increased for partially interface debonding, and the time for complete interface debonding decreased; the time-dependent fraction of the interface oxidation increased for partially interface debonding, and the time for complete interface oxidation decreased; the time-dependent fraction of the fiber broken increased.

References 1 Li, L.B. (2018). Damage, Fracture and Fatigue of Ceramic-Matrix Composites. Singapore: Springer Nature. 2 Li, L.B. (2018). Fatigue life prediction of ceramic-matrix composites. Aircr. Eng. Aerosp. Technol. 90: 720–726. https://doi.org/10.1108/AEAT-01-2016-0014. 3 Li, L.B. (2019). Failure analysis of long-fiber-reinforced ceramic-matrix composites subjected to in-phase thermomechanical and isothermal cyclic loading. Eng. Fail. Anal. 104: 856–872. https://doi.org/10.1016/j.engfailanal.2019.06.082. 4 Naslain, R. (2004). Design, preparation and properties of non-oxide CMCs for application in engines and nuclear reactors: an overview. Compos. Sci. Technol. 64: 155–170. https://doi.org/10.1016/S0266-3538(03)00230-6. 5 Fantozzi, G. and Reynaud, P. (2009). Mechanical hysteresis in ceramic matrix composites. Mater. Sci. Eng., A 521–522: 18–23. https://doi.org/10.1016/j.msea .2008.09.128.

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7 Vibration Damping of Ceramic-Matrix Composites at Elevated Temperature 7.1

Introduction

Ceramic-matrix composites (CMCs) possess high specific strength and modulus and high temperature resistance and are the candidate materials for hot-section components of aerospace vehicles, aeroengines, and thermal protection systems (TPS) [1–3]. Silicon carbide fiber-reinforced silicon carbide CMCs (SiC/SiC) and oxide fiber-reinforced oxide CMCs (Oxide/Oxide) have already been used in aeroengines. SiC/SiC composite consists of the SiC fiber with a diameter of 12 μm, the interface layer with a thickness of 0.2–0.5 μm, and the SiC matrix. SiC/SiC composite has high oxidation resistance, lightweight (i.e. density 2.1–2.8 g/cm3 ), and high-temperature (1200–1400 ∘ C) gas lifetime up to thousands of hours and is the most ideal material for the hot-section structures of aeroengines [4–6]. In 2010, General Electric (GE) aviation carried out turbine rotor blade test with an F414 engine as the verification platform to build a ceramic aeroengine [6]. In 2016, SiC/SiC composite material was first used in the turbine outer ring of Leading Edge Aviation Propulsion (LEAP) aeroengine and has been produced in batches. Then, this material was applied in the combustion chamber, guide vane, turbine outer ring, and other structures of the new GE9x commercial aeroengine. The fuel consumption rate was 10% lower than that of GE90-115B [7, 8]. However, in the above applications, failure analysis shows that approximately two-thirds of the failures are related to vibration and noise, leading to reduced operational control accuracy, structural fatigue damage, and shortened safety life [9, 10]. Therefore, studying the damping performance of fiber-reinforced CMCs and improving their reliability in the service environment is an important guarantee for the safe service of CMCs in various fields. Compared with metals and alloys, CMCs have many unique damping mechanisms because of internal structure and complex damage mechanisms [11–15]. Hao et al. [16] performed computational and experimental analysis on the modal parameters and vibration response of C/SiC bolted fastenings. The composite vibration damping affects the vibration amplitude of CMC components. Zhang [17] investigated the vibration characteristics of a CMC panel subjected to high temperature and large gradient thermal environment and performed damping measurement experiments of CMC panels in the thermal environment. The effect of thermal environment on High Temperature Mechanical Behavior of Ceramic-Matrix Composites, First Edition. Longbiao Li. © 2021 WILEY-VCH GmbH. Published 2021 by WILEY-VCH GmbH.

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the natural frequency and vibration damping of CMCs was analyzed. Huang and Wu [18] performed natural frequencies and acoustic emission testing of 2D SiC/SiC composite subjected to tensile loading. The natural frequencies decrease with increasing tensile stress because of internal damages. Natural frequency and vibration damping are affected by the damage state of CMCs [19]. Energy dissipation of frictional sliding between matrix cracks is the main mechanism for the damping of unidirectional or cross-ply CMCs at room temperature [20]. The damping of CMCs is also affected by the fabrication method [21], property of the fiber [21], internal damages inside of CMCs [12, 22], and interphase thickness [23]. At elevated temperature, the temperature-dependent damage mechanisms affect the mechanical behavior and vibration damping of CMCs [24–29]. In this chapter, a micromechanical temperature-dependent vibration damping model of fiber-reinforced CMCs is developed. Temperature-dependent damage mechanisms of matrix cracking, interface debonding and slip, and fiber fracture contribute to the vibration damping of damaged CMCs. Temperature-dependent fiber and matrix strain energy and dissipated energy density are formulated of composite constituent properties and damage-related microparameters of matrix crack spacing, interface debonding and slip length, and broken fiber fraction. Relationships between temperature-dependent composite damping, temperature-dependent damage mechanisms, temperature, and oxidation duration are established. Effects of composite constituent properties and composite damage state on temperature-dependent composite vibration damping of SiC/SiC and C/SiC composites are analyzed. Experimental temperature-dependent composite vibration damping of 2D SiC/SiC and C/SiC composites are predicted.

7.2

Temperature-Dependent Vibration Damping of CMCs

7.2.1

Theoretical Models

Temperature-dependent vibration damping Ωx (T) of a composite without damage subject to axial loading is [30] Ωx (T) =

Ef (T)Vf Ωf + Em (T)Vm Ωm Ef (T)Vf + Em (T)Vm

(7.1)

where V f and V m are the volume of the fiber and the matrix, Ωf and Ωm are the damping of the fiber and matrix materials, and Ef (T) and Em (T) are the temperature-dependent fiber and matrix elastic modulus. When temperature-dependent damage mechanisms of matrix cracking, interface debonding and slip, and fiber failure occur, the temperature-dependent vibration damping Ωy (T) is [20] Ωy (T) =

Ud (T) 2𝜋U(T)

(7.2)

where U d (T) is the temperature-dependent energy-dissipated density per cycle of motion, and U(T) is the temperature-dependent maximum strain energy density

7.2 Temperature-Dependent Vibration Damping of CMCs

during a cycle. Ud (T) = Ud

unloading (T)

+ Ud

reloading (T)

(7.3) (7.4)

U(T) = Uf (T) + Um (T)

where U d_unloading (T) and U d_reloading (T) are the temperature-dependent dissipated energy density upon unloading and reloading, respectively, and U f (T) and U m (T) are the temperature-dependent fiber and matrix strain energy, respectively. [ ] 𝜏 (T) 8 Ec (T)𝜏i (T) 3 (Φ(T) − ΦU (T))l2y (T) − ly (T) Ud unloading (T) = 2𝜋rf i Ef (T) 3 rf Vm Em (T) (7.5) [ ] 𝜏 (T) 8 Ec (T)𝜏i (T) 3 (Φ(T) − ΦR (T))l2z (T) − l (T) (7.6) Ud reloading (T) = 2𝜋rf i Ef (T) 3 rf Vm Em (T) z {

( ) 2 2 𝜎fo (T) lc (T) 𝜏 (T)Φ(T) 2 Φ2 (T) 4 𝜏i (T) 3 l (T) + (T) ld (T) − 2 i ld (T) + − l d Ef (T) rf Ef (T) 3 r 2 Ef (T) d Ef 2 f )( [ ]) ( l (T) l (T)∕2 − ld (T) r 𝜎 (T) 𝜏 (T) 1 − exp −𝜌 c Φ(T) − 𝜎fo (T) − 2 d + 2 f fo 𝜌Ef (T) rf i rf ( [ ])} )2 ( lc (T)∕2 − ld (T) ld (T) rf Φ(T) − 𝜎fo (T) − 2 1 − exp −2𝜌 𝜏i (T) + 2𝜌Ef (T) rf rf

Uf (T) = 𝜋rf2

(7.7) ) 2 2 𝜎 2 (T) lc (T) 4 Vf 𝜏i (T) 3 Um (T) = 𝜋rf2 ld (T) + mo − ld (T) 2 2 3 r Vm Em (T) Em (T) 2 f ][ [ ]] [ l (T) 𝜌(l (T)∕2 − ld (T)) r V 𝜎 (T) 1 − exp − c Φ(T) − 𝜎fo (T) − 2𝜏i (T) d − 2 f f mo 𝜌Vm Em (T) rf rf ] [ [ ]]} [ rf Vf2 𝜌(lc (T)∕2 − ld (T)) ld (T) 2 1 − exp −2 Φ(T) − 𝜎 (T) − 2𝜏 (T) + fo i rf rf 2𝜌Vm2 Em (T) {

(

(7.8)

where r f is the fiber radius, 𝜏 i (T) is temperature-dependent interface shear stress, ld (T) is temperature-dependent interface debonding length, ly (T) and lz (T) are temperature-dependent counter slip length upon unloading and new slip length upon reloading, respectively, F(T) is temperature-dependent intact fiber stress, FU (T) and FR (T) are temperature-dependent unloading and reloading intact fiber stress, 𝜌 is shear-lag model parameter, and 𝜎 fo (T) and 𝜎 mo (T) are temperature-dependent fiber and matrix axial stress in the interface bonding region, respectively. Ef (T) 𝜎 + Ef (T)(𝛼lc (T) − 𝛼lf (T))ΔT Ec (T) E (T) 𝜎 + Em (T)(𝛼lc (T) − 𝛼lm (T))ΔT 𝜎mo (T) = m Ec (T) 𝜎fo (T) =

(7.9) (7.10)

where 𝛼 lc (T) is temperature-dependent composite thermal expansion coefficient and ΔT denotes temperature difference between testing temperature (T) and fabricated temperature (T 0 ).

309

310

7 Vibration Damping of Ceramic-Matrix Composites at Elevated Temperature

Temperature-dependent damping of CMCs with matrix cracking, interface debonding, and fiber failure is (7.11)

Ωc (T) = Ωx (T) + Ωy (T)

Two parameters of interface debonding fraction 𝜂 and interface slip fraction 𝛿 are introduced to characterize the interface damage state inside of CMCs: 𝜂(T) = 2

7.2.2

ld (T) , lc

𝛿(T) = 2

ly (T) lc

(7.12)

Results and Discussion

Effects of material properties and damage state on temperature-dependent vibration damping of SiC/SiC composite are analyzed. Material properties of SiC/SiC composite are given by V f = 0.3, r f = 7 μm, T 0 = 1248 K, 𝛼 rf = 2.9 × 10−6 K−1 , and 𝛼 lf = 3.9 × 10−6 K−1 . 7.2.2.1 Effect of Fiber Volume on Temperature-Dependent Vibration Damping of SiC/SiC Composite

Figures 7.1 and 7.2 and Table 7.1 show the effect of fiber volume (i.e. V f = 20% and 30%) on temperature-dependent composite vibration damping Ωc , the fraction of interface debonding 𝜂, and the fraction of interface slip 𝛿 versus temperature curves of SiC/SiC composite under 𝜎 = 30, 40, and 50 MPa. When temperature increases, the interface shear stress of SiC/SiC composite decreases because of the misfit between the fiber and the matrix radial thermal expansional coefficient [27], the composite vibration damping increases to the peak value and then decreases, and the interface debonding and slip fraction increases with the temperature. When the fiber volume increases, the interface debonding and slip range decrease, the composite vibration damping decreases, and the temperature for the peak value of the composite damping increases, mainly because of the decrease of the fraction of the interface debonding and slip range. Under the same fiber volume, the composite damping increases with the vibration stress, as the interface debonding and slip range increases with vibration stress, and the temperature for the peak of composite damping decreases with increasing temperature. To improve the vibration damping of SiC/SiC composite, the fiber volume should be low, in order to increase the interface debonding and slip range at the vibration stress. However, low fiber volume decreases the proportional limit stress [31, 32], and it is necessary to get the balance design of CMCs between the design stress and damping capacity. When V f = 20% and 𝜎 = 30 MPa, the fiber broken fraction is q = 0.05%, the temperature-dependent composite vibration damping increases from Ωc (T = 293 K) = 0.006 to the peak value Ωc (T = 1303 K) = 0.0425 and then decreases to Ωc (T = 1400 K) = 0.0406; the fraction of the interface debonding increases from 𝜂 (T = 293 K) = 0.18 to 𝜂 (T = 1400 K) = 0.35, and the fraction of the interface slip increases from 𝛿 (T = 293 K) = 0.18 to 𝛿 (T = 1400 K) = 0.22. Under 𝜎 = 40 MPa, the broken fiber fraction is q = 0.16%, the temperature-dependent composite

7.2 Temperature-Dependent Vibration Damping of CMCs

Figure 7.1 (a) Composite damping versus temperature curves; (b) the fraction of the interface debonding versus temperature curves; and (c) the fraction of the interface slip versus temperature curves of SiC/SiC composite under different vibration stress when V f = 20%.

(a)

(b)

(c)

311

312

7 Vibration Damping of Ceramic-Matrix Composites at Elevated Temperature

Figure 7.2 (a) Composite damping versus temperature curves; (b) the fraction of interface debonding versus temperature curves; and (c) the fraction of the interface slip versus temperature curves of SiC/SiC composite under different vibration stress when V f = 30%.

