1 Introduction

Ceramic-matrix composites (CMCs) possess high strength-to-weight ratio at elevated temperatures, and are being designed in the hot section components of commercial aero engine, i.e., CFM 56 − 5B and LEAP turbofan engine (CFM International, Cincinnati, OH USA) [1]. The Civil Aviation Administration of China (CAAC) issues and enforces regulations and minimum standards covering the safe manufacture, operation and maintenance of civil aircraft. As new materials, these ceramic composite components will have to meet the certification regulations of the CAAC for airworthiness. The CMC producer and user will have to show the CAAC the component is well designed for operation, safety and durability, and the failure risk is within the accepted level. However, during cyclic loading at elevated temperatures, the composite strength degrade with flight hours or cycles, which increases the failure risk level.

Under cyclic fatigue loading, the material performance of CMCs degrade with applied cycles due to damage mechanisms of matrix cracking, interface debonding, interface wear and fibers fracture. Lee et al. [2] investigated the tension − tension fatigue behavior of 2D SiC/[Si-N-C] composite at room and elevated temperatures. At room temperature, the fatigue limit was about 75% tensile strength; and at 1000 °C in air, the fatigue run-out stress level of 110 MPa was 50 MPa lower than that at room temperature, however, 35 MPa higher above the proportional limit. The area of the stress − strain loops was calculated to generate hysteresis energy density (HED) values. The fatigue hysteresis energy density decreases with applied cycles. Bertrand et al. [3] investigated the tension − tension fatigue behavior of Sylramic-iBN/BN/SiC composite at elevated temperature in a simulated combustion condition. After prior fatigue for 90,000 cycles, the tensile strength reduced by 70% under high peak stress level of 125 MPa for 1250 °C, and 35% under low peak stress level of 88 MPa for 1350 °C. The strength degradation depends not only on the test temperature but also on the applied stress level. Ruggles-Wrenn et al. [4] investigated tension − tension fatigue behavior of 2D Nextel™ 720/Alumina composite at 1200 °C in air and in steam environment under the loading frequency of 0.1 and 10 Hz. In air environment, the fatigue life appears to be independent of the loading frequency; and in steam atmosphere, the fatigue limit, number of applied cycles to failure and failure time all decrease as the loading frequency decreases, and the strength and stiffness degradation increases with decreasing frequency of prior fatigue. Mehrman et al. [5] investigated the influence of hold times on the tension − tension fatigue behavior of 2D Nextel™ 720/Alumina composite at 1200 °C in air and in steam condition. In air environment, the fatigue lives with hold times were shorter than those obtained in fatigue; and in steam atmosphere, the fatigue lives with hold times reduced significantly, and were less than those obtained in air. Ruggles-Wrenn and Lanser [6] investigated the tension − compression fatigue behavior of 2D Nextel™ 720/Alumina composite at 1200 °C in air and in steam condition. The tensile strength degraded approximately 40% after prior fatigue in steam condition, and the tension-compression cycling is much more damaging than tension − tension fatigue, due to fiber micro-buckling during compression portion of the cycle. For the non-oxide CMCs, the degradation under cyclic loading involves interface wear at room temperature, and oxidation of interphase and fibers at elevated temperature