Vibration Behavior in Ceramic-Matrix Composites 9811978379, 9789811978371

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Advanced Ceramics and Composites 5 Series Editor: Longbiao Li

Longbiao Li

Vibration Behavior in Ceramic-Matrix Composites

Advanced Ceramics and Composites Volume 5

Series Editor Longbiao Li , College of Civil Aviation, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu, China

The book series “Advanced Ceramics and Composites” publishes insights and latest research results on advanced ceramics and composites, as well as the applications of these materials. The intent is to cover all the technical contents, applications, and multidisciplinary aspects of Advanced Ceramics and Composites. The objective of the book series is to publish monographs, reference works, selected contributions from specialized conferences, and textbooks with high quality in the field advanced ceramics and composite materials. The series provides valuable references to a wide audience in research community in materials science, research and development personnel of ceramic and composite materials, industry practitioners and anyone else who are looking to expand their knowledge of ceramics and composites.

Longbiao Li

Vibration Behavior in Ceramic-Matrix Composites

Longbiao Li College of Civil Aviation Nanjing University of Aeronautics and Astronautics Nanjing, Jiangsu, China

ISSN 2662-9305 ISSN 2662-9313 (electronic) Advanced Ceramics and Composites ISBN 978-981-19-7837-1 ISBN 978-981-19-7838-8 (eBook) https://doi.org/10.1007/978-981-19-7838-8 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

To My Son Shengning Li

Preface

Ceramic-matrix composites (CMCs) possess a great potential for the application in hot-section components of aeroengines due to their low density, high-temperature resistance, and oxidation resistance. The high-temperature components in aeroengines mainly include combustion chambers, turbines, and nozzles. Currently, the temperature resistance limit of high-temperature alloys is maintained around 1100 o C, but the application of CMCs increases the temperature resistance of hotsection components to 1200–1350 o C, and the mass of CMC components is usually 1/4–1/3 of the mass of nickel-based alloy components, which can reduce the specific fuel consumption by increasing the operating temperature and reducing the mass of components. This book focuses on the vibration behavior in CMCs, as follows: • The vibration natural frequency of 2D SiC/SiC composite under tensile loading was investigated. Nonlinear damage and fracture are mainly attributed to damage mechanisms of matrix cracking, interface debonding, and fibers fracture. Under cyclic loading/unloading, hysteresis loops appeared due to internal frictional slip between the fiber and the matrix and the natural frequency were obtained for different peak stress. A micromechanical tensile and cyclic loading/unloading constitutive model was adopted to predict the tensile curves. Microdamage parameters of interface debonding ratio and broken fibers fraction were used to characterize tensile damage and fracture. Relationships between natural frequency, interface debonding, and fibers fracture were established. • The vibration damping in fiber-reinforced CMCs was investigated considering fibers debonding and fracture. Micromehcanical vibration damping models were developed considering multiple damage mechanisms. Relationships between the damping of CMCs, damping of fiber and the matrix, damping caused by frictional slip between the fiber and the matrix, and fiber debonding and fracture were established. Effects of fiber volume, matrix crack spacing, interface shear stress, interface debonding energy, fiber strength and fiber Weibull modulus on the damping of CMCs, interface debonding and slip between the fiber and the matrix, and fiber broken fraction were analyzed. Experimental damping of 2D C/SiC composite was predicted.

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• The temperature-dependent vibration damping in C/SiC composites with different fiber preforms under different vibration frequencies was investigated. A micromechanical temperature-dependent vibration damping model was developed to establish the relationship between composite damping, material properties, internal damage mechanisms, and temperature. Effects of fiber volume, matrix crack spacing, and interface properties on temperature-dependent vibration damping of CMCs and interface damage were analyzed. Experimental temperature-dependent composite damping of 2D and 3D C/SiC composites was predicted for different loading frequencies. • A time-dependent vibration damping model of fiber-reinforced CMCs was developed. 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 were 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 were discussed. • A cyclic-dependent vibration damping model of fiber-reinforced CMCs was developed. Combining cyclic-dependent damage mechanisms, damage models, and dissipated energy model, relationships between composite vibration damping, cyclic-dependent damage mechanisms, vibration stress, and applied cycle number were established. Effects of material properties and damage state on composite vibration damping were analyzed for different applied cycle number and vibration stress. Experimental composite vibration damping of 2D and 3D C/SiC composites without/with coating was predicted for different vibration frequencies and applied cycle number. I hope this book can help the material scientists and engineering designers to understand and master the vibration behavior of ceramic-matrix composites. Nanjing, China September 2022

Longbiao Li

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Application Background of Ceramic-Matrix Composites . . . . . . . . . 1.2 Vibration Behavior in Ceramic-Matrix Composites . . . . . . . . . . . . . . 1.2.1 Vibration Natural Frequency of CMCs . . . . . . . . . . . . . . . . . . 1.2.2 Vibration Damping of CMCs . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 3 5 11 14

2 Vibration Natural Frequency of Ceramic-Matrix Composites . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Materials and Experimental Procedures . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Theoretical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 15 16 16 17 19 20

3 Vibration Damping of Ceramic-Matrix Composites Considering Fiber Debonding and Fracture . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Micromechanical Damping Models of Ceramic-Matrix Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Damping in Intact Ceramic-Matrix Composites . . . . . . . . . . . 3.2.2 Damping in Damaged Ceramic-Matrix Composites . . . . . . . 3.3 Result and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Effect of Fiber Volume on Damping of Ceramic-Matrix Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Effect of Matrix Crack Spacing on Damping of Ceramic-Matrix Composites . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Effect of Interface Shear Stress on Damping of Ceramic-Matrix Composites . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Effect of Interface Debonding Energy on Damping of Ceramic-Matrix Composites . . . . . . . . . . . . . . . . . . . . . . . .

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3.3.5 Effect of Fiber Strength on Damping of Ceramic-Matrix Composites . . . . . . . . . . . . . . . . . . . . . . . . 3.3.6 Effect of Fiber Weibull Modulus on Damping of Ceramic-Matrix Composites . . . . . . . . . . . . . . . . . . . . . . . . 3.3.7 Comparison of Damping of Ceramic-Matrix Composites without/with Considering Fiber Failure . . . . . . . 3.4 Experimental Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Temperature-Dependent Vibration Damping of Ceramic-Matrix Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Temperature-Dependent Micromechanical Vibration Damping Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Effect of Fiber Volume on Temperature-Dependent Damping of C/SiC Composite . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Effect of Matrix Crack Spacing on Temperature-Dependent Damping of C/SiC Composite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Effect of Interface Debonding Energy on Temperature-Dependent Damping of C/SiC Composite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Effect of Steady-State Interface Shear Stress on Temperature-Dependent Damping of C/SiC Composite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Experimental Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 2D C/SiC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 3D C/SiC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Time-Dependent Vibration Damping of Ceramic-Matrix Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Time-Dependent Micromechanical Vibration Damping Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Effect of Fiber Volume on Time-Dependent Vibration Damping of C/SiC Composite . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Effect of Vibration Stress on Time-Dependent Vibration Damping of C/SiC Composite . . . . . . . . . . . . . . . . . 5.2.3 Effect of Matrix Crack Spacing on Time-Dependent Vibration Damping of C/SiC Composite . . . . . . . . . . . . . . . . .

33 35 37 39 41 41 43 43 44 47 48

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5.2.4 Effect of Interface Shear Stress on Time-Dependent Vibration Damping of C/SiC Composite . . . . . . . . . . . . . . . . . 85 5.2.5 Effect of Temperature on Time-Dependent Vibration Damping of C/SiC Composite . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.3 Experimental Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.3.1 t = 2 h at T = 700, 1000, and 1300 °C . . . . . . . . . . . . . . . . . . 91 5.3.2 t = 5 h at T = 700, 1000, and 1300 °C . . . . . . . . . . . . . . . . . . 94 5.3.3 t = 10 h at T = 700, 1000, and 1300 °C . . . . . . . . . . . . . . . . . 97 5.4 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.5 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6 Cyclic-Dependent Vibration Damping of Ceramic-Matrix Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Cyclic-Dependent Micromechanical Vibration Damping Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Cyclic-Dependent Damping in Intact CMCs . . . . . . . . . . . . . 6.2.2 Cyclic-Dependent Damping in Damaged CMCs . . . . . . . . . . 6.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Effect of Fiber Volume on Cyclic-Dependent Vibration Damping of CMCs . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Effect of Matrix Crack Spacing on Cyclic-Dependent Vibration Damping of CMCs . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Effect of Interface Debonding Energy on Cyclic-Dependent Vibration Damping of CMCs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Experimental Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 2D C/SiC Composite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 3D C/SiC Composite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

105 105 106 107 107 109 110 112

114 115 117 117 122 123 123

Chapter 1

Introduction

Abstract Ceramic-matrix composites (CMCs) possess great potential for the application in hot-section components of aero engines due to their low density, hightemperature resistance, and oxidation resistance. In this chapter, the application of ceramic-matrix composites (CMCs) in hot-section components of aerospace was analyzed. The vibration behavior in CMCs including vibration natural frequency and vibration damping was also discussed. Effects of temperature, time, and cycle on the vibration damping in CMCs were analyzed. Keywords Ceramic-matrix composites (CMCs) · Hot-section components · Vibration

1.1 Application Background of Ceramic-Matrix Composites Ceramic-matrix composites (CMCs) possess great potential for the application in hotsection components of aeroengines due to their low density, high-temperature resistance, and oxidation resistance. The high-temperature components in aeroengines mainly include combustion chambers, turbines, and nozzles. Currently, the temperature resistance limit of high-temperature alloys is maintained around 1100 °C, but the application of CMCs increases the temperature resistance of hot-section components to 1200–1350 °C, and the mass of CMC components is usually 1/4–1/3 of the mass of nickel-based alloy components, which can reduce the specific fuel consume by increasing the operating temperature and reducing mass of components (Naslain 2004). Since the 1950s, many countries have carried out research on the application of CMC on the hot-section components of aeroengines. In 1990s, SNECMA company developed CERASEP CMC and successfully applied the CMC on the nozzle flaps of M-88 aeroengine. SNECMA cooperated with P&W company to develop CMC nozzle flap/sealing components and completed tests on F100 engine with 1300 h cumulative operation. SNECMA company upgraded CERASEP CMC by introducing self-healing matrix to improve the high-temperature mechanical properties and oxidation resistance and fabricated the combustion liner and other aeroengine © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 L. Li, Vibration Behavior in Ceramic-Matrix Composites, Advanced Ceramics and Composites 5, https://doi.org/10.1007/978-981-19-7838-8_1

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1 Introduction

components. The CERASEP-A410 flame stabilizer passed 143 h test at 1180 °C, and the CERASEP-A415 combustion liner passed 180 h bench test with no defects produced. The full-size combustion chamber test was conducted on the platform of CFM56 aeroengine, and the cooling air volume was reduced by 35% compared with the high-temperature alloy (Li 2020). GE has successfully applied CMC materials on the hot-section components of F414 and GEnx aeroengines, including the turbine outer rings, nozzles, combustion flame barrels, and turbine guide vanes, resulting in a significant improvement in engine performance. On February 10, 2015, GE Aviation introduced SiC/SiC blades to the worst engine conditions for the first time, successfully verifying the SiC/SiC turbine blades on the F414 engine. The test results showed that the SiC/SiC blades were still strong enough to withstand the high turbine temperatures and rotational stresses after 500 severe cycle tests at 1650 °C (DiCarlo and Roode 2006). Under the support of the UEET program, NASA carried out research work on SiC/SiC turbine guide vanes with 15–25% reduction in cooling air consumption and inlet turbine temperature exceeding 1650 °C. NASA Greene Research Center conducted comparative tests for high-temperature alloys and SiC/SiC turbine guide vanes in a real gas turbine environment. During the tests, 102 thermal cycle tests were conducted after 50 h of steady-state high-temperature examination. The results showed that the SiC/SiC blades in service under extreme thermal conditions demonstrated superior durability and reliability than the high-temperature alloy blades. IHI Corporation designed and tested the CMC low-pressure turbine guide vane, a SiC/SiC composite, which uses three-dimensional braiding technology. The guide vane leads the swirling airflow to axial flow, consisting of airfoil section and edge section. To evaluate the durability of CMC low-pressure turbine guide vane under high-temperature gas flow, a special testing device was designed for the model turbine guide vane. The test temperature is 1150–1300 °C, and the test cycle number is 1000. The results show that no damage occurs, which proves that the CMC turbine guide vane has enough durability (Tamura and Nakamura 2005). The Continuous Lower Energy, Emissions, and Noise (CLEEN) program is the FAA’s principal NextGen environmental effort to develop and demonstrate new technologies, procedures, and sustainable alternative jet fuels (Petervary and Steyer 2012). Under the CLEEN program, Boeing is teaming with partners ATK/COI-C, Albany Engineered Composites (AEC) to develop and demonstrate an acoustic CMC (oxide) exhaust nozzle on a Rolls Royce Trent 1000 engine. The CMC technology will reduce weight and increase the temperature capability of future engine nozzle, reducing aircraft fuel burn by 1%. Liu et al. (2022) performed the investigation on the design, fabrication, and experimental verification of SiC/SiC turbine blisk. The Spider Web Structure (SWS) SiC preform was used as the reinforcement in turbine blisk. The BN interphase and SiC matrix were deposited on the surface of the SWS SiC fiber preform, respectively. The SiC/SiC turbine blisk was machined by “on-line processing” to form a turbine blisk that meets the design requirements. The environmental barrier coatings (EBCs) were prepared on the surface of SiC/SiC turbine blisk using the atmospheric plasma

1.2 Vibration Behavior in Ceramic-Matrix Composites

3

Fig. 1.1 CT detection of SiC/SiC turbine blisk

spraying (APS) method. The CT scan was conducted to characterize the internal defects in SiC/SiC turbine blisk (Fig. 1.1). The mechanical performance evaluation, over rotation test and engine bench test were performed, respectively. The maximum failure strength approached 300 MPa. During the over rotation test, when the rotating speed reached n = 104,166 r/min, the blade in the turbine blisk broke; and when the rotating speed reached n = 108,072 r/min, the disk in the turbine blisk broke. During the engine bench test, low cyclic fatigue of N = 994 cyclic with maximum speed nmax = 60,000 r/min and N = 100 cyclic with maximum speed nmax = 70,000 r/min were completed. On January 1, 2022, the SiC/SiC turbine blisk successfully completed the first flight test in Zhuzhou China, which verified the feasibility of SiC/SiC turbine rotor application in gas turbine engines.

1.2 Vibration Behavior in Ceramic-Matrix Composites 1.2.1 Vibration Natural Frequency of CMCs Figure 1.2 shows experimental and predicted cyclic loading/unloading hysteresis loops, degradation rate of composite modulus, interface debonding ratio, and broken fiber fraction versus degradation rate of natural frequency curves of 2D SiC/SiC composite.

4 Fig. 1.2 a Experimental and predicted cyclic loading/unloading hysteresis loops; b the interface debonding ratio versus degradation rate of natural frequency curve; and c the broken fiber fraction versus degradation rate of natural frequency curve of 2D SiC/SiC composite

1 Introduction

1.2 Vibration Behavior in Ceramic-Matrix Composites

5

• When the peak stress increases to σ max = 213.2 MPa, the natural frequency decreases from f 0 = 3415.6 Hz to 3271.5, and the degradation rate of natural frequency is φ = 0.042; the interface debonding ratio increases to ψ d = 1.0; and the broken fiber fraction increases to q = 0.034. • When the degradation rate of natural frequency is φ = 0.01, matrix cracking and interface debonding occur, and however, fiber failure does not appear; when the degradation rate of natural frequency is φ = 0.04, the composite modulus decreases approximately 47%, the interface debonding ratio approaches ψ d = 0.8, and the broken fiber fraction is approximately q = 2.2%.

1.2.2 Vibration Damping of CMCs 1.2.2.1

Vibration Damping of C/SiC at Room Temperature

Experimental and predicted composite damping ηc , interface debonding ratio ψ d , and interface slip ratio ψ s versus amplitude of stress curves of C/SiC composite were shown in Fig. 1.3. Experimental composite damping ηc was ηc = 0.0106 at T = 50 °C, corresponding to σ = 30 MPa, the interface shear stress range τ i = 9 and 10 MPa, the interface debonding ratio range ψ d = 0.716 and 0.798, the interface slip ratio range ψ s = 0.45 and 0.5, and the broken fiber fraction P = 0.0002.

1.2.2.2

Temperature-Dependent Vibration Damping of C/SiC

Experimental and predicted temperature-dependent composite damping (ηc ), interface debonding ratio (ψ d ), and interface slip ratio (ψ s ) versus temperature curves of the 2D C/SiC composite at the vibration frequency f = 1 Hz were shown in Fig. 1.4. The predicted peak composite damping agreed with experimental data, and the predicted corresponding temperature for the peak composite damping was a little lower than the experimental data. Experimental composite damping increased from ηc = 0.01 at room temperature to the peak value ηc = 0.019 at T = 283 °C and then decreased to ηc = 0.014 at T = 400 °C. The theoretical predicted composite damping increased from ηc = 0.008 at room temperature to the peak value ηc = 0.019 at T = 279 °C and then decreased to ηc = 0.015 at T = 400 °C. The interface debonding ratio (ψ d ) decreased from ψ d = 0.337 at room temperature to ψ d = 0.114 at T = 400 °C, and the interface slip ratio (ψ s ) decreased from ψ s = 0.248 at room temperature to ψ s = 0.091 at T = 400 °C. Experimental and predicted temperature-dependent composite damping (ηc ), interface debonding ratio (ψ d ), and interface slip ratio (ψ s ) versus temperature curves of 3D C/SiC composite at the vibration frequency f = 1 Hz were shown in Fig. 1.5. Experimental composite damping (ηc ) increased from ηc = 0.009 at room temperature to the peak value ηc = 0.0165 at T = 325 °C and then decreased to ηc = 0.015 at T = 400 °C. The theoretical predicted composite damping (ηc ) increased from

6 Fig. 1.3 a ηc ; b ψ d ; c ψ s ; and d P of C/SiC composite

1 Introduction

1.2 Vibration Behavior in Ceramic-Matrix Composites

7

Fig. 1.3 (continued)

ηc = 0.009 at room temperature to the peak value ηc = 0.0163 at T = 308 °C and then decreased to ηc = 0.015 at T = 400 °C. The interface debonding ratio (ψ d ) decreased from ψ d = 0.455 at room temperature to ψ d = 0.245 at T = 400 °C, and the interface slip ratio (ψ s ) decreased from ψ s = 0.449 at room temperature to ψ s = 0.245 at temperature T = 400 °C.

1.2.2.3

Time-Dependent Vibration Damping of C/SiC

Experimental and predicted time-dependent composite vibration damping (ηc ), the interface debonding ratio (ψ d = 2ld /l c ), interface oxidation ratio (ψ o = ζ /l d ), and interface slip ratio (ψ s = 2ly /l c ) of C/SiC composite after oxidation duration t = 2 h at oxidation temperature T = 700, 1000, and 1300 °C were shown in Fig. 1.6. When the oxidation duration was t = 2 h, 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.

1.2.2.4

Cyclic-Dependent Vibration Damping of C/SiC

Experimental and predicted cyclic-dependent composite vibration damping, ratio of interface debonding, and slip versus the vibration stress amplitude curves of 2D C/SiC composite without/with CVD-SiC coating for different vibration frequencies and applied cycle number were shown in Figs. 1.7 and 1.8. For 2D C/SiC without SiC coating, experimental composite vibration damping ηc was ηc = 0.01057, 0.00927, and 0.00791 at f = 1, 2, and 5 Hz corresponding to σ = 30 MPa, as shown in Fig. 1.7a; theoretical results of cyclic-dependent composite

8 Fig. 1.4 a Experimental and predicted temperature-dependent composite damping (ηc ) versus temperature curves; b the temperature-dependent interface debonding ratio (ψ d ) versus temperature curves; and c the temperature-dependent interface slip ratio (ψ s ) versus temperature curves of 2D C/SiC composite at the vibration frequency f = 1 Hz

1 Introduction

1.2 Vibration Behavior in Ceramic-Matrix Composites Fig. 1.5 a Experimental and predicted temperature-dependent composite damping (ηc ) versus temperature curves; b the temperature-dependent interface debonding ratio (ψ d ) versus temperature curves; and c the temperature-dependent interface slip ratio (ψ s ) versus temperature curves of 3D C/SiC composite at the vibration frequency f = 1 Hz

9

10 Fig. 1.6 a Experimental and predicted ηc versus T curves; b time-dependent ψ d versus T curves; c time-dependent ψ o versus T curves; and d time-dependent ψ s versus T curves of C/SiC composite after oxidation duration t = 2 h at T = 700, 1000, and 1300 °C

1 Introduction

1.3 Summary and Conclusions

11

Fig. 1.6 (continued)

vibration damping, interface debonding, and slip fraction for cycle number N = 1, 102 , and 103 were shown in Fig. 1.7a–c. For 2D C/SiC with SiC coating, experimental composite vibration damping ηc was ηc = 0.00871, 0.00766, and 0.00625 at f = 1, 2, and 5 Hz corresponding to σ = 30 MPa, as shown in Fig. 1.8a; theoretical results of cyclic-dependent composite vibration damping, interface debonding, and slip fraction for cycle number N = 1, 300, and 500 were shown in Fig. 1.8a–c.

1.3 Summary and Conclusions This chapter focused on the introduction on the application background of CMCs in hot-section components in aero-engines. The vibration behavior in CMCs including vibration natural frequency and vibration damping was also discussed. Effects of temperature, time, and cycle on the vibration damping in CMCs were analyzed.

12 Fig. 1.7 a Experimental and predicted ηc versus σ curves; b theoretical ψ d versus σ curves; and c theoretical ψ s versus σ curves of 2D C/SiC composite without coating for different vibration frequencies and applied cycle number

1 Introduction

1.3 Summary and Conclusions Fig. 1.8 a Experimental and predicted ηc versus σ curves; b theoretical ψ d versus σ curves; and c theoretical ψ s versus σ curves of 2D C/SiC composite with coating for different vibration frequencies and applied cycle number

13

14

1 Introduction

References DiCarlo JA, Roode M (2006) Ceramic composite development for gas turbine hot section components. In: Proceedings of the ASME turbo expo: power for land, sea and air, vol 2, pp 221−231 Li LB (2020) Durability of ceramic matrix composites. Elsevier, Oxford, UK Liu X, Xu Y, Li J, Luo X, Guo X, Hu X, Cao X, Li LB, Liu C, Dong N, Liu Y (2022) Design, fabrication and testing of ceramic-matrix composite turbine blisk. Acta Materiae Compositae Sinica, Online. https://doi.org/10.13801/j.cnki.fhclxb.20220407.00 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 Petervary M, Steyer T (2012) The Boeing Company. Ceramic matrix composites for structural aerospace applications. In: The 4th international congress on ceramics, Chicago, IL, USA, 17 July 2012 Tamura T, Nakamura T (2005) Research of CMC application to turbine components. IHI Eng Rev 38:58–62

Chapter 2

Vibration Natural Frequency of Ceramic-Matrix Composites

Abstract In this chapter, the vibration natural frequency of 2D SiC/SiC composite under tensile loading was investigated. Nonlinear damage and fracture are mainly attributed to damage mechanisms of matrix cracking, interface debonding, and fiber fracture. Under cyclic loading/unloading, hysteresis loops appeared due to internal frictional slip between the fiber and the matrix, and the natural frequency was obtained for different peak stress. A micromechanical tensile and cyclic loading/unloading constitutive model was adopted to predict the tensile curves. Microdamage parameters of interface debonding ratio and broken fiber fraction were used to characterize tensile damage and fracture. Relationships between natural frequency, interface debonding, and fiber fracture were established. Keywords Ceramic-matrix composites (CMCs) · Natural frequency · Matrix cracking · Interface debonding · Fiber failure

2.1 Introduction During application of CMC components, there exist internal process defects and damages caused by external environment and load (Naslain 2004; Li 2018, 2020, 2019a, 2019b, 2019c). Wang et al. (2008) investigated cyclic loading/unloading tensile behavior of 2D C/SiC composite and obtained the interface shear stress and thermal residual stress in the composite through hysteresis analysis. Li et al. (2014) investigated the tensile loading/unloading stress–strain behavior of 2D SiC/SiC composite and analyzed the matrix cracking evolution during tensile loading. It is necessary to develop damage detecting method to determine internal defects and damages. There are many types of nondestructive testing (NDT) methods applied to the internal defects and environmental damage of CMCs, including X-ray, infrared thermal imaging, microanalysis, laser holography, CT, acoustic emission (AE), electrical resistance (ER), ultrasonic, etc. The natural frequency is one of the main characteristics of vibration characteristics (Guo et al. 2021). Bai et al. (2005) investigated the natural frequency of delaminated advanced grid stiffened composite plates by hump resonance method. Li (2019d) performed vibration analysis of damaged CMC

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 L. Li, Vibration Behavior in Ceramic-Matrix Composites, Advanced Ceramics and Composites 5, https://doi.org/10.1007/978-981-19-7838-8_2

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2 Vibration Natural Frequency of Ceramic-Matrix Composites

structures. When the natural frequency decreased approximately 5.47%, the CMC structure was seriously damaged and in a dangerous state. The objective of this chapter was to investigate tensile damage and fracture of 2D SiC/SiC composites using natural frequency. Nonlinear damage and fracture were mainly attributed to damage mechanisms of matrix cracking, interface debonding, and fiber fracture. Under cyclic loading/unloading, hysteresis loops appear due to internal frictional slip between the fiber and the matrix, and the natural frequency and composite modulus were obtained for different peak stress. A micromechanical tensile and cyclic loading/unloading constitutive model was adopted to predict the hysteresis curves. Microdamage parameters of interface debonding ratio and broken fiber fraction were used to characterize tensile damage and fracture. Relationships between natural frequency, composite modulus, interface debonding, and fiber fracture were established.