(a)

(b)

(c)

7.2 Temperature-Dependent Vibration Damping of CMCs

Table 7.1 Effect of fiber volume on composite damping, interface debonding, and slip fraction of SiC/SiC composite under different vibration stress. 𝝈 (MPa)

q (%)

T (K)

𝛀c

𝜼

𝜹

0.05

293

0.0060

0.18

0.18

1303

0.0425

0.324

0.219

1400

0.0406

0.35

0.22

V f = 20% 30

40

50

0.16

0.39

293

0.0140

0.31

0.26

1266

0.0428

0.46

0.29

1400

0.041

0.5

0.29

293

0.024

0.44

0.32

1185

0.043

0.58

0.36

1400

0.041

0.65

0.37

293

0.0005

0.04

0.04

1326

0.023

0.16

0.13

1400

0.021

0.18

0.132

293

0.002

0.12

0.12

1325

0.0263

0.25

0.174

1400

0.025

0.27

0.176

V f = 30% 30

40

50

0.01

0.03

0.07

293

0.006

0.2

0.19

1324

0.0287

0.34

0.21

1400

0.0282

0.36

0.22

vibration damping increases from Ωc (T = 293 K) = 0.014 to the peak value Ωc (T = 1266 K) = 0.0428 and then decreases to Ωc (T = 1400 K) = 0.041; the fraction of the interface debonding increases from 𝜂 (T = 293 K) = 0.31 to 𝜂 (T = 1400 K) = 0.5, and the fraction of the interface slip increases from 𝛿 (T = 293 K) = 0.26 to 𝛿 (T = 1400 K) = 0.29. Under 𝜎 = 50 MPa, the broken fiber fraction is q = 0.39%, the temperature-dependent composite vibration damping increases from Ωc (T = 293 K) = 0.024 to the peak value Ωc (T = 1185 K) = 0.0434 and then decreases to Ωc (T = 1400 K) = 0.0415; the fraction of the interface debonding increases from 𝜂 (T = 293 K) = 0.44 to 𝜂 (T = 1400 K) = 0.65, and the fraction of the interface slip increases from 𝛿 (T = 293 K) = 0.32 to 𝛿 (T = 1400 K) = 0.37. When V f = 30% and 𝜎 = 30 MPa, the broken fiber fraction is q = 0.01%, the temperature-dependent composite vibration damping increases from Ωc (T = 293 K) = 0.0005 to the peak value Ωc (T = 1326 K) = 0.023 and then decreases to Ωc (T = 1400 K) = 0.021; the fraction of the interface debonding increases from 𝜂 (T = 293 K) = 0.04 to 𝜂 (T = 1400 K) = 0.18, and the fraction of the interface slip increases from 𝛿 (T = 293 K) = 0.04 to 𝛿 (T = 1400 K) = 0.13. Under

313

314

7 Vibration Damping of Ceramic-Matrix Composites at Elevated Temperature

𝜎 = 40 MPa, the broken fiber fraction is q = 0.03%, the temperature-dependent composite vibration damping increases from Ωc (T = 293 K) = 0.002 to the peak value Ωc (T = 1325 K) = 0.0263 and then decreases to Ωc (T = 1400 K) = 0.025; the fraction of the interface debonding increases from 𝜂 (T = 293 K) = 0.12 to 𝜂 (T = 1400 K) = 0.27, and the fraction of the interface slip increases from 𝛿 (T = 293 K) = 0.12 to 𝛿 (T = 1400 K) = 0.17. Under 𝜎 = 50 MPa, the broken fiber fraction is q = 0.07%, the temperature-dependent composite vibration damping increases from Ωc (T = 293 K) = 0.006 to the peak value Ωc (T = 1324 K) = 0.0287 and then decreases to Ωc (T = 1400 K) = 0.0282; the fraction of the interface debonding increases from 𝜂 (T = 293 K) = 0.2 to 𝜂 (T = 1400 K) = 0.36, and the fraction of the interface slip increases from 𝛿 (T = 293 K) = 0.19 to 𝛿 (T = 1400 K) = 0.22. 7.2.2.2 Effect of Matrix Crack Spacing on Temperature-Dependent Vibration Damping of SiC/SiC Composite

Figures 7.3 and 7.4 and Table 7.2 show the effect of matrix crack spacing (i.e. lc = 300 and 400 μm) on temperature-dependent composite vibration damping, the fraction of interface debonding, and slip versus temperature curves of SiC/SiC composite under 𝜎 = 30, 40, and 50 MPa. When the matrix crack spacing increases, the interface debonding and slip occupy few range between the matrix crack spacing, and the composite vibration damping decreases, and the temperature for the peak damping increases, mainly because of the decrease of the interface debonding and sliding range. To improve the composite vibration damping, it is better to improve the matrix cracking density of CMCs at the same vibration stress. When lc = 300 μm and 𝜎 = 30 MPa, the broken fiber fraction is q = 0.01%; the temperature-dependent composite vibration damping increases from Ωc (T = 293 K) = 0.0003 to the peak value Ωc (T = 1326 K) = 0.0175 and then decreases to Ωc (T = 1400 K) = 0.0163; the fraction of the interface debonding increases from 𝜂 (T = 293 K) = 0.029 to 𝜂 (T = 1400 K) = 0.124, and the fraction of the interface slip increases from 𝛿 (T = 293 K) = 0.029 to 𝛿 (T = 1400 K) = 0.088. Under 𝜎 = 40 MPa, the broken fiber fraction is q = 0.03%, the temperature-dependent composite vibration damping increases from Ωc (T = 293 K) = 0.001 to the peak value Ωc (T = 1341 K) = 0.0206 and then decreases to Ωc (T = 1400 K) = 0.0201; the fraction of the interface debonding increases from 𝜂 (T = 293 K) = 0.081 to 𝜂 (T = 1400 K) = 0.183, and the fraction of the interface slip increases from 𝛿 (T = 293 K) = 0.081 to 𝛿 (T = 1400 K) = 0.117. Under 𝜎 = 50 MPa, the broken fiber fraction is q = 0.07%, the temperature-dependent composite damping increases from Ωc (T = 293 K) = 0.004 to the peak value Ωc (T = 1350 K) = 0.023 and then decreases to Ωc (T = 1400 K) = 0.0228; the fraction of the interface debonding increases from 𝜂 (T = 293 K) = 0.133 to 𝜂 (T = 1400 K) = 0.242, and the fraction of the interface slip increases from 𝛿 (T = 293 K) = 0.129 at T = 293 K to 𝛿 (T = 1400 K) = 0.146. When lc = 400 μm and 𝜎 = 30 MPa, the broken fiber fraction is q = 0.01%, the temperature-dependent composite damping increases from Ωc (T = 293 K) = 0.0003 to the peak value Ωc (T = 1328 K) = 0.014 and then decreases to Ωc (T = 1400 K) = 0.013; the fraction of the interface debonding increases from 𝜂 (T = 293 K) = 0.022

7.2 Temperature-Dependent Vibration Damping of CMCs

Figure 7.3 (a) Composite damping versus temperature curves; (b) the fraction of interface debonding versus temperature curves; and (c) the fraction of interface slip versus temperature curves of SiC/SiC composite under different vibration stress when lc = 300 μm.

(a)

(b)

(c)

315

316

7 Vibration Damping of Ceramic-Matrix Composites at Elevated Temperature

Figure 7.4 (a) Composite damping versus temperature curves; (b) the fraction of interface debonding versus temperature curves; and (c) the fraction of interface slip versus temperature curves of SiC/SiC composite under different vibration stress when lc = 400 μm.

(a)

(b)

(c)

7.2 Temperature-Dependent Vibration Damping of CMCs

Table 7.2 Effect of matrix crack spacing on composite vibration damping, interface debonding, and slip fraction of SiC/SiC composite under different vibration stress. 𝝈 (MPa)

q (%)

T (K)

𝛀c

𝜼

𝜹

0.01

293

0.0003

0.029

0.029

1326

0.0175

0.110

0.087

1400

0.0163

0.124

0.088

lc = 300 𝜇m 30

40

50

0.03

0.07

293

0.001

0.081

0.081

1341

0.0206

0.171

0.116

1400

0.0201

0.183

0.117

293

0.0042

0.133

0.129

1350

0.0230

0.231

0.145

1400

0.0228

0.242

0.146

lc = 400 𝜇m 30

40

50

0.01

0.03

0.07

293

0.0003

0.022

0.022

1328

0.0140

0.083

0.065

1400

0.0131

0.093

0.066

293

0.001

0.061

0.061

1348

0.0169

0.129

0.087

1400

0.0166

0.137

0.088

293

0.003

0.1

0.09

1363

0.0193

0.175

0.109

1400

0.0191

0.181

0.11

to 𝜂 (T = 1400 K) = 0.093, and the fraction of the interface slip increases from 𝛿 (T = 293 K) = 0.022 to 𝛿 (T = 1400 K) = 0.066. Under 𝜎 = 40 MPa, the broken fiber fraction is q = 0.03%, the temperature-dependent composite damping increases from Ωc (T = 293 K) = 0.001 to the peak value Ωc (T = 1348 K) = 0.0169 and then decreases to Ωc (T = 1400 K) = 0.0166; the fraction of the interface debonding increases from 𝜂 (T = 293 K) = 0.061 to 𝜂 (T = 1400 K) = 0.137, and the fraction of the interface slip increases from 𝛿 (T = 293 K) = 0.061 to 𝛿 (T = 1400 K) = 0.088. Under 𝜎 = 50 MPa, the broken fiber fraction is q = 0.07%, the temperature-dependent composite damping increases from Ωc (T = 293 K) = 0.003 to the peak value Ωc (T = 1363 K) = 0.0193 and then decreases to Ωc (T = 1400 K) = 0.0191; the fraction of the interface debonding increases from 𝜂 (T = 293 K) = 0.1 to 𝜂 (T = 293 K) = 0.181, and the fraction of the interface slip increases from 𝛿 (T = 293 K) = 0.09 to 𝛿 (T = 1400 K) = 0.11. 7.2.2.3 Effect of Interface Debonding Energy on Temperature-Dependent Vibration Damping of SiC/SiC Composite

Figures 7.5 and 7.6 and Table 7.3 show the effect of interface debonding energy (i.e. Γd = 0.05 and 0.15 J/m2 ) on the temperature-dependent composite vibration

317

318

7 Vibration Damping of Ceramic-Matrix Composites at Elevated Temperature

Figure 7.5 (a) Composite damping versus temperature curves; (b) the fraction of the interface debonding versus temperature curves; and (c) the fraction of the interface slip versus temperature curves of SiC/SiC composite under different vibration stress when Γd = 0.05 J/m2 .

(a)

(b)

(c)

7.2 Temperature-Dependent Vibration Damping of CMCs

Figure 7.6 (a) Composite damping versus temperature curves; (b) the fraction of the interface debonding versus temperature curves; and (c) the fraction of the interface slip versus temperature curves of SiC/SiC composite under different vibration stress when Γd = 0.15 J/m2 .

(a)

(b)

(c)

319

320

7 Vibration Damping of Ceramic-Matrix Composites at Elevated Temperature

Table 7.3 Effect of interface debonding energy on composite damping, interface debonding, and slip fraction of SiC/SiC composite under different vibration stress. 𝝈 (MPa)

q (%)

T (K)

𝛀c

𝜼

𝜹

0.01

293

0.001

0.092

0.092

1324

0.0224

0.185

0.131

1400

0.021

0.201

0.132

𝛤 d = 0.05 J/m2 30

40

50

0.03

0.07

293

0.003

0.17

0.155

1337

0.0256

0.27

0.174

1400

0.0251

0.28

0.176

293

0.007

0.248

0.194

1338

0.028

0.363

0.218

1400

0.027

0.377

0.22

293

0.0001

0.0069

0.0069

1317

0.0238

0.146

0.13

1400

0.022

0.176

0.132

293

0.0016

0.0847

0.0847

1320

0.0269

0.236

0.174

1400

0.0260

0.264

0.176

𝛤 d = 0.15 J/m2 30

40

50

0.01

0.03

0.07

293

0.005

0.162

0.162

1322

0.0292

0.321

0.218

1400

0.028

0.352

0.220

damping, the fraction of the interface debonding and slip fraction versus the temperature curves of SiC/SiC composite under 𝜎 = 30, 40, and 50 MPa. When the interface debonding energy increases, the energy needs to debonding at the interface increases, and the interface debonding and slip length at the same applied stress decreases, and the composite damping decreases, and the temperature for peak damping also decreases because of the decrease of the fraction of the interface debonding and slip. To improve the damping capacity of the composite, it is better to design the interface as weak bonding between the fiber and the matrix. When Γd = 0.05 J/m2 and 𝜎 = 30 MPa, the broken fiber fraction is q = 0.01%, the temperature-dependent composite vibration damping increases from Ωc (T = 293 K) = 0.0003 to the peak value Ωc (T = 1326 K) = 0.0175 and then decreases to Ωc (T = 1400 K) = 0.0163; the fraction of the interface debonding increases from 𝜂 (T = 293 K) = 0.029 to 𝜂 (T = 1400 K) = 0.124, and the fraction of the interface slip increases from 𝛿 (T = 293 K) = 0.029 to 𝛿 (T = 1400 K) = 0.088. Under 𝜎 = 40 MPa, the broken fiber fraction is q = 0.03%, the temperature-dependent

7.2 Temperature-Dependent Vibration Damping of CMCs

composite vibration damping increases from Ωc (T = 293 K) = 0.003 to the peak value Ωc (T = 1337 K) = 0.0256 and then decreases to Ωc (T = 1400 K) = 0.0251; the fraction of the interface debonding increases from 𝜂 (T = 293 K) = 0.17 to 𝜂 (T = 1400 K) = 0.28, and the fraction of the interface slip increases from 𝛿 (T = 293 K) = 0.155 to 𝛿 (T = 1400 K) = 0.176. Under 𝜎 = 50 MPa, the broken fiber fraction is q = 0.07%, the temperature-dependent composite vibration damping increases from Ωc (T = 293 K) = 0.007 to the peak value Ωc (T = 1338 K) = 0.028 and then decreases to Ωc (T = 1400 K) = 0.027; the fraction of the interface debonding increases from 𝜂 (T = 293 K) = 0.248 to 𝜂 (T = 1400 K) = 0.377, and the fraction of the interface slip increases from 𝛿 (T = 293 K) = 0.194 to 𝛿 (T = 1400 K) = 0.22. When Γd = 0.15 J/m2 and 𝜎 = 30 MPa, the broken fiber fraction is q = 0.01%, the temperature-dependent composite damping increases from Ωc (T = 293 K) = 0.0001 to the peak value Ωc (T = 1317 K) = 0.0238 and then decreases to Ωc (T = 1400 K) = 0.022; the fraction of the interface debonding increases from 𝜂 (T = 293 K) = 0.0069 to 𝜂 (T = 1400 K) = 0.176, and the fraction of the interface slip increases from 𝛿 (T = 293 K) = 0.0069 to 𝛿 (T = 1400 K) = 0.132. Under 𝜎 = 40 MPa, the broken fiber fraction is q = 0.03%, the temperature-dependent composite vibration damping increases from Ωc (T = 293 K) = 0.0016 to the peak value Ωc (T = 1320 K) = 0.0269 and then decreases to Ωc (T = 1400 K) = 0.026; the fraction of the interface debonding increases from 𝜂 (T = 293 K) = 0.084 to 𝜂 (T = 1400 K) = 0.264, and the fraction of the interface slip increases from 𝛿 (T = 293 K) = 0.084 to 𝛿 (T = 1400 K) = 0.176. Under 𝜎 = 50 MPa, the broken fiber fraction is q = 0.07%, the temperature-dependent composite damping increases from Ωc (T = 293 K) = 0.005 to the peak value Ωc (T = 1322 K) = 0.029 and then decreases to Ωc (T = 1400 K) = 0.028; the fraction of the interface debonding increases from 𝜂 (T = 293 K) = 0.162 to 𝜂 (T = 1400 K) = 0.352, and the fraction of the interface slip increases from 𝛿 (T = 293 K) = 0.162 to 𝛿 (T = 1400 K) = 0.22. 7.2.2.4 Effect of Steady-State Interface Shear Stress on Temperature-Dependent Vibration Damping of SiC/SiC Composite

Figures 7.7 and 7.8 and Table 7.4 show the effect of steady-state interface shear stress (i.e. 𝜏 0 = 10 and 30 MPa) on temperature-dependent composite vibration damping, the fraction of the interface debonding, and slip fraction versus temperature curves of SiC/SiC composite under 𝜎 = 30, 40, and 50 MPa. When the steady-state interface shear stress increases, the energy needed for debonding at the interface between the fiber and the matrix increases, and the interface debonding and slip length decreases at the same applied stress, and the composite vibration damping decreases, and the temperature for peak damping increases. To improve the damping capacity of the composite, it is better to decrease the steady-state interface shear stress between the fiber and the matrix. However, low interface shear stress also decreases the proportional limit stress of the composite [32], which would decrease the design stress of composite. It needs to find the balance between the design stress and capacity of the composite. When 𝜏 0 = 10 MPa and 𝜎 = 30 MPa, the broken fiber fraction is q = 0.01%, the temperature-dependent composite vibration damping increases from

321

322

7 Vibration Damping of Ceramic-Matrix Composites at Elevated Temperature

Figure 7.7 (a) Composite damping versus temperature curves; (b) the fraction of the interface debonding versus temperature curves; and (c) the fraction of the interface slip versus temperature curves of SiC/SiC composite under different vibration stress when 𝜏 0 = 10 MPa.

(a)

(b)

(c)

7.2 Temperature-Dependent Vibration Damping of CMCs

Figure 7.8 (a) Composite damping versus temperature curves; (b) the fraction of the interface debonding versus temperature curves; and (c) the fraction of the interface slip versus temperature curves of SiC/SiC composite under different vibration stress when 𝜏 0 = 30 MPa.

(a)

(b)

(c)

323

324

7 Vibration Damping of Ceramic-Matrix Composites at Elevated Temperature

Table 7.4 Effect of steady-state interface shear stress on composite damping, interface debonding, and slip fraction of SiC/SiC composite under different vibration stress. 𝝈 (MPa)

q (%)

T (K)

𝛀c

𝜼

𝜹

0.01

293

0.001

0.097

0.097

1297

0.034

0.343

0.262

1400

0.031

0.41

0.269

𝜏 0 = 10 MPa 30

40

50

0.03

0.07

293

0.006

0.237

0.237

1273

0.0363

0.505

0.348

1400

0.0346

0.589

0.359

293

0.013

0.377

0.349

1217

0.0383

0.649

0.429

1400

0.0364

0.769

0.449

293

0.002

0.0233

0.0233

1327

0.0176

0.1

0.087

1400

0.0165

0.114

0.087

293

0.0017

0.077

0.077

1344

0.0208

0.161

0.116

1400

0.0204

0.172

0.116

𝜏 0 = 30 MPa 30

40

50

0.01

0.03

0.07

293

0.004

0.131

0.131

1353

0.0232

0.221

0.145

1400

0.023

0.230

0.145

Ωc (T = 293 K) = 0.001 to the peak value Ωc (T = 1297 K) = 0.034 and then decreases to Ωc (T = 1400 K) = 0.031; the fraction of the interface debonding increases from 𝜂 (T = 293 K) = 0.097 to 𝜂 (T = 1400 K) = 0.41, and the fraction of the interface slip increases from 𝛿 (T = 293 K) = 0.097 to 𝛿 (T = 1400 K) = 0.269. Under 𝜎 = 40 MPa, the broken fiber fraction is q = 0.03%, the temperature-dependent composite damping increases from Ωc (T = 293 K) = 0.006 to the peak value Ωc (T = 1273 K) = 0.036 and then decreases to Ωc (T = 1400 K) = 0.034; the fraction of the interface debonding increases from 𝜂 (T = 293 K) = 0.237 to 𝜂 (T = 1400 K) = 0.589, and the fraction of the interface slip increases from 𝛿 (T = 293 K) = 0.237 to 𝛿 (T = 1400 K) = 0.359. Under 𝜎 = 50 MPa, the broken fiber fraction is q = 0.07%, the temperature-dependent composite damping increases from Ωc (T = 293 K) = 0.013 to the peak value Ωc (T = 1217 K) = 0.038 and then decreases to Ωc (T = 1400 K) = 0.036; the fraction of the interface debonding increases from 𝜂 (T = 293 K) = 0.377 to 𝜂 (T = 1400 K) = 0.769, and the fraction of the interface slip increases from 𝛿 (T = 293 K) = 0.349 to 𝛿 (T = 1400 K) = 0.449.

7.2 Temperature-Dependent Vibration Damping of CMCs

When 𝜏 0 = 30 MPa and 𝜎 = 30 MPa, the broken fiber fraction is q = 0.01%, the temperature-dependent composite vibration damping increases from Ωc (T = 293 K) = 0.0002 to the peak value Ωc (T = 1327 K) = 0.0176 and then decreases to Ωc (T = 1400 K) = 0.016; the fraction of the interface debonding increases from 𝜂 (T = 293 K) = 0.023 to 𝜂 (T = 1400 K) = 0.114, and the fraction of the interface slip increases from 𝛿 (T = 293 K) = 0.023 to 𝛿 (T = 1400 K) = 0.087. Under 𝜎 = 40 MPa, the broken fiber fraction is q = 0.03%, the temperature-dependent composite vibration damping increases from Ωc (T = 293 K) = 0.0017 to the peak value Ωc (T = 1344 K) = 0.0208 and then decreases to Ωc (T = 1400 K) = 0.0204; the fraction of the interface debonding increases from 𝜂 (T = 293 K) = 0.077 to 𝜂 (T = 1400 K) = 0.172, and the fraction of the interface slip increases from 𝛿 (T = 293 K) = 0.077 to 𝛿 (T = 1400 K) = 0.116. Under 𝜎 = 50 MPa, the broken fiber fraction is q = 0.07%, the temperature-dependent composite vibration damping increases from Ωc (T = 293 K) = 0.004 to the peak value Ωc (T = 1344 K) = 0.0232 and then decreases to Ωc (T = 1400 K) = 0.023; the fraction of the interface debonding increases from 𝜂 (T = 293 K) = 0.131 to 𝜂 (T = 1400 K) = 0.23, and the fraction of the interface slip increases from 𝛿 (T = 293 K) = 0.131 to 𝛿 (T = 1400 K) = 0.145. 7.2.2.5 Effect of Interface Frictional Coefficient on Temperature-Dependent Vibration Damping of SiC/SiC Composite

Figures 7.9 and 7.10 and Table 7.5 show the effect of the interface frictional coefficient (i.e. 𝜇 = 0.03 and 0.05) on the temperature-dependent composite vibration damping, the fraction of the interface debonding, and slip versus temperature curves of SiC/SiC composite under 𝜎 = 30, 40, and 50 MPa. When the interface frictional coefficient increases, the composite damping increases, and the temperature for peak damping also increases. When 𝜇 = 0.03 and 𝜎 = 30 MPa, the broken fiber fraction is q = 0.01%, the temperature-dependent composite vibration damping increases from Ωc (T = 293 K) = 0.003 to the peak value Ωc (T = 1327 K) = 0.0234 and then decreases to Ωc (T = 1400 K) = 0.022; the fraction of the interface debonding increases from 𝜂 (T = 293 K) = 0.031 to 𝜂 (T = 1400 K) = 0.196, and the fraction of the interface slip increases from 𝛿 (T = 293 K) = 0.031 to 𝛿 (T = 1400 K) = 0.137. Under 𝜎 = 40 MPa, the broken fiber fraction is q = 0.03%, the temperature-dependent composite damping increases from Ωc (T = 293 K) = 0.002 to the peak value Ωc (T = 1340 K) = 0.0266 and then decreases to Ωc (T = 1400 K) = 0.0261; the fraction of the interface debonding increases from 𝜂 (T = 293 K) = 0.095 to 𝜂 (T = 1400 K) = 0.288, and the fraction of the interface slip increases from 𝛿 (T = 293 K) = 0.095 to 𝛿 (T = 1400 K) = 0.183. Under 𝜎 = 50 MPa, the broken fiber fraction is q = 0.07%, the temperature-dependent composite damping increases from Ωc (T = 293 K) = 0.005 to the peak value Ωc (T = 1344 K) = 0.029 and then decreases to Ωc (T = 1400 K) = 0.028; the fraction of the interface debonding increases from 𝜂 (T = 293 K) = 0.158 to 𝜂 (T = 1400 K) = 0.38, and the fraction of the interface slip increases from 𝛿 (T = 293 K) = 0.158 to 𝛿 (T = 1400 K) = 0.229. When 𝜇 = 0.05 and 𝜎 = 30 MPa, the broken fiber fraction is q = 0.01%, the temperature-dependent composite vibration damping increases from

325

326

7 Vibration Damping of Ceramic-Matrix Composites at Elevated Temperature

Figure 7.9 (a) Composite damping versus temperature curves; (b) the fraction of the interface debonding versus temperature curves; and (c) the fraction of the interface slip versus temperature curves of SiC/SiC composite under different vibration stress when 𝜇 = 0.03.

(a)

(b)

(c)

7.2 Temperature-Dependent Vibration Damping of CMCs

Figure 7.10 (a) Composite damping versus temperature curves; (b) the fraction of the interface debonding versus temperature curves; and (c) the fraction of the interface slip versus temperature curves of SiC/SiC composite under different vibration stress when 𝜇 = 0.05.

(a)

(b)

(c)

327

328

7 Vibration Damping of Ceramic-Matrix Composites at Elevated Temperature

Table 7.5 Effect of interface frictional coefficient on composite damping, interface debonding, and slip fraction of SiC/SiC composite under different vibration stress. 𝝈 (MPa)

q (%)

T (K)

𝛀c

𝜼

𝜹

0.01

293

0.003

0.031

0.031

1327

0.0234

0.169

0.133

1400

0.0222

0.196

0.137

𝜇 = 0.03 30

40

50

0.03

0.07

293

0.002

0.095

0.095

1340

0.0266

0.263

0.179

1400

0.0261

0.288

0.183

293

0.005

0.158

0.158

1344

0.029

0.355

0.224

1400

0.028

0.38

0.229

293

0.027

0.023

0.023

1335

0.0238

0.177

0.137

1400

0.0227

0.206

0.143

293

0.0017

0.076

0.076

1353

0.027

0.276

0.185

1400

0.026

0.301

0.191

𝜇 = 0.05 30

40

50

0.01

0.03

0.07

293

0.004

0.13

0.13

1362

0.0293

0.374

0.233

1400

0.0292

0.397

0.239

Ωc (T = 293 K) = 0.027 to the peak value Ωc (T = 1335 K) = 0.0238 and then decreases to Ωc (T = 1400 K) = 0.0227; the fraction of the interface debonding increases from 𝜂 (T = 293 K) = 0.023 to 𝜂 (T = 1400 K) = 0.206, and the fraction of the interface slip increases from 𝛿 (T = 293 K) = 0.023 to 𝛿 (T = 1400 K) = 0.143. Under 𝜎 = 40 MPa, the broken fiber fraction is q = 0.03%, the temperature-dependent composite damping increases from Ωc (T = 293 K) = 0.0017 to the peak value Ωc (T = 1353 K) = 0.027 and then decreases to Ωc (T = 1400 K) = 0.026; the fraction of the interface debonding increases from 𝜂 (T = 293 K) = 0.076 to 𝜂 (T = 1400 K) = 0.301, and the fraction of the interface slip increases from 𝛿 (T = 293 K) = 0.076 to 𝛿 (T = 1400 K) = 0.191. Under 𝜎 = 50 MPa, the broken fiber fraction is q = 0.07%, the temperature-dependent composite vibration damping increases from Ωc (T = 293 K) = 0.004 to the peak value Ωc (T = 1362 K) = 0.0293 and then decreases to Ωc (T = 1400 K) = 0.0292; the fraction of the interface debonding increases from 𝜂 (T = 293 K) = 0.13 to 𝜂 (T = 1400 K) = 0.397, and the fraction of the interface slip increases from 𝛿 (T = 293 K) = 0.13 to 𝛿 (T = 1400 K) = 0.239.

7.3 Time-Dependent Vibration Damping of CMCs

7.2.3

Experimental Comparisons

Sato et al. [21] investigated the vibration damping of 2D chemical vapor infiltration (CVI) SiC/SiC composite from T = 293 to 1373 K. Figure 7.11 shows the experimental and predicted composite vibration damping versus the temperature curves. The experimental composite vibration damping increases from Ωc (T = 293 K) = 0.0001 to Ωc (T = 1373 K) = 0.0017. Under 𝜎 = 5 MPa, the temperature-dependent composite vibration damping (Ωc ) increases from Ωc (T = 293 K) = 0.0001 to the peak value Ωc (T = 1400 K) = 0.0018; the fraction of the interface debonding increases from 𝜂 (T = 293 K) = 0.023 to 𝜂 (T = 1400 K) = 0.035, and the fraction of the interface slip increases from 𝛿 (T = 293 K) = 0.023 to 𝛿 (T = 1400 K) = 0.035. Under 𝜎 = 6 MPa, the temperature-dependent composite vibration damping increases from Ωc (T = 293 K) = 0.000 17 to the peak value Ωc (T = 1400 K) = 0.004; the fraction of the interface debonding increases from 𝜂 (T = 293 K) = 0.04 to 𝜂 (T = 1400 K) = 0.055, and the fraction of the interface slip increases from 𝛿 (T = 293 K) = 0.04 to 𝛿 (T = 1400 K) = 0.055. Under 𝜎 = 7 MPa, the temperature-dependent composite vibration damping increases from Ωc (T = 293 K) = 0.0002 to the peak value Ωc (T = 1400 K) = 0.006; the fraction of the interface debonding increases from 𝜂 (T = 293 K) = 0.058 to 𝜂 (T = 1400 K) = 0.075, and the fraction of the interface slip increases from 𝛿 (T = 293 K) = 0.058 to 𝛿 (T = 1400 K) = 0.0069.

7.3

Time-Dependent Vibration Damping of CMCs

Considering time- and temperature-dependent interface damages of oxidation, debonding, and slip, relationships between composite vibration damping, material properties, internal damages, oxidation duration, and temperature are established. Effects of material properties, vibration stress, damage state, and oxidation temperature on time-dependent composite vibration damping and interface damages of C/SiC composite are discussed. Experimental composite vibration damping and internal damages of 2D C/SiC composite for different oxidation durations t = 2, 5, and 10 hours at T = 700, 1000, and 1300 ∘ C are predicted.

7.3.1

Theoretical Models

The vibration damping of fiber-reinforced CMCs can be divided into two sections, as shown in Eqs. (7.1, 7.2). Time-dependent fiber and matrix strain energy density per cycle can be determined by Eqs. (7.13, 7.14). 1 𝜎 (t)𝜀f (t) dV 2 ∫V f { 2 𝜏2 Δ𝜎𝜏f 2 Δ𝜎 2 4 𝜏f 3 ld (t) − 2 𝜁 (t) + 𝜁 (t) + 4 2 f 𝜁 2 (t)[ld (t) − 𝜁(t)] =𝜋rf2 2 2 rf Vf Ef 3 r Ef Vf Ef rf Ef f

Uf (𝜎, t) =

+

2 Δ𝜎𝜏f Δ𝜎𝜏i 4 𝜏i [l (t) − 𝜁(t)]3 − 4 𝜁(t)[ld (t) − 𝜁(t)] − 2 [l (t) − 𝜁(t)]2 2 3 r Ef d rf Vf Ef rf Vf Ef d f

329

330

7 Vibration Damping of Ceramic-Matrix Composites at Elevated Temperature

Figure 7.11 (a) Experimental and predicted composite damping versus temperature curves; (b) the fraction of the interface debonding versus temperature curves; and (c) the fraction of the interface slip versus temperature curves of 2D SiC/SiC composite under different vibration stress.