2.2 Materials and Experimental Procedures 2D woven SiC/SiC composite was provided by Central South University. The Boxiang™ III SiC fiber was woven into two-dimensional fiber preform. The PyC interphase was deposited on the SiC surface using chemical vapor infiltration (CVI) method, and the interphase thickness was approximately 150 nm. The fabrication of SiC/SiC composite was using the precursor infiltration process (PIP) method. After fabrication, the dog-bone-shaped specimens, with dimensions of 111.86 mm in length, 3.42 mm thick, and 6.32 mm width in the gauge section, were cut from composite panes using wire-electrode cutting. The fiber volume of the composite was approximately 40%. Cyclic loading/unloading tensile tests of 2D SiC/SiC composite were conducted on an INSTRON 8872 servo hydraulic load frame (INSTRON Systems Corp., Boston, Massachusetts, USA). During tensile tests, the clip-on extensometer was used to obtain the composite strain. PSV-500 laser vibration meter and piezoelectric ceramics were used to test the natural frequency of the sample.

2.3 Theoretical Models Under cyclic loading/unloading tensile, multiple damage mechanisms of matrix cracking, interface debonding, and fiber failure contribute to the nonlinear tensile and hysteresis loops of CMCs. Upon unloading/reloading, the hysteresis loop constitutive relationship for the interface partial is (Bai et al. 2005). εunloading

( )( ) ΦU τi l y2 τi 2l y − ld 2l y + ld − lc = +4 − − (αc − αf )ΔT (2.1) Ef E f rf lc Ef rf lc

2.4 Results and Discussions

17

( )2 ΦR τi l z2 τi l y − 2l z −4 +4 Ef E rl Ef rf lc ( f fc )( ) τi ld − 2l y + 2l z ld + 2l y − 2l z − lc +2 − (αc − αf )ΔT Ef rf lc

εreloading =

(2.2)

where ΦU and ΦR are the intact fiber stress upon unloading and reloading, and ly and lz are the interface counter slip length and interface new slip length. When the interface complete debonding occurs, the hysteresis loop constitutive relationship is (Bai et al. 2005). ( )2 ΦU τi l y2 τi 2l y − lc /2 +4 −2 − (αc − αf )ΔT εunloading = Ef E f rf lc Ef rf lc ( )2 ΦR τi l z2 τi l y − 2l z −4 +4 εreloading = Ef E f r f lc Ef rf lc ( )2 τi lc /2 − 2l y + 2l z −2 − (αc − αf )ΔT Ef rf lc

(2.3)

(2.4)

In the present analysis, the degradation rate of natural frequency (φ) and the interface debonding ratio (η) are defined as: φ =1− ψd =

f (σ ) f0 2ld lc

(2.5) (2.6)

where f 0 denotes the composite initial natural frequency.

2.4 Results and Discussions Figure 2.1 shows experimental and predicted cyclic loading/unloading hysteresis loops, degradation rate of composite modulus, interface debonding ratio, and broken fiber fraction versus degradation rate of natural frequency curves. • Under σ max = 48.5 and 72.7 MPa, the interface debonding length is less than the matrix crack spacing with the interface debonding ψ d = 0.01 and 0.07. Upon unloading and reloading, the interface counter slip and new slip lengths approach the interface debonding tip, which indicates that the hysteresis loops under σ max = 48.5 and 72.7 MPa correspond to the interface partial debonding and the fiber sliding completely relative to the matrix in the interface debonding region.

18 Fig. 2.1 a Experimental and predicted cyclic loading/unloading hysteresis loops; b the interface debonding ratio versus degradation rate of natural frequency curve; and c the broken fiber fraction versus degradation rate of natural frequency curve of 2D SiC/SiC composite

2 Vibration Natural Frequency of Ceramic-Matrix Composites

2.5 Summary and Conclusions

19

• Under σ max = 87.2, 96.9, 111.5, 121.1, 133.3, 169.6, 193.8, and 198.7 MPa, the interface debonding length and interface debonding ratio increase with peak stress, i.e., from ψ d = 0.11 at σ max = 87.2 MPa to ψ d = 0.83 at σ max = 198.7 MPa. Upon unloading and reloading, the interface counter slip and new slip lengths are less than the interface debonding length, which indicates that the hysteresis loops under σ max = 87.2, 96.9, 111.5, 121.1, 133.3, 169.6, 193.8, and 198.7 MPa correspond to the interface partial debonding and the fiber sliding partial relative to the matrix in the interface debonding region. • Under σ max = 203.5, 208.4, and 213.2 MPa, the interface debonding length approaches the matrix crack spacing, and the interface debonding ratio is ψ d = 1.0. Upon unloading and reloading, the interface counter slip and new slip lengths are less than the matrix crack spacing, which indicates that the hysteresis loops under σ max = 203.5, 208.4, and 213.2 MPa correspond to the interface complete debonding and the fiber sliding partial relative to the matrix in the interface debonding region. • When the peak stress increases to σ max = 213.2 MPa, the natural frequency decreases from f 0 = 3415.6 Hz to 3271.5, and the degradation rate of natural frequency is φ = 0.042; the interface debonding ratio increases to ψ d = 1.0; and the broken fiber fraction increases to q = 0.034. • When the degradation rate of natural frequency is φ = 0.01, matrix cracking and interface debonding occur, and however, fiber failure does not appear; when the degradation rate of natural frequency is φ = 0.04, the composite modulus decreases approximately 47%, the interface debonding ratio approaches ψ d = 0.8, and the broken fiber fraction is approximately q = 2.2%.

2.5 Summary and Conclusions In this chapter, the natural frequency of 2D SiC/SiC composites under tensile loading was investigated. Nonlinear damage and fracture were mainly attributed to damage mechanisms of matrix cracking, interface debonding, and fiber fracture. Monotonic tensile stress–strain curves were divided into three stages based on the analysis of AE count, amplitude, and energy. Under cyclic loading/unloading, hysteresis loops appeared due to internal frictional slip between the fiber and the matrix, and the natural frequency and composite modulus were obtained for different peak stress. A micromechanical tensile and cyclic loading/unloading constitutive model was adopted to predict the tensile curves. Microdamage parameters of interface debonding ratio and broken fiber fraction were used to characterize tensile damage and fracture. Relationships between natural frequency, composite modulus, interface debonding, and fiber fracture were established. When the degradation rate of natural frequency was φ = 0.01, matrix cracking and interface debonding occurred, and however, fiber failure did not appear; when the degradation rate of natural frequency was φ = 0.04, the interface debonding ratio approached ψ d = 0.8 and the broken fiber fraction was approximately q = 2.2%.

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2 Vibration Natural Frequency of Ceramic-Matrix Composites

References Bai R, Chu J, Wang M, Chen H (2005) Numerical prediction of frequency of delaminated advanced grid stiffened plate. J Chongqing Univ 28:119–124 Guo XJ, Wu JW, Li J, Zeng Y, Huang X, Li LB (2021) Damage monitoring of 2D SiC/SiC composites under monotonic and cyclic loading/unloading using acoustic emission and natural frequency. Ceramics-Silikaty 65:125–131 Li P, Wang B, Zhen W, Jiao G (2014) Tensile loading/unloading stress-strain behavior of 2D-SiC/SiC composites. Acta Materiae Compositae Sinica 31:676–682 Li LB (2018) Damage, fracture and fatigue of ceramic-matrix composites. Springer Nature, Singapore Li LB (2019a) Thermomechanical fatigue of ceramic-matrix composites. Wiley-VCH, Weinheim, Germany Li LB (2019b) Modeling matrix multi-cracking evolution of fiber-reinforced ceramic-matrix composites considering fiber fracture. Ceramics-Silikaty 63:21–31 Li LB (2019c) Temperature-dependent proportional limit stress of carbon fiber-reinforced silicon carbide ceramic-matrix composites. Ceramics-Silikaty 63:330–337 Li H (2019d) Analysis of vibration and damage test of ceramic matrix composites structure. Master thesis, Nanchang Hangkong University, Nanchang, China Li LB (2020) Durability of ceramic-matrix composites. Woodhead Publishing, Oxford, UK 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 Wang Y, Zhang L, Cheng L, Mei H, Ma J (2008) Characterization of tensile behavior of a twodimensional woven carbon/silicon carbide composite fabricated by chemical vapor infiltration. Mater Sci Eng A 497:295–300

Chapter 3

Vibration Damping of Ceramic-Matrix Composites Considering Fiber Debonding and Fracture

Abstract In this chapter, the vibration damping in fiber-reinforced ceramic-matrix composites (CMCs) was investigated considering fiber debonding and fracture. Micromechanical vibration damping models were developed considering multiple damage mechanisms. Relationships between the damping of CMCs, damping of fiber and the matrix, damping caused by frictional slip between the fiber and the matrix, and fiber debonding and fracture were established. Effects of fiber volume, matrix crack spacing, interface shear stress, interface debonding energy, fiber strength and fiber Weibull modulus on the damping of CMCs, interface debonding and slip between the fiber and the matrix, and fiber broken fraction were analyzed. Experimental damping of 2D C/SiC composite was predicted. Keywords C/SiC · Damping · Matrix cracking · Interface debonding

3.1 Introduction Ceramic-matrix composites (CMCs) have the advantages of low density, high specific strength, high specific modulus, high-temperature resistance, etc., and are the candidate materials for hot-section components of aerospace vehicles, high thrust-toweight-ratio aeroengine, satellite attitude control engine, ramjet, thermal protection system, and so on (Naslain 2004; Li 2020a, 2019a, 2018a). However, in the above applications, there exist vibration and noise problems. Failure analysis of rockets and satellites shows that about two-thirds of the failures are related to vibration and noise, leading to reduced operational control accuracy, structural fatigue damage, and shortened safety life (Min et al. 2011). Therefore, studying the damping performance of CMCs and improving their reliability in the service environment of vibration and noise is an important guarantee for the safe service of CMCs in various fields. Compared with metals and alloys, CMCs have many unique damping mechanisms due to internal structure and complex damage mechanisms (Zhang et al. 1993; Li 2020b). Birman and Byrd (2002) investigated the effect of matrix cracks on damping in unidirectional and cross-ply CMCs at room temperature. The energy dissipation in unidirectional CMCs with bridging matrix cracks was considered. Sato et al. (2003) compared the internal friction and elastic modulus of SiC/SiC composites © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 L. Li, Vibration Behavior in Ceramic-Matrix Composites, Advanced Ceramics and Composites 5, https://doi.org/10.1007/978-981-19-7838-8_3

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3 Vibration Damping of Ceramic-Matrix Composites Considering Fiber …

fabricated by polymer impregnation and pyrolysis (PIP), hot press (HP), and chemical vapor infiltration (CVI). The internal friction of CVI SiC/SiC composite was found to be largely affected by the property of the fiber and the microstructure of the matrix. Wang et al. (2006) investigated the effects of coating, testing frequency, and thermal treatment on the damping capacity of 2D C/SiC composite fabricated by CVI. It was found that the damping mechanism of CVI C/SiC composite includes microcracking propagation, interface debonding, and damping of reinforcement fiber and interphase. Zhang et al. (2007) investigated the effect of interphase thickness on damping behavior of 2D C/SiC composite from room temperature to 400 °C in air atmosphere. The damping increased gradually with temperature and then decreased after damping peak appeared in the temperature range of 250–300 °C. Damping capacity and peak value decreased with increasing frequency and accompanied with a shift of damping peak toward lower temperatures. With increasing PyC interphase thickness at the range of 90–296 nm, damping peak of the composite increased and the temperature corresponding to the peak shifted to the lower temperature. Hong et al. (2013) investigated the relationship between oxidation and internal friction of C/SiC composite at 1300 °C in air atmosphere. The internal frictional behavior of C/SiC is controlled by the oxidation of PyC interphase. Li (2019b) investigated the natural frequency and damage of CMCs and established the relationship between the natural frequency and internal damages. Li (2019c, 2019d, 2019e, 2020c, 2018a, 2018b, 2019f) investigated matrix cracking, hysteresis loops, damage evolution, and life prediction of different fiber-reinforced CMCs. However, in the researches mentioned above, the effect of interface debonding on vibration damping behavior of CMCs has not been investigated. In this chapter, the vibration damping of CMCs is investigated considering fiber debonding and fracture. Relationships between the damping of CMCs, damping of fiber and the matrix, damping caused by frictional slip between the fiber and the matrix, and fiber debonding and fracture are established. Effects of fiber volume, matrix crack spacing, interface shear stress, interface debonding energy, fiber strength and fiber Weibull modulus on the damping of CMCs, interface debonding and slip between the fiber and the matrix, and fiber broken fraction are analyzed. The damping of unidirectional CMCs with and without considering fiber failure is discussed. The experimental damping of 2D C/SiC composite is predicted using the present analysis.

3.2 Micromechanical Damping Models of Ceramic-Matrix Composites The loss factor of the material η is η=

Ud 2πU

(3.1)

3.2 Micromechanical Damping Models of Ceramic-Matrix Composites Fig. 3.1 Effect of fiber volume on a ηc ; b ψ d ; c ψ s ; and d P of unidirectional SiC/CAS composite

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3 Vibration Damping of Ceramic-Matrix Composites Considering Fiber …

Fig. 3.1 (continued)

where U d is a density of energy dissipated per cycle of motion (energy per unit volume per cycle) and U is the maximum strain energy density during the cycle. Relationship between the specific damping capacity (ψ), the logarithmic decrement (δ), and the damping ratio (ζ ) is ψ = 2δ = 4π ζ = 2π η

(3.2)

Under vibration of CMCs, the composite applied stress is σc = σ (1 + sin ωt)

(3.3)

where ω is the vibration frequency.

3.2.1 Damping in Intact Ceramic-Matrix Composites The loss factor (ηa ) for a unidirectional composite without damage subject to axial loading is ηa =

E f Vf ηf + E m Vm ηm E f Vf + E m Vm

(3.4)

where ηf and ηm denote the loss factors for the fiber and matrix materials, respectively. When matrix cracking and interface debonding, the effective matrix elastic modulus (E m ) is

3.2 Micromechanical Damping Models of Ceramic-Matrix Composites

Em =

τi Vm

1 τi Ec

+

rf Δσ 4lc E f

(

Vm E m Vf E c

)2 −

Vf E f Vm

25

(3.5)

where τ i is the interface shear stress; r f is the fiber radius; V f and V m are the fiber and matrix volume fraction, respectively; E f and E m are the fiber and matrix elastic modulus, respectively; E c is the longitudinal modulus of the intact composite material; l c is the matrix crack spacing; and Δσ is the range of applied stress (Δσ = 2σ ).

3.2.2 Damping in Damaged Ceramic-Matrix Composites When matrix cracking and interface debonding occur in CMCs, the loss factor (ηb ) is ηb =

Ud 2πU

(3.6)

where Ud = Ud_ unloading + Ud_ reloading

(3.7)

U = Uf + Um

(3.8)

where U d_unloading and U d_reloading are the dissipated energy upon unloading and reloading, respectively, and U f and U m are the fiber and matrix strain energy, respectively. [ ] τi 8 E c τi 3 2 l Ud_ unloading = 2πrf (Φ − ΦU )ly − Ef 3 rf Vm E m y [ ] τi 8 E c τi 3 Ud_ reloading = 2πrf lz (Φ − Φ R )lz2 − Ef 3 rf Vm E m { 2 ( ) τi Φ 2 4 τi2 3 σfo2 lc 2 Φ − ld Uf = πrf ld − 2 l + l + Ef rf E f d 3 rf2 E f d Ef 2 ( )( [ ]) lc /2 − ld ld rf σfo Φ − σfo − 2 τi 1 − exp −ρ +2 ρ Ef rf rf ( )2 ( [ ])} lc /2 − ld rf ld Φ − σfo − 2 τi 1 − exp −2ρ + 2ρ E f rf rf

(3.9) (3.10)

(3.11)

26

3 Vibration Damping of Ceramic-Matrix Composites Considering Fiber …

Fig. 3.2 Effect of matrix crack spacing on a ηc ; b ψ d ; c ψ s ; and d P of unidirectional SiC/CAS composite

3.2 Micromechanical Damping Models of Ceramic-Matrix Composites

27

Fig. 3.2 (continued)

{

) 2 ( 4 Vf2 τi2 3 σmo lc Um = l + − ld 3 rf2 Vm2 E m d Em 2 [ ][ [ ]] ld ρ(lc /2 − ld ) rf Vf σmo Φ − σfo − 2τi 1 − exp − −2 ρVm E m rf rf [ ]2 [ [ ]]} 2 rf Vf ld ρ(lc /2 − ld ) + Φ − σfo − 2τi 1 − exp −2 2ρVm2 E m rf rf πrf2

(3.12)

where l d is the interface debonding length; ly and l z are the counter slip length upon unloading and new slip length upon reloading, respectively; ρ is the shear-lag model parameter; and σ fo and σ mo are the fiber and matrix axial stress in the interface bonding region, respectively. ( ) / ( )2 ( ) rf Vm E m 1 rf2 Vf Vm E f E m Φ σ rf Vm E m E f rf Φ− − ┌i − Φ− + ld = 2 E c τi ρ 2ρ Vf 4E c2 τi2 E c τi2 (3.13) [ ( { ) 1 rf Vm E m 1 ΦU − ld − ly = 2 2 E c τi ρ ⎤⎫ /( ) ( ) 2 2 r Vf Vm E f E m ΦU σ rf rf Vm E m E f ⎦⎬ − − f − ┌i Φ + (3.14) U 2 ⎭ 2ρ Vf 4E c2 τi E c τi2 [ ( { ) 1 rf Vm E m 1 ΦR − ld − 2 2 E c τi ρ ⎤⎫ /( ) ( ) 2 2 r Vf Vm E f E m Φ R σ rf rf Vm E m E f ⎦⎬ − − f − ┌i Φ + R 2 ⎭ 2ρ Vf 4E c2 τi E c τi2

lz = ly −

(3.15)

28

3 Vibration Damping of Ceramic-Matrix Composites Considering Fiber …

}   ( )( )m f +1  ( σ )m f +1 Φ σ Φ − ΦU 1 c = 2Φ exp − − 1 + P(Φ) Vf Φ 2Φ σc 2   ( )( )m f +1  ( σ )m f +1 Φ σ Φm c = 2Φ exp − Vf Φ 2Φ σc }  ( )( )m f +1  Φ ΦR − Φ + Φm 1 − exp − + P(Φ) 2Φ σc 2 σfo = σmo =

(3.16)

(3.17)

Ef σ + E f (αc − αf )ΔT Ec

(3.18)

Em σ + E m (αc − αm )ΔT Ec

(3.19)

where ┌ i is the interface debonding energy; α f , α m , and α c are the fiber, matrix, and composite thermal expansion coefficient, respectively; and ΔT is the temperature difference between testing and fabricated temperature.

3.3 Result and Discussion The damping of CMCs with matrix cracking and interface debonding is ηc = ηa + ηb

(3.20)

Birman and Byrd (2002) investigated the effect of matrix cracks on damping in unidirectional SiC/CAS composite. The material properties of SiC/CAS composite are given by Birman and Byrd (2002): E f = 200 GPa, E m = 97 GPa, V f = 0.35, r f = 8 µm, ┌ i = 0.01 J/m2 , τ i = 17 MPa, l c = 125 µm, ηf = 0.002, ηm = 0.001, α f = 3.3 × 10–6 /°C, α m = 4.6 × 10–6 /°C, ΔT = –1000 °C, σ c = 2.0 GPa, and mf = 3. Effects of fiber volume, matrix crack spacing, interface shear stress, interface debonding energy, fiber strength, and fiber Weibull modulus on damping of CMCs were analyzed. The comparison analysis of damping with and without considering fiber failure was also discussed.

3.3.1 Effect of Fiber Volume on Damping of Ceramic-Matrix Composites Effect of fiber volume (V f = 35 and 40%) on the composite damping (ηc ), interface debonding ratio (ψ d = 2ld /l c ), interface slip ratio (ψ s = 2l y /l c ), and the broken fiber

3.3 Result and Discussion Fig. 3.3 Effect of interface shear stress on a ηc ; b ψ d ; c ψ s ; and d P of unidirectional SiC/CAS composite

29

30

3 Vibration Damping of Ceramic-Matrix Composites Considering Fiber …

Fig. 3.3 (continued)

fraction P versus amplitude of stress curves of unidirectional SiC/CAS composite was shown in Fig. 3.1. When the fiber volume increases, the composite damping decreases, due to the decrease of the interface debonding and slip ratio and the broken fiber fraction. • When V f = 35%, the composite damping increased from ηc = 0.00153 at σ = zero MPa to ηc = 0.01735 at σ = 100 MPa; the interface debonding ratio ψ d increased from ψ d = zero at σ = zero MPa to ψ d = 0.92 at σ = 100 MPa; the interface slip ratio ψ s increased from ψ s = zero at σ = zero MPa to ψ s = 0.52 at σ = 100 MPa; and the broken fiber fraction increased from P = zero at σ = zero MPa to P = 0.0067 at σ = 100 MPa. • When V f = 40%, the composite damping ηc increased from ηc = 0.0016 at σ = zero MPa to ηc = 0.01351 at σ = 100 MPa; the interface debonding ratio ψ d increased from ψ d = zero at σ = zero MPa to ψ d = 0.69 at σ = 100 MPa; the interface slip ratio ψ s increased from ψ s = zero at σ = zero to ψ s = 0.40 at σ = 100 MPa; and the broken fiber fraction P increases from P = zero at σ = zero MPa to P = 0.0039 at σ = 100 MPa.

3.3.2 Effect of Matrix Crack Spacing on Damping of Ceramic-Matrix Composites Effect of matrix crack spacing on the composite damping (ηc ), interface debonding ratio (ψ d = 2ld /l c ), interface slip ratio (ψ s = 2ly /l c ), and the broken fiber fraction P versus amplitude of stress curves of unidirectional SiC/CAS composite was shown in Fig. 3.2. When the matrix crack spacing increased, the composite damping decreased, due to the decrease of the interface debonding and slip range.

3.3 Result and Discussion

31

• When l c = 150 µm, the composite damping ηc increases from ηc = 0.00153 at σ = zero MPa to ηc = 0.016 at σ = 100 MPa; the interface debonding ratio ψ d increased from ψ d = zero at σ = zero MPa to ψ d = 0.77 at σ = 100 MPa; the interface slip ratio ψ s increased from ψ s = zero at σ = zero MPa to ψ s = 0.43 at σ = 100 MPa; and the broken fiber fraction P increased from P = zero at σ = zero MPa to P = 0.00673 at σ = 100 MPa. • When l c = 200 µm, the composite damping ηc increased from ηc = 0.00153 at σ = zero MPa to ηc = 0.013 at σ = 100 MPa; the interface debonding ratio ψ d increased from ψ d = zero at σ = zero MPa to ψ d = 0.58 at σ = 100 MPa; the interface slip ratio ψ s increased from ψ s = zero at σ = zero MPa to ψ s = 0.327 at σ = 100 MPa; and the broken fiber fraction P increased from P = zero at σ = zero MPa to P = 0.00673 at σ = 100 MPa.

3.3.3 Effect of Interface Shear Stress on Damping of Ceramic-Matrix Composites Effect of interface shear stress on the composite damping ηc , interface debonding ratio (ψ d = 2ld /l c ), interface slip ratio (ψ s = 2l y /l c ), and the broken fiber fraction P versus amplitude of stress curves of unidirectional SiC/CAS composite was shown in Fig. 3.3. When the interface shear stress increased, the composite damping decreased, due to the decrease of the interface debonding and slip ratio. • When τ i = 15 MPa, the composite damping ηc increased from ηc = 0.00153 at σ = zero MPa to ηc = 0.01847 at σ = 100 MPa; the interface debonding ratio ψ d increased from ψ d = zero at σ = zero MPa to ψ d = 1 at σ = 100 MPa; the interface slip ratio ψ s increased from ψ s = zero at σ = zero MPa to ψ s = 0.59 at σ = 100 MPa; and the broken fiber fraction P increased from P = zero at σ = zero MPa to P = 0.00673 at σ = 100 MPa. • When τ i = 20 MPa, the composite damping ηc increased from ηc = 0.00153 at σ = zero MPa to ηc = 0.01621 at σ = 100 MPa; the interface debonding ratio ψ d increased from ψ d = zero at σ = zero MPa to ψ d = 0.78 at σ = 100 MPa; the interface slip ratio ψ s increased from ψ s = zero at σ = zero MPa to ψ s = 0.44 at σ = 100 MPa; and the broken fiber fraction P increased from P = zero at σ = zero MPa to P = 0.00673 at σ = 100 MPa.

3.3.4 Effect of Interface Debonding Energy on Damping of Ceramic-Matrix Composites Effect of interface debonding energy on the composite damping ηc , interface debonding ratio (ψ d = 2ld /l c ), interface slip ratio (ψ s = 2l y /l c ), and the broken fiber fraction P versus amplitude of stress curves of unidirectional SiC/CAS composite was shown in Fig. 3.4. When the interface debonding energy increased, the composite

32

3 Vibration Damping of Ceramic-Matrix Composites Considering Fiber …

Fig. 3.4 Effect of interface debonding energy on a ηc ; b ψ d ; c ψ s ; and d P of unidirectional SiC/CAS composite

3.3 Result and Discussion

33

Fig. 3.4 (continued)

damping decreased first and then increased, due to the decrease of the interface debonding length and the first decrease and then increase of interface slip length. • When ┌ i = 0.01 J/m2 , the composite damping ηc increased from ηc = 0.00153 at σ = zero MPa to ηc = 0.01735 at σ = 100 MPa; the interface debonding ratio ψ d increased from ψ d = zero at σ = zero MPa to ψ d = 0.927 at σ = 100 MPa; the interface slip ratio ψ s increased from ψ s = zero at σ = zero to ψ s = 0.523 at σ = 100 MPa; and the broken fiber fraction P increased from P = zero at σ = zero MPa to P = 0.00673 at σ = 100 MPa. • When ┌ i = 0.03 J/m2 , the composite damping ηc increased from ηc = 0.00153 at σ = zero MPa to ηc = 0.01789 at σ = 100 MPa; the interface debonding ratio ψ d increased from ψ d = zero at σ = zero MPa to ψ d = 0.85 at σ = 100 MPa; the interface slip ratio ψ s increased from ψ s = zero at σ = zero MPa to ψ s = 0.517 at σ = 100 MPa; and the broken fiber fraction P increased from P = zero at σ = zero MPa to P = 0.00673 at σ = 100 MPa.

3.3.5 Effect of Fiber Strength on Damping of Ceramic-Matrix Composites Effect of fiber strength on the composite damping ηc , interface debonding ratio (ψ d = 2ld /l c ), interface slip ratio (ψ s = 2ly /l c ), and the broken fiber fraction P versus amplitude of stress curves of unidirectional SiC/CAS composite was shown in Fig. 3.5. When the fiber strength increased, the composite damping increased, due to the decrease of the interface debonding length, interface slip length, and broken fiber fraction. • When σ c = 2.0 GPa, the composite damping ηc increased from ηc = 0.00153 at σ = zero MPa to ηc = 0.01735 at σ = 100 MPa; the interface debonding ratio ψ d

34

3 Vibration Damping of Ceramic-Matrix Composites Considering Fiber …

Fig. 3.5 Effect of fiber strength on a ηc ; b ψ d ; c ψ s ; and d P of unidirectional SiC/CAS composite

3.3 Result and Discussion

35

Fig. 3.5 (continued)

increased from ψ d = zero MPa at σ = zero MPa to ψ d = 0.927 at σ = 100 MPa; the interface slip ratio ψ s increased from ψ s = zero at σ = zero MPa to ψ s = 0.523 at σ = 100 MPa; and the broken fiber fraction P increases from P = zero at σ = zero MPa to P = 0.00673 at σ = 100 MPa. • When σ c = 2.5 GPa, the composite damping ηc increased from ηc = 0.00153 at σ = zero MPa to ηc = 0.01753 at σ = 100 MPa; the interface debonding ratio ψ d increased from ψ d = zero at σ = zero MPa to ψ d = 0.91 at σ = 100 MPa; the interface slip ratio ψ s increased from ψ s = zero at σ = zero MPa to ψ s = 0.514 at σ = 100 MPa; and the broken fiber fraction P increased from P = zero at σ = zero MPa to P = 0.00274 at σ = 100 MPa.

3.3.6 Effect of Fiber Weibull Modulus on Damping of Ceramic-Matrix Composites Effect of fiber Weibull modulus on the composite damping ηc , interface debonding ratio (ψ d = 2ld /l c ), interface slip ratio (ψ s = 2ly /l c ), and broken fiber fraction versus amplitude of stress curves of unidirectional SiC/CAS composite was shown in Fig. 3.6. When the fiber Weibull modulus increased, the composite damping increased, due to the decrease of the interface debonding length, interface slip length, and broken fiber fraction. • When mf = 3, the composite damping ηc increased from ηc = 0.00153 at σ = zero MPa to ηc = 0.01735 at σ = 100 MPa; the interface debonding ratio ψ d increased from ψ d = zero at σ = zero MPa to ψ d = 0.927 at σ = 100 MPa; the interface slip ratio ψ s increased from ψ s = zero at σ = zero MPa to ψ s = 0.523 at σ = 100 MPa; and the broken fiber fraction P increased from P = zero at σ = zero MPa to P = 0.00673 at σ = 100 MPa.

36

3 Vibration Damping of Ceramic-Matrix Composites Considering Fiber …

Fig. 3.6 Effect of fiber Weibull modulus on a ηc ; b ψ d ; c ψ s ; and d P of unidirectional SiC/CAS composite

3.3 Result and Discussion

37

Fig. 3.6 (continued)

• When mf = 5, the composite damping ηc increased from ηc = 0.00153 at σ = zero MPa to ηc = 0.0176 at σ = 100 MPa; the interface debonding ratio ψ d increased from ψ d = zero at σ = zero MPa to ψ d = 0.9 at σ = 100 MPa; the interface slip ratio ψ s increased from ψ s = zero at σ = zero MPa to ψ s = 0.51 at σ = 100 MPa; and the broken fiber fraction P increased from P = zero at σ = zero MPa to P = 0.0005 at σ = 100 MPa.

3.3.7 Comparison of Damping of Ceramic-Matrix Composites without/with Considering Fiber Failure Comparisons of the composite damping ηc , interface debonding ratio (ψ d = 2ld /l c ), interface slip ratio (ψ s = 2l y /l c ) with and without considering fiber failure, and the broken fiber fraction P versus amplitude of stress curves of unidirectional SiC/CAS composite were shown in Fig. 3.7. Considering fiber failure, the composite damping decreased, due to the increase of fiber broken fraction. • When the fiber failure was not considered, the composite damping ηc increased from ηc = 0.00153 at σ = zero MPa to ηc = 0.01762 at σ = 100 MPa; the interface debonding ratio ψ d increased from ψ d = zero at σ = zero MPa to ψ d = 0.89 at σ = 100 MPa; and the interface slip ratio ψ s increased from ψ s = zero at σ = zero MPa to ψ s = 0.509 at σ = 100 MPa. • When the fiber failure was considered, the composite damping ηc increased from ηc = 0.00153 at σ = zero MPa to ηc = 0.01735 at σ = 100 MPa; the interface debonding ratio ψ d increased from ψ d = zero at σ = zero MPa to ψ d = 0.927 at σ = 100 MPa; the interface slip ratio ψ s increased from ψ s = zero at σ = zero MPa to ψ s = 0.523 at σ = 100 MPa; and the broken fiber fraction P increased from P = zero at σ = zero MPa to P = 0.00673 at σ = 100 MPa.

38

3 Vibration Damping of Ceramic-Matrix Composites Considering Fiber …

Fig. 3.7 a ηc ; b ψ d ; c ψ s ; and d P of unidirectional SiC/CAS composite

3.4 Experimental Comparison

39

Fig. 3.7 (continued)

3.4 Experimental Comparison Wang et al. (2006) investigated the damping behavior of 2D plain-weave CVI T300™ C/SiC composite at different temperatures. Material properties of C/SiC composite were given by Wang et al. (2006): E f = 230 GPa, E m = 350 GPa, V f = 40%, r f = 3.5 µm, ┌ i = 0.01 J/m2 , l c = 100 µm, ηf = 0.002, ηm = 0.005, α f = −0.7 × 10–6 /°C, α m = 4.6 × 10–6 /°C, ΔT = –1000 °C, σ c = 2.5 GPa, and mf = 3. Experimental and predicted composite damping ηc , interface debonding ratio ψ d , and interface slip ratio ψ s versus amplitude of stress curves of C/SiC composite were shown in Fig. 3.8. Experimental composite damping ηc was ηc = 0.0106 at T = 50 °C, corresponding to σ = 30 MPa, the interface shear stress range τ i = 9 and 10 MPa, the interface debonding ratio range ψ d = 0.716 and 0.798, the interface slip ratio range ψ s = 0.45 and 0.5, and the broken fiber fraction P = 0.0002.

40

3 Vibration Damping of Ceramic-Matrix Composites Considering Fiber …

Fig. 3.8 a ηc ; b ψ d ; c ψ s ; and d P of C/SiC composite

References

41

Fig. 3.8 (continued)

3.5 Summary and Conclusion In this chapter, vibration damping of CMCs was investigated considering fiber debonding and fracture. Relationships between the vibration damping of composite, damping of fiber and the matrix, damping caused by frictional slip between the fiber and the matrix, and fiber debonding and fracture were established. Effects of fiber volume, matrix crack spacing, interface shear stress, interface debonding energy, fiber strength and fiber Weibull modulus on the damping of CMCs, interface debonding and slip between the fiber and the matrix, and fiber broken fraction were analyzed. Damping of unidirectional CMCs with and without considering fiber failure was discussed. Experimental damping of 2D C/SiC composite was also predicted. When the fiber volume increased, the composite damping decreased, due to the decrease of the interface debonding and slip ratio and the broken fiber fraction. Considering fiber failure, the composite damping decreased, due to the increase of fiber broken fraction. When the fiber strength and fiber Weibull modulus increased, the composite damping increased, due to the decrease of the interface debonding ratio, interface slip ratio, and broken fiber fraction.

References Birman V, Byrd LW (2002) Effect of matrix cracks on damping in unidirectional and cross-ply ceramic matrix composites. J Compos Mater 36:1859–1877 Hong ZL, Cheng LF, Zhao CN, Zhang LT, Wang YG (2013) Effect of oxidation on internal friction behavior of C/SC composites. Acta Materiae Compositae Sinica 30:93–100 Li LB (2018) Modeling for monotonic and cyclic tensile stress−strain behavior of 2D and 2.5D woven C/SiC ceramic−matrix composites. Mech Compos Mater 54:165–178 Li LB (2018a) Damage, fracture and fatigue of ceramic-matrix composites. Springer Nature, Singapore

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Li LB (2018b) Damage monitor 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 Li LB (2019a) Thermomechanical fatigue of ceramic-matrix composites. Wiley-VCH, Germany Li LB (2019c) Temperature-dependent proportional limit stress of carbon fiber-reinforced silicon carbide ceramic-matrix composites. Ceramics-Silikaty 63:330–337 Li LB (2019d) Modeling matrix multicracking development of fiber-reinforced ceramic-matrix composites considering fiber debonding. Int J Appl Ceram Technol 16:97–107 Li LB (2019e) Time-dependent matrix multi-fracture of SiC/SiC ceramic-matrix composites considering interface oxidation. Ceramics-Silikaty 63:131–148 Li LB (2019f) Synergistic effects of frequency and temperature on fatigue hysteresis of cross-ply SiC/MAS composite under tension-tension loading. Ceramics-Silikaty 63:51–66 Li LB (2020a) Durability of ceramic matrix composites. Elsevier, Oxford, UK Li LB (2020b) Synergistic effects of fiber debonding and fracture on vibration damping in fiberreinforced ceramic-matrix composites. Ceramics-Silikaty 64:387–397 Li LB (2020c) Modeling tensile damage and fracture processes of fiber-reinforced ceramic-matrix composites under the effect of pre-exposure at elevated temperatures. Ceramics-Silikaty 64:50–62 Li HQ (2019b) Analysis of vibration and damage test of ceramic matrix composites structure. Master thesis, Nanchang Hongkong University, Nanchang, China Min JB, Harris DL, Ting JM (2011) Advances in ceramic matrix composite blade damping characteristics for aerospace turbomachinery applications. In: 52nd AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics and materials conference, 4–7 April 2011, Denver, Colorado 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 Sato S, Serizawa H, Araki H, Node T, Kohyama A (2003) Temperature dependence of internal friction and elastic modulus of SiC/SiC composites. J Alloys Compounds 355:142–147 Wang W, Cheng LF, Zhang LT, Xu YD, Wu WM (2006) Study on damping capacity of two dimensional carbon fiber reinforced silicon carbide (2D C/SiC) composites. J Solid Rocket Technol 29:455–459 Zhang J, Perez RJ, Lavernia EJ (1993) Documentation of damping capacity of metallic ceramic and metal-matrix composite materials. J Mater Sci 28:2395–2404 Zhang Q, Cheng LF, Wang W, Wei X, Zhang LT, Xu YD (2007) Effect of interphase thickness on damping behavior of 2D C/SiC composites. Mater Sci Forum 546–549:1531–1534

Chapter 4

Temperature-Dependent Vibration Damping of Ceramic-Matrix Composites

Abstract In this chapter, the temperature-dependent vibration damping in C/SiC composites with different fiber preforms under different vibration frequencies was investigated. A micromechanical temperature-dependent vibration damping model was developed to establish the relationship between composite damping, material properties, internal damage mechanisms, and temperature. Effects of fiber volume, matrix crack spacing, and interface properties on temperature-dependent vibration damping of CMCs and interface damage were analyzed. Experimental temperaturedependent composite damping of 2D and 3D C/SiC composites was predicted for different loading frequencies. Keywords Ceramic-matrix composites (CMCs) · C/SiC · Damping · Temperature-dependent · Matrix cracking · Interface debonding

4.1 Introduction Ceramic-matrix composites (CMCs) are the candidate materials for hot-section components of aerospace vehicles, high thrust-to-weight-ratio aeroengines, satellite attitude control engines, ramjets, and thermal protection systems (Naslain 2004; Li 2020a). However, in the above applications, there exist vibration and noise problems. Failure analysis of rockets and satellites shows that about two-thirds of the failures are related to vibration and noise, leading to reduced operational control accuracy, structural fatigue damage, and shortened safety life (Min et al. 2011). Therefore, studying the damping performance of CMCs and improving their reliability in the service environment of vibration and noise is an important guarantee for the safe service of CMCs in various fields (Momon et al. 2010). Compared with metals and alloys, CMCs have many unique damping mechanisms due to their internal structure and complex damage mechanisms (Zhang et al. 1993; Chandra et al. 1999; Birman and Byrd 2003; Melo and Radford 2005). The damping properties of composites are usually much more complicated than homogenous material. Temperature, moisture, loading frequency, and wave form affect the damping of composites (Li 2020b; Patel et al. 2007). During manufacturing and service, cracks

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 L. Li, Vibration Behavior in Ceramic-Matrix Composites, Advanced Ceramics and Composites 5, https://doi.org/10.1007/978-981-19-7838-8_4

43

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4 Temperature-Dependent Vibration Damping of Ceramic-Matrix …

might occur in the matrix, fiber, and interface both between fiber/matrix and neighboring plies (Sato et al. 2003). Friction slip in the interface debonding region among matrix crack space consumes energy (Gowayed et al. 2015). The internal friction of CMCs is affected by fabrication method (Patel et al. 2007), interphase thickness (Birman and Byrd 2002), oxidation (Zhang et al. 2007), coating, and heat treatment (Hong et al. 2013; Wang 2005). Holmes and Cho (Zhang et al. 2008) developed an analytical model for predicting energy dissipation of SiC/CAS-II during a cycle based on the interfacial friction slip mechanisms. The energy dissipation corresponding to each cycle depends on stress level, matrix crack spacing, and interface frictional shear stress. Li (Holmes and Cho 1992) investigated internal frictional behavior of C/SiC considering fiber failure and developed temperature- and time-dependent damage models for matrix cracking (Li 2014, 2019a). The dynamic properties extracted from vibration response of damaged composites can be used for damage monitoring, and these include natural frequencies, mode shape, and damping. Li (2019b) established a relationship between natural frequency, critical rotation speed, and internal damage inside CMCs. Kyriazoglou et al. (Li 2019c) measured and analyzed the specific damping capacity (SDC) of composite beams in flexure before and after quasi-static loading or fatigue damage. Zhang and Hartwig (Kyriazoglou et al. 2004) detected a damping plateau from fatigue cycles in epoxy composites due to energy balance between fatigue load input and damage dissipation. In this chapter, a micromechanical vibration damping model was developed to analyze the temperature-dependent damping of C/SiC composites with different fiber preforms under different loading frequencies. Relationships between composite damping, internal damage, and temperature were established considering different material properties and damage states. Experimental temperaturedependent damping of 2D and 3D C/SiC under vibration frequencies of f = 1, 2, 5, and 10 Hz was predicted.

4.2 Temperature-Dependent Micromechanical Vibration Damping Models When a solid vibrates, its kinetic and strain energies transform mutually. The largest strain energy, equaling the entire energy driving vibration, determines the intensities of deformation or vibration of the structure. The proportion of energy consumed during one vibration cycle is directly associated with the vibration attenuation rate, which is also known as damping. The composite damping is given by Birman and Byrd (2003) η=

Ud 2πU

(4.1)

where U d and U are dissipated energy density and maximum strain energy per cycle, respectively.

4.2 Temperature-Dependent Micromechanical Vibration Damping Models

45

For CMCs without damage, the temperature-dependent composite damping (ηa ) is obtained as ηa (T ) =

E f (T )Vf ηf + E m (T )Vm ηm E f (T )Vf + E m (T )Vm

(4.2)

where ηf and ηm denote fiber and matrix damping, respectively; V f and V m are the volume of fiber and matrix, respectively; and E f (T ) and E m (T ) are the temperaturedependent elastic modulus of fiber and matrix, respectively. When damage occurs inside of CMCs, the effective temperature-dependent matrix elastic modulus (E m (T )) is obtained as E m (T ) =

τi (T ) Vm

1 τi (T ) E c (T )

+

rf Δσ 4lc (T ) E f (T )

(

Vm E m (T ) Vf E c (T )

)2 −

Vf E f (T ) Vm

(4.3)

where τ i (T ) is the temperature-dependent interface shear stress; r f is fiber radius; E c (T ) is the temperature-dependent longitudinal modulus of intact composite material; lc (T ) is the temperature-dependent matrix crack spacing; and Δσ is applied stress range (Δσ = 2σ ). σc = σ (1 + sin ωt)

(4.4)

where ω is a vibration frequency. For damaged CMCs, the energy dissipation during each vibration cycle contributes to the composite damping (ηb ), which is given by ηb (T ) =

Ud (T ) 2πU (T )

(4.5)

where Ud (T ) = Ud_ u (T ) + Ud_ r (T )

(4.6)

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

(4.7)

where U d_u (T ) and U d_r (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 density, respectively. ] Δσ E c (T ) τi (T ) 3 8 2 l (T ) − l (T ) Ud_ u (T ) = 2πrf τi (T ) Vf E f (T ) y 3 Vm E f (T )E m (T ) rf y ] [ Δσ E c (T ) τi (T ) 3 8 lz2 (T ) − lz (T ) Ud_ r (T ) = 2πrf τi (T ) Vf E f (T ) 3 Vm E f (T )E m (T ) rf [

(4.8) (4.9)

46

4 Temperature-Dependent Vibration Damping of Ceramic-Matrix … {

) ( σ 2 (T ) lc (T ) σ2 σ τi (T ) 2 4 τi2 (T ) 3 ld (T ) − 2 l (T ) + fo l (T ) + − ld (T ) rf Vf E f (T ) d 3 r 2 E f (T ) d E f (T ) 2 Vf2 E f (T ) f )( ( )) ( 2rf σfo (T ) Vm lc (T )/2 − ld (T ) ld (T ) + σmo (T ) − 2 τi (T ) 1 − exp −ρ ρ E f (T ) Vf rf rf ( ( ))} )2 ( ld (T ) rf Vm lc (T )/2 − ld (T ) + σmo (T ) − 2 τi (T ) (4.10) 1 − exp −2ρ 2ρ E f (T ) Vf rf rf { ) ( 4 Vf2 τi2 (T ) 3 lc (T ) σ2 Um (T ) = πrf2 − ld (T ) l (T ) + mo 2 E (T ) d 3 r 2 Vm Em 2 m f ( )[ [ ]] 2r σmo (T ) V l (T ) ρ(lc (T )/2 − ld (T )) − f σmo (T ) − 2τi (T ) f d 1 − exp − ρ E m (T ) Vm rf rf ( ) [ [ ]]} V l (T ) 2 rf ρ(lc (T )/2 − ld (T )) σmo (T ) − 2τi (T ) f d + 1 − exp −2 (4.11) 2ρ E m (T ) Vm rf rf Uf (T ) = πrf2

where ld (T ), ly (T ), and l z (T ) are temperature-dependent interface debonding length, counter slip length, and new slip length, respectively; 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. ( ) /( )2 rf 1 rf Vm E m (T )E f (T ) rf Vm E m (T )σ − − ld (T ) = + ┌i (T ) (4.12) 2 Vf E c (T )τi (T ) ρ 2ρ E c (T )τi2 (T ) { ) [ ( 1 rf Vm E m (T )σ 1 ld (T ) − − ly (T ) = 2 2 Vf E c (T )τi (T ) ρ ⎤⎫ /( ) 2 ⎬ rf rf Vm E m (T )E f (T ) ⎦ − ┌ + (4.13) (T ) i ⎭ 2ρ E c (T )τi2 (T ) { ) [ ( 1 1 rf Vm E m (T )σ ld (T ) − − lz (T ) = ly (T ) − 2 2 Vf E c (T )τi (T ) ρ ⎫ ⎤ /( ) ⎬ rf 2 rf Vm E m (T )E f (T ) ⎦ − + ┌ (T ) i ⎭ 2ρ E c (T )τi2 (T ) σfo (T ) = σmo (T ) =

(4.14)

E f (T ) σ + E f (T )(αlc (T ) − αlf (T ))ΔT E c (T )

(4.15)

E m (T ) σ + E m (T )(αlc (T ) − αlm (T ))ΔT E c (T )

(4.16)

where ┌ i (T ) denotes temperature-dependent interface debonding energy; α lf (T ), α lm (T ), and α lc (T ) are temperature-dependent fiber, matrix, and composite axial thermal expansion coefficient, respectively; and ΔT denotes temperature difference between testing temperature (T ) and fabricated temperature (T 0 ). The total temperature-dependent composite damping (ηc ) can be determined as ηc = ηa + ηb

(4.17)

4.3 Results and Discussions

47

where ηa and ηb can be determined by Eqs. (4.2) and (4.5), respectively.