(a)

(b)

(c)

7.3 Time-Dependent Vibration Damping of CMCs

+4

𝜏f 𝜏i rf2 Ef

𝜁(t)[ld (t) − 𝜁(t)]2 +

2 𝜎fo

Ef

[

]

lc − ld (t) 2

][ ( )] rf 𝜎fo Vm l ∕2 − ld (t) 𝜏 𝜏 𝜎mo − 2 f 𝜁(t) − 2 i (ld (t) − 𝜁(t)) 1 − exp −𝜌 c 𝜌Ef Vf rf rf rf ]2 [ [ ( )]} l ∕2 − ld (t) 𝜏 𝜏 Vm r 𝜎mo − 2 f 𝜁(t) − 2 i (ld (t) − 𝜁(t)) 1 − exp −2𝜌 c + f 2𝜌Ef Vf rf rf rf [

+2

(7.13) 1 𝜎 (t)𝜀m (t) dV 2 ∫V m { V 2𝜏2 V 2 𝜏f 𝜏i Vf 𝜏f =𝜋rf2 𝜁 2 (t) + 4 2 f 2 f 𝜁 2 (t)[ld (t) − 𝜁(t)] + 4 2 f 2 𝜁(t)[ld (t) − 𝜁(t)]2 rf Vm Em rf Vm Em rf Vm Em ] 2 2 2 [ lc 𝜎mo 4 Vf 𝜏i 3 + [l (t) − 𝜁(t)] + (t) − l d 3 r 2 Vm2 Em d Em 2 f [ )] ][ ( r𝜎 l ∕2 − ld (t) V𝜏 V𝜏 − 2 f mo 𝜎mo − 2 f f 𝜁(t) − 2 f i (ld (t) − 𝜁(t)) 1 − exp −𝜌 c 𝜌Em rf Vm rf Vm rf [ ( )]} ]2 [ lc ∕2 − ld (t) Vf 𝜏f Vf 𝜏i 1 rf 𝜎 −2 1 − exp −2𝜌 + 𝜁(t) − 2 (l (t) − 𝜁(t)) 2 𝜌Em mo rf Vm rf Vm d rf

Um (𝜎, t) =

(7.14)

and U(t) = Uf (t) + Um (t)

(7.15)

Upon unloading and reloading, the energy is dissipated through frictional slip at the fiber/matrix interface, and the time-dependent dissipated energy upon unloading and reloading can be determined by Eqs. (7.16, 7.17). [ ] 8 E c 𝜏f Δ𝜎 2 3 l (t) − l (t) (7.16) Ud u (t) = 2𝜋rf 𝜏f Vf Ef y 3 rf Vm Ef Em y [ ] 8 E c 𝜏f Δ𝜎 2 Ud r (t) = 2𝜋rf 𝜏f lz (t) − l3z (t) (7.17) Vf Ef 3 rf Vm Ef Em where Δ𝜎 is the range of applied stress and Ud (t) = Ud u (t) + Ud r (t)

7.3.2

(7.18)

Results and Discussion

Effects of material properties, vibration stress, damage state, and temperature on time-dependent composite vibration damping of C/SiC composite are discussed. 7.3.2.1 Effect of Fiber Volume on Time-Dependent Vibration Damping of C/SiC Composite

Figure 7.12 and Table 7.6 show the time-dependent composite vibration damping, the ratio of interface debonding, oxidation, and slip of C/SiC composite after oxidation duration t = 2 hours at T = 800 ∘ C for different fiber volumes.

331

332

7 Vibration Damping of Ceramic-Matrix Composites at Elevated Temperature

Figure 7.12 Effect of fiber volume on (a) time-dependent composite vibration damping versus temperature curves; (b) time-dependent interface debonding ratio versus temperature curves; (c) time-dependent interface oxidation ratio versus temperature curves; and (d) time-dependent interface slip ratio versus temperature curves of C/SiC composite after oxidation duration t = 2 hours at T = 800 ∘ C. (a)

(b)

(c)

7.3 Time-Dependent Vibration Damping of CMCs

Figure 7.12

(Continued)

(d)

Table 7.6 Effect of fiber volume on time-dependent composite vibration damping and the ratio of interface debonding, slip, and oxidation of C/SiC composite after oxidation duration t = 2 hours at T = 800 ∘ C. T (∘ C)

𝛀c

𝜼

𝝎

𝜹

20

0.005 07

0.345

0.017

0.345

100

0.005 09

0.323

0.018

0.323

200

0.005 69

0.311

0.019

0.311

300

0.006 59

0.303

0.019 6

0.303

400

0.007 94

0.295

0.020 5

0.295

500

0.009 99

0.289

0.020 5

0.289

20

0.001 69

0.144 6

0.041 21

0.144 6

100

0.001 67

0.129 6

0.045 96

0.129 6

200

0.001 75

0.126 6

0.047 05

0.126 6

300

0.001 88

0.125 3

0.047 54

0.125 3

400

0.002 1

0.124 9

0.047 71

0.124 9

500

0.002 46

0.125 1

0.047 62

0.125 1

20

0.001 38

0.048 5

0.122 9

0.048 5

100

0.001 37

0.037 5

0.158 7

0.037 5

200

0.001 37

0.038 4

0.155

0.038 4

300

0.001 39

0.040 6

0.146

0.040 6

400

0.001 41

0.043 3

0.137

0.043 3

500

0.001 46

0.046 4

0.128

0.046 4

V f = 0.2

V f = 0.3

V f = 0.4

333

334

7 Vibration Damping of Ceramic-Matrix Composites at Elevated Temperature

When fiber volume increased, the composite damping decreased because of the decrease of frictional energy dissipated through interface slip. At low fiber volume, i.e. V f = 0.2–0.3, the ratio of interface debonding and slip decreased with temperature, and the ratio of interface oxidation increased with temperature; however, at high fiber volume, i.e. V f = 0.4, the ratio of interface debonding and slip decreased first and then increased, and the ratio of interface debonding increased first and then decreased. When fiber volume increased, the stress transfer between the fiber and the matrix increased, the ratio of interface debonding decreased, and the ratio of interface oxidation increased. The vibration damping of damaged CMCs mainly depended on the energy dissipated in the slip region, and the composite vibration damping decreased with the increase of fiber volume. When V f = 0.2, the composite damping increased from Ωc = 0.005 07 at T = 20 ∘ C to Ωc = 0.009 99 at T = 500 ∘ C; the interface debonding ratio decreased from 𝜂 = 0.345 at T = 20 ∘ C to 𝜂 = 0.289 at T = 500 ∘ C; the interface oxidation ratio increased from 𝜔 = 0.017 at T = 20 ∘ C to 𝜔 = 0.0205 at T = 500 ∘ C, and the interface slip ratio decreased from 𝛿 = 0.345 at T = 20 ∘ C to 𝛿 = 0.289 at T = 500 ∘ C. When V f = 0.3, the composite damping increased from Ωc = 0.001 69 at T = 20 ∘ C to Ωc = 0.002 46 at T = 500 ∘ C; the interface debonding ratio decreased from 𝜂 = 0.1446 at T = 20 ∘ C to 𝜂 = 0.1251 at T = 500 ∘ C; the interface oxidation ratio increased from 𝜔 = 0.041 at T = 20 ∘ C to 𝜔 = 0.047 at T = 500 ∘ C, and the interface slip ratio decreased from 𝛿 = 0.1446 at T = 20 ∘ C to 𝛿 = 0.1251 at T = 500 ∘ C. When V f = 0.4, the composite damping increased from Ωc = 0.001 38 at T = 20 ∘ C to Ωc = 0.001 46 at T = 500 ∘ C; the interface debonding ratio decreased from 𝜂 = 0.0485 at T = 20 ∘ C to 𝜂 = 0.0384 at T = 200 ∘ C and increased to 𝜂 = 0.0464 at T = 500 ∘ C; the interface oxidation ratio increased from 𝜔 = 0.1229 at T = 20 ∘ C to 𝜔 = 0.1587 at T = 100 ∘ C and decreased to 𝜔 = 0.128, and the interface slip ratio decreased from 𝛿 = 0.0485 at T = 20 ∘ C to 𝛿 = 0.0384 at T = 200 ∘ C and increased to 𝛿 = 0.0464 at T = 500 ∘ C. 7.3.2.2 Effect of Vibration Stress on Time-Dependent Vibration Damping of C/SiC Composite

Figure 7.13 and Table 7.7 show the time-dependent composite vibration damping, ratio of interface debonding, oxidation, and slip of C/SiC composite after oxidation duration t = 2 hours at T = 800 ∘ C for different vibration stress. When vibration stress increased, the time-dependent composite vibration damping increased because of the increasing frictional dissipated energy caused by the interface frictional slip. Under low vibration stress, i.e. 𝜎 = 50 MPa, the time-dependent ratio of interface debonding and slip decreased first and then increased with temperature, and the time-dependent ratio of interface oxidation increased first and then decreased with temperature. However, under high vibration stress, i.e. 𝜎 = 70 and 90 MPa, the time-dependent ratio of interface debonding and slip decreased with temperature and the ratio of interface oxidation increased with temperature. The ratio of interface debonding increased with vibration stress, and the ratio of interface oxidation decreased with vibration stress. However, the total frictional slip between the fiber and the matrix increased with vibration stress, and

7.3 Time-Dependent Vibration Damping of CMCs

Figure 7.13 Effect of vibration stress on (a) time-dependent composite vibration damping versus temperature curves; (b) time-dependent interface debonding ratio versus temperature curves; (c) time-dependent interface oxidation ratio versus temperature curves; and (d) time-dependent interface slip ratio versus temperature curves of C/SiC composite after oxidation duration t = 2 hours at T = 800 ∘ C. (a)

(b)

(c)

335

336

7 Vibration Damping of Ceramic-Matrix Composites at Elevated Temperature

Figure 7.13

(Continued)

(d)

Table 7.7 Effect of vibration stress on time-dependent composite vibration damping and the ratio of interface debonding, oxidation, and slip of C/SiC composite after oxidation duration t = 2 hours at T = 800 ∘ C. T (∘ C)

𝛀c

𝜼

𝝎

𝜹

20

0.001 69

0.144 6

0.041 2

0.144 6

100

0.001 67

0.129 6

0.045 9

0.129 6

200

0.001 75

0.126 6

0.047 0

0.126 6

300

0.001 88

0.125 3

0.047 5

0.125 3

400

0.002 1

0.124 9

0.047 7

0.124 9

500

0.002 4

0.125 1

0.047 6

0.125 1

0.003 61

0.275 8

0.021 6

0.275 8

𝜎 = 50 MPa

𝜎 = 70 MPa 20 100

0.003 62

0.256 2

0.023 2

0.256 2

200

0.004 0

0.247 8

0.024 0

0.247 8

300

0.004 5

0.241 5

0.024 6

0.241 5

400

0.005 4

0.236 4

0.025 2

0.236 4

500

0.006 7

0.232 4

0.025 6

0.232 4

𝜎 = 90 MPa 20

0.008 29

0.407 0

0.014 6

0.407 0

100

0.008 3

0.382 8

0.015 5

0.382 8

200

0.009 1

0.369

0.016 1

0.369

300

0.010 3

0.357 7

0.016 6

0.357 7

400

0.011 8

0.348 0

0.017 1

0.348 0

500

0.013 9

0.339 7

0.017 5

0.339 7

7.3 Time-Dependent Vibration Damping of CMCs

the ratio of energy dissipated in the low interface shear stress region decreased. The energy dissipated in the slip region is the main part for the vibration damping of damaged CMCs. The total composite vibration damping increased with vibration stress because of the increase of the interface slip range. When 𝜎 = 50 MPa, the composite damping increased from Ωc = 0.001 69 at T = 20 ∘ C to Ωc = 0.0024 at T = 500 ∘ C; the interface debonding ratio decreased from 𝜂 = 0.1446 at T = 20 ∘ C to 𝜂 = 0.1249 at T = 400 ∘ C and increased to 𝜂 = 0.1251 at T = 500 ∘ C; the interface oxidation ratio increased from 𝜔 = 0.0412 at T = 20 ∘ C to 𝜔 = 0.0477 at T = 400 ∘ C and decreased to 𝜔 = 0.0476 at T = 500 ∘ C, and the interface slip ratio decreased from 𝛿 = 0.1446 at T = 20 ∘ C to 𝛿 = 0.1249 at T = 400 ∘ C, and increased to 𝛿 = 0.1251 at T = 500 ∘ C. When 𝜎 = 70 MPa, the composite damping increased from Ωc = 0.003 61 at T = 20 ∘ C to Ωc = 0.0067 at T = 500 ∘ C; the interface debonding ratio decreased from 𝜂 = 0.2758 at T = 20 ∘ C to 𝜂 = 0.2324 at T = 500 ∘ C; the interface oxidation ratio increased from 𝜔 = 0.0216 at T = 20 ∘ C to 𝜔 = 0.0256 at T = 500 ∘ C, and the interface slip ratio decreased from 𝛿 = 0.2758 at T = 20 ∘ C to 𝛿 = 0.2324 at T = 500 ∘ C. When 𝜎 = 90 MPa, the composite damping increased from Ωc = 0.008 29 at T = 20 ∘ C to Ωc = 0.0139 at T = 500 ∘ C; the interface debonding ratio decreased from 𝜂 = 0.407 at T = 20 ∘ C to 𝜂 = 0.3397 at T = 500 ∘ C; the interface oxidation ratio increased from 𝜔 = 0.0146 at T = 20 ∘ C to 𝜔 = 0.0175 at T = 500 ∘ C; and the interface slip ratio decreased from 𝛿 = 0.407 at T = 20 ∘ C to 𝛿 = 0.3397 at T = 500 ∘ C. 7.3.2.3 Effect of Matrix Crack Spacing on Time-Dependent Vibration Damping of C/SiC Composite

Figure 7.14 and Table 7.8 show the time-dependent composite vibration damping, ratio of interface debonding, oxidation, and slip of C/SiC composite after oxidation duration t = 2 hours at T = 800 ∘ C for different matrix crack spacings. When matrix crack spacing increased, the time-dependent composite vibration damping decreased because of the decrease of the ratio of interface debonding and slip between the matrix crack spacing. However, for partial interface debonding, the ratio of interface oxidation was not affected by matrix crack spacing. The ratio of interface debonding depended on the matrix crack spacing and vibration stress. Under the same vibration stress, the increase of matrix crack spacing decreased the ratio of interface slip, and the ratio between the interface oxidation length and interface debonding length remained unchanged, and the composite vibration damping decreased because of the decreasing ratio of the frictional dissipated energy in the total energy dissipated. When lc = 100 μm, the composite damping increased from Ωc = 0.002 36 at T = 20 ∘ C to Ωc = 0.003 99 at T = 500 ∘ C; the interface debonding ratio decreased from 𝜂 = 0.2893 at T = 20 ∘ C to 𝜂 = 0.2498 at T = 400 ∘ C and increased to 𝜂 = 0.2503 at T = 500 ∘ C; the interface oxidation ratio increased from 𝜔 = 0.0412 at T = 20 ∘ C to 𝜔 = 0.0477 at T = 400 ∘ C and decreased to 𝜔 = 0.0476 at T = 500 ∘ C, and the interface slip ratio decreased from 𝛿 = 0.2893 at T = 20 ∘ C to 𝛿 = 0.2498 at T = 400 ∘ C and increased to 𝛿 = 0.2503 at T = 500 ∘ C.

337

338

7 Vibration Damping of Ceramic-Matrix Composites at Elevated Temperature

Figure 7.14 Effect of matrix crack spacing on (a) time-dependent composite vibration damping versus temperature curves; (b) time-dependent interface debonding ratio versus temperature curves; (c) time-dependent interface oxidation ratio versus temperature curves; and (d) time-dependent interface slip ratio versus temperature curves of C/SiC composite after oxidation duration t = 2 hours at T = 800 ∘ C. (a)

(b)

(c)

7.3 Time-Dependent Vibration Damping of CMCs

Figure 7.14

(Continued)

(d)

Table 7.8 Effect of matrix crack spacing on time-dependent composite vibration damping and the ratio of interface debonding, oxidation, and slip of C/SiC composite after oxidation duration t = 2 hours at T = 800 ∘ C. T (∘ C)

𝛀c

𝜼

𝝎

𝜹

20

0.002 36

0.289 3

0.041 2

0.289 3

100

0.002 28

0.259 3

0.045 9

0.259 3

200

0.002 46

0.253 3

0.047 0

0.253 3

300

0.002 76

0.250 7

0.047 5

0.250 7

400

0.003 23

0.249 8

0.047 7

0.249 8

500

0.003 99

0.250 3

0.047 6

0.250 3

20

0.001 69

0.144 6

0.041 2

0.144 6

100

0.001 67

0.129 6

0.045 9

0.129 6

200

0.001 75

0.126 6

0.047 0

0.126 6

lc = 100 𝜇m

lc = 200 𝜇m

300

0.001 88

0.125 3

0.047 5

0.125 3

400

0.002 1

0.124 9

0.047 7

0.124 9

500

0.002 46

0.125 1

0.047 6

0.125 1

20

0.001 52

0.096 4

0.041 2

0.096 4

100

0.001 51

0.086 4

0.045 9

0.086 4

200

0.001 56

0.084 4

0.047 0

0.084 4

300

0.001 64

0.083 5

0.047 5

0.083 5

400

0.001 78

0.083 2

0.047 7

0.083 2

500

0.002 02

0.083 4

0.047 6

0.083 4

lc = 300 𝜇m

339

340

7 Vibration Damping of Ceramic-Matrix Composites at Elevated Temperature

When lc = 200 μm, the composite damping increased from Ωc = 0.001 69 at T = 20 ∘ C to Ωc = 0.002 46 at T = 500 ∘ C; the interface debonding ratio decreased from 𝜂 = 0.1446 at T = 20 ∘ C to 𝜂 = 0.1249 at T = 400 ∘ C and increased to 𝜂 = 0.1251 at T = 500 ∘ C; the interface oxidation ratio increased from 𝜔 = 0.0412 at T = 20 ∘ C to 𝜔 = 0.0477 at T = 400 ∘ C and decreased to 𝜔 = 0.0476 at T = 500 ∘ C, and the interface slip ratio decreased from 𝛿 = 0.1446 at T = 20 ∘ C to 𝛿 = 0.1249 at T = 400 ∘ C and increased to 2ly /lc = 0.1251 at T = 500 ∘ C. When lc = 300 μm, the composite damping increased from Ωc = 0.001 52 at T = 20 ∘ C to Ωc = 0.002 02 at T = 500 ∘ C; the interface debonding ratio decreased from 𝜂 = 0.0964 at T = 20 ∘ C to 𝜂 = 0.0832 at T = 400 ∘ C and increased to 𝜂 = 0.0834 at T = 500 ∘ C; the interface oxidation ratio increased from 𝜔 = 0.0412 at T = 20 ∘ C to 𝜔 = 0.0477 at T = 400 ∘ C and decreased to 𝜔 = 0.0476 at T = 500 ∘ C, and the interface slip ratio decreased from 𝛿 = 0.0964 at T = 20 ∘ C to 𝛿 = 0.0832 at T = 400 ∘ C and increased to 2ly /lc = 0.0834 at T = 500 ∘ C. 7.3.2.4 Effect of Interface Shear Stress on Time-Dependent Vibration Damping of C/SiC Composite

Figure 7.15 and Table 7.9 show the time-dependent composite vibration damping, ratio of the interface debonding, oxidation, and slip of C/SiC composite after oxidation duration t = 2 hours at T = 800 ∘ C for different interface shear stress. When the interface shear stress in the oxidation region increased, the time-dependent composite vibration damping increased because of the increase of frictional dissipated energy along the interface oxidation region. Interface shear stress transferred the stress between the fiber and the matrix. High interface shear stress decreased the interface debonding length between the fiber and the matrix. However, the interface oxidation length was time dependent and not affected by the interface shear stress in the oxidation region. The ratio between the interface oxidation length and the interface slip length increased, leading to the high energy dissipated in the oxidation region, and the time-dependent composite vibration damping increased. When 𝜏 f = 3 MPa, the composite damping increased from Ωc = 0.001 87 at T = 20 ∘ C to Ωc = 0.002 97 at T = 500 ∘ C; the interface debonding ratio decreased from 𝜂 = 0.1442 at T = 20 ∘ C to 𝜂 = 0.1245 at T = 400 ∘ C and increased to 𝜂 = 0.1248 at T = 500 ∘ C; the interface oxidation ratio increased from 𝜔 = 0.0413 at T = 20 ∘ C to 𝜔 = 0.0478 at T = 400 ∘ C and decreased to 𝜔 = 0.0477 at T = 500 ∘ C, and the interface slip ratio decreased from 𝛿 = 0.1442 at T = 20 ∘ C to 𝛿 = 0.1245 at T = 400 ∘ C and increased to 𝛿 = 0.1248 at T = 500 ∘ C. When 𝜏 f = 5 MPa, the composite damping increased from Ωc = 0.002 15 at T = 20 ∘ C to Ωc = 0.003 84 at T = 500 ∘ C; the interface debonding ratio decreased from 𝜂 = 0.1433 at T = 20 ∘ C to 𝜂 = 0.1238 at T = 400 ∘ C and increased to 𝜂 = 0.1241 at T = 500 ∘ C; the interface oxidation ratio increased from 𝜔 = 0.0415 at T = 20 ∘ C to 𝜔 = 0.0481 at T = 400 ∘ C and decreased to 𝜔 = 0.048 at T = 500 ∘ C, and the interface slip ratio decreased from 𝛿 = 0.1433 at T = 20 ∘ C to 𝛿 = 0.1238 at T = 400 ∘ C and increased to 𝛿 = 0.1241 at T = 500 ∘ C.

7.3 Time-Dependent Vibration Damping of CMCs

Figure 7.15 Effect of interface shear stress on (a) time-dependent composite vibration damping versus temperature curves; (b) time-dependent interface debonding ratio versus temperature curves; (c) time-dependent interface oxidation ratio versus temperature curves; and (d) time-dependent interface slip ratio versus temperature curves of C/SiC composite after oxidation duration t = 2 hours at T = 800 ∘ C. (a)

(b)

(c)

341

342

7 Vibration Damping of Ceramic-Matrix Composites at Elevated Temperature

Figure 7.15

(Continued)

(d)

Table 7.9 Effect of interface shear stress on time-dependent composite vibration damping and the ratio of interface debonding, oxidation, and slip of C/SiC composite after oxidation time t = 2 hours at T = 800 ∘ C. T (∘ C)

𝛀c

𝜼

𝝎

𝜹

20

0.001 87

0.144 2

0.041 3

0.144 2

100

0.001 84

0.129 2

0.046 1

0.129 2

200

0.001 95

0.126 2

0.047 2

0.126 2

300

0.002 14

0.125

0.047 6

0.125

400

0.002 45

0.124 5

0.047 8

0.124 5

500

0.002 97

0.124 8

0.047 7

0.124 8

20

0.002 15

0.143 3

0.041 5

0.143 3

100

0.002 12

0.128 4

0.046 4

0.128 4

200

0.002 29

0.125 5

0.047 5

0.125 5

300

0.002 59

0.124 2

0.047 9

0.124 2

400

0.003 05

0.123 8

0.048 1

0.123 8

500

0.003 84

0.124 1

0.048 0

0.124 1

20

0.002 36

0.142 5

0.041 8

0.142 5

100

0.002 33

0.127 6

0.046 7

0.127 6

200

0.002 56

0.124 7

0.047 8

0.124 7

300

0.002 93

0.123 4

0.048 2

0.123 4

400

0.003 52

0.123 1

0.048 4

0.123 1

500

0.004 51

0.123 4

0.048 3

0.123 4

𝜏 f = 3 MPa

𝜏 f = 5 MPa

𝜏 f = 7 MPa

7.3 Time-Dependent Vibration Damping of CMCs

When 𝜏 f = 7 MPa, the composite damping increased from Ωc = 0.002 36 at T = 20 ∘ C to Ωc = 0.004 51 at T = 500 ∘ C; the interface debonding ratio decreased from 𝜂 = 0.1425 at T = 20 ∘ C to 𝜂 = 0.1231 at T = 400 ∘ C and increased to 𝜂 = 0.1234 at T = 500 ∘ C; the interface oxidation ratio increased from 𝜔 = 0.0418 at T = 20 ∘ C to 𝜔 = 0.0484 at T = 400 ∘ C and decreased to 𝜔 = 0.0483 at T = 500 ∘ C, and the interface slip ratio decreased from 𝛿 = 0.1425 at T = 20 ∘ C to 𝛿 = 0.1231 at T = 400 ∘ C and increased to 𝛿 = 0.1234 at T = 500 ∘ C. 7.3.2.5 Effect of Temperature on Time-Dependent Vibration Damping of C/SiC Composite

Figure 7.16 and Table 7.10 show the time-dependent composite vibration damping, the ratio of interface debonding, oxidation, and slip of C/SiC composite after oxidation duration t = 2 hours for different oxidation temperatures. When temperature increased, the temperature-dependent composite vibration damping increased because of the increase of the interface debonding and slip range between the matrix crack spacing. The interface oxidation length is temperature dependent and increased with temperature. The ratio of interface debonding and oxidation increased with temperature, and the energy dissipated of frictional slip between the fiber and the matrix increased because of the increase of interface slip range and the time-dependent composite vibration damping increased. When T = 600 ∘ C, the composite damping increased from Ωc = 0.001 67 at T = 20 ∘ C to Ωc = 0.002 37 at T = 500 ∘ C; the interface debonding ratio decreased from 𝜂 = 0.1403 at T = 20 ∘ C to 𝜂 = 0.1204 at T = 400 ∘ C and increased to 𝜂 = 0.1207 at T = 500 ∘ C; the interface oxidation ratio increased from 𝜔 = 0.0064 at T = 20 ∘ C to 𝜔 = 0.007 46 at T = 400 ∘ C and decreased to 𝜔 = 0.007 45 at T = 500 ∘ C, and the interface slip ratio decreased from 𝛿 = 0.1403 at T = 20 ∘ C to 𝛿 = 0.1204 at T = 400 ∘ C and increased to 𝛿 = 0.1207 at T = 500 ∘ C. When T = 900 ∘ C, the composite damping increased from Ωc = 0.001 72 at T = 20 ∘ C to Ωc = 0.002 56 at T = 500 ∘ C; the interface debonding ratio decreased from 𝜂 = 0.1498 at T = 20 ∘ C to 𝜂 = 0.1302 at T = 400 ∘ C and increased to 𝜂 = 0.1305 at T = 500 ∘ C; the interface oxidation ratio increased from 𝜔 = 0.0804 at T = 20 ∘ C to 𝜔 = 0.0924 at T = 400 ∘ C and decreased to 𝜔 = 0.0923 at T = 500 ∘ C, and the interface slip ratio decreased from 𝛿 = 0.1498 at T = 20 ∘ C to 𝛿 = 0.1302 at T = 400 ∘ C and increased to 𝛿 = 0.1305 at T = 500 ∘ C. When T = 1100 ∘ C, the composite damping increased from Ωc = 0.001 86 at T = 20 ∘ C to Ωc = 0.003 at T = 500 ∘ C; the interface debonding ratio decreased from 𝜂 = 0.1705 at T = 20 ∘ C to 𝜂 = 0.1514 at T = 400 ∘ C and increased to 𝜂 = 0.1518 at T = 500 ∘ C; the interface oxidation ratio increased from 𝜔 = 0.2121 at T = 20 ∘ C to 𝜔 = 0.2389 at T = 400 ∘ C and decreased to 𝜔 = 0.2383 at T = 500 ∘ C, and the interface slip ratio decreased from 𝛿 = 0.1705 at T = 20 ∘ C to 𝛿 = 0.1514 at T = 400 ∘ C and increased to 𝛿 = 0.1518 at T = 500 ∘ C.