4.3 Results and Discussions Material properties of the C/SiC composite were given by: V f = 0.3, r f = 3.5 μm, ┌ i = 0.1 J/m2 , ηf = 0.002, ηm = 0.001, and T 0 = 1000 °C, and the temperaturedependent constituent properties were given by Zhang and Hartwig (2002); Sauder et al. 2004; Snead et al. 2007; Pradere and Sauder 2008; Reynaud et al. 1998) )] ( [ T + 273 , T < 2000 ◦ C E f (T ) = 230 1 − 2.86 × 10−4 exp (4.18) 324 [ ( )] [ ] 962 350 460 − 0.04(T + 273) exp − , T ∈ 27◦ C, 1500◦ C E m (T ) = 460 T + 273 (4.19) αlf (T ) = 2.529 × 10−2 − 1.569 × 10−4 (T + 273) + 2.228 × 10−7 (T + 273)2 [ ] − 1.877 × 10−11 (T + 273)3 − 1.288 × 10−14 (T + 273)4 , T ∈ 27◦ C, 2227◦ C

(4.20) αrf (T ) = −1.86 × 10−1 + 5.85 × 10−4 (T + 273) − 1.36 × 10−8 (T + 273)2 [ ] + 1.06 × 10−22 (T + 273)3 , T ∈ 27◦ C, 2500◦ C (4.21) ⎧ × 10−5 (T + 273)2 ⎪ ⎨ −1.8276 + 0.0178(T + 273) − 1.5544 [ ] αlm (T ) = αrm (T ) = +4.5246 × 10−9 (T + 273)3 , T ∈ 0◦ C, 1000◦ C ⎪ ⎩ 5.0 × 10−6 , T > 1000◦ C (4.22) τi (T ) = τ0 + μ

|αrf (T ) − αrm (T )|(T0 − T ) A

(4.23)

where τ0 was the steady-state interface shear stress; μ was the interface frictional coefficient; α rf and α rm denoted the temperature-dependent fiber and matrix radial thermal expansion coefficient, respectively; and A was a constant depending on the elastic properties of the matrix and the fiber. [ ┌i (T ) = ┌ir 1 −

{T

Tr C P (T )dT { T0 Tr C P (T )dT

] (4.24)

where T r denoted the reference temperature; T 0 denoted the fabricated temperature; and ┌ ir denoted the interface debonding energy at the reference temperature of T r .

48

4 Temperature-Dependent Vibration Damping of Ceramic-Matrix …

Effects of material properties and damage state on temperature-dependent composite damping and interface damage of the C/SiC composite were analyzed.

4.3.1 Effect of Fiber Volume on Temperature-Dependent Damping of C/SiC Composite Effect of fiber volume on temperature-dependent composite damping (ηc ), interface debonding ratio (ψ d = 2l d /l c ), and interface slip ratio (ψ s = 2l y /l c ) versus temperature curves of C/SiC composite was analyzed for the temperature range from room temperature (T = 20 °C) to elevated temperature T = 400 °C, as shown in Fig. 4.1 and Table 4.1. At the temperature range from room temperature (T = 20 °C) to elevated temperature T = 400 °C, the temperature-dependent composite damping of the C/SiC composite increased with temperature to the peak value first and then decreases with temperature. When the fiber volume increased from V f = 30 to 35%, the temperaturedependent peak value damping of the C/SiC composite (ηc ) decreased from ηc = 0.00752 to ηc = 0.00431, and the corresponding temperature for the peak value damping of the C/SiC composite decreased from T = 262 °C to T = 250 °C, mainly attributed to the decrease of interface debonding and slip ratio. • When V f = 30%, the temperature-dependent composite damping (ηc ) increased from ηc = 0.00306 at T = 20 °C to the peak value ηc = 0.00752 at T = 262 °C and decreased to ηc = 0.00527 at T = 400 °C; the temperature-dependent interface debonding ratio (ψ d ) decreased from ψ d = 0.152 at T = 20 °C to ψ d = 0.048 at T = 400 °C; and the temperature-dependent interface slip length (ψ s ) decreased from ψ s = 0.15 at T = 20 °C to ψ s = 0.048 at T = 400 °C. • When V f = 35 %, the temperature-dependent composite damping (ηc ) increases from ηc = 0.00223 at T = 20 °C to peak value ηc = 0.00431 at T = 250 °C and decreases to ηc = 0.00301 at T = 400 °C; the temperature-dependent interface debonding length (2l d /l c ) decreases from 2l d /l c = 0.1 at T = 20 °C to 2l d /l c = 0.03 at T = 400 °C; and the temperature-dependent interface slip length (2ly /l c ) decreases from 2ly /l c = 0.1 at T = 20 °C to 2ly /l c = 0.03 at T = 400 °C.

4.3.2 Effect of Matrix Crack Spacing on Temperature-Dependent Damping of C/SiC Composite Effect of matrix crack spacing on temperature-dependent composite damping (ηc ), interface debonding ratio (ψd ), and interface slip ratio (ψs ) versus temperature curves of the C/SiC composite was analyzed for the temperature range from room temperature T = 20 °C to elevated temperature T = 400 °C, as shown in Fig. 4.2 and Table 4.2. When the matrix crack spacing increased from lc = 300 to 400 μm, the peak

4.3 Results and Discussions Fig. 4.1 Effect of fiber volume on a the temperature-dependent composite damping (ηc ) versus temperature curves; b the temperature-dependent interface debonding ratio (ψ d ) versus temperature curves; and c the temperature-dependent interface slip ratio (ψ s ) versus temperature curves of C/SiC composite

49

50

4 Temperature-Dependent Vibration Damping of Ceramic-Matrix …

Table 4.1 Temperature-dependent composite damping, interface debonding, and slip ratio of C/SiC composite for different fiber volumes V f /(%)

T /(°C)

ηc

ψd

ψs

30

20

0.00306

0.152

0.15

262

0.00752

0.056

0.056

400

0.00527

0.048

0.048

20

0.00223

0.1

0.1

250

0.00431

0.036

0.036

400

0.00301

0.03

0.03

35

damping of the C/SiC composite decreased from ηc = 0.0056 to ηc = 0.00458, and the interface debonding and slip ratio at the same temperature also decreased. • When lc = 300 μm, the temperature-dependent composite damping (ηc ) increased from ηc = 0.00238 at T = 20 °C to the peak value ηc = 0.0056 at T = 263 °C and decreased to ηc = 0.00397 at T = 400 °C; the temperature-dependent interface debonding ratio (2ld /l c ) decreased from ψd = 0.101 at T = 20 °C to ψd = 0.032 at T = 400 °C; and the temperature-dependent interface slip ratio (ψs ) decreased from ψs = 0.1 at T = 20 °C to ψs = 0.032 at T = 400 °C. • When lc = 400 μm, the temperature-dependent composite damping (ηc ) increased from ηc = 0.00207 at T = 20 °C to the peak value ηc = 0.00458 at T = 263 °C and decreased to ηc = 0.00207 at T = 400 °C; the temperature-dependent interface debonding ratio (ψd ) decreased from ψd = 0.0759 at T = 20 °C to ψd = 0.0243 at T = 400 °C; and the temperature-dependent interface slip ratio (ψs ) decreased from ψs = 0.0751 at T = 20 °C to ψs = 0.0243 at T = 400 °C.

4.3.3 Effect of Interface Debonding Energy on Temperature-Dependent Damping of C/SiC Composite Effect of interface debonding energy (┌ i = 0.2 and 0.3 J/m2 ) on temperaturedependent composite damping (ηc ), interface debonding ratio (ψ d ), and interface slip ratio (ψ s ) versus temperature curves of the C/SiC composite was analyzed for the temperature range from room temperature T = 20 °C to elevated temperature T = 400 °C, as shown in Fig. 4.3 and Table 4.3. When interface debonding energy increased from ┌ i = 0.2 to 0.3 J/m2 , the peak damping of the C/SiC composite decreased from ηc = 0.00478 to 0.0022, and the corresponding temperature for peak damping decreased from T = 256 to 245 °C, and the interface debonding and slip ratio at the same temperature also decreased. • When ┌ i = 0.2 J/m2 , the temperature-dependent composite damping (ηc ) increased from ηc = 0.00245 at T = 20 °C to the peak value ηc = 0.00478

4.3 Results and Discussions Fig. 4.2 Effect of matrix crack spacing on a the temperature-dependent composite damping (ηc ) versus temperature curves; b the temperature-dependent interface debonding ratio (ψd ) versus temperature curves; and c the temperature-dependent interface slip ratio (ψs ) versus temperature curves of C/SiC composite

51

52

4 Temperature-Dependent Vibration Damping of Ceramic-Matrix …

Table 4.2 Temperature-dependent composite damping, interface debonding, and slip ratio of C/SiC composite for different matrix crack spacing l c /(μm)

T /(°C)

ηc

ψd

ψs

300

20

0.00238

0.101

0.1

263

0.0056

0.037

0.037

400

0.0039

0.032

0.032

20

0.00207

0.0759

0.0751

263

0.00458

0.0281

0.0281

400

0.00207

0.0243

0.0243

400

at T = 256 °C and decreased to ηc = 0.00337 at T = 400 °C; the temperaturedependent interface debonding ratio (ψ d ) decreased from ψ d = 0.0949 at T = 20 °C to ψ d = 0.0279 at T = 400 °C; and the temperature-dependent interface slip ratio (ψ s ) decreased from ψ s = 0.0949 at T = 20 °C to ψ s = 0.0279 at T = 400 °C. • When ┌ i = 0.3 J/m2 , the temperature-dependent composite damping (ηc ) increased from ηc = 0.00172 at T = 20 °C to the peak value ηc = 0.0022 at T = 245 °C and then decreased to ηc = 0.00174 at T = 400 °C; the temperaturedependent interface debonding ratio (ψ d ) decreased from ψ d = 0.0511 at T = 20 °C to ψ d = 0.0118 at T = 400 °C; and the temperature-dependent interface slip ratio (ψ s ) decreased from ψ s = 0.0511 at T = 20 °C to ψ s = 0.0118 at T = 400 °C.

4.3.4 Effect of Steady-State Interface Shear Stress on Temperature-Dependent Damping of C/SiC Composite Effect of steady-state interface shear stress (τ 0 = 40 and 50 MPa) on temperaturedependent composite damping (ηc ), interface debonding ratio (ψd ), and interface slip ratio (ψs ) versus temperature curves of the C/SiC composite was analyzed for the temperature range from room temperature T = 20 °C to elevated temperature T = 400 °C, as shown in Fig. 4.4 and Table 4.4. When the steady-state interface shear stress increased from τ 0 = 40 to 50 MPa, the peak damping of the C/SiC composite decreased from ηc = 0.00627 to ηc = 0.00535, and the interface debonding and slip ratio at the same temperature also decreased. • When τ 0 = 40 MPa, the temperature-dependent composite damping (ηc ) increased from ηc = 0.00235 at T = 20 °C to the peak value ηc = 0.00627 at T = 264 °C and decreased to ηc = 0.00448 at T = 400 °C; the interface debonding ratio (ψd ) decreased from ψd = 0.094 at T = 20 °C to ψd = 0.038 at T = 400 °C; and the

4.3 Results and Discussions Fig. 4.3 Effect of interface debonding energy on a the temperature-dependent composite damping (ηc ) versus temperature curves; b the temperature-dependent interface debonding ratio (ψd ) versus temperature curves; and c the temperature-dependent interface slip ratio (ψd ) versus temperature curves of C/SiC composite

53

54

4 Temperature-Dependent Vibration Damping of Ceramic-Matrix …

Table 4.3 Temperature-dependent composite damping, interface debonding, and slip ratio of C/SiC composite for different interface debonding energy ┌ i /(J/m2 )

T /(°C)

ηc

ψd

ψs

0.2

20

0.00245

0.0949

0.0949

256

0.00478

0.0332

0.0332

400

0.00337

0.0279

0.0279

20

0.00172

0.0511

0.0511

245

0.0022

0.0154

0.0154

400

0.00174

0.0118

0.0118

0.3

interface slip ratio (ψs ) decreased from ψs = 0.094 at T = 20 °C to ψs = 0.038 at T = 400 °C. • When τ 0 = 50 MPa, the temperature-dependent composite damping (ηc ) increased from ηc = 0.00202 at T = 20 °C to the peak value ηc = 0.00535 at T = 265 °C and decreased to ηc = 0.00389 at T = 400 °C; the temperature-dependent interface debonding ratio (ψd ) decreased from ψd = 0.0663 at T = 20 °C to ψd = 0.0309 at T = 400 °C; and the temperature-dependent interface slip ratio (ψs ) decreases from ψs = 0.0663 at T = 20 °C to ψs = 0.0309 at T = 400 °C.

4.4 Experimental Comparisons Wang et al. (Hong et al. 2013) investigated damping capacity of 2D and 3D C/SiC composites at different vibration frequencies. The 2D C/SiC composite was prepared by laminating 1 k T300™ woven carbon fabrics, and the 3D C/SiC composite was prepared by braiding 3 k T300™ carbon fibers in a four-step method. The volume of fiber was about 40%, and the fiber’s diameter was 7.0 μm. The C/SiC with the PyC interphase was fabricated using the chemical vapor infiltration (CVI). A Dynamical Mechanical Analyzer (DMA 2980) made by TA company, USA, was used for damping measurements of the C/SiC composite. All of the measurements were performed in air atmosphere from room temperature to 400 °C, and the testing frequencies were f = 1, 2, 5, and 10 Hz.

4.4.1 2D C/SiC Experimental and predicted temperature-dependent composite damping (ηc ), interface debonding ratio (ψd ), and interface slip ratio (ψs ) versus temperature curves of the 2D C/SiC composite at the vibration frequency f = 1 Hz were shown in Fig. 4.5 and Table 4.5. The predicted peak composite damping agreed with experimental

4.4 Experimental Comparisons Fig. 4.4 Effect of steady-state interface shear stress on a the temperature-dependent composite damping (ηc ) versus temperature curves; b the temperature-dependent interface debonding ratio (ψd ) versus temperature curves; and c the temperature-dependent interface slip ratio (ψs ) versus temperature curves of C/SiC composite

55

56

4 Temperature-Dependent Vibration Damping of Ceramic-Matrix …

Table 4.4 Temperature-dependent composite damping, interface debonding, and slip ratio of C/SiC composite for different steady-state interface shear stress τ0 /(MPa)

T /(°C)

ηc

ψd

ψs

40

20

0.00235

0.094

0.094

264

0.00627

0.043

0.043

400

0.00448

0.038

0.038

20

0.00202

0.0663

0.0663

265

0.00535

0.0345

0.0345

400

0.00389

0.0309

0.0309

50

data, and the predicted corresponding temperature for the peak composite damping was a little lower than the experimental data. Experimental composite damping increased from ηc = 0.01 at room temperature to the peak value ηc = 0.019 at T = 283 °C and then decreased to ηc = 0.014 at T = 400 °C. The theoretical predicted composite damping increased from ηc = 0.008 at room temperature to the peak value ηc = 0.019 at T = 279 °C and then decreased to ηc = 0.015 at T = 400 °C. The interface debonding ratio (ψd ) decreased from ψd = 0.337 at room temperature to ψd = 0.114 at T = 400 °C, and the interface slip ratio (ψs ) decreased from ψs = 0.248 at room temperature to ψs = 0.091 at T = 400 °C. Experimental and predicted temperature-dependent composite damping (ηc ), interface debonding ratio (ψd ), and interface slip ratio (ψs ) versus temperature curves of the 2D C/SiC composite at the vibration frequency of f = 2 Hz were shown in Fig. 4.6 and Table 4.5. Experimental composite damping (ηc ) increased from ηc = 0.009 at T = 150 °C to the peak value ηc = 0.015 at T = 266 °C and then decreased to ηc = 0.012 at T = 400 °C. The theoretical predicted composite damping increased from ηc = 0.006 at room temperature to the peak value ηc = 0.0144 at T = 283 °C and then decreased to ηc = 0.012 at T = 400 °C. The interface debonding ratio (ψd ) decreased from ψd = 0.2 at room temperature to ψd = 0.05 at T = 400 °C, and the interface slip ratio (ψs ) decreased from ψs = 0.16 at room temperature to ψs = 0.048 at T = 400 °C. Experimental and predicted temperature-dependent composite damping (ηc ), interface debonding ratio (ψd ), and interface slip ratio (ψs ) versus temperature curves of 2D C/SiC composite at the vibration frequency f = 5 Hz were shown in Fig. 4.7 and Table 4.5. Experimental composite damping (ηc ) increased from ηc = 0.007 at T = 136 °C to the peak value ηc = 0.0106 at T = 261 °C and then decreased to ηc = 0.008 at T = 400 °C. The theoretical predicted composite damping increased from ηc = 0.0048 at room temperature to the peak value ηc = 0.0101 at T = 263 °C and then decreased to ηc = 0.007 at T = 400 °C. The interface debonding ratio (ψd ) decreased from ψd = 0.186 at room temperature to ψd = 0.029 at T = 400 °C, and the interface slip ratio (ψs ) decreased from ψs = 0.174 at room temperature to ψs = 0.029 at T = 400 °C. Experimental and predicted temperature-dependent composite damping (ηc ), interface debonding ratio (ψd ), and interface slip ratio (ψs ) versus temperature curves

4.4 Experimental Comparisons Fig. 4.5 a Experimental and predicted temperature-dependent composite damping (ηc ) versus temperature curves; b the temperature-dependent interface debonding ratio (ψd ) versus temperature curves; and c the temperature-dependent interface slip ratio (ψs ) versus temperature curves of 2D C/SiC composite at the vibration frequency f = 1 Hz

57

58

4 Temperature-Dependent Vibration Damping of Ceramic-Matrix …

Table 4.5 Experimental and predicted peak value of composite damping and corresponding temperature of 2D C/SiC composite under the vibration frequencies of f = 1, 2, 5, and 10 Hz at temperature range from room temperature to 400 °C Frequency/Hz 1

Experiment (Hong et al. 2013)

Theory

Peak damping

Temperature/(o C)

Peak damping

Temperature/(o C)

0.019

283

0.019

279

2

0.015

266

0.014

283

5

0.0106

261

0.0101

263

10

0.010

258

0.0095

256

of the 2D C/SiC composite at the vibration frequency of f = 10 Hz were shown in Fig. 4.8 and Table 4.5. Experimental composite damping (ηc ) decreased from ηc = 0.0085 at room temperature to ηc = 0.0068 at T = 125 °C, then increased to the peak value ηc = 0.01 at T = 258 °C, and then decreased to ηc = 0.007 at T = 400 °C. The theoretical predicted composite damping (ηc ) increased from ηc = 0.0045 at room temperature to the peak value ηc = 0.0095 at T = 256 °C and then decreased to ηc = 0.0058 at T = 400 °C. The interface debonding ratio (ψd ) decreased from ψd = 0.169 at room temperature to ψd = 0.025 at T = 400 °C, and the interface slip ratio (ψs ) decreased from ψs = 0.165 at room temperature to ψs = 0.0258 at T = 400 °C. Under a high loading frequency f = 10 Hz, the damage mechanism of CMCs including matrix cracking and interface debonding is affected by the loading frequency. Sorensen and Holmes (Wang et al. 2015) investigated the effect of loading rate on tensile behavior of a SiC/CAS-II composite. It was found that the saturation matrix crack spacing increased with loading rate, and dynamic frictional coefficient also increases. However, in the present analysis, the effect of temperature on dynamic loading damage of CMCs (i.e., matrix cracking and interface damage) is not considered. The predicted composite damping is different from the experimental result at low temperature.

4.4.2 3D C/SiC Experimental and predicted temperature-dependent composite damping (ηc ), interface debonding ratio (ψd ), and interface slip ratio (ψs ) versus temperature curves of 3D C/SiC composite at the vibration frequency f = 1 Hz were shown in Fig. 4.9 and Table 4.6. Experimental composite damping (ηc ) increased from ηc = 0.009 at room temperature to the peak value ηc = 0.0165 at T = 325 °C and then decreased to ηc = 0.015 at T = 400 °C. The theoretical predicted composite damping (ηc ) increased from ηc = 0.009 at room temperature to the peak value ηc = 0.0163 at T = 308 °C and then decreased to ηc = 0.015 at T = 400 °C. The interface debonding ratio (ψd )

4.4 Experimental Comparisons Fig. 4.6 a Experimental and predicted temperature-dependent composite damping (ηc ) versus temperature curves; b the temperature-dependent interface debonding ratio (ψd ) versus temperature curves; and c the temperature-dependent interface slip ratio (ψs ) versus temperature curves of 2D C/SiC composite at the vibration frequency of f = 2 Hz

59

60

4 Temperature-Dependent Vibration Damping of Ceramic-Matrix …

Fig. 4.7 a Experimental and predicted temperature-dependent composite damping (ηc ) versus temperature curves; b the temperature-dependent interface debonding ratio (ψd ) versus temperature curves; and c the temperature-dependent interface slip ratio (ψs ) versus temperature curves of 2D C/SiC composite at the vibration frequency of f = 5 Hz

4.4 Experimental Comparisons Fig. 4.8 a Experimental and predicted temperature-dependent composite damping (ηc ) versus temperature curves; b the temperature-dependent interface debonding ratio (ψd ) versus temperature curves; and c the temperature-dependent interface slip ratio (ψs ) versus temperature curves of 2D C/SiC composite at the vibration frequency f = 10 Hz

61

62

4 Temperature-Dependent Vibration Damping of Ceramic-Matrix …

Fig. 4.9 a Experimental and predicted temperature-dependent composite damping (ηc ) versus temperature curves; b the temperature-dependent interface debonding ratio (ψd ) versus temperature curves; and c the temperature-dependent interface slip ratio (ψs ) versus temperature curves of 3D C/SiC composite at the vibration frequency f = 1 Hz

4.4 Experimental Comparisons

63

Table 4.6 Experimental and predicted peak value of composite damping and corresponding temperature of 3D C/SiC composite under the vibration frequencies of f = 1, 2, 5, and 10 Hz at temperature range from room temperature to 400 °C Frequency/Hz 1

Experiment (Hong et al. 2013)

Theory

Peak damping

Temperature/(° C)

Peak damping

Temperature/(° C)

0.0165

325

0.0163

308

2

0.0135

370

0.0136

360

5

0.0095

300

0.0095

300

10

0.009

295

0.0087

300

decreased from ψd = 0.455 at room temperature to ψd = 0.245 at T = 400 °C, and the interface slip ratio (ψs ) decreased from ψs = 0.449 at room temperature to ψs = 0.245 at temperature T = 400 °C. Experimental and predicted temperature-dependent composite damping (ηc ), interface debonding ratio (ψd ), and interface slip ratio (ψs ) versus temperature curves of the 3D C/SiC composite at the vibration frequency f = 2 Hz were shown in Fig. 4.10 and Table 4.6. Experimental composite damping (ηc ) increased from ηc = 0.0083 at room temperature to the peak value ηc = 0.0135 at T = 370 °C and then decreased to ηc = 0.0134 at T = 400 °C. The theoretical predicted composite damping (ηc ) increased from ηc = 0.0086 at room temperature to the peak value ηc = 0.0136 at T = 360 °C and then decreased to ηc = 0.0135 at T = 400 °C. The interface debonding ratio (ψd ) decreased from ψd = 0.445 at room temperature to ψd = 0.204 at T = 400 °C, and the interface slip ratio (ψs ) decreased from ψs = 0.445 at room temperature to ψs = 0.204 at T = 400 °C. Experimental and predicted temperature-dependent composite damping (ηc ), interface debonding ratio (ψd ), and interface slip ratio (ψs ) versus temperature curves of the 3D C/SiC composite at the vibration frequency f = 5 Hz were shown in Fig. 4.11 and Table 4.6. Experimental composite damping (ηc ) increased from ηc = 0.008 at room temperature to the peak value ηc = 0.0095 at T = 300 °C and then decreased to ηc = 0.009 at T = 400 °C. The theoretical predicted composite damping (ηc ) decreased from ηc = 0.0097 at room temperature to ηc = 0.007 at T = 86 °C, then increased to the peak value ηc = 0.0095 at T = 300 °C, and then decreased to ηc = 0.0092 at T = 400 °C. The interface debonding ratio (ψd ) decreased from ψd = 0.507 at room temperature to ψd = 0.138 at T = 400 °C, and the interface slip ratio (ψs ) decreased from ψs = 0.507 at room temperature to ψs = 0.138 at T = 400 °C. Experimental and predicted temperature-dependent composite damping (ηc ), interface debonding ratio (ψd ), and interface slip ratio (ψs ) versus temperature curves of the 3D C/SiC composite at the vibration frequency f = 10 Hz were shown in Fig. 4.12 and Table 4.6. Experimental composite damping (ηc ) decreases from ηc = 0.0084 at room temperature to ηc = 0.0075 at T = 125 °C, then increased to the peak value ηc

64

4 Temperature-Dependent Vibration Damping of Ceramic-Matrix …

Fig. 4.10 a Experimental and predicted temperature-dependent composite damping (ηc ) versus temperature curves; b the temperature-dependent interface debonding ratio (ψd ) versus temperature curves; and c the temperature-dependent interface slip ratio (ψs ) versus temperature curves of 3D C/SiC composite at the vibration frequency f = 2 Hz

4.4 Experimental Comparisons Fig. 4.11 a Experimental and predicted temperature-dependent composite damping (ηc ) versus temperature curves; b the temperature-dependent interface debonding ratio (ψd ) versus temperature curves; and c the temperature-dependent interface slip ratio (ψs ) versus temperature curves of 3D C/SiC composite at the vibration frequency f = 5 Hz

65

66

4 Temperature-Dependent Vibration Damping of Ceramic-Matrix …

Fig. 4.12 a Experimental and predicted temperature-dependent composite damping (ηc ) versus temperature curves; b the temperature-dependent interface debonding ratio (ψd ) versus temperature curves; and c the temperature-dependent interface slip ratio (ψs ) versus temperature curves of 3D C/SiC composite at the vibration frequency f = 10 Hz

4.5 Discussions

67

= 0.009 at T = 295 °C, and then decreased to ηc = 0.0084 at T = 400 °C. The theoretical predicted composite damping (ηc ) decreased from ηc = 0.0084 at room temperature to ηc = 0.0064 at T = 96 °C, then increased to the peak value ηc = 0.0087 at T = 300 °C, and then decreased to ηc = 0.0085 at T = 400 °C. The interface debonding ratio (ψd ) decreased from ψd = 0.472 at room temperature to ψd = 0.127 at T = 400 °C, and the interface slip ratio (ψs ) decreased from ψs = 0.472 at room temperature to ψs = 0.127 at T = 400 °C.

4.5 Discussions Due to temperature-dependent material properties and especially the interface properties (i.e., the interface shear stress (τi (T ))), the composite damping, interface debonding, and slip state of C/SiC were temperature-dependent. For 2D and 3D C/SiC, the temperature-dependent composite vibration damping increased with temperature to the peak value and then decreased, and the temperature-dependent interface debonding and slip ratio decreased with temperature. Experimental and predicted composite damping peak values of 2D and 3D C/SiC under vibration frequencies f = 1, 2, 5, and 10 Hz from room temperature to 400 °C were shown in Tables 4.5 and 4.6. For 2D C/SiC, the composite damping peak value decreased with vibration frequency, i.e., from ηc = 0.019 at a vibration frequency f = 1 Hz to ηc = 0.01 at a vibration frequency f = 10 Hz, and the corresponding temperature for peak composite damping also decreased, i.e., from T = 283 °C at a vibration frequency f = 1 Hz to T = 258 °C at a vibration frequency f = 10 Hz. For 3D C/SiC, the composite damping peak value decreases with vibration frequency, i.e., from ηc = 0.0165 at a vibration frequency f = 1 Hz to ηc = 0.009 at a vibration frequency f = 10 Hz, and the corresponding temperature for peak composite damping also decreased, i.e., from T = 325 °C at a vibration frequency f = 1 Hz to T = 295 °C at a vibration frequency f = 10 Hz. For C/SiC, the fiber and matrix damping contributes little to composite damping. However, the frictional dissipated energy caused by frictional slip in the debonding region mainly contributes to the composite damping. For 2D C/SiC, when the vibration frequency increases, the dynamic frictional slip range decreases, which decreases the energy dissipated through frictional slip and composite damping. For C/SiC with weak interface bonding, the interface debonding occurs when matrix cracking propagates to the fiber/matrix interphase. The frictional slip between the fiber and the matrix or between fiber and fiber causes the energy dissipation, which contributes to the damping of C/SiC. However, when the interface slip range or interface debonding/slip length decreases, the composite damping decreases. For C/SiC, the composite damping of 2D C/SiC is higher than that of 3D C/SiC, mainly due to the damage mechanisms of matrix cracking and interface debonding. For 3D C/SiC, the fiber volume along the longitudinal loading direction is higher than

68

4 Temperature-Dependent Vibration Damping of Ceramic-Matrix …

that of 2D C/SiC, leading to higher matrix cracking density, low interface debonding length, and low composite damping.

4.6 Summary and Conclusions In this chapter, a micromechanical temperature-dependent vibration damping model of a C/SiC composite was developed. The composite damping was divided into damping of the fiber and the matrix and the damping caused by frictional dissipated energy. Relationships between composite damping, composite internal damage, and temperature were established for different material properties and damage states. Experimental temperature-dependent damping of 2D and 3D C/SiC was predicted for different vibration frequencies. • For C/SiC, the temperature-dependent composite vibration damping increased with temperature to the peak value and then decreased, and the temperaturedependent interface debonding and slip ratio decreased with temperature. • For C/SiC, when the vibration frequency increases, the dynamic frictional slip range decreased, which decreased the energy dissipated through frictional slip and composite damping. • For 3D C/SiC, the fiber volume along the longitudinal loading direction was higher than that of 2D C/SiC, leading to higher matrix cracking density, low interface debonding length, and low composite damping. • When the fiber volume and interface debonding energy increased, the peak value of composite damping and the corresponding temperature both decreased. • When the matrix crack spacing and steady-state interface shear stress increased, the peak value of composite damping decreased, and the corresponding temperature for peak damping changed a little.

References Birman V, Byrd LW (2002) Effect of matrix cracks on damping in unidirectional and cross-ply ceramic matrix composites. J Compos Mater 36:1859–1877 Birman V, Byrd LW (2003) Damping in ceramic matrix composites with matrix cracks. Int J Solids Struct 40:4239–4256 Chandra R, Singh SP, Gupta K (1999) Damping studies in fiber-reinforced composites—a review. Compos Struct 46:41–51 Gowayed Y, Ojard G, Santhosh U, Jefferson G (2015) Modeling of crack density in ceramic matrix composites. J Compos Mater 49:2285–2294 Holmes JW, Cho C (1992) Experimental observation of frictional heating in fiber-reinforced ceramics. J Am Ceram Soc 75:929–938 Hong ZL, Cheng LF, Zhao CN, Zhang LT, Wang YG (2013) Effect of oxidation on internal friction behavior of C/SC composites. Acta Mater Compos Sin 30:93–100

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69

Kyriazoglou C, Page BH, Guild FJ (2004) Vibration damping for crack detection in composite laminates. Compos Part A 35:945–953 Li LB (2014) Modeling fatigue hysteresis behavior of unidirectional C/SiC ceramic-matrix composites. Compos Part B 66:466–474 Li LB (2019a) Modeling matrix multicracking development of fiber-reinforced ceramic-matrix composites considering fiber debonding. Int J Appl Ceram Technol 16:97–107 Li LB (2019b) Time-dependent damage and fracture of fiber-reinforced ceramic-matrix composites at elevated temperatures. Compos Interf 26:963–988 Li HQ (2019c) Analysis of vibration and damage test of ceramic matrix composites structure. Master’s Thesis, Nanchang Hongkong University, Nanchang, China Li LB (2020a) Durability of ceramic-matrix composites. Elsevier, Oxford, UK Li LB (2020b) Modeling temperature-dependent vibration damping in C/SiC fiber-reinforced ceramic-matrix composites. Materials 13:1633 Melo JD, Radford DW (2005) Time and temperature dependence of the viscoelastic properties of CFRP by dynamic mechanical analysis. Compos Struct 70:240–253 Min JB, Harris DL, Ting JM (2011) Advances in ceramic matrix composite blade damping characteristics for aerospace turbomachinery applications. In: Proceedings of the 52nd AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics and materials conference, Denver, Colorado, 4−7 April 2011 Momon S, Moevus M, Godin N, R’Mili M, Reynaud P, Fantozzi G, Fayolle G (2010) Acoustic emission and lifetime prediction during static fatigue tests on ceramic-matrix-composite at high temperature under air. Compos Part A 41:913–918 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 Patel RK, Bhattacharya B, Basu S (2007) A finite element based investigation on obtaining high material damping over a larger frequency range in viscoelastic composites. J Sound Vib 303:753– 766 Pradere C, Sauder C (2008) Transverse and longitudinal coefficient of thermal expansion of carbon fibers at high temperatures (300–2500K). Carbon 46:1874–1884 Reynaud P, Douby D, 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 Sato S, Serizawa H, Araki H, Node T, Kohyama A (2003) Temperature dependence of internal friction and elastic modulus of SiC/SiC composites. J Alloy Compd 355:142–147 Sauder C, Lamon J, Pailler R (2004) The tensile behavior of carbon fibers at high temperatures up to 2400 °C. Carbon 42:715–725 Snead LL, Nozawa T, Katoh Y, Byun TS, Kondo S, Petti DA (2007) Handbook of SiC properties for fuel performance modeling. J Nucl Mater 371:329–377 Sorensen BF, Holmes JW (1996) Effect of loading rate on the monotonic tensile behavior of a continuous-fiber-reinforced glass-ceramic matrix composite. J Am Ceram Soc 79:313–320 Wang W (2005) Study on damping capacity of C/SiC composites. Master’s Thesis, Northwestern Polytechnical University, Xi’an, China Wang RZ, Li WG, Li DY, Fang DNA (2015) new temperature dependent fracture strength model for the ZrB2 -SiC composites. J Eur Ceram Soc 35:2957–2962 Zhang J, Perez RJ, Lavernia EJ (1993) Documentation of damping capacity of metallic ceramic and metal-matrix composite materials. J Mater Sci 28:2395–2404 Zhang Z, Hartwig G (2002) Relation of damping and fatigue damage of unidirectional fiber composites. Int J Fatigue 24:713–718 Zhang Q, Cheng LF, Wang W, Wei X, Zhang LT, Xu YD (2007) Effect of interphase thickness on damping behavior of 2D C/SiC composites. Mater Sci Forum 546–549:1531–1534 Zhang Q, Cheng L, Wang W, Zhang L, Xu Y (2008) Effect of SiC coating and heat treatment on damping behavior of C/SiC composites. Mater Sci Eng A 473:254–258

Chapter 5

Time-Dependent Vibration Damping of Ceramic-Matrix Composites

Abstract In this chapter, a time-dependent vibration damping model of fiberreinforced ceramic-matrix composites (CMCs) was developed. Considering timeand temperature-dependent interface damages of oxidation, debonding, and slip, relationships between composite vibration damping, material properties, internal damages, oxidation duration, and temperature were established. Effects of material properties, vibration stress, damage state, and oxidation temperature on timedependent composite vibration damping and interface damages of C/SiC composite were discussed. Keywords Ceramic-matrix composites (CMCs) · Damping · Time-dependent · Matrix cracking · Interface debonding

5.1 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 and thermal protection system (TPS) (Naslain 2004; Wing and Halloran 2018; Misra et al. 2020; Li 2020a). However, vibration and noise may cause structural fatigue damage and final failure of CMC components (Min et al. 2011). Damping is a key parameter to reflect the vibration behavior of materials (Momon et al. 2010). For CMCs, the damping mechanism was much different from that of traditional metal and alloy materials and mainly includes three aspects as follows: (1) damping of matrix and fiber itself; (2) interface damping caused by friction slip between the fiber and the matrix under external mechanical vibration when the interface debonding occurs; and (3) damping caused by energy dissipation such as matrix cracking and fiber fracture (Zhang et al. 1993; Chandra et al. 1994; Birman and Byrd 2003; Melo and Radford 2005; Li 2020b]. Temperature, moisture, loading frequency, and wave form affect the vibration damping of composites (Patel et al. 2007; Sato et al. 2003). During manufacturing and service, cracks might happen in matrix, fiber, and interface both between fiber/matrix and neighboring plies (Gowayed et al. 2015). These cracks cause friction between the fiber and the matrix in the interface debonding region and © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 L. Li, Vibration Behavior in Ceramic-Matrix Composites, Advanced Ceramics and Composites 5, https://doi.org/10.1007/978-981-19-7838-8_5

71

72

5 Time-Dependent Vibration Damping of Ceramic-Matrix Composites

consume energy (Birman and Byrd 2002). The internal friction of CMCs is affected by the fabrication method (Sato et al. 2003), interphase thickness (Zhang et al. 2007), oxidation (Zhang et al. 2014; Hong et al. 2013), coating and heat treatment (Wang 2005; Zhang et al. 2008), interface damage and stress level (Holmes and Cho 1992), environmental temperature and duration (Li 2014, 2019a, 2019b). During service, the natural frequency, internal structure characteristics and defects, and internal damages of CMCs change due to service environment and loading, which causes the change of the mechanical energy and the ability of absorbing and dissipating vibration of the materials and the damping behavior (Li 2019c; Kyriazoglou et al. 2004; Zhang and Hartwig 2002; Lamouroux et al. 1999). The change of composite damping has an important impact on the application and service life of the materials (Li et al. 2019). At the same time, the research on the damping behavior of CMCs in the service environment is helpful to monitor and characterize the microstructure which changes the composite in the service process. At elevated temperature in oxidative environment, the oxidation of the interphase decreases the interface shear stress, which affects the vibration damping behavior of fiber-reinforced CMCs (Zhang et al. 2014; Hong et al. 2013; Li 2020c; 2019d, 2020d, 2020e, 2020f). In this chapter, a time-dependent vibration damping model of fiber-reinforced CMCs was developed considering time-dependent internal damage development. Relationships between composite vibration damping, internal damages, oxidation duration, and temperature were established for different material properties. Experimental time-dependent composite vibration damping of 2D C/SiC composite for different oxidation duration t = 2, 5, and 10 h at T = 700, 1000, and 1300 °C was predicted.

5.1.1 Time-Dependent Micromechanical Vibration Damping Models The vibration damping of fiber-reinforced CMCs can be divided into two sections, as shown in Eq. (5.1) (Birman and Byrd 2003). ηc = ηa + ηb

(5.1)

where ηa is the vibration damping of CMCs without damage, and ηb is the vibration damping of CMCs with time-dependent damages. (Birman and Byrd 2003) ηa =

E f Vf ηf + E m Vm ηm E f Vf + E m Vm

ηb (t) =

Ud (t) 2πU (t)

(5.2) (5.3)

5.1 Introduction

73

where V f , E f , and ηf are the volume, elastic modulus, and damping of the fiber, respectively; and V m , E m , and ηm are the volume, elastic modulus, and damping of the matrix, respectively, U d (t) and U(t) are time-dependent energy dissipated density and maximum strain energy density per cycle, respectively At elevated temperature, the time-dependent damage mechanisms change the microstress filed of the damaged fiber-reinforced CMCs. A shear-lag model is adopted to describe the stress distribution in different damage regions. The timedependent fiber and matrix axial stress upon first loading can be determined by Eqs. (5.4) and (5.5). ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

σ 2τf − x, x ∈ [0, ζ (t)] Vf rf σ 2τf 2τi − ζ (t) − (x − ζ (t)), x ∈ [ζ (t), ld (t)] σf (x) = V r rf ⎪ f f ⎪ ⎪ ] ( ] ) [ [ ⎪ ⎪ x − ld (t) lc Vm τf τi ⎪ ⎪ ⎩ σfo + , x ∈ ld (t), σmo − 2 ζ (t) − 2 (ld (t) − ζ (t)) exp −ρ Vf rf rf rf 2

(5.4)

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

(5.5)

where τ f and τ i are the fiber/matrix interface shear stress in the oxidation region and slip region, respectively; ζ (t) and l d (t) are the time-dependent interface oxidation and debonding length. The time-dependent fiber and matrix strain energy density per cycle can be determined by Eqs. (5.6) and (5.7). Uf (σ, t) = { = πrf2

1 2

{ σf (t)εf (t)d V V

Δσ 2 Vf2 E f

ld (t) − 2

τf2 2 Δσ τf 2 4 τf2 3 + 4 ζ (t) + ζ ζ (t)[ld (t) − ζ (t)] (t) rf Vf E f 3 r 2 Ef rf2 E f f

4 τi2 Δσ τi Δσ τf ζ (t)[ld (t) − ζ (t)] − 2 [ld (t) − ζ (t)]3 − 4 [ld (t) − ζ (t)]2 3 r 2 Ef rf Vf E f rf Vf E f f [ ] σ 2 lc τf τi − ld (t) + 4 2 ζ (t)[ld (t) − ζ (t)]2 + fo Ef 2 rf E f ][ ( [ )] lc /2 − ld (t) τf τi rf σfo Vm σmo − 2 ζ (t) − 2 (ld (t) − ζ (t)) 1 − exp −ρ +2 ρ E f Vf rf rf rf [ ( )]} ]2 [ Vm lc /2 − ld (t) rf τf τi 1 − exp −2ρ + σmo − 2 ζ (t) − 2 (ld (t) − ζ (t)) 2ρ E f Vf rf rf rf +

(5.6)

74

5 Time-Dependent Vibration Damping of Ceramic-Matrix Composites

Um (σ, t) =

1 2

{ σm (t)εm (t)d V V

{

V 2τ 2 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 E m rf Vm E m rf Vm E m ] 2 [ 4 Vf2 τi2 3 + σmo lc − l (t) − ζ + [l (t) (t)] d d 3 r 2 Vm2 E m Em 2 f [ )] ][ ( lc /2 − ld (t) Vf τf Vf τi rf σmo σmo − 2 ζ (t) − 2 −2 (ld (t) − ζ (t)) 1 − exp −ρ ρ Em rf Vm rf Vm rf ]2 [ [ ( )]} 1 rf lc /2 − ld (t) Vf τf Vf τi + σmo − 2 1 − exp −2ρ ζ (t) − 2 (ld (t) − ζ (t)) 2 ρ Em rf Vm rf Vm rf

(5.7)

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

(5.8)

Upon unloading, the time-dependent fiber axial stress distribution in different damage regions can be determined by Eq. (5.9). ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

[ ] σ 2τf + x, x ∈ 0, ly (t) Vf rf ) [ ] 2τf ) σ 2ly (t) − x , x ∈ ly (t), ζ (t) + Vf rf σf (x) = ) 2τi 2τf ) σ ⎪ ⎪ ⎪ 2ly (t) − ζ (t) − + (x − ζ (t)), x ∈ [ζ (t), ld (t)] ⎪ ⎪ Vf rf rf ⎪ ⎪ ] ( ] ) [ [ ⎪ ⎪ ) 2τi ⎪ x − ld (t) 2τf ) Vm lc ⎪ ⎩ σfo + , x ∈ ld (t), 2ly (t) − ζ (t) − σmo + (ld (t) − ζ (t)) exp −ρ Vf rf rf rf 2

(5.9) where l y (t) is the time-dependent interface counter slip length. Upon reloading, the time-dependent fiber axial stress distribution in different damage regions can be determined by Eq. (5.10). ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

σ Vf σ Vf σ Vf σ Vf

2τf x, x ∈ [0, lz (t)] rf [ ] 2τf − (2l z (t) − x), x ∈ l z (t), ly (t) rf ) [ ] 2τf ) 2l y (t) − 2l z (t) − x , x ∈ l y (t), ζ (t) + σf (x) = r ⎪ f ⎪ ⎪ ⎪ ⎪ ) 2τi 2τf ) ⎪ ⎪ 2l y (t) − 2l z (t) − ζ (t) − + (x − ζ (t)), x ∈ [ζ (t), ld (t)] ⎪ ⎪ ⎪ r rf f ⎪ ⎪ [ ] ( ] ) [ ⎪ ⎪ ) 2τi Vm 2τf ) x − ld (t) lc ⎪ ⎪ ⎩ σfo + 2ly (t) − 2l z (t) − ζ (t) − σmo + , x ∈ ld (t), (ld (t) − ζ (t)) exp −ρ Vf rf rf rf 2 −

(5.10) where l z (t) is the time-dependent interface counter slip length. Upon unloading and reloading, the energy is dissipated through frictional slip at the fiber/matrix interface, and the time-dependent dissipated energy upon unloading

5.2 Results and Discussion

75

and reloading can be determined by Eqs. (5.11) and (5.12). ] Δσ 2 E c τf 8 3 Ud_ u (t) = 2πrf τf l (t) − l (t) Vf E f y 3 rf Vm E f E m y ] [ Δσ 2 E c τf 8 3 Ud_ r (t) = 2πrf τf l (t) − l (t) Vf E f z 3 rf Vm E f E m z [

(5.11) (5.12)

where Δσ is the range of applied stress and Ud (t) = Ud_ u (t) + Ud_ r (t)

(5.13)

5.2 Results and Discussion Effects of material properties, vibration stress, damage state, and temperature on time-dependent composite vibration damping of C/SiC composite were discussed. The material properties of C/SiC composite were given by V f = 30%, r f = 3.5 μm, ┌ i = 0.1 J/m2 , ηf = 0.002, ηm = 0.001, T 0 = 1000 °C, and the temperature-dependent constituent properties of fiber and matrix elastic modulus and thermal expansional coefficient, interface shear stress, and interface debonding energy are given by )] ( [ T , T < 2273 K E f (T ) = 230 1 − 2.86 × 10−4 exp 324 [ ( )] 962 350 460 − 0.04T exp − , T ∈ [300 K 1773 K] E m (T ) = 460 T

(5.14) (5.15)

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]

(5.16)

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] ⎧ ⎨ −1.8276 + 0.0178T − 1.5544 × 10−5 T 2 αlm (T ) = αrm (T ) = +4.5246 × 10−9 T 3 , T ∈ [125 K 1273 K] ⎩ 5.0 × 10−6 /K, T > 1273 K τi (T ) = τ0 + μ

|αrf (T ) − αrm (T )|(T0 − T ) A

(5.17)

(5.18)

(5.19)

76

5 Time-Dependent Vibration Damping of Ceramic-Matrix Composites

[ ┌i (T ) = ┌ir 1 −

{T

Tr C P (T )dT { T0 Tr C P (T )dT

] (5.20)

where T r denotes the reference temperature; T 0 denotes the fabricated temperature; ┌ ir denotes the interface debonding energy at the reference temperature T r ; and C P (T ) can be described using the following equation: C P (T ) = 76.337 + 109.039 × 10−3 T − 6.535 × 105 T −2 − 27.083 × 10−6 T 2 (5.21)

5.2.1 Effect of Fiber Volume on Time-Dependent Vibration Damping of C/SiC Composite Time-dependent composite vibration damping (ηc ), the interface debonding ratio (ψ d = 2ld /l c ), interface oxidation ratio (ψ o = ζ /l d ), and interface slip ratio (ψ s = 2ly /l c ) of C/SiC composite after oxidation duration t = 2 h at T = 800 °C for different fiber volumes were shown in Fig. 5.1 and Table 5.1. When fiber volume increased, the composite damping decreased due to the decrease of frictional energy dissipated through interface slip. At low fiber volume, i.e., V f = 20−30%, 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 = 40%, the ratio of interface debonding and slip decreased first and then increased, and the ratio of interface debonding increased first and then decreased. When the 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 = 20%, the composite damping increased from ηc = 0.00507 at T = 20 °C to ηc = 0.00999 at T = 500 °C; the interface debonding ratio (ψ d ) decreased from ψ d = 0.345 at T = 20 °C to ψ d = 0.289 at T = 500 °C; the interface oxidation ratio (ψ o ) increased from ψ o = 0.017 at T = 20 °C to ψ o = 0.0205 at T = 500 °C; and the interface slip ratio (ψ s ) decreased from ψ s = 0.345 at T = 20 °C to ψ s = 0.289 at T = 500 °C. • When V f = 30%, the composite damping increased from ηc = 0.00169 at T = 20 °C to ηc = 0.00246 at T = 500 °C; the interface debonding ratio (ψ d ) decreased from ψ d = 0.1446 at T = 20 °C to ψ d = 0.1251 at T = 500 °C; the interface oxidation ratio (ψ o ) increased from ψ o = 0.041 at T = 20 °C to ψ o = 0.047 at T = 500 °C; and the interface slip ratio (ψ s ) decreased from ψ s = 0.1446 at T = 20 °C to ψ s = 0.1251 at T = 500 °C.