7.3.3

Experimental Comparisons

Zhang et al. [33] performed experimental investigation on the effect of oxidation on vibration damping behavior of 2D C/SiC composite at elevated temperature T = 700,

343

344

7 Vibration Damping of Ceramic-Matrix Composites at Elevated Temperature

Figure 7.16 Effect of oxidation temperature on (a) time-dependent composite vibration damping versus temperature curves; (b) time-dependent interface debonding ratio versus temperature curves; (c) time-dependent interface oxidation ratio versus temperature curves; and (d) time-dependent interface slip ratio versus temperature curves of C/SiC composite after oxidation duration t = 2 hours. (a)

(b)

(c)

7.3 Time-Dependent Vibration Damping of CMCs

Figure 7.16

(Continued)

(d)

Table 7.10 Effect of temperature on time-dependent composite vibration damping and the ratio of interface debonding, oxidation, and slip of C/SiC composite after oxidation duration t = 2 hours. T (∘ C)

𝛀c

𝜼

𝝎

𝜹

20

0.001 67

0.140 3

0.00 64

0.140 3

100

0.001 64

0.125 3

0.007 18

0.125 3

200

0.001 72

0.122 3

0.007 35

0.122 3

300

0.001 84

0.120 9

0.007 43

0.120 9

400

0.002 04

0.120 4

0.007 46

0.120 4

500

0.002 37

0.120 7

0.007 45

0.120 7

20

0.001 72

0.149 8

0.080 4

0.149 8

100

0.001 7

0.134 9

0.089 3

0.134 9

200

0.001 79

0.131 9

0.091 3

0.131 9

300

0.001 93

0.130 6

0.092 2

0.130 6

400

0.002 17

0.130 2

0.092 4

0.130 2

500

0.002 56

0.130 5

0.092 3

0.130 5

T = 600 ∘ C

T = 900 ∘ C

T = 1100 ∘ C 20

0.001 86

0.170 5

0.212 1

0.170 5

100

0.001 85

0.155 7

0.232 3

0.155 7

200

0.001 97

0.152 8

0.236 6

0.152 8

300

0.002 1

0.151 7

0.238 4

0.151 7

400

0.002 4

0.151 4

0.238 9

0.151 4

500

0.003

0.151 8

0.238 3

0.151 8

345

346

7 Vibration Damping of Ceramic-Matrix Composites at Elevated Temperature

1000, and 1300 ∘ C for different oxidation durations t = 2, 5, and 10 hours. The composite was fabricated using the CVI. The volume of fiber was approximately 40%. A pyrocarbon (PyC) layer as the interphase was deposited on the surface of the carbon fiber. Dynamical Mechanical Analyzer (DMA 2980) made by TA company of USA was used for measurements of damping of C/SiC composite. All of the measurements were performed in air atmosphere from room temperature to 400 ∘ C, and the testing frequency was f = 1 Hz. 7.3.3.1

t = 2 hours at T = 700, 1000, and 1300 ∘ C

Figure 7.17 and Table 7.11 show the experimental and predicted time-dependent composite vibration damping, and the ratio of interface debonding, oxidation, and slip of C/SiC composite after oxidation duration t = 2 hours at oxidation temperature T = 700, 1000, and 1300 ∘ C. When the oxidation duration was t = 2 hours, the time-dependent ratio of interface debonding and slip decreased when temperature increased from T = 700 ∘ C to T = 1000 ∘ C first and increased when the temperature increased to T = 1300 ∘ C; the ratio of interface oxidation increased with temperature, and the composite vibration damping decreased with temperature. After oxidation at T = 700 ∘ C for t = 2 hours, the experimental time-dependent composite damping increased from Ωc = 0.014 77 at T = 20 ∘ C to Ωc = 0.022 49 at T = 400 ∘ C; the predicted time-dependent composite vibration damping increased from Ωc = 0.015 73 at T = 20 ∘ C to Ωc = 0.024 13 at T = 500 ∘ C; the time-dependent interface debonding ratio decreased from 𝜂 = 0.5378 at T = 20 ∘ C to 𝜂 = 0.4463 at T = 500 ∘ C; the time-dependent interface oxidation ratio increased from ω = 0.0047 at T = 20 ∘ C to ω = 0.0057 at T = 500 ∘ C, and the time-dependent interface slip ratio decreased from 𝛿 = 0.5378 at T = 20 ∘ C to 𝛿 = 0.4463 at T = 500 ∘ C. After oxidation at T = 1000 ∘ C for t = 2 hours, the experimental time-dependent composite vibration damping increases from Ωc = 0.014 79 at T = 20 ∘ C to Ωc = 0.0211 at T = 400 ∘ C; the predicted time-dependent composite vibration damping increased from Ωc = 0.014 14 at T = 20 ∘ C to Ωc = 0.022 63 at T = 500 ∘ C; the time-dependent interface debonding ratio decreased from 𝜂 = 0.5218 at T = 20 ∘ C to 𝜂 = 0.4367 at T = 500 ∘ C; the time-dependent interface oxidation ratio increased from 𝜔 = 0.041 at T = 20 ∘ C to 𝜔 = 0.049 at T = 500 ∘ C; and the time-dependent interface slip ratio decreased from 𝛿 = 0.5218 at T = 20 ∘ C to 𝛿 = 0.4367 at T = 500 ∘ C. After oxidation at T = 1300 ∘ C for t = 2 hours, the experimental composite vibration damping increased from Ωc = 0.013 79 at T = 20 ∘ C to Ωc = 0.02 at T = 400 ∘ C; the predicted time-dependent composite vibration damping increased from Ωc = 0.014 53 at T = 20 ∘ C to Ωc = 0.023 51 at T = 500 ∘ C; the time-dependent interface debonding ratio decreased from 𝜂 = 0.5408 at T = 20 ∘ C to 𝜂 = 0.4632 at T = 500 ∘ C; the time-dependent interface oxidation ratio increased from 𝜔 = 0.1515 at T = 20 ∘ C to 𝜔 = 0.1769 at T = 500 ∘ C; and the time-dependent interface slip ratio decreased from 𝛿 = 0.5408 at T = 20 ∘ C to 𝛿 = 0.4632 at T = 500 ∘ C. 7.3.3.2

t = 5 hours at T = 700, 1000, and 1300 ∘ C

Figure 7.18 and Table 7.12 show the experimental and predicted time-dependent composite vibration damping, and the ratio of interface debonding, oxidation, and

7.3 Time-Dependent Vibration Damping of CMCs

Figure 7.17 (a) Experimental and predicted time-dependent composite vibration damping versus temperature curves; (b) time-dependent interface debonding ratio versus temperature curves; (c) time-dependent interface oxidation ratio versus temperature curves; and (d) time-dependent interface slip ratio versus temperature curves of C/SiC composite after oxidation duration t = 2 hours at T = 700, 1000, and 1300 ∘ C. (a)

(b)

(c)

347

348

7 Vibration Damping of Ceramic-Matrix Composites at Elevated Temperature

Figure 7.17

(Continued)

(d)

Table 7.11 Experimental and predicted time-dependent composite vibration damping and the ratio of interface debonding, oxidation, and slip of C/SiC composite after oxidation duration t = 2 hours at T = 700, 1000, and 1300 ∘ C. T (∘ C)

𝛀c (experiment)

𝛀c (theory)

𝜼

𝝎

𝜹

T = 700 ∘ C 20

0.014 77

0.015 73

0.537 8

0.004 7

0.537 8

100

0.015 64

0.015 54

0.508 3

0.005 0

0.508 3

200

0.016 43

0.016 76

0.489 08

0.005 2

0.489 08

300

0.018 86

0.018 57

0.472 89

0.005 4

0.472 89

400

0.022 49

0.021

0.458 76

0.005 56

0.458 76

500

—

0.024 13

0.446 39

0.005 72

0.446 39

20

0.014 79

0.014 14

0.521 83

0.041 79

0.521 83

100

0.015 82

0.014 05

0.493 53

0.044 18

0.493 53

200

0.017 21

0.015 28

0.475 75

0.045 83

0.475 75

300

0.022 96

0.017 07

0.460 9

0.047 31

0.460 9

400

0.021 11

0.019 5

0.447 99

0.048 67

0.447 99

500

—

0.022 63

0.436 76

0.049 93

0.436 76

20

0.013 79

0.014 53

0.540 8

0.151 5

0.540 8

100

0.015 68

0.014 52

0.513 99

0.159 5

0.513 99

200

0.016 73

0.015 87

0.497 89

0.164 66

0.497 89

300

0.017 80

0.017 79

0.484 59

0.169 18

0.484 59

400

0.020 12

0.020 31

0.473 12

0.173 28

0.473 12

500

—

0.023 51

0.463 21

0.176 99

0.463 21

T = 1000 ∘ C

T = 1300 ∘ C

7.3 Time-Dependent Vibration Damping of CMCs

Figure 7.18 (a) Experimental and predicted time-dependent composite vibration damping versus temperature curves; (b) time-dependent interface debonding ratio versus temperature curves; (c) time-dependent interface oxidation ratio versus temperature curves; and (d) time-dependent interface slip ratio versus temperature curves of C/SiC composite after oxidation duration t = 5 hours at T = 700, 1000, and 1300 ∘ C. (a)

(b)

(c)

349

350

7 Vibration Damping of Ceramic-Matrix Composites at Elevated Temperature

Figure 7.18

(Continued)

(d)

Table 7.12 Experimental and predicted time-dependent composite vibration damping and the ratio of the interface debonding, oxidation, and slip of C/SiC composite after oxidation duration t = 5 hours at T = 700, 1000, and 1300 ∘ C. T (∘ C)

𝛀c (experiment)

𝛀c (theory)

𝜼

𝝎

𝜹

T = 700 ∘ C 20

0.011 69

0.014 23

0.519 4

0.012 28

0.519 4

100

0.014 1

0.014 11

0.490 7

0.013

0.490 7

200

0.016 2

0.015 32

0.472 38

0.013 5

0.472 38

300

0.022 7

0.017 09

0.457 04

0.013 96

0.457 04

400

0.023 0

0.019 5

0.443 68

0.014 38

0.443 68

500

—

0.022 64

0.432 02

0.014 77

0.432 02

20

0.015 24

0.011 78

0.495 64

0.109 94

0.495 64

100

0.015 51

0.011 82

0.469 42

0.116 08

0.469 42

200

0.015 09

0.013 03

0.454 05

0.120 01

0.454 05

300

0.016 52

0.014 77

0.441 42

0.123 45

0.441 42

400

0.017 73

0.017 14

0.430 57

0.126 56

0.430 57

500

—

0.020 27

0.421 23

0.129 37

0.421 23

20

—

0.014 58

0.569 44

0.358 24

0.569 44

100

0.017 66

0.014 79

0.545 88

0.373 7

0.545 88

200

0.019 55

0.016 42

0.533 61

0.382 29

0.533 61

300

0.019 8

0.018 63

0.523 83

0.389 43

0.523 83

400

0.020 4

500

—

T = 1000 ∘ C

T = 1300 ∘ C

0.021 48

0.515 61

0.395 64

0.515 61

0.025

0.508 7

0.401 01

0.508 7

7.3 Time-Dependent Vibration Damping of CMCs

slip ratio of C/SiC composite after oxidation duration t = 5 hours at T = 700, 1000, and 1300 ∘ C. When the oxidation duration was t = 5 hours, the time-dependent ratio of interface debonding and slip decreased when temperature increased from T = 700 ∘ C to T = 1000 ∘ C first and increased when the temperature increased to T = 1300 ∘ C; the time-dependent ratio of interface oxidation increased with temperature; and the composite vibration damping decreased when temperature increased from T = 700 ∘ C to T = 1000 ∘ C first and increased when the temperature increased to T = 1300 ∘ C. After oxidation at T = 700 ∘ C for t = 5 hours, the experimental time-dependent composite vibration damping increased from Ωc = 0.011 69 at T = 20 ∘ C to Ωc = 0.023 at T = 400 ∘ C; the predicted time-dependent composite vibration damping increased from Ωc = 0.014 23 at T = 20 ∘ C to Ωc = 0.022 64 at T = 500 ∘ C; the time-dependent ratio of the interface debonding decreased from 𝜂 = 0.5194 at T = 20 ∘ C to 𝜂 = 0.432 at T = 500 ∘ C; the time-dependent ratio of the interface oxidation increased from 𝜔 = 0.0122 at T = 20 ∘ C to 𝜔 = 0.0147 at T = 500 ∘ C; and the time-dependent ratio of the interface slip decreased from 𝛿 = 0.5194 at T = 20 ∘ C to 𝛿 = 0.432 at T = 500 ∘ C. After oxidation at T = 1000 ∘ C for t = 5 hours, the experimental time-dependent composite vibration damping increased from Ωc = 0.015 24 at T = 20 ∘ C to Ωc = 0.0177 at T = 400 ∘ C; the predicted time-dependent composite vibration damping increased from Ωc = 0.011 78 at T = 20 ∘ C to Ωc = 0.0202 at T = 500 ∘ C; the time-dependent ratio of the interface debonding decreased from 𝜂 = 0.4956 at T = 20 ∘ C to 𝜂 = 0.4212 at T = 500 ∘ C; the time-dependent ratio of the interface oxidation increased from 𝜔 = 0.109 at T = 20 ∘ C to 𝜔 = 0.129 at T = 500 ∘ C; and the time-dependent ratio of the interface slip decreased from 𝛿 = 0.495 at T = 20 ∘ C to 𝛿 = 0.421 at T = 500 ∘ C. After oxidation at T = 1300 ∘ C for t = 5 hours, the experimental time-dependent composite vibration damping increased from Ωc = 0.017 66 at T = 100 ∘ C to Ωc = 0.0204 at T = 400 ∘ C; the predicted time-dependent composite damping increased from Ωc = 0.014 58 at T = 20 ∘ C to Ωc = 0.025 at T = 500 ∘ C; the time-dependent ratio of the interface debonding decreased from 𝜂 = 0.569 at T = 20 ∘ C to 𝜂 = 0.508 at T = 500 ∘ C; the time-dependent ratio of the interface oxidation increased from 𝜔 = 0.358 at T = 20 ∘ C to 𝜔 = 0.401 at T = 500 ∘ C; and the time-dependent ratio of the interface slip decreased from 𝛿 = 0.569 at T = 20 ∘ C to 𝛿 = 0.508 at T = 500 ∘ C. 7.3.3.3

t = 10 hours at T = 700, 1000, and 1300 ∘ C

Figure 7.19 and Table 7.13 show the experimental and predicted time-dependent composite vibration damping, and the ratio of interface debonding, oxidation, and slip of C/SiC composite after oxidation duration t = 10 hours at oxidation temperature of T = 700, 1000, and 1300 ∘ C. When the oxidation duration was t = 10 hours, the ratio of interface debonding and slip decreased when the temperature increased from T = 700 ∘ C to T = 1000 ∘ C first and increased when the temperature increased to T = 1300 ∘ C; the ratio of interface oxidation increased with temperature; and the composite vibration damping decreased when the temperature increased from

351

352

7 Vibration Damping of Ceramic-Matrix Composites at Elevated Temperature

Figure 7.19 (a) Experimental and predicted time-dependent composite vibration damping versus temperature curves; (b) time-dependent interface debonding ratio versus temperature curves; (c) time-dependent interface oxidation ratio versus temperature curves; and (d) time-dependent interface slip ratio versus temperature curves of C/SiC composite after oxidation duration t = 10 hours at T = 700, 1000, and 1300 ∘ C. (a)

(b)

(c)

7.3 Time-Dependent Vibration Damping of CMCs

Figure 7.19

(Continued)

(d)

Table 7.13 Experimental and predicted time-dependent composite vibration damping and the ratio of interface debonding, oxidation, and slip of C/SiC composite after oxidation duration t = 10 hours at T = 700, 1000, and 1300 ∘ C. T (∘ C)

𝛀c (experiment)

𝛀c (theory)

𝜼

𝝎

𝜹

T = 700 ∘ C 20

0.011 05

0.014 53

0.524 92

0.024 3

0.524 92

100

0.012 3

0.014 41

0.496 19

0.025 71

0.496 19

200

0.014 0

0.015 63

0.477 91

0.026 69

0.477 91

300

0.016 8

0.017 43

0.462 6

0.027 58

0.462 6

400

0.019 6

0.019 85

0.449 28

0.028 4

0.449 28

500

—

0.022 98

0.437 65

0.029 15

0.437 65

20

—

0.011 18

0.498 9

0.218 32

0.498 9

100

0.015 4

0.011 31

0.474 48

0.229 56

0.474 48

200

0.015 4

0.012 59

0.461 2

0.236 16

0.461 2

300

0.015 5

0.014 41

0.450 51

0.241 77

0.450 51

400

0.016 4

0.016 87

0.441 44

0.246 74

0.441 44

500

—

0.020 1

0.433 74

0.251 11

0.433 74

T = 1000 ∘ C

T = 1300 ∘ C 20

0.018 9

0.015 85

0.633

0.639 54

0.633

100

0.018 3

0.016 44

0.614 28

0.659 03

0.614 28

200

0.016 8

0.018 69

0.607 62

0.666 25

0.607 62

300

0.018 9

0.021 64

0.603 03

0.671 32

0.603 03

400

0.020 7

0.025 35

0.599 58

0.675 18

0.599 58

500

—

0.029 85

0.597 09

0.678

0.597 09

353

354

7 Vibration Damping of Ceramic-Matrix Composites at Elevated Temperature

T = 700 ∘ C to T = 1000 ∘ C first and increased when the temperature increased to T = 1300 ∘ C. After oxidation at T = 700 ∘ C for t = 10 hours, the time-dependent experimental composite damping increased from Ωc = 0.011 05 at T = 20 ∘ C to Ωc = 0.0196 at T = 400 ∘ C; the predicted time-dependent composite vibration damping increased from Ωc = 0.014 53 at T = 20 ∘ C to Ωc = 0.022 98 at T = 500 ∘ C; the time-dependent ratio of interface debonding decreased from 𝜂 = 0.5249 at T = 20 ∘ C to 𝜂 = 0.4376 at T = 500 ∘ C; the time-dependent ratio of interface oxidation increased from 𝜔 = 0.0243 at T = 20 ∘ C to 𝜔 = 0.0291 at T = 500 ∘ C; and the ratio of time-dependent interface slip decreased from 𝛿 = 0.5249 at T = 20 ∘ C to 𝛿 = 0.4376 at T = 500 ∘ C. After oxidation at T = 1000 ∘ C for t = 10 hours, the experimental time-dependent composite vibration damping increased from Ωc = 0.0154 at T = 100 ∘ C to Ωc = 0.0164 at T = 400 ∘ C; the predicted time-dependent composite vibration damping increased from Ωc = 0.011 18 at T = 20 ∘ C to Ωc = 0.0201 at T = 500 ∘ C; the time-dependent ratio of interface debonding decreased from 𝜂 = 0.4989 at T = 20 ∘ C to 𝜂 = 0.4337 at T = 500 ∘ C; the time-dependent ratio of interface oxidation increased from 𝜔 = 0.2183 at T = 20 ∘ C to 𝜔 = 0.2511 at T = 500 ∘ C; and the time-dependent ratio of interface slip decreased from 𝛿 = 0.4989 at T = 20 ∘ C to 𝛿 = 0.4337 at T = 500 ∘ C. After oxidation at T = 1300 ∘ C for t = 10 hours, the experimental time-dependent composite vibration damping increased from Ωc = 0.0189 at T = 20 ∘ C to Ωc = 0.0207 at T = 400 ∘ C; the predicted time-dependent composite vibration damping increased from Ωc = 0.015 85 at T = 20 ∘ C to Ωc = 0.029 85 at T = 500 ∘ C; the time-dependent ratio of interface debonding decreased from 𝜂 = 0.633 at T = 20 ∘ C to 𝜂 = 0.597 at T = 500 ∘ C; the time-dependent ratio of interface oxidation increased from 𝜔 = 0.639 at T = 20 ∘ C to 𝜔 = 0.678 at T = 500 ∘ C; and the time-dependent ratio of interface slip decreased from 𝛿 = 0.633 at T = 20 ∘ C to 𝛿 = 0.597 at T = 500 ∘ C. 7.3.3.4

Discussion

Because of temperature-dependent material properties and especially the interface properties (i.e. the interface shear stress), the composite vibration damping and interface damages of C/SiC composite is time and temperature dependent. For 2D C/SiC, the time-dependent composite vibration damping increases with temperature, the interface debonding and slip ratio decreases with temperature, and the interface oxidation ratio increases with temperature. When the oxidation duration was short (i.e. t = 2 hours), the time-dependent composite vibration damping decreased with temperature; however, when the oxidation duration was long (i.e. t = 5 and 10 hours), the time-dependent composite vibration damping decreased with temperature below 1000 ∘ C and increased with temperature to 1300 ∘ C. The time-dependent interface oxidation ratio increased with temperature, and the time-dependent interface debonding and slip ratio decreased with temperature below 1000 ∘ C and increased with temperature to 1300 ∘ C. When the oxidation temperature was T = 700 ∘ C, the time-dependent composite damping increased for short oxidation duration (i.e. t = 2 and 5 hours) and decreased for long oxidation duration (i.e. t = 10 hours). When the oxidation temperature

7.3 Time-Dependent Vibration Damping of CMCs

Figure 7.20 Experimental and predicted time-dependent composite vibration damping versus temperature curves for (a) t = 2, 5, 10 hours at T = 700 ∘ C; (b) t = 2, 5, 10 hours at T = 1000 ∘ C; and, (c) t = 2, 5, 10 hours at T = 1300 ∘ C.

(a)

(b)

(c)

355

356

7 Vibration Damping of Ceramic-Matrix Composites at Elevated Temperature

was T = 1000 ∘ C, the time-dependent composite damping decreased with increasing oxidation duration. When the oxidation temperature was T = 1300 ∘ C, the time-dependent composite damping increased with oxidation duration, as shown in Figure 7.20. In the present analysis, the time-dependent damage mechanism of the interface oxidation is considered in the analysis. However, at elevated temperature, the oxidation inside of CMCs may degrade the fiber strength, and the effect of time-dependent fiber failure on vibration damping of CMCs has not been considered, which may be the reason for the difference between the experimental data and predicted results of present model.

7.4

Conclusion

In this chapter, temperature-dependent vibration damping of fiber-reinforced CMCs is investigated using the micromechanical approach. Relationships between composite’s vibration damping, damage mechanisms, and environmental temperature are established. Experimental temperature-dependent vibration damping of 2D SiC/SiC and C/SiC composites is predicted. ●

●

Experimental vibration damping of 2D SiC/SiC composite increases with temperature, and the theoretical predicted composite vibration damping for different vibration stress of 𝜎 = 5, 6, and 7 MPa are obtained and agreed with experimental data, and the predicted interface debonding and slip fraction increases with temperature. For 2D C/SiC, the time-dependent composite vibration damping increases with temperature, the interface debonding and slip ratio decreases with temperature, and the interface oxidation ratio increases with temperature.

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359

Index a ACK matrix cracking stress 80 ACK model 19, 20, 106 acoustic emission testing of 2D SiC/SiC composite 9, 308 adjacent fractured fibers 190 AK model 19

b BHE model 19 broken fiber fraction 7, 159, 161–162, 174–178, 190–191, 193–194, 197–201, 203–206, 223, 226, 228–229, 233–238, 240–241, 243–244, 246–247, 251–256, 259–268, 321, 325, 328

c carbide matrix 2, 39 carbon matrix 2 ceramic-matrix composites (CMCs) 19, 75 application 1 crack bridging effect 1 density of 1 fatigue behavior 6–7 vs. polymer matrix composites 1 stress-rupture behavior 7–9 vs. superalloy 1 tensile behavior 2–5 vibration behavior 9–10 chemical vapor infiltration (CVI) 346 process 6, 187 SiC matrix minicomposites 212 Chiang model 19 Chiang-Wang-Chou model 19

composite vibration damping 9–10, 307–308, 310, 313–314, 317, 320, 321, 325, 328–329, 331–356 composite’s proportional limit stress 80, 106 crack bridging effect 1 critical matrix strain energy (CMSE) 80, 105 criterion 5, 76, 147 model 102 parameter 80 critical stress 80, 105, 106 cross-ply CMCs 8, 9, 213, 308 C/SiC bolted fastenings 9, 307 C/SiC composite 173 temperature-dependent matrix cracking stress of for different fiber/matrix interface frictional coefficient 25 different fiber volumes 23 for different interface debonding energy 26 for different interface shear stress 24 matrix fracture energy effect 27 theoretical models 20–21 temperature-dependent matrix multiple cracking evolution of ACK matrix cracking stress 80 critical matrix strain energy 80 critical matrix strain energy parameter 80 for different fiber/matrix interface debonding energy 84–85 for different interface shear stress 82–83

High Temperature Mechanical Behavior of Ceramic-Matrix Composites, First Edition. Longbiao Li. © 2021 WILEY-VCH GmbH. Published 2021 by WILEY-VCH GmbH.

360

Index

C/SiC composite (contd.) for different matrix fracture energy 85–88 energy balance criterion 80 experimental comparisons 88–89 fiber/matrix interface debonding 78–79 interface debonding ratio 80 matrix cracking density 80 matrix multiple cracking evolution 80 temperature-dependent carbon fiber axial and radial thermal expansion coefficient 81 temperature-dependent carbon fiber elastic modulus 81 temperature dependent fiber/matrix interface debonding energy 81 temperature-dependent matrix fracture energy 80 temperature-dependent matrix strain energy 79 temperature-dependent stress analysis 77–78 temperature-dependent vibration damping fiber volume effect 334 interface shear stress effect 340–343 matrix crack spacing effect 337–340 temperature effect 343, 344–345 vibration stress effect 334–337 time-dependent matrix cracking stress of for different fiber volumes 42, 43–44 for different interface debonding energy 53–56 for different interface frictional coefficient 50–53 for different interface shear stress 42, 45–50 different matrix fracture energy 56–59 experimental comparisons 59, 60 theoretical models 39–41

time-dependent matrix multiple cracking of for different interface debonding energy 111–113 for different interface frictional coefficient 108–111 for different matrix fracture energy 113–114 experimental comparisons 114–116 time-dependent interface debonding 103–104 time-dependent stress analysis 102–103 cyclic-dependent damage parameters of internal friction 7, 188, 191 cyclic-dependent dissipated energy 199, 200, 203 cyclic-dependent fatigue damage evolution 187 cyclic-dependent internal damage parameter 190 cyclic-dependent unloading composite hysteresis strain 189, 190 cyclic-dependent unloading intact fiber stress 189, 190

d Danchaivijit–Shetty model 19 1D C/SiC composite 173 2D C/SiC 20, 28, 29, 59, 175, 176, 329, 343, 354, 356 2D C/SiC composite 20, 28, 29, 59, 175, 176, 329, 343 2D SiC/SiC composite 3, 4, 9, 30, 36, 68, 71, 146, 147, 330, 356 2D Sylramic-iBN SiC/SiC composite 4, 147 2D woven Hi-NicalonTM SiC/[SiC-B4 C] composite 7, 188, 191, 206 2D woven melt-infiltrated (MI) Hi-NicalonTM SiC/SiC composite 7, 212 2D woven self-healing Hi-NicalonTM SiC/[SiC-B4 C] composite 199, 204, 206 at T = 1200o C in air and in steam atmosphere 199–203 2.5D C/SiC composite 173, 177, 178

Index

2.5D NicalonTM NL202 SiC/[Si-B-C] 6, 188 2.5D woven Hi-NicalonTM SiC/[Si-B-C] at 600o C in air atmosphere 191–199 2.5D woven self-healing Hi-NicalonTM SiC/[Si-B-C] composite 206 at T = 1200o C in air atmosphere 193–199, 204 3D C/SiC composite 3, 146, 173, 177, 180 3D needled C/SiC composite 3, 146 debonding fraction at the interface 215, 217–219, 221, 223–227, 229–231, 233, 235, 237, 239 dissipated energy 7, 188, 191, 199–201, 203, 206, 308, 309, 334, 337, 340 dwell-fatigue loading 8, 213 Dynamical Mechanical Analyzer (DMA 2980) 346

e energy balance approach 4, 5, 19, 20, 30, 76, 147 energy balance criterion 80 energy balance failure criteria 4, 147 energy dissipation of frictional sliding 9, 308

f fatigue behavior 6–7, 188 of ceramic-matrix composites experimental comparisons 191–205 theoretical analysis 189–191 fiber axial stress 77, 102, 214, 215 fiber debonding 1, 4, 146 fiber failure probability 160, 161, 166, 173, 179, 181, 277–280, 283, 285, 287, 288, 290, 292–295 fiber radius 78, 149, 151, 152, 179, 190, 214, 309 fiber stress 148, 149, 189, 190, 214, 272, 309 fiber thermal expansion coefficient 214 fiber Young’s modulus 148, 190, 214

fiber-reinforced ceramic-matrix composites (CMCs) 1, 5, 75–139, 187 micromechanical creep-oxidation model 213 fiber/matrix interface debonded length 76, 92, 93, 105, 139 fiber/matrix interface debonding 5, 19, 20, 25, 26, 29–36, 38, 76–85, 87, 89–93, 95, 97, 99, 101, 102, 111, 139 fiber/matrix interface debonding energy 5, 26, 76, 81, 111, 139 fiber/matrix interface frictional coefficient 25, 77, 127 fiber/matrix interface shear stress 5, 20, 40, 41, 76, 77, 92, 139, 236 first matrix cracking stress 3, 5, 8, 19, 20, 71, 83–101, 139, 146, 212 fracture mechanics approach 8, 78, 212, 214 fracture statistics-based model 8, 213 fracture strain 2, 3, 145, 146, 149, 215, 217, 218, 219, 223, 228, 229 fracture toughness 1, 6, 145, 215

g GE9x commercial aeroengine 307

212,

h Hi-Nicalon SiC/PyC/SiC composite 4, 147 Hi-NicalonTM SiC/[SiC-B4 C] composites 7, 188, 189, 191, 206 Hi-NicalonTM SiC/BN/SiC 36 Hi-NicalonTM SiC/SiC 6, 149, 150, 151–154, 156–159, 161–164, 188

i initial matrix crack spacing 80, 106 intact fiber stress 149, 189, 190, 214, 272, 309 interface bonding region 77, 78, 148, 214, 309 interface debonding energy 21, 102, 149 fraction 310 length 148, 214 ratio 80, 337