5.2 Results and Discussion Fig. 5.1 Effect of fiber volume on a ηc versus T curves; b ψ d versus T curves; c ψ o versus T curves; d ψ s versus T curves of C/SiC composite after oxidation duration t = 2 h at T = 800 °C

77

78

5 Time-Dependent Vibration Damping of Ceramic-Matrix Composites

Fig. 5.1 (continued)

Table 5.1 Effect of fiber volume on time-dependent ηc , ψ d , ψ o , and ψ s of C/SiC composite after oxidation duration t = 2 h at T = 800 °C ηc

ψd

ψo

ψs

20

0.00507

0.345

0.017

0.345

100

0.00509

0.323

0.018

0.323

200

0.00569

0.311

0.019

0.311

300

0.00659

0.303

0.0196

0.303

400

0.00794

0.295

0.0205

0.295

500

0.00999

0.289

0.0205

0.289

20

0.00169

0.1446

0.04121

0.1446

100

0.00167

0.1296

0.04596

0.1296

200

0.00175

0.1266

0.04705

0.1266

300

0.00188

0.1253

0.04754

0.1253

400

0.0021

0.1249

0.04771

0.1249

500

0.00246

0.1251

0.04762

0.1251

20

0.00138

0.0485

0.1229

0.0485

100

0.00137

0.0375

0.1587

0.0375

200

0.00137

0.0384

0.155

0.0384

300

0.00139

0.0406

0.146

0.0406

400

0.00141

0.0433

0.137

0.0433

500

0.00146

0.0464

0.128

0.0464

T /(°C) V f = 20%

V f = 30%

V f = 40%

5.2 Results and Discussion

79

• When V f = 40%, the composite damping increased from ηc = 0.00138 at T = 20 °C to ηc = 0.00146 at T = 500 °C; the interface debonding ratio (ψ d ) decreased from ψ d = 0.0485 at T = 20 °C to ψ d = 0.0384 at T = 200 °C and increased to ψ d = 0.0464 at T = 500 °C; the interface oxidation ratio (ψ o ) increased from ψ o = 0.1229 at T = 20 °C to ψ o = 0.1587 at T = 100 °C and decreased to ψ o = 0.128; and the interface slip ratio (ψ s ) decreased from ψ s = 0.0485 at T = 20 °C to ψ s = 0.0384 at T = 200 °C and increased to ψ s = 0.0464 at T = 500 °C.

5.2.2 Effect of Vibration Stress on Time-Dependent Vibration Damping of C/SiC Composite Time-dependent composite vibration damping (ηc ), the interface debonding ratio (ψ d = 2ld /l c ), interface oxidation ratio (ψ o = ζ /l d ), and interface slip ratio (ψ s = 2ly /l c ) of C/SiC composite after oxidation duration t = 2 h at T = 800 °C for different vibration stress were shown in Fig. 5.2 and Table 5.2. When vibration stress increased, the timedependent composite vibration damping increased due to 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 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 due to the increase of the interface slip range. • When σ = 50 MPa, the composite damping (ηc ) increased from ηc = 0.00169 at T = 20 °C to ηc = 0.0024 at T = 500 °C; the interface debonding ratio (ψ d ) decreased from ψ d = 0.1446 at T = 20 °C to ψ d = 0.1249 at T = 400 °C and increased to ψ d = 0.1251 at T = 500 °C; the interface oxidation ratio (ψ o ) increased from ψ o = 0.0412 at T = 20 °C to ψ o = 0.0477 at T = 400 °C and decreased to ψ o = 0.0476 at T = 500 °C; and the interface slip ratio (ψ s ) decreased from 2l y /l c = 0.1446 at T = 20 °C to 2l y /l c = 0.1249 at T = 400 °C and increased to 2l y /l c = 0.1251 at T = 500 °C. • When σ = 70 MPa, the composite damping (ηc ) increased from ηc = 0.00361 at T = 20 °C to ηc = 0.0067 at T = 500 °C; the interface debonding ratio (ψ d ) decreased from ψ d = 0.2758 at T = 20 °C to ψ d = 0.2324 at T = 500 °C; the interface oxidation ratio (ψ o ) increased from ψ o = 0.0216 at T = 20 °C to ψ o

80

5 Time-Dependent Vibration Damping of Ceramic-Matrix Composites

Fig. 5.2 Effect of vibration stress on a ηc versus T curves; b ψ d versus T curves; c ψ o versus T curves; d ψ s versus T curves of C/SiC composite after oxidation duration t = 2 h at T = 800 °C

5.2 Results and Discussion

81

Fig. 5.2 (continued)

Table 5.2 Effect of vibration stress on time-dependent ηc , ψ d , ψ o , and ψ s of C/SiC composite after oxidation duration t = 2 h at T = 800 °C ηc

ψd

ψo

ψs

20

0.00169

0.1446

0.0412

0.1446

100

0.00167

0.1296

0.0459

0.1296

200

0.00175

0.1266

0.0470

0.1266

300

0.00188

0.1253

0.0475

0.1253

400

0.0021

0.1249

0.0477

0.1249

500

0.0024

0.1251

0.0476

0.1251

20

0.00361

0.2758

0.0216

0.2758

100

0.00362

0.2562

0.0232

0.2562

200

0.0040

0.2478

0.0240

0.2478

300

0.0045

0.2415

0.0246

0.2415

400

0.0054

0.2364

0.0252

0.2364

500

0.0067

0.2324

0.0256

0.2324

20

0.00829

0.4070

0.0146

0.4070

100

0.0083

0.3828

0.0155

0.3828

200

0.0091

0.369

0.0161

0.369

300

0.0103

0.3577

0.0166

0.3577

400

0.0118

0.3480

0.0171

0.3480

500

0.0139

0.3397

0.0175

0.3397

T /(°C) σ = 50 MPa

σ = 70 MPa

σ = 90 MPa

82

5 Time-Dependent Vibration Damping of Ceramic-Matrix Composites

= 0.0256 at T = 500 °C; and the interface slip ratio (ψ s ) decreased from ψ s = 0.2758 at T = 20 °C to ψ s = 0.2324 at T = 500 °C. • When σ = 90 MPa, the composite damping increased from ηc = 0.00829 at T = 20 °C to ηc = 0.0139 at T = 500 °C; the interface debonding ratio (ψ d ) decreased from ψ d = 0.407 at T = 20 °C to ψ d = 0.3397 at T = 500 °C; the interface oxidation ratio (ψ o ) increased from ψ o = 0.0146 at T = 20 °C to ψ o = 0.0175 at T = 500 °C; and the interface slip ratio (ψ s ) decreased from ψ s = 0.407 at T = 20 °C to ψ s = 0.3397 at T = 500 °C.

5.2.3 Effect of Matrix Crack Spacing on Time-Dependent Vibration Damping of C/SiC Composite Time-dependent composite vibration damping (ηc ), the interface debonding ratio (ψ d = 2ld /l c ), interface oxidation ratio (ψ o = ζ /l d ), and interface slip ratio (ψ s = 2ly /l c ) of C/SiC composite after oxidation duration t = 2 h at T = 800 °C for different matrix crack spacing were shown in Fig. 5.3 and Table 5.3. When matrix crack spacing increased, the time-dependent composite vibration damping decreased due to 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, the ratio between the interface oxidation length and interface debonding length remained unchanged, and the composite vibration damping decreased due to the decreasing ratio of the frictional dissipated energy in the total energy dissipated. • When lc = 100 μm, the composite damping (ηc ) increased from ηc = 0.00236 at T = 20 °C to ηc = 0.00399 at T = 500 °C; the interface debonding ratio (ψ d ) decreased from ψ d = 0.2893 at T = 20 °C to ψ d = 0.2498 at T = 400 °C and increased to ψ d = 0.2503 at T = 500 °C; the interface oxidation ratio (ψ o ) increased from ψ o = 0.0412 at T = 20 °C to ψ o = 0.0477 at T = 400 °C and decreased to ψ o = 0.0476 at T = 500 °C; and the interface slip ratio (ψ s ) decreased from ψ s = 0.2893 at T = 20 °C to ψ s = 0.2498 at T = 400 °C and increased to ψ s = 0.2503 at T = 500 °C. • When l c = 200 μm, the composite damping (ηc ) increased from ηc = 0.00169 at T = 20 °C to ηc = 0.00246 at T = 500 °C; the interface debonding ratio (ψ d ) decreased from ψ d = 0.1446 at T = 20 °C to ψ d = 0.1249 at T = 400 °C and increased to ψ d = 0.1251 at T = 500 °C; the interface oxidation ratio (ψ o ) increased from ψ o = 0.0412 at T = 20 °C to ψ o = 0.0477 at T = 400 °C and decreased to ψ o = 0.0476 at T = 500 °C; and the interface slip ratio (ψ s ) decreased from ψ s = 0.1446 at T = 20 °C to ψ s = 0.1249 at T = 400 °C and increased to ψ s = 0.1251 at T = 500 °C.

5.2 Results and Discussion Fig. 5.3 Effect of matrix crack spacing on a ηc versus T curves; b ψ d versus T curves; c ψ o versus T curves; d ψ s versus T curves of C/SiC composite after oxidation duration t = 2 h at T = 800 °C

83

84

5 Time-Dependent Vibration Damping of Ceramic-Matrix Composites

Fig. 5.3 (continued)

Table 5.3 Effect of matrix crack spacing on time-dependent ηc , ψ d , ψ o , and ψ s of C/SiC composite after oxidation duration t = 2 h at T = 800 °C ηc

ψd

ψo

ψs

20

0.00236

0.2893

0.0412

0.2893

100

0.00228

0.2593

0.0459

0.2593

200

0.00246

0.2533

0.0470

0.2533

300

0.00276

0.2507

0.0475

0.2507

400

0.00323

0.2498

0.0477

0.2498

500

0.00399

0.2503

0.0476

0.2503

20

0.00169

0.1446

0.0412

0.1446

100

0.00167

0.1296

0.0459

0.1296

200

0.00175

0.1266

0.0470

0.1266

300

0.00188

0.1253

0.0475

0.1253

400

0.0021

0.1249

0.0477

0.1249

500

0.00246

0.1251

0.0476

0.1251

20

0.00152

0.0964

0.0412

0.0964

100

0.00151

0.0864

0.0459

0.0864

200

0.00156

0.0844

0.0470

0.0844

300

0.00164

0.0835

0.0475

0.0835

400

0.00178

0.0832

0.0477

0.0832

500

0.00202

0.0834

0.0476

0.0834

T /(°C) l c = 100 μm

l c = 200 μm

l c = 300 μm

5.2 Results and Discussion

85

• When lc = 300 μm, the composite damping (ηc ) increased from ηc = 0.00152 at T = 20 °C to ηc = 0.00202 at T = 500 °C; the interface debonding ratio (ψ d ) decreased from ψ d = 0.0964 at T = 20 °C to ψ d = 0.0832 at T = 400 °C and increased to ψ d = 0.0834 at T = 500 °C; the interface oxidation ratio (ψ o ) increased from ψ o = 0.0412 at T = 20 °C to ψ o = 0.0477 at T = 400 °C and decreased to ψ o = 0.0476 at T = 500 °C; and the interface slip ratio (ψ s ) decreased from ψ s = 0.0964 at T = 20 °C to ψ s = 0.0832 at T = 400 °C and increased to ψ s = 0.0834 at T = 500 °C.

5.2.4 Effect of Interface Shear Stress on Time-Dependent Vibration Damping of C/SiC Composite Time-dependent composite vibration damping (ηc ), the interface debonding ratio (ψ d = 2ld /l c ), interface oxidation ratio (ψ o = ζ /l d ), and interface slip ratio (ψ s = 2ly /l c ) of C/SiC composite after oxidation duration t = 2 h at T = 800 °C for different interface shear stress were shown in Fig. 5.4 and Table 5.4. When the interface shear stress in the oxidation region increased, the time-dependent composite vibration damping increased due to 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 (ηc ) increased from ηc = 0.00187 at T = 20 °C to ηc = 0.00297 at T = 500 °C; the interface debonding ratio (ψ d ) decreased from ψ d = 0.1442 at T = 20 °C to ψ d = 0.1245 at T = 400 °C and increased to ψ d = 0.1248 at T = 500 °C; the interface oxidation ratio (ψ o ) increased from ψ o = 0.0413 at T = 20 °C to ψ o = 0.0478 at T = 400 °C and decreased to ψ o = 0.0477 at T = 500 °C; and the interface slip ratio (ψ s ) decreased from ψ s = 0.1442 at T = 20 °C to ψ s = 0.1245 at T = 400 °C and increased to ψ s = 0.1248 at T = 500 °C. • When τ f = 5 MPa, the composite damping (ηc ) increased from ηc = 0.00215 at T = 20 °C to ηc = 0.00384 at T = 500 °C; the interface debonding ratio (ψ d ) decreased from ψ d = 0.1433 at T = 20 °C to ψ d = 0.1238 at T = 400 °C and increased to ψ d = 0.1241 at T = 500 °C; the interface oxidation ratio (ψ o ) increased from ψ o = 0.0415 at T = 20 °C to ψ o = 0.0481 at T = 400 °C and decreased to ψ o = 0.048 at T = 500 °C; and the interface slip ratio (ψ s ) decreased from ψ s = 0.1433 at T = 20 °C to ψ s = 0.1238 at T = 400 °C and increased to ψ s = 0.1241 at T = 500 °C.

86

5 Time-Dependent Vibration Damping of Ceramic-Matrix Composites

Fig. 5.4 Effect of interface shear stress on a ηc versus T curves; b ψ d versus T curves; c ψ o versus T curves; d ψ s versus T curves of C/SiC composite after oxidation duration t = 2 h at T = 800 °C

5.2 Results and Discussion

87

Fig. 5.4 (continued)

Table 5.4 Effect of interface shear stress on time-dependent ηc , ψ d , ψ o , and ψ s of C/SiC composite after oxidation time t = 2 h at T = 800 °C ηc

ψd

ψo

ψs

20

0.00187

0.1442

0.0413

0.1442

100

0.00184

0.1292

0.0461

0.1292

200

0.00195

0.1262

0.0472

0.1262

300

0.00214

0.125

0.0476

0.125

400

0.00245

0.1245

0.0478

0.1245

500

0.00297

0.1248

0.0477

0.1248

20

0.00215

0.1433

0.0415

0.1433

100

0.00212

0.1284

0.0464

0.1284

200

0.00229

0.1255

0.0475

0.1255

300

0.00259

0.1242

0.0479

0.1242

400

0.00305

0.1238

0.0481

0.1238

500

0.00384

0.1241

0.0480

0.1241

20

0.00236

0.1425

0.0418

0.1425

100

0.00233

0.1276

0.0467

0.1276

200

0.00256

0.1247

0.0478

0.1247

300

0.00293

0.1234

0.0482

0.1234

400

0.00352

0.1231

0.0484

0.1231

500

0.00451

0.1234

0.0483

0.1234

T /(°C) τ f = 3 MPa

τ f = 5 MPa

τ f = 7 MPa

88

5 Time-Dependent Vibration Damping of Ceramic-Matrix Composites

• When τ f = 7 MPa, the composite damping (ηc ) increased from ηc = 0.00236 at T = 20 °C to ηc = 0.00451 at T = 500 °C; the interface debonding ratio (ψ d ) decreased from ψ d = 0.1425 at T = 20 °C to 2ld /l c = 0.1231 at T = 400 °C and increased to ψ d = 0.1234 at T = 500 °C; the interface oxidation ratio (ψ o ) increased from ψ o = 0.0418 at T = 20 °C to ψ o = 0.0484 at T = 400 °C and decreased to ψ o = 0.0483 at T = 500 °C; and the interface slip ratio (ψ s ) decreased from ψ s = 0.1425 at T = 20 °C to ψ s = 0.1231 at T = 400 °C and increased to ψ s = 0.1234 at T = 500 °C.

5.2.5 Effect of Temperature on Time-Dependent Vibration Damping of C/SiC Composite Time-dependent composite vibration damping (ηc ), the interface debonding ratio (ψ d = 2ld /l c ), interface oxidation ratio (ψ o = ζ /l d ), and interface slip ratio (ψ s = 2ly /l c ) of C/SiC composite after oxidation duration t = 2 h for different oxidation temperatures were shown Fig. 5.5 and Table 5.5. When temperature increased, the temperature-dependent composite vibration damping increased due to 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, the energy dissipated of frictional slip between the fiber and the matrix increased due to the increase of interface slip range, and the time-dependent composite vibration damping increased. • When T = 600 °C, the composite damping (ηc ) increased from ηc = 0.00167 at T = 20 °C to ηc = 0.00237 at T = 500 °C; the interface debonding ratio (ψ d ) decreased from ψ d = 0.1403 at T = 20 °C to ψ d = 0.1204 at T = 400 °C and increased to ψ d = 0.1207 at T = 500 °C; the interface oxidation ratio (ψ o ) increased from ψ o = 0.0064 at T = 20 °C to ψ o = 0.00746 at T = 400 °C and decreased to ψ o = 0.00745 at T = 500 °C; and the interface slip ratio (ψ s ) decreased from ψ s = 0.1403 at T = 20 °C to ψ s = 0.1204 at T = 400 °C and increased to ψ s = 0.1207 at T = 500 °C. • When T = 900 °C, the composite damping (ηc ) increased from ηc = 0.00172 at T = 20 °C to ηc = 0.00256 at T = 500 °C; the interface debonding ratio (ψ d ) decreased from ψ d = 0.1498 at T = 20 °C to ψ d = 0.1302 at T = 400 °C and increased to ψ d = 0.1305 at T = 500 °C; the interface oxidation ratio (ψ o ) increased from ψ o = 0.0804 at T = 20 °C to ψ o = 0.0924 at T = 400 °C and decreased to ψ o = 0.0923 at T = 500 °C; and the interface slip ratio (ψ s ) decreased from ψ s = 0.1498 at T = 20 °C to ψ s = 0.1302 at T = 400 °C and increased to ψ s = 0.1305 at T = 500 °C. • When T = 1100 °C, the composite damping (ηc ) increased from ηc = 0.00186 at T = 20 °C to ηc = 0.003 at T = 500 °C; the interface debonding ratio (ψ d ) decreased from ψ d = 0.1705 at T = 20 °C to ψ d = 0.1514 at T = 400 °C and increased

5.2 Results and Discussion Fig. 5.5 Effect of oxidation temperature on a ηc versus T curves; b ψ d versus T curves; c ψ o versus T curves; d ψ s versus T curves of C/SiC composite after oxidation duration t = 2 h

89

90

5 Time-Dependent Vibration Damping of Ceramic-Matrix Composites

Fig. 5.5 (continued)

Table 5.5 Effect of temperature on time-dependent ηc , ψ d , ψ o , and ψ s of C/SiC composite after oxidation duration t = 2 h ηc

ψd

ψo

ψs

20

0.00167

0.1403

0.0064

0.1403

100

0.00164

0.1253

0.00718

0.1253

200

0.00172

0.1223

0.00735

0.1223

300

0.00184

0.1209

0.00743

0.1209

400

0.00204

0.1204

0.00746

0.1204

500

0.00237

0.1207

0.00745

0.1207

20

0.00172

0.1498

0.0804

0.1498

100

0.0017

0.1349

0.0893

0.1349

200

0.00179

0.1319

0.0913

0.1319

300

0.00193

0.1306

0.0922

0.1306

400

0.00217

0.1302

0.0924

0.1302

500

0.00256

0.1305

0.0923

0.1305

20

0.00186

0.1705

0.2121

0.1705

100

0.00185

0.1557

0.2323

0.1557

200

0.00197

0.1528

0.2366

0.1528

300

0.0021

0.1517

0.2384

0.1517

400

0.0024

0.1514

0.2389

0.1514

500

0.003

0.1518

0.2383

0.1518

T /(°C) T = 600 °C

T = 900 °C

T = 1100 °C

5.3 Experimental Comparisons

91

to ψ d = 0.1518 at T = 500 °C; the interface oxidation ratio (ψ o ) increased from ψ o = 0.2121 at T = 20 °C to ψ o = 0.2389 at T = 400 °C and decreased to ψ o = 0.2383 at T = 500 °C; and the interface slip ratio (ψ s ) decreased from ψ s = 0.1705 at T = 20 °C to ψ s = 0.1514 at T = 400 °C and increased to ψ s = 0.1518 at T = 500 °C.

5.3 Experimental Comparisons Zhang et al. (2014) performed experimental investigation on the effect of oxidation on vibration damping behavior of 2D C/SiC composite at T = 700, 1000, and 1300 °C for different oxidation duration t = 2, 5, and 10 h. The composite was fabricated using the chemical vapor infiltration (CVI). The volume of fiber was approximately 40%. A 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 measurements were performed in air atmosphere from room temperature to 400 °C, and the testing frequencies were f = 1 Hz.

5.3.1 t = 2 h at T = 700, 1000, and 1300 °C Experimental and predicted time-dependent composite vibration damping (ηc ), the interface debonding ratio (ψ d = 2ld /l c ), interface oxidation ratio (ψ o = ζ /l d ), and interface slip ratio (ψ s = 2ly /l c ) of C/SiC composite after oxidation duration t = 2 h at oxidation temperature T = 700, 1000, and 1300 °C were shown in Fig. 5.6 and Table 5.6. When the oxidation duration was t = 2 h, 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 h, the experimental time-dependent composite damping (ηc ) increased from ηc = 0.01477 at T = 20 °C to ηc = 0.02249 at T = 400 °C; the predicted time-dependent composite vibration damping increased from ηc = 0.01573 at T = 20 °C to ηc = 0.02413 at T = 500 °C; the time-dependent interface debonding ratio (ψ d ) decreased from ψ d = 0.5378 at T = 20 °C to ψ d = 0.4463 at T = 500 °C; the time-dependent interface oxidation ratio (ψ o ) increased from ψ o = 0.0047 at T = 20 °C to ψ o = 0.0057 at T = 500 °C; and the time-dependent interface slip ratio (ψ s ) decreased from ψ s = 0.5378 at T = 20 °C to ψ s = 0.4463 at T = 500 °C. • After oxidation at T = 1000 °C for t = 2 h, the experimental time-dependent composite vibration damping (ηc ) increased from ηc = 0.01479 at T = 20 °C to

92

5 Time-Dependent Vibration Damping of Ceramic-Matrix Composites

Fig. 5.6 a Experimental and predicted ηc versus T curves; b time-dependent ψ d versus T curves; c time-dependent ψ o versus T curves; and d time-dependent ψ s versus T curves of C/SiC composite after oxidation duration t = 2 h at T = 700, 1000, and 1300 °C

5.3 Experimental Comparisons

93

Fig. 5.6 (continued)

Table 5.6 Experimental and predicted time-dependent ηc , ψ d , ψ o , and ψ s of C/SiC composite after oxidation duration t = 2 h at T = 700, 1000, and 1300 °C ηc (Experiment)

ηc (Theory)

ψd

ψo

ψs

20

0.01477

0.01573

0.5378

0.0047

0.5378

100

0.01564

0.01554

0.5083

0.0050

0.5083

200

0.01643

0.01676

0.48908

0.0052

0.48908

300

0.01886

0.01857

0.47289

0.0054

0.47289

400

0.02249

0.021

0.45876

0.00556

0.45876

500

—

0.02413

0.44639

0.00572

0.44639

20

0.01479

0.01414

0.52183

0.04179

0.52183

100

0.01582

0.01405

0.49353

0.04418

0.49353

200

0.01721

0.01528

0.47575

0.04583

0.47575

300

0.02296

0.01707

0.4609

0.04731

0.4609

400

0.02111

0.0195

0.44799

0.04867

0.44799

500

—

0.02263

0.43676

0.04993

0.43676

20

0.01379

0.01453

0.5408

0.1515

0.5408

100

0.01568

0.01452

0.51399

0.1595

0.51399

200

0.01673

0.01587

0.49789

0.16466

0.49789

300

0.01780

0.01779

0.48459

0.16918

0.48459

400

0.02012

0.02031

0.47312

0.17328

0.47312

500

—

0.02351

0.46321

0.17699

0.46321

T /(°C) T = 700 °C

T = 1000 °C

T = 1300 °C

94

5 Time-Dependent Vibration Damping of Ceramic-Matrix Composites

ηc = 0.0211 at T = 400 °C; the predicted time-dependent composite vibration damping (ηc ) increased from ηc = 0.01414 at T = 20 °C to ηc = 0.02263 at T = 500 °C; the time-dependent interface debonding ratio (ψ d ) decreased from ψ d = 0.5218 at T = 20 °C to ψ d = 0.4367 at T = 500 °C; the time-dependent interface oxidation ratio (ψ o ) increased from ψ o = 0.041 at T = 20 °C to ψ o = 0.049 at T = 500 °C; and the time-dependent interface slip ratio (ψ s ) decreased from ψ s = 0.5218 at T = 20 °C to ψ s = 0.4367 at T = 500 °C. • After oxidation at T = 1300 °C for t = 2 h, the experimental composite vibration damping (ηc ) increased from ηc = 0.01379 at T = 20 °C to ηc = 0.02 at T = 400 °C; the predicted time-dependent composite vibration damping (ηc ) increased from ηc = 0.01453 at T = 20 °C to ηc = 0.02351 at T = 500 °C; the time-dependent interface debonding ratio (ψ d ) decreased from ψ d = 0.5408 at T = 20 °C to ψ d = 0.4632 at T = 500 °C; the time-dependent interface oxidation ratio (ψ o ) increased from ψ o = 0.1515 at T = 20 °C to ψ o = 0.1769 at T = 500 °C; and the time-dependent interface slip ratio (ψ s ) decreased from ψ s = 0.5408 at T = 20 °C to ψ s = 0.4632 at T = 500 °C.

5.3.2 t = 5 h at T = 700, 1000, and 1300 °C Experimental and predicted time-dependent composite vibration damping (ηc ), the interface debonding ratio (ψ d = 2ld /l c ), interface oxidation ratio (ψ o = ζ /l d ), and interface slip ratio (ψ s = 2ly /l c ) of C/SiC composite after oxidation duration t = 5 h at T = 700, 1000, and 1300 °C were shown in Fig. 5.7 and Table 5.7. When oxidation duration was t = 5 h, 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 h, the experimental time-dependent composite vibration damping (ηc ) increased from ηc = 0.01169 at T = 20 °C to ηc = 0.023 at T = 400 °C; the predicted time-dependent composite vibration damping (ηc ) increased from ηc = 0.01423 at T = 20 °C to ηc = 0.02264 at T = 500 °C; the time-dependent ratio of the interface debonding (ψ d ) decreased from ψ d = 0.5194 at T = 20 °C to ψ d = 0.432 at T = 500 °C; the time-dependent ratio of the interface oxidation (ψ o ) increased from ψ o = 0.0122 at T = 20 °C to ψ o = 0.0147 at T = 500 °C; and the time-dependent ratio of the interface slip (ψ s ) decreased from ψ s = 0.5194 at T = 20 °C to ψ s = 0.432 at T = 500 °C. • After oxidation at T = 1000 °C for t = 5 h, the experimental time-dependent composite vibration damping (ηc ) increased from ηc = 0.01524 at T = 20 °C to ηc = 0.0177 at T = 400 °C; the predicted time-dependent composite vibration damping (ηc ) increased from ηc = 0.01178 at T = 20 °C to ηc = 0.0202 at T =

5.3 Experimental Comparisons Fig. 5.7 a Experimental and predicted ηc versus T curves; b time-dependent ψ d versus T curves; c time-dependent ψ o versus T curves; and d time-dependent ψ s versus T curves of C/SiC composite after oxidation duration t = 5 h at T = 700, 1000, and 1300 °C

95

96

5 Time-Dependent Vibration Damping of Ceramic-Matrix Composites

Fig. 5.7 (continued)

Table 5.7 Experimental and predicted time-dependent ηc , ψ d , ψ o , and ψ s of C/SiC composite after oxidation duration t = 5 h at T = 700, 1000, and 1300 °C ηc (Experiment)

ηc (Theory)

ψd

ψo

ψs

20

0.01169

0.01423

0.5194

0.01228

0.5194

100

0.0141

0.01411

0.4907

0.013

0.4907

200

0.0162

0.01532

0.47238

0.0135

0.47238

300

0.0227

0.01709

0.45704

0.01396

0.45704

400

0.0230

0.0195

0.44368

0.01438

0.44368

500

—

0.02264

0.43202

0.01477

0.43202

20

0.01524

0.01178

0.49564

0.10994

0.49564

100

0.01551

0.01182

0.46942

0.11608

0.46942

200

0.01509

0.01303

0.45405

0.12001

0.45405

300

0.01652

0.01477

0.44142

0.12345

0.44142

400

0.01773

0.01714

0.43057

0.12656

0.43057

500

—

0.02027

0.42123

0.12937

0.42123

20

—

0.01458

0.56944

0.35824

0.56944

100

0.01766

0.01479

0.54588

0.3737

0.54588

200

0.01955

0.01642

0.53361

0.38229

0.53361

300

0.0198

0.01863

0.52383

0.38943

0.52383

400

0.0204

0.02148

0.51561

0.39564

0.51561

500

—

0.025

0.5087

0.40101

0.5087

T /(°C) T = 700 °C

T = 1000 °C

T = 1300 °C

5.3 Experimental Comparisons

97

500 °C; the time-dependent ratio of the interface debonding (ψ d ) decreased from ψ d = 0.4956 at T = 20 °C to ψ d = 0.4212 at T = 500 °C; the time-dependent ratio of the interface oxidation (ψ o ) increased from ψ o = 0.109 at T = 20 °C to ψ o = 0.129 at T = 500 °C; and the time-dependent ratio of the interface slip (ψ s ) decreased from ψ s = 0.495 at T = 20 °C to ψ s = 0.421 at T = 500 °C. • After oxidation at T = 1300 °C for t = 5 h, the experimental time-dependent composite vibration damping (ηc ) increased from ηc = 0.01766 at T = 100 °C to ηc = 0.0204 at T = 400 °C; the predicted time-dependent composite damping (ηc ) increased from ηc = 0.01458 at T = 20 °C to ηc = 0.025 at T = 500 °C; the time-dependent ratio of the interface debonding (ψ d ) decreased from ψ d = 0.569 at T = 20 °C to ψ d = 0.508 at T = 500 °C; the time-dependent ratio of the interface oxidation (ψ o ) increased from ψ o = 0.358 at T = 20 °C to ψ o = 0.401 at T = 500 °C; and the time-dependent ratio of the interface slip (ψ s ) decreased from ψ s = 0.569 at T = 20 °C to ψ s = 0.508 at T = 500 °C.

5.3.3 t = 10 h at T = 700, 1000, and 1300 °C 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 h at oxidation temperature of T = 700, 1000, and 1300 °C were shown in Fig. 5.8 and Table 5.8. When oxidation duration was t = 10 h, the 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 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 = 10 h, the time-dependent experimental composite damping (ηc ) increased from ηc = 0.01105 at T = 20 °C to ηc = 0.0196 at T = 400 °C; the predicted time-dependent composite vibration damping (ηc ) increased from ηc = 0.01453 at T = 20 °C to ηc = 0.02298 at T = 500 °C; the time-dependent ratio of interface debonding (ψd ) decreased from ψd = 0.5249 at T = 20 °C to ψd = 0.4376 at T = 500 °C; the time-dependent ratio of interface oxidation (ψo ) increased from ψo = 0.0243 at T = 20 °C to ψo = 0.0291 at T = 500 °C; and the ratio of time-dependent interface slip (ψs ) decreased from ψs = 0.5249 at T = 20 °C to ψs = 0.4376 at T = 500 °C. • After oxidation at T = 1000 °C for t = 10 h, the experimental time-dependent composite vibration damping (ηc ) increased from ηc = 0.0154 at T = 100 °C to ηc = 0.0164 at T = 400 °C; the predicted time-dependent composite vibration damping (ηc ) increased from ηc = 0.01118 at T = 20 °C to ηc = 0.0201 at T = 500 °C; the time-dependent ratio of interface debonding (ψd ) decreased from ψd = 0.4989 at T = 20 °C to ψd = 0.4337 at T = 500 °C; the time-dependent ratio of interface oxidation (ψo ) increased from ζ /l d = 0.2183 at T = 20 °C to ζ /ld =

98

5 Time-Dependent Vibration Damping of Ceramic-Matrix Composites

Fig. 5.8 a Experimental and predicted ηc versus T curves; b time-dependent ψ d versus T curves; c time-dependent ψ o versus T curves; and d time-dependent ψ s versus T curves of C/SiC composite after oxidation duration t = 10 h at T = 700, 1000, and 1300 °C

5.3 Experimental Comparisons

99

Fig. 5.8 (continued)

Table 5.8 Experimental and predicted time-dependent ηc , ψ d , ψ o , and ψ s of C/SiC composite after oxidation duration t = 10 h at T = 700, 1000, and 1300 °C ηc (Experiment)

ηc (Theory)

ψd

ψo

ψs

20

0.01105

0.01453

0.52492

0.0243

0.52492

100

0.0123

0.01441

0.49619

0.02571

0.49619

200

0.0140

0.01563

0.47791

0.02669

0.47791

300

0.0168

0.01743

0.4626

0.02758

0.4626

400

0.0196

0.01985

0.44928

0.0284

0.44928

500

—

0.02298

0.43765

0.02915

0.43765

20

—

0.01118

0.4989

0.21832

0.4989

100

0.0154

0.01131

0.47448

0.22956

0.47448

200

0.0154

0.01259

0.4612

0.23616

0.4612

300

0.0155

0.01441

0.45051

0.24177

0.45051

400

0.0164

0.01687

0.44144

0.24674

0.44144

500

—

0.0201

0.43374

0.25111

0.43374

20

0.0189

0.01585

0.633

0.63954

0.633

100

0.0183

0.01644

0.61428

0.65903

0.61428

200

0.0168

0.01869

0.60762

0.66625

0.60762

300

0.0189

0.02164

0.60303

0.67132

0.60303

400

0.0207

0.02535

0.59958

0.67518

0.59958

500

—

0.02985

0.59709

0.678

0.59709

T /(°C) T = 700 °C

T = 1000 °C

T = 1300 °C

100

5 Time-Dependent Vibration Damping of Ceramic-Matrix Composites

0.2511 at T = 500 °C; and the time-dependent ratio of interface slip decreased from ψo = 0.4989 at T = 20 °C to ψo = 0.4337 at T = 500 °C. • After oxidation at T = 1300 °C for t = 10 h, the experimental time-dependent composite vibration damping (ηc ) increased from ηc = 0.0189 at T = 20 °C to ηc = 0.0207 at T = 400 °C; the predicted time-dependent composite vibration damping (ηc ) increased from ηc = 0.01585 at T = 20 °C to ηc = 0.02985 at T = 500 °C; the time-dependent ratio of interface debonding (ψd ) decreased from ψd = 0.633 at T = 20 °C to ψd = 0.597 at T = 500 °C; the time-dependent ratio of interface oxidation (ψo ) increased from ψo = 0.639 at T = 20 °C to ψo = 0.678 at T = 500 °C; and the time-dependent ratio of interface slip (ψs ) decreased from ψs = 0.633 at T = 20 °C to ψs = 0.597 at T = 500 °C.

5.4 Discussions Due to 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 are time- and temperature-dependent. For 2D C/SiC, the time-dependent composite vibration damping increases with temperature, the interface debonding and slip ratio decrease with temperature, and the interface oxidation ratio increases with temperature. When the oxidation duration was short (i.e., t = 2 h), the time-dependent composite vibration damping decreased with temperature; however, when the oxidation duration was long (i.e., t = 5 and 10 h), 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 h) and decreased for long oxidation duration (i.e., t = 10 h). When the oxidation 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 Fig. 5.9.

5.5 Summary and Conclusion In this chapter, a time-dependent vibration damping model of fiber-reinforced CMCs was developed considering time-dependent interface damages. Relationships between composite vibration damping, material properties, internal damages, oxidation duration, and temperature were established. Effects of material properties, vibration stress, damage state, and oxidation temperature on time-dependent composite

5.5 Summary and Conclusion Fig. 5.9 Experimental and predicted time-dependent composite vibration damping versus temperature curves for a t = 2, 5, 10 h at T = 700 °C; b t = 2, 5, 10 h at T = 1000 °C; and, c t = 2, 5, 10 h at T = 1300 °C

101

102

5 Time-Dependent Vibration Damping of Ceramic-Matrix Composites

vibration damping of C/SiC composite were discussed. Experimental time-dependent composite vibration damping and internal damages of 2D C/SiC composite for different oxidation duration t = 2, 5, and 10 h at elevated temperature T = 700, 1000, and 1300 °C were predicted. • When the fiber volume increased, the time-dependent composite vibration damping decreased due to the decreases of frictional energy dissipated through interface slip. • When the vibration stress increased, the time-dependent composite vibration damping increased due to the increasing frictional dissipated energy caused by interface frictional slip. • When the matrix crack spacing increased, the time-dependent composite damping decreased due to the decrease of ratio of interface debonding and slip. However, for partial interface debonding, the ratio of interface oxidation was not affected by matrix crack spacing. • When the interface shear stress in the oxidation region increased, the timedependent composite damping increased due to the increase of frictional dissipated energy along the interface oxidation region.

References Birman V, Byrd LW (2002) Effect of matrix cracks on damping in unidirectional and cross-ply ceramic matrix composites. J Compos Mater 36:1859–1877 Birman V, Byrd LW (2003) Damping in ceramic matrix composites with matrix cracks. Int J Solids Struct 40:4239–4256 Chandra R, Singh SP, Gupta K (1994) Damping studies in fiber-reinforced composites—a review. Compos Struct 46:41–51 Gowayed Y, Ojard G, Santhosh U, Jefferson G (2015) Modeling of crack density in ceramic matrix composites. J Compos Mater 49:2285–2294 Holmes JW, Cho C (1992) Experimental observation of frictional heating in fiber-reinforced ceramics. J Am Ceram Soc 75:929–938 Hong ZL, Cheng LF, Zhao CN, Zhang LT, Wang YG (2013) Effect of oxidation on internal friction behavior of C/SC composites. Acta Materiae Compositae Sinica 30:93–100 Kyriazoglou C, Page BH, Guild FJ (2004) Vibration damping for crack detection in composite laminates. Compos Part A 35:945–953 Lamouroux F, Bertrand S, Pailler R, Naslain R, Cataldi M (1999) Oxidation-resistance carbonfiber-reinforced ceramic-matrix composites. Compos Sci Technol 59:1073–1085 Li LB (2014) Modeling fatigue hysteresis behavior of unidirectional C/SiC ceramic-matrix composites. Compos Part B 66:466–474 Li LB (2019a) Modeling matrix multicracking development of fiber-reinforced ceramic-matrix composites considering fiber debonding. Int J Appl Ceram Technol 16:97–107 Li LB (2019b) Time-dependent damage and fracture of fiber-reinforced ceramic-matrix composites at elevated temperatures. Compos Interf 26:963–988 Li H (2019c) Analysis of vibration and damage test of ceramic matrix composites structure. Master thesis, Nanchang Hongkong University, Nanchang, China. Li LB (2019d) Time-dependent proportional limit stress of carbon fiber-reinforced silicon carbide ceramic-matrix composites considering interface oxidation. J Ceram Soc Japan 127:279–287

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Li LB (2020a) Durability of ceramic-matrix composites. Woodhead Publishing, Amsterdam Li LB (2020b) A time-dependent vibration damping model of fiber-reinforced ceramic-matrix composites at elevated temperature. Ceram Int 46:27031–27045 Li LB (2020c) Modeling temperature-dependent vibration damping in C/SiC fiber-reinforced ceramic-matrix composites. Materials 13:1633 Li LB (2020d) Synergistic effects of stochastic loading stress and time on stress-rupture damage evolution and lifetime of fiber-reinforced ceramic-matrix composites at intermediate temperatures. Ceram Int 46:7792–7812 Li LB (2020e) Time-dependent matrix fracture of carbon fiber-reinforced silicon carbide ceramicmatrix composites considering interface oxidation. Compos Interfaces 27:551–567 Li LB (2020f) Time-dependent mechanical behavior of ceramic-matrix composites at elevated temperatures. Springer Nature Singapore Pte Ltd., Singapore Li X, Fan Y, Zhao X, Ma R, Du A, Cao X, Ban H (2019) Damping capacity and storage modulus of SiC matrix composite infiltrated by AlSi alloy. Metals 9:1195 Melo JD, Radford DW (2005) Time and temperature dependence of the viscoelastic properties of CFRP by dynamic mechanical analysis. Compos Struct 70:240–253 Min JB, Harris DL, Ting JM (2011) Advances in ceramic matrix composite blade damping characteristics for aerospace turbomachinery applications. In: 52nd AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics and materials conference, 4–7 April 2011, Denver, Colorado Misra D, Nemane V, Mukhopahyay S, Chatterjee S (2020) Effect of hBN and SiC addition on laser assisted processing of ceramic matrix composite coatings. Ceram Int 46:9758–9764 Momon S, Moevus M, Godin N, R’Mili M, Reynaud P, Fantozzi G, Fayolle G (2010) Acoustic emission and lifetime prediction during static fatigue tests on ceramic-matrix-composite at high temperature under air. Compos Part A 41:913–918 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 Patel RK, Bhattacharya B, Basu S (2007) A finite element based investigation on obtaining high material damping over a larger frequency range in viscoelastic composites. J. Sound Vibration 303:753–766 Pradere C, Sauder C (2008) Transverse and longitudinal coefficient of thermal expansion of carbon fibers at high temperatures (300–2500K). Carbon 46:1874–1884 Reynaud P, Douby D, 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 Sato S, Serizawa H, Araki H, Node T, Kohyama A (2003) Temperature dependence of internal friction and elastic modulus of SiC/SiC composites. J Alloys Compounds 355:142–147 Sauder C, Lamon J, Pailler R (2004) The tensile behavior of carbon fibers at high temperatures up to 2400 °C. Carbon 42:715–725 Snead LL, Nozawa T, Katoh Y, Byun TS, Kondo S, Petti DA (2007) Handbook of SiC properties for fuel performance modeling. J Nucl Mater 371:329–377 Wang W (2005) Study on damping capacity of C/SiC composites. Master Thesis, Northwestern Polytechnical University, Xi’an, China Wang RZ, Li WG, Li DY, Fang DN (2015) A new temperature dependent fracture strength model for the ZrB2 -SiC composites. J Eur Ceram Soc 35:2957–2962 Wing BL, Halloran JW (2018) Subsurface oxidation of boron nitride coatings on silicon carbide fibers in SiC/SiC ceramic matrix composites. Ceram Int 44:17499–17505 Zhang Z, Hartwig G (2002) Relation of damping and fatigue damage of unidirectional fiber composites. Int J Fatigue 24:713–718 Zhang J, Perez RJ, Lavernia EJ (1993) Documentation of damping capacity of metallic ceramic and metal-matrix composite materials. J Mater Sci 28:2395–2404 Zhang Q, Cheng LF, Wang W, Wei X, Zhang LT, Xu YD (2007) Effect of interphase thickness on damping behavior of 2D C/SiC composites. Mater Sci Forum 546–549:1531–1534 Zhang Q, Cheng L, Wang W, Zhang L, Xu Y (2008) Effect of SiC coating and heat treatment on damping behavior of C/SiC composites. Mater Sci Eng A 473:254–258

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Zhang Q, Cheng L, Wang F, Zhang L (2014) Effect of oxidation on the damping behavior of 2D C/SiC composites. New Carbon Mater 29:522–528

Chapter 6

Cyclic-Dependent Vibration Damping of Ceramic-Matrix Composites

Abstract Under cyclic fatigue loading, cyclic-dependent damage mechanisms affected the vibration damping of fiber-reinforced ceramic-matrix composites (CMCs). In this chapter, a cyclic-dependent vibration damping model of fiberreinforced CMCs was developed. Combining cyclic-dependent damage mechanisms, damage models, and dissipated energy model, relationships between composite vibration damping, cyclic-dependent damage mechanisms, vibration stress, and applied cycle number were established. Effects of material properties and damage state on composite vibration damping were analyzed for different applied cycle number and vibration stress. Experimental composite vibration damping of 2D and 3D C/SiC composites without/with coating was predicted for different vibration frequencies and applied cycle number. Keywords Ceramic-matrix composites (CMCs) · Damping · Cyclic-dependent · Matrix cracking · Interface debonding · Interface wear

6.1 Introduction Ceramic-matrix composites (CMCs) possess high specific strength, high specific modulus, and high-temperature resistance and are the candidate materials for hotsection components of aerospace vehicles and thermal protection system (TPS) (Naslain 2004; Li 2018a, 2019a, 2020). However, vibration and noise cause structural fatigue damage and fracture of CMC components (Min et al. 2011). Damping is a key parameter to reflect the vibration behavior of materials. Due to complex damage mechanisms, damping of CMCs is much different from metals or alloys (Zhang et al. 1993). Fabrication method, frequency, thermal treatment, and coating affect the damping behavior of fiber-reinforced CMCs (Sato et al. 2003; Wang et al. 2006). For CMCs with cracks in the matrix, Birman and Byrd (2002) developed a micromechanical approach to analyze the damping of unidirectional and cross-ply CMCs and related the composite damping with energy dissipation through interface friction between the fiber and the matrix (Birman and Byrd 2002). The internal friction of CMCs was found to be largely affected by the property of the fiber, the microstructure © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 L. Li, Vibration Behavior in Ceramic-Matrix Composites, Advanced Ceramics and Composites 5, https://doi.org/10.1007/978-981-19-7838-8_6

105

106

6 Cyclic-Dependent Vibration Damping of Ceramic-Matrix Composites

of the matrix (Sato et al. 2003), microcracking propagation (Wang et al. 2006), interphase thickness (Zhang et al. 2007), temperature (Zhang et al. 2007), oxidation (Hong et al. 2013), and natural frequency (Li 2019b). Under cyclic fatigue loading, cyclic-dependent damage mechanisms appear (Li 2018b, 2018c, 2019c, 2021). For SiC/CAS composite, initial fiber/matrix interface shear stress was approximately τ o = 10–12 MPa, cyclically steady-state interface shear stress was approximately τ s = 5–7 MPa, the ratio between initial and steady-state interface shear stress was τ s /τ o ≈ 1/2; and for SiC/MAS composite, initial fiber/matrix interface shear stress was approximately τ o = 20–25 MPa, after N = 40,000 cycles, the interfacial shear stress was approximately τ s = 7–8 MPa, and ratio of the interface shear stress was τ s /τ o ≈ 1/3 (McNulty and Zok 1999). Under cyclic fatigue loading, the interface shear stress degraded with increasing applied cycle number. The interface debonding length and interface slip range increased with applied cycles. The objective of this chapter was to develop a cyclic-dependent vibration damping model of fiber-reinforced CMCs. Relationships between composite vibration damping, cyclic-dependent damage mechanisms, vibration stress, and applied cycle number were established. Effects of material properties and damage state on composite vibration damping were analyzed for different applied cycle number and vibration stress. Experimental composite vibration damping of 2D and 3D C/SiC composites without/with coating was predicted for different vibration frequencies and applied cycle number.

6.2 Cyclic-Dependent Micromechanical Vibration Damping Models Damping mechanism of fiber-reinforced CMCs is much different from that of traditional metal and alloy materials (Chandra et al. 1999; Birman and Byrd 2003; Melo and Radford 2005). Its damping mechanism mainly includes three aspects as follows: (1) Damping of matrix and fiber itself. (2) Interface damping caused by friction slip between the fiber and the matrix under external mechanical vibration when the interface debonding occurs. (3) Damping caused by energy dissipation such as matrix cracking and fiber fracture.

6.2 Cyclic-Dependent Micromechanical Vibration Damping Models

107

6.2.1 Cyclic-Dependent Damping in Intact CMCs For CMCs without damage, the composite vibration damping is given by Eq. (6.1). (Birman and Byrd 2002) ηa =

E f Vf ηf + E m Vm ηm E f Vf + E m Vm

(6.1)

where V f and V m are the volume of the fiber and the matrix, E f and E m are the elastic modulus of the fiber and the matrix, and ηf and ηm are the vibration damping of the fiber and matrix materials, respectively. When fatigue damage mechanisms occur in CMCs, the effective matrix elastic modulus is determined by Eq. (6.2). E m (N ) =

τi (N ) Vm

1 τi (N ) Ec

+

rf Δσ 4lc E f

(

Vm E m Vf E c

)2 −

Vf E f Vm

(6.2)

where r f is the fiber radius, E c is the longitudinal modulus of the intact composite material, lc is the matrix crack spacing, Δσ is the range of vibration stress, and N is the cycle number.