361

362

Index

Li model 19 long matrix crack spacing 76 low-cycle fatigue damage mechanism 8, 212

matrix crack spacing 2, 9, 76, 78, 80, 105–106, 145, 148, 152, 155, 179, 189, 214, 236, 251–255, 271, 274, 280–285, 308, 314–317, 337–339 matrix cracking 75, 92 density 80 saturation stress 83, 84, 86, 92 stress 2, 92 matrix fracture energy 5, 20–22, 27, 34–36, 39, 41, 59–60, 71, 76–77, 80–82, 89, 98, 102, 106, 113, 117, 133, 139 matrix interaction cracking model 76 matrix microcracking 4, 146 matrix multi-cracking evolution 76, 106 matrix multiple cracking evolution 5, 75–139 matrix statistical cracking model 76 matrix strain energy 5, 9, 76, 79–80, 105–106, 147, 308–309, 329 maximum stress criterion 4, 75, 147 MC model 19 McCartney model 19 MCE model 19–20 medium matrix crack spacing 76 micro matrix cracking 19 micro stress field analysis 5, 76–77, 139, 214 micro-cracks 2, 39–40, 102 micromechanical temperature-dependent vibration damping model 9, 308 mini-SiC/SiC composite 139 minicomposite 3–4, 7–8, 139, 145–147, 150–151, 153–154, 156–159, 161–164, 166–167, 169, 171, 212–213 monolithic ceramic 1, 145 multiple loading sequence 274–302 multiple matrix cracking 8, 75, 191, 212

m

n

material properties of SiC/SiC composite 163, 215–216, 310 matrix axial stress 21, 41, 77–79, 102, 105, 309

natural frequency, CMCs 308 NicalonTM SiC/[Si-B-C] composite 6 NicalonTM SiC/PyC/SiC 36 non-oxidized matrix 2

interface debonding (contd.) region 77 and slip length 190 interface frictional coefficient 5, 20, 22, 25, 33, 40, 50–53, 60, 77, 93–95, 102, 108, 117, 127–130, 325–328 interface frictional slip 334 interface oxidation length 40, 102, 105, 148, 160, 214, 284, 337, 340, 343 interface oxidation model parameters 149 interface oxidation ratio 106–118, 120–121, 123–124, 126–128, 130–131, 136–139, 149–162, 166–167, 173–179, 238–241, 247–248, 251–256, 259–261, 263–264, 267–268, 271–272, 278–279, 282–285, 334–335, 337–338, 340–341, 343–344, 346–347, 352, 354, 356 interface shear stress 24–25, 30–33, 42–50, 62–66, 77, 82–84, 92–93, 102, 106–108, 120–126, 155, 190, 191, 221–227, 253–261, 285–292, 321–325, 340–343 interface slip fraction 310 interface slip ratio 332, 334, 335, 337, 338, 340, 341, 343, 344, 346, 347, 349, 352 internal time-dependent damage evolution 214

k Kachanov’s damage parameter 7, 188, 190, 191, 193, 194, 197, 198, 206 Kuo–Chou model 19

l

Index

non-steady first matrix cracking stress 8, 212

o oxidation 4–5, 10, 38–42, 47, 101–102, 106–118, 146–150, 187–189, 211–215, 307–308 oxidation fraction at the interface 214–215, 217–219, 221, 223–227, 229–235, 237 oxide fiber reinforced oxide ceramic-matrix composites (oxide/oxide) 8, 211, 212, 307 oxide matrix 2

p parabolic rate constant 215 predicted matrix cracking stress 39, 60, 68, 70–71, 115 prepreg infiltration process 211 processing-induced micro-cracks 40 proportional limit stress (PLS) 2, 4, 20, 29, 80, 106, 145–146, 160, 310, 321 pyrocarbon (PyC) 6, 346 pyrolytic carbon (PyC) interphase 6, 28, 59, 188, 212

r reloading composite hysteresis strain 189–190 reloading intact fiber stress 189–190, 309

s saturated stress of matrix cracks 2 saturation matrix crack spacing 155, 179, 251–252, 254, 280, 284 saturation matrix cracking density 92–93, 96, 139 saturation matrix cracking stress 90, 98 self-healing 2.5D NicalonTM NL202 SiC/[Si-B-C] 6 self-healing 2.5D woven Hi-NicalonTM SiC/[Si-B-C] 7, 188 self-healing Hi-NicalonTM SiC/[Si-B-C] composite 6, 7, 192, 194–197, 199, 204

self-healing Hi-NicalonTM SiC/SiC composite 6, 188, 192, 194–197, 199–202, 204–205 self-sealing matrix 6, 187 shear-lag model parameter 78, 149, 214, 309 short matrix crack spacing 76 short matrix cracking stress 19 SiCf /SiC composite 187 SiC/SiC composite(s) material properties of 215 stress-rupture of for different environmental temperature 229, 232, 233 for different fiber volumes 215–218 for different fiber Weibull modulus 227–229, 230–231 for different interface shear stress 221–227 for different peak stress levels 218–221 for different saturation spaces between matrix cracking 221, 222 experimental comparisons 230–234 stress-rupture under multiple loading sequence for different environment temperature 292–295 for different fiber volume 275–280 for different interface shear stress 285–292 for different matrix crack spacing 280–285 experimental comparisons 295–300 stress-rupture under stochastic loading for different environmental temperature 261–264, 265 for different fiber volume 247–251 for different interface shear stress 253–261 for different matrix crack spacing 251–253 for different stochastic loading time interval 240–247

363

364

Index

SiC/SiC composite(s) (contd.) experimental comparisons 264, 266–274 temperature-dependent matrix cracking stress of for different fiber volumes 30 for different interface debonding energy 34 for different interface frictional coefficient 33–34 for different interface shear stress 30–33 for different matrix fracture energy 34–36 experimental comparisons 36–39 temperature-dependent matrix multiple cracking evolution of different fiber volumes 90–92 for different interface debonding energy 95–98 different interface frictional coefficient 93–95 for different matrix fracture energy 98–100 experimental comparison 100–101 for interface shear stress 92–93 temperature-dependent vibration damping effect of fiber volume 310–314 interface debonding energy effect 317–321 interface frictional coefficient effect 325–328 matrix crack spacing effect 314–317 steady-state interface shear stress effect 321–325 time-dependent matrix cracking stress of for different fiber volume 60–62 for different interface debonding energy 66–68 for different interface shear stress 62–66 for different matrix fracture energy 68, 69–70 experimental comparisons 68, 70–71

time-dependent matrix multiple cracking of for different fiber volume 117–120 for different interface debonding energy 130–133 for different interface shear stress 120–126 mini-SiC/SiC composite 139 unidirectional SiC/SiC composite 136–138 time-dependent tensile behavior of for different fiber radius 149, 151–152 for different fiber strength 159–160 for different fiber volumes 149, 150 for different fiber Weibull modulus 160, 161 for different interface debonding energy 155, 158–159 for different interface shear stress 155, 157 for different matrix cracking characteristic strength 152, 154 for different matrix cracking saturation spacing 155, 156 for different matrix Weibull modulus 152, 153 for different oxidation duration 160–161, 162 experimental comparisons 161, 163–179 SiC/SiC minicomposite(s) 3–4, 7, 139, 145–147, 150–151, 153–154, 156–159, 161–164, 166–167, 169, 171, 212 silicon carbide fiber reinforced silicon carbide ceramic-matrix composites (SiC/SiC) 211, 307 silicon nitride matrix 2 single crystal alloy 187 single dominant cracking 80, 147 single tow Hi-Nicalon SiC/PyC/SiC composite 4, 147 single tow Hi-NicalonTM 4, 147, 212 single-phase ceramics 19 slow-crack-growth mechanism 4, 147

Index

steady state strain energy release rate 76 steady-state first matrix cracking stress 19 steady-state matrix cracking 20, 76 stochastic loading stress 234–236, 260 stress analysis, time-dependent 102 stress intensity factor method 19 stress-rupture behavior 4, 7–9, 147, 212, 230, 235, 264, 288, 291, 295 stress-rupture lifetime 8–9, 212–213, 215, 218–219, 221, 223, 225, 227–230, 235–237, 240–241, 244, 247–248, 253, 255–256, 261, 264, 267–268, 271–272, 275, 280, 285, 288, 292, 302 stress-rupture of SiC/SiC composite for different environmental temperature 229, 232, 233 for different fiber volumes 215–218 for different fiber Weibull modulus 227–229, 230–231 for different interface shear stress 221–227 for different peak stress levels 218–221 for different saturation spaces between matrix cracking 221, 222 experimental comparisons 230–234 under multiple loading sequence for different environment temperature 292–295 for different fiber volume 275–280 for different interface shear stress 285–292 for different matrix crack spacing 280–285 experimental comparisons 295–300 under stochastic loading for different environmental temperature 261–264, 265 for different fiber volume 247–251 for different interface shear stress 253–261

for different matrix crack spacing 251–253 for different stochastic loading time interval 240–247 experimental comparisons 264, 266–274 stress-strain curve 2, 5, 19, 28, 36–38, 75, 80, 145, 149–162, 164, 167, 169, 171, 173–180 of CMCs 75 superalloy 1, 187, 211 Sutcu–Hilling model 19 Sylramic-iBN/BN/SiC composite 4, 147

t temperature resistance 75, 187, 307 temperature-dependent carbon fiber axial and radial thermal expansion coefficient 22, 81 elastic modulus 22, 81 temperature-dependent composite thermal expansion coefficient 309 temperature-dependent composite vibration damping 10, 308, 310, 313–314, 320–321, 325, 328–329, 343 temperature-dependent counter slip length 309 temperature-dependent critical stress 80, 106 temperature-dependent energy dissipated density per cycle of motion 308–309 temperature-dependent fiber axial stress 77, 102 temperature-dependent fiber/matrix axial stress 78 temperature-dependent fiber/matrix interface debonded length 105 temperature dependent fiber/matrix interface debonding energy 81 temperature-dependent fiber/matrix interface shear stress 5, 40, 76–77, 139 temperature-dependent intact fiber stress 309

365

366

Index

temperature-dependent interface debonding energy 22 temperature-dependent interface debonding length 21, 41, 78, 104, 309 temperature-dependent interface oxidation length 105 temperature-dependent interface shear stress 78, 102, 309 temperature-dependent matrix axial stress 79, 105 temperature-dependent matrix crack spacing 105 temperature-dependent matrix cracking stress of C/SiC composites different fiber volumes 23 for different fiber/matrix interface frictional coefficient 25 for different interface debonding energy 26 for different interface shear stress 24 experimental comparisons 28 theoretical models 20–21 of SiC/SiC composite for different fiber volumes 30 for different interface debonding energy 34 for different interface frictional coefficient 33–34 for different interface shear stress 30–33 for different matrix fracture energy 34–36 experimental comparisons 36–39 temperature-dependent matrix fracture energy 21, 80, 106 temperature-dependent matrix modulus 105 temperature-dependent matrix multiple cracking evolution of C/SiC composites ACK matrix cracking stress 80 critical matrix strain energy 80 critical matrix strain energy parameter 80 for different fiber/matrix interface debonding energy 84–85

for different interface shear stress 82–83 for different matrix fracture energy 85–88 energy balance criterion 80 experimental comparisons 88–89 fiber/matrix interface debonding 78–79 interface debonding ratio 80 matrix cracking density 80 matrix fracture energy 81 matrix multiple cracking evolution 80 temperature-dependent carbon fiber axial and radial thermal expansion coefficient 81 temperature-dependent carbon fiber elastic modulus 81 temperature dependent fiber/matrix interface debonding energy 81 temperature-dependent matrix fracture energy 80 temperature-dependent matrix strain energy 79 temperature-dependent SiC matrix elastic modulus 81 temperature-dependent stress analysis 77–78 of SiC/SiC composites different fiber volumes 90–92 for different interface debonding energy 95–98 different interface frictional coefficient 93–95 for different matrix fracture energy 98–100 experimental comparison 100–101 for interface shear stress 92–93 temperature-dependent matrix strain energy 79, 105 temperature-dependent maximum strain energy density 308–309 temperature-dependent SiC matrix axial and radial thermal expansion coefficient 22, 81 temperature-dependent SiC matrix elastic modulus 22, 81

Index

temperature-dependent stress analysis 77–78 temperature-dependent unloading and reloading intact fiber stress 309 temperature-dependent vibration damping of CMCs effect of fiber volume on SiC/SiC composite 310–314 experimental comparisons 329, 330, 343, 346–356 fiber volume effect, C/SiC composite 331–334 interface debonding energy effect, SiC/SiC composite 317–321 interface frictional coefficient effect, SiC/SiC composite 325–328 interface shear stress effect, C/SiC composite 340–343 matrix crack spacing effect on SiC/SiC composite 314–317 matrix crack spacing effect, C/SiC composite 337–340 steady-state interface shear stress effect, SiC/SiC composite 321–325 temperature effect, C/SiC composite 343, 344–345 theoretical models 308–310, 329, 331 vibration stress effect, C/SiC composite 334–337 tensile behavior 2–5, 10, 28, 36, 59, 68, 145–181 tensile strength 2–3, 28, 59, 145–146, 149, 160, 163, 166, 173, 175, 177, 179, 181 thermal expansion coefficient 22, 39–40, 77–78, 80–81, 102, 148, 190, 214, 309 Thouless–Evans model 19 T300TM -C/SiC composite 59 time-dependent composite damage mechanism 214 time-dependent composite deformation 9, 213, 215–216, 221, 227, 229–230, 302 time-dependent composite strain 148–149, 214–215, 221, 226, 230–231, 233, 292, 296–302

time-dependent debonding fraction 215, 217, 219, 223–225, 227, 229–231, 233, 235, 237, 239 time-dependent failure probability 217, 224 time-dependent fiber and matrix strain energy density per cycle 329 time-dependent fiber axial stress 214–215 time-dependent fiber/matrix interface oxidation length 40 time-dependent interface debonding 103–104, 150–151, 153–154, 156–158, 162, 164, 167, 169, 171, 332, 335, 338, 341, 344, 346–347, 349, 352, 354 time-dependent interface oxidation length 102 time-dependent matrix cracking stress of C/SiC composites for different fiber volumes 42, 43–44 for different interface debonding energy 53–56 for different interface frictional coefficient 50–53 for different interface shear stress 42, 45–50 different matrix fracture energy 56–59 experimental comparisons 59, 60 theoretical models 39–41 of SiC/SiC composites for different fiber volume 60–62 for different interface debonding energy 66–68 for different interface shear stress 62–66 for different matrix fracture energy 68, 69–70 experimental comparisons 68, 70–71 time-dependent matrix multiple cracking of C/SiC composite for different interface debonding energy 111–113 for different interface frictional coefficient 108–111

367

368

Index

time-dependent matrix multiple cracking (contd.) for different interface shear stress 106–108 for different matrix fracture energy 113–114 experimental comparisons 114–116 time-dependent interface debonding 103–104 time-dependent stress analysis 102–103 of SiC/SiC composite for different fiber volume 117–120 for different interface debonding energy 130–133 for different interface frictional coefficient 127–130 for different interface shear stress 120–126 for different matrix fracture energy 133–136 mini-SiC/SiC composite 139 unidirectional SiC/SiC composite 136–138 time-dependent oxidation fraction 215, 217, 219, 223–225, 227, 230–233, 235, 237, 239 time-dependent proportional limit stress 4, 146 time-dependent stress analysis 102–103 time-dependent tensile behavior CMCs theoretical analysis 148–149 of SiC/SiC composite for different fiber radius 149, 151–152 for different fiber strength 159–160 for different fiber volumes 149, 150 for different fiber Weibull modulus 160 for different interface debonding energy 155, 158–159 for different interface shear stress 155, 157

for different matrix cracking characteristic strength 152, 154 for different matrix cracking saturation spacing 155, 156 for different matrix Weibull modulus 152, 153 for different oxidation duration 160–163 experimental comparisons 161, 163–173

u unidirectional C/SiC composite 3, 76, 88–89, 102, 114, 116, 139, 146, 174 unidirectional Hi-NicalonTM SiC/SiC minicomposite 150–151, 153–154, 156–159, 161, 163 unidirectional Hi-NicalonTM Type S SiC/SiC minicomposite 166, 167 unidirectional SiC/SiC composite 89, 100, 101, 136–138 unidirectional SiC/SiC minicomposites 161 unidirectional TyrannoTM SA3 SiC/SiC minicomposite 166 unidirectional TyrannoTM ZMI SiC/SiC minicomposite 166 unidirectional/cross-ply CMCs 9, 308

v vibration behavior 2, 9–10 vibration characteristics of CMC panel 9, 307

w Weibull modulus of the fiber 215, 227–230 woven SiC/BN/SiC composite 4, 147 woven Sylramic-iBN/BN/SiC composite 4, 147

y Young’s modulus of matrix and fibers 76