6.2.2 Cyclic-Dependent Damping in Damaged CMCs For CMCs with fatigue damages, the cyclic-dependent composite vibration damping is given by Eq. (6.3) (Birman and Byrd 2002) ηb =

Ud (N ) 2πU (N )

(6.3)

where U d (N) is a cyclic-dependent energy dissipated density per cycle of motion, and U(N) is the cyclic-dependent maximum strain energy density during the cycle. Ud (N ) = Ud_ unloading (N ) + Ud_ reloading (N )

(6.4)

U (N ) = Uf (N ) + Um (N )

(6.5)

where ( ) Δσ 2 Ec τi (N ) 3 8 Ud_ unloading (N ) = 2πrf τi (N ) l (N ) − l (N ) Vf E f y 3 Vm E f E m rf y

(6.6)

108

6 Cyclic-Dependent Vibration Damping of Ceramic-Matrix Composites

) Δσ 2 Ec τi (N ) 3 8 (6.7) lz (N ) − lz (N ) Vf E f 3 Vm E f E m rf ) { 2 ( σ σ τi (N ) 2 4 τi2 (N ) 3 σfo2 lc 2 − ld (N ) Uf (N ) = πrf ld (N ) − 2 l (N ) + l (N ) + rf Vf E f d 3 rf2 E f d Ef 2 Vf2 E f )( ( ( )) lc /2 − ld (N ) 2rf σfo Vm ld (N ) + σmo − 2 τi (N ) 1 − exp −ρ ρ Ef Vf rf rf )2 ( ( ( ))} rf Vm ld (N ) lc /2 − ld (N ) + σmo − 2 τi (N ) 1 − exp −2ρ 2ρ E f Vf rf rf (6.8) ) { ( 2 4 Vf2 τi2 (N ) 3 lc σmo Um (N ) = πrf2 l + − ld (N ) (N ) d 2 2 3 rf Vm E m Em 2 ( )[ [ ]] ρ(lc /2 − ld (N )) 2rf σmo Vf ld (N ) σmo − 2τi (N ) 1 − exp − − ρ Em Vm rf rf ( )2 [ [ ]]} rf Vf ld (N ) ρ(lc /2 − ld (N )) + σmo − 2τi (N ) 1 − exp −2 2ρ E m Vm rf rf (6.9) (

Ud_ reloading (N ) = 2πrf τi (N )

where τ i (N) is the cyclic-dependent interface shear stress (Evans et al. 1995) [ ( )] τi (N ) = τio + 1 − exp −α N β (τi min − τio )

(6.10)

where τ io is the initial interface shear stress, τi min is the steady-state interface shear stress, and α and β are empirical constants. For unidirectional SiC/CAS composite under σ max = 280 MPa at room temperature, the interface shear stress degraded from τ io = 22 MPa to τi min = 5 MPa at the first 100 applied cycles, as shown in Fig. 6.1. l d (N) is the cyclic-dependent interface debonding length, ly (N) is the cyclicdependent counter slip length upon unloading, and lz (N) is the cyclic-dependent new slip length upon reloading. ( ) /( )2 rf Vm E m σ 1 rf Vm E m E f rf − − ┌i ld (σ, N ) = + 2 Vf E c τi (N ) ρ 2ρ E c τi2 (N ) 1 {ld (σmax , N ) 2⎡ ⎤⎫ ( ) /( )2 ⎬ r r V 1 E σ V E E r f f m m f m m f ⎦ − −⎣ − ┌ + i ⎭ 2 Vf E c τi (N ) ρ 2ρ E c τi2 (N )

(6.11)

ly (σ, N ) =

(6.12)

6.3 Results and Discussion

109

Fig. 6.1 Experimental and predicted interface shear stress versus applied cycle number curves of unidirectional SiC/CAS composite

1 lz (σ, N ) = ly (σmin , N ) − {ld (σmax , N ) 2 ⎡ ⎤⎫ ( ) /( )2 ⎬ r r V E σ V E E 1 r m m f m m f f f ⎦ − −⎣ − + ┌ i ⎭ 2 Vf E c τi (N ) ρ 2ρ E c τi2 (N )

(6.13)

where σ max is the vibration peak stress, σ min is the vibration valley stress, and ┌ i is the interface debonding energy. It should be noted that in Eqs. (6.12) and (6.13), the interface debonding length ld is obtained by Eq. (6.11) with σ = σ max , and in Eq. (6.13), the interface counter slip length ly is determined by Eq. (6.12) with σ = σ min . σ fo and σ mo are the fiber and matrix axial stress in the interface bonding region. σfo = σmo =

Ef σ + E f (αc − αf )ΔT Ec

(6.14)

Em σ + E m (αc − αm )ΔT Ec

(6.15)

where α f , α m , and α c are the fiber, matrix, and composite thermal expansion coefficient, respectively, and ΔT is the temperature difference between testing and fabricated temperature.

6.3 Results and Discussion The cyclic-dependent composite vibration damping ηc of CMCs with fatigue damages can be determined by Eq. (6.16).

110

6 Cyclic-Dependent Vibration Damping of Ceramic-Matrix Composites

ηc = ηa + ηb

(6.16)

Effect of material properties and damage state on cyclic-dependent vibration damping of SiC/CAS composite were analyzed for different applied cycle number and vibration stress. Pryce and Smith (Pryce and Smith 1993) performed experimental investigations on tensile and fatigue behavior of unidirectional SiC/CAS composite. The first matrix cracking stress of SiC/CAS composite was approximately σ mc = 85 MPa, and the composite fatigue limit stress was around the matrix cracking stress. In the present analysis, the vibration amplitude stress between σ = zero and 100 MPa was used for vibration damping and interface debonding and slip analysis.

6.3.1 Effect of Fiber Volume on Cyclic-Dependent Vibration Damping of CMCs Cyclic-dependent composite vibration damping (ηc ), ratio of interface debonding (ψ d = 2l d /l c ), and slip (ψ s ) versus vibration stress amplitude curves of unidirectional SiC/CAS composite for different fiber volume were shown in Fig. 6.2. When the fiber volume increased, the stress transfer between the fiber and the matrix increased, the interface debonding length and interface slip length decreased, and the cyclicdependent composite vibration damping decreased, due to the decrease of cyclicdependent ratio of interface debonding and interface slip. With increasing applied cycle number, the interface shear stress between the fiber and the matrix decreased, leading to the decrease of load transfer capacity between the fiber and the matrix, and the interface debonding and slip length increased, and the cyclic-dependent composite vibration damping increased, due to the increase of cyclic-dependent ratio of interface debonding and slip. In the present analysis, two values of fiber volume V f = 30 and 35% were used for the analysis of damping, interface debonding, and slip. However, the evolution of composite damping, interface debonding, and slip ratio at higher fiber volume were similar with the present analysis. The composite damping, interface debonding, and slip ratio were very sensitivity to the fiber volume and fatigue cycle. During application of CMC components, high damping level is required, corresponding to low fiber volume. However, the composite proportional limit stress, fatigue limit stress, and tensile strength decrease with decreasing fiber volume. It is necessary to perform optimization analysis for high damping and high mechanical behavior of CMCs during applications. • When the fiber volume is V f = 30% at N = 1, the cyclic-dependent composite damping ηc increased from ηc = 0.00145 at σ = zero MPa to ηc = 0.01297 at σ = 100 MPa; the interface debonding ratio ψ d increased from ψ d = zero at σ = zero MPa to ψ d = 0.3326 at σ = 100 MPa; and the interface slip ratio (ψ s = 2ly /l c ) increases from ψ s = zero at σ = zero MPa to ψ s = 0.1887 at σ = 100 MPa.

6.3 Results and Discussion Fig. 6.2 a Cyclic-dependent ηc versus σ curves; b cyclic-dependent ψ d versus σ curves; c cyclic-dependent ψ s versus σ curves of unidirectional SiC/CAS composite corresponding to N = 1, 102 , and 103 when V f = 30 and 35%

111

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6 Cyclic-Dependent Vibration Damping of Ceramic-Matrix Composites

When the cycle number is N = 102 , the cyclic-dependent composite damping increases from ηc = 0.00145 at σ = zero to ηc = 0.01322 at σ = 100 MPa; the interface debonding ratio ψ d increases from ψ d = zero at σ = zero MPa to ψ d = 0.3465 at σ = 100 MPa; and the interface slip ratio increases from ψ s = zero at σ = zero MPa to ψ s = 0.1961 at σ = 100 MPa. When the cycle number is N = 103 , the cyclic-dependent composite damping increased from ηc = 0.00145 at σ = zero MPa to ηc = 0.01796 at σ = 100 MPa; the interface debonding ratio ψ d increased from ψ d = zero at σ = zero MPa to ψ d = 0.7268 at σ = 100 MPa; and the interface slip ratio ψ s increased from ψ s = zero at σ = zero MPa to ψ s = 0.3983 at σ = 100 MPa. • When the fiber volume is V f = 35% at N = 1, the cyclic-dependent composite damping increased from ηc = 0.00153 at σ = zero MPa to ηc = 0.00886 at σ = 100 MPa; the interface debonding ratio ψ d increased from ψ d = zero at σ = zero MPa to ψ d = 0.2483 at σ = 100 MPa; and the interface slip ratio increased from ψ s = zero at σ = zero MPa to ψ s = 0.144 at σ = 100 MPa. When N = 102 , the cyclic-dependent composite damping increased from ηc = 0.00153 at σ = zero MPa to ηc = 0.00907 at σ = 100 MPa; the interface debonding ratio ψ d increased from ψ d = zero at σ = zero MPa to ψ d = 0.2587 at σ = 100 MPa; and the interface slip ratio increased from ψ s = 0 at σ = zero MPa to ψ s = 0.15 at σ = 100 MPa. When N = 103 , the cyclic-dependent composite damping increased from ηc = 0.00153 at σ = zero MPa to ηc = 0.01366 at σ = 100 MPa; the interface debonding ratio ψ d increases from ψ d = zero at σ = zero MPa to ψ d = 0.5451 at σ = 100 MPa; and the interface slip ratio ψ s increased from ψ s = zero at σ = zero MPa to ψ s = 0.3 at σ = 100 MPa.

6.3.2 Effect of Matrix Crack Spacing on Cyclic-Dependent Vibration Damping of CMCs Cyclic-dependent composite vibration damping (ηc ), ratio of interface debonding (ψ d = 2ld /l c ), and slip (ψ s ) versus vibration stress amplitude curves of unidirectional SiC/CAS composite for different matrix crack spacing were shown in Fig. 6.3. When the matrix crack spacing increased, the ratio between the interface debonding or slip length and matrix crack spacing decreased, the fraction of frictional dissipated energy in the total energy dissipated decreased, and the composite vibration damping decreased, due to the decrease of the cyclic-dependent ratio of interface debonding and slip. With increasing applied cycle number, the interface debonding and slip length increased, and the cyclic-dependent composite vibration damping increased, due to the increase of cyclic-dependent ratio of interface debonding and slip. In the present analysis, two values of matrix crack spacing lc = 200 and 300 µm were used for the analysis of damping, interface debonding, and slip. However, the evolution of composite damping, interface debonding, and slip fraction at higher matrix crack spacing were similar with the present analysis. The composite damping, interface debonding, and slip fraction were very sensitivity to the matrix crack spacing

6.3 Results and Discussion Fig. 6.3 a Cyclic-dependent ηc versus σ curves; b cyclic-dependent ψ d versus σ curves; c cyclic-dependent ψ s versus σ curves of unidirectional SiC/CAS composite corresponding to N = 1, 100, and 1000 when l c = 200 and 400 µm

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and fatigue cycle. During application of CMC components, high damping level was required, corresponding to short matrix crack spacing or high matrix cracking density. • When l c = 200 µm at N = 1, the cyclic-dependent composite damping ηc increased from ηc = 0.00145 at σ = zero MPa to ηc = 0.01581 at σ = 100 MPa; the interface debonding ratio ψ d increased from ψ d = zero at σ = zero MPa to ψ d = 0.499 at σ = 100 MPa; and the interface slip ratio ψ s increased from ψ s = zero at σ = zero MPa to ψ s = 0.283 at σ = 100 MPa. When N = 102 , the cyclic-dependent composite damping ηc increased from ηc = 0.00145 at σ = zero MPa to ηc = 0.01607 at σ = 100 MPa; the interface debonding ratio ψ d increased from ψ d = zero at σ = zero MPa to ψ d = 0.519 at σ = 100 MPa; and the interface slip ratio ψ s increased from ψ s = zero at σ = zero MPa to ψ s = 0.294 at σ = 100 MPa. When N = 103 , the cyclic-dependent composite damping ηc increased from ηc = 0.00145 at σ = zero MPa to ηc = 0.0215 at σ = 100 MPa; the interface debonding ratio ψ d increased from ψ d = zero at σ = zero MPa to ψ d = 1 at σ = 100 MPa; and the interface slip ratio ψ s increased from ψ s = zero at σ = zero MPa to ψ s = 0.597 at σ = 100 MPa. • When l c = 400 µm at N = 1, the cyclic-dependent composite damping ηc increased from ηc = 0.00145 at σ = zero MPa to ηc = 0.01106 at σ = 100 MPa; the interface debonding ratio ψ d increased from ψ d = zero at σ = zero MPa to ψ d = 0.249 at σ = 100 MPa; and the interface slip ψ s ratio increased from ψ s = zero at σ = zero MPa to ψ s = 0.141 at σ = 100 MPa. When N = 102 , the cyclic-dependent composite damping ηc increased from ηc = 0.00145 at σ = zero MPa to ηc = 0.0113 at σ = 100 MPa; the interface debonding ratio ψ d increased from ψ d = zero at σ = zero MPa to ψ d = 0.259 at σ = 100 MPa; and the interface slip ratio ψ s increased from ψ s = zero at σ = zero MPa to ψ s = 0.147 at σ = 100 MPa. When N = 103 , the cyclic-dependent composite damping ηc increased from ηc = 0.00145 at σ = zero MPa to ηc = 0.016 at σ = 100 MPa; the interface debonding ratio ψ d increased from ψ d = zero at σ = zero MPa to ψ d = 0.545 at σ = 100 MPa; and the interface slip ratio ψ s increased from ψ s = zero at σ = zero MPa to ψ s = 0.298 at σ = 100 MPa.

6.3.3 Effect of Interface Debonding Energy on Cyclic-Dependent Vibration Damping of CMCs Cyclic-dependent composite vibration damping (ηc ), ratio of interface debonding (ψ d ), and slip (ψ s ) versus the vibration stress amplitude (σ ) curves of unidirectional SiC/CAS composite for different interface debonding energy were shown in Fig. 6.4. When the interface debonding energy increased, the cyclic-dependent composite vibration damping decreased first and then increased, due to the decrease of cyclicdependent interface debonding length, and the first decrease and then increase of interface slip length. With increasing applied cycle number, the interface debonding and slip length increased, and the cyclic-dependent composite damping increased,

6.4 Experimental Comparisons

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due to the increase of cyclic-dependent ratio of interface debonding and interface slip. In the present analysis, two values of interface debonding energy ┌ i = 0.01 and 0.03 J/m2 were used for the analysis of damping, interface debonding, and slip. However, the evolution of composite damping, interface debonding, and slip fraction at higher interface debonding energy were similar with the present analysis. The composite damping, interface debonding, and slip fraction were low sensitivity to the interface debonding energy. During application of CMC components, high damping level was required, corresponding to high interface debonding energy. • When ┌ i = 0.01 J/m2 at N = 1, the cyclic-dependent composite damping ηc increased from ηc = 0.00153 at σ = zero MPa to ηc = 0.00886 at σ = 100 MPa; the interface debonding ratio ψ d increases from ψ d = zero at σ = zero MPa to ψ d = 0.248 at σ = 100 MPa; and the interface slip ratio ψ s increased from ψ s = zero at σ = zero MPa to ψ s = 0.144 at σ = 100 MPa. When N = 102 , the cyclic-dependent composite damping ηc increased from ηc = 0.00153 at σ = zero MPa to ηc = 0.00907 at σ = 100 MPa; the interface debonding ratio ψ d increased from ψ d = zero at σ = zero MPa to ψ d = 0.258 at σ = 100 MPa; and the interface slip ratio ψ s increased from ψ s = zero at σ = zero MPa to ψ s = 0.15 at σ = 100 MPa. When N = 103 , the cyclic-dependent composite damping ηc increased from ηc = 0.00153 at σ = zero MPa to ηc = 0.01366 at σ = 100 MPa; the interface debonding ratio ψ d increased from ψ d = zero at σ = zero MPa to ψ d = 0.545 at σ = 100 MPa; and the interface slip ratio ψ s increased from ψ s = zero at σ = zero MPa to ψ s = 0.3 at σ = 100 MPa. • When ┌ i = 0.03 J/m2 at N = 1, the cyclic-dependent composite damping ηc increased from ηc = 0.00153 at σ = zero MPa to ηc = 0.00894 at σ = 100 MPa; the interface debonding ratio ψ d increased from ψ d = zero at σ = zero MPa to ψ d = 0.232 at σ = 100 MPa; and the interface slip ratio ψ s increased from ψ s = zero at σ = zero MPa to ψ s = 0.144 at σ = 100 MPa. When N = 102 , the cyclic-dependent composite damping ηc increased from ηc = 0.00153 at σ = zero MPa to ηc = 0.00916 at σ = 100 MPa; the interface debonding ratio ψ d increased from ψ d = zero at σ = zero MPa to ψ d = 0.242 at σ = 100 MPa; and the interface slip ratio ψ s increased from ψ s = zero at σ = zero MPa to ψ s = 0.15 at σ = 100 MPa. When N = 103 , the cyclic-dependent composite damping ηc increases from ηc = 0.00153 at σ = zero MPa to ηc = 0.01392 at σ = 100 MPa; the interface debonding ratio ψ d increased from ψ d = zero at σ = zero MPa to ψ d = 0.51 at σ = 100 MPa; and the interface slip ratio ψ s increased from ψ s = zero at σ = zero MPa to ψ s = 0.3 at σ = 100 MPa.

6.4 Experimental Comparisons Wang et al. (2006) investigated the vibration damping behavior of 2D and 3D CVI T300™ C/SiC composites with and without CVD-SiC coating at different loading frequencies. The dimension of the testing specimen is 3 mm × 12 mm × 50 mm.

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Fig. 6.4 a Cyclic-dependent ηc versus σ curves; b cyclic-dependent ψ d versus σ curves; c cyclic-dependent ψ s versus σ curves of unidirectional SiC/CAS composite corresponding to N = 1, 100, and 1000 when ┌ i = 0.01 and 0.03 J/m2

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The Dynamical Mechanical Analyzer 2980 (TA Instruments, USA) was used for vibration damping testing. Three vibration frequencies are considered for damping testing including f = 1, 2, and 5 Hz. The fiber volume is in the range of 40–45%. The 2D and 3D preforms were infiltrated with both pyrolysis carbon (PyC) as interphase.

6.4.1 2D C/SiC Composite Experimental and predicted cyclic-dependent composite vibration damping, ratio of interface debonding, and slip versus the vibration stress amplitude curves of 2D C/SiC composite without/with CVD-SiC coating for different vibration frequencies and applied cycle number were shown in Figs. 6.5 and 6.6. For 2D C/SiC without SiC coating, experimental composite vibration damping ηc was ηc = 0.01057, 0.00927, and 0.00791 at f = 1, 2, and 5 Hz corresponding to σ = 30 MPa, as shown in Fig. 6.5a; theoretical results of cyclic-dependent composite vibration damping, interface debonding, and slip fraction for cycle number N = 1, 102 , and 103 were shown in Fig. 6.5a–c. • When N = 1, cyclic-dependent composite vibration damping ηc was ηc = 0.00701 with ψ d = 0.64 and ψ s = 0.409. • When N = 102 , cyclic-dependent composite vibration damping ηc was ηc = 0.00757 with ψ d = 0.674 and ψ s = 0.425. • When N = 500, cyclic-dependent composite vibration damping ηc was ηc = 0.0185 with ψ d = 0.873 and ψ s = 0.547. For 2D C/SiC with SiC coating, experimental composite vibration damping ηc was ηc = 0.00871, 0.00766, and 0.00625 at f = 1, 2, and 5 Hz corresponding to σ = 30 MPa, as shown in Fig. 6.6a; theoretical results of cyclic-dependent composite vibration damping, interface debonding, and slip fraction for cycle number N = 1, 300, and 500 were shown in Fig. 6.6a–c. • When N = 1, cyclic-dependent composite vibration damping ηc was ηc = 0.00528 with ψ d = 0.54 and ψ s = 0.34. • When N = 300, cyclic-dependent composite vibration damping ηc was ηc = 0.0068 with ψ d = 0.639 and ψ s = 0.403. • When N = 500, the cyclic-dependent composite vibration damping ηc was ηc = 0.0095 with ψ d = 0.74 and ψ s = 0.46.

6.4.2 3D C/SiC Composite Experimental and predicted cyclic-dependent composite vibration damping, ratio of interface debonding, and slip versus the vibration stress amplitude curves of 3D C/SiC composite without/with CVD-SiC coating for different vibration frequencies and applied cycles were shown in Figs. 6.7 and 6.8.

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Fig. 6.5 a Experimental and predicted ηc versus σ curves; b theoretical ψ d versus σ curves; c theoretical ψ s versus σ curves of 2D C/SiC composite without coating for different vibration frequencies and applied cycle number

6.4 Experimental Comparisons Fig. 6.6 a Experimental and predicted ηc versus σ curves; b theoretical ψ d versus σ curves; c theoretical ψ s versus σ curves of 2D C/SiC composite with coating for different vibration frequencies and applied cycle number

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Fig. 6.7 a Experimental and predicted ηc versus σ curves; b theoretical ψ d versus σ curves; c theoretical ψ s versus σ curves of 3D C/SiC composite without coating for different vibration frequencies and applied cycle number

6.4 Experimental Comparisons Fig. 6.8 a Experimental and predicted ηc versus σ curves; b theoretical ψ d versus σ curves; c theoretical ψ s versus σ curves of 3D C/SiC composite with coating for different vibration frequencies and applied cycle number

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For 3D C/SiC without SiC coating, experimental composite vibration damping ηc was ηc = 0.01069, 0.00983, and 0.00845 at f = 1, 2, and 5 Hz corresponding to σ = 30 MPa; theoretical results of cyclic-dependent composite vibration damping, interface debonding, and slip fraction for cycle number N = 1 and 500 were shown in Fig. 6.7a–c. • When N = 1, the cyclic-dependent composite vibration damping ηc was ηc = 0.0053 with ψ d = 0.7 and ψ s = 0.58. • When N = 500, the cyclic-dependent composite vibration damping ηc was ηc = 0.022 with ψ d = 0.92 and ψ s = 0.76. For 3D C/SiC with SiC coating, experimental composite vibration damping was ηc = 0.00885, 0.00807, and 0.00664 at f = 1, 2, and 5 Hz corresponding to σ = 30 MPa, as shown in Fig. 6.8a; theoretical results of cyclic-dependent composite vibration damping, interface debonding, and slip fraction for cycle number N = 1 and 500 were shown in Fig. 6.8a–c. • When N = 1, the cyclic-dependent composite vibration damping ηc was ηc = 0.008 with ψ d = 0.82 and ψ s = 0.67. • When N = 500, the cyclic-dependent composite vibration damping ηc was ηc = 0.042 with ψ d = 1 and ψ s = 0.88.

6.5 Discussions Figure 6.9 shows theoretical composite vibration damping versus amplitude of stress curves of 2D and 3D C/SiC composites without and with coating. For C/SiC with coating, the composite vibration damping was less than that of C/SiC without coating, as shown in Fig. 6.9, as the coating improved the surface density of C/SiC (Wang et al. 2006) and decreased the fiber volume inside of CMCs. In the present analysis, the effect of coating on the composite vibration damping was considered in the micromechanical vibration damping model by decreasing the fiber volume. For 2D and 3D fiber architecture, the effective coefficient of fiber volume along the loading direction (ECFL) was considered in the micromechanical damping model. For 2D C/SiC composite, ECFL is 0.5, and for 3D C/SiC composite, ECFL is 0.93. Substituting the ECFL in the micromechanical damping model, the effect of fiber architecture on composite vibration damping is considered. In the present analysis, experimental composite vibration damping data of 2D and 3D C/SiC composite all correspond to single vibration amplitude of stress σ = 30 MPa. When the vibration stress increases, damages in the composite of matrix cracking and interface debonding have already considered in the micromechanical vibration damping models. The author will perform more experimental tests in future to obtain more results for different vibration stresses.

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Fig. 6.9 Theoretical composite vibration damping versus amplitude of vibration stress curves of 2D and 3D C/SiC composites without and with coating

6.6 Summary and Conclusion In this chapter, a cyclic-dependent vibration damping model of fiber-reinforced CMCs was developed. Relationships between composite vibration damping, cyclicdependent damage mechanisms, vibration stress, and applied cycle number were established. Effects of material properties and damage state on composite vibration damping were analyzed for different applied cycle number and vibration stress. Experimental vibration damping of 2D and 3D C/SiC composites without/with coating was predicted for different vibration frequencies and applied cycle number.

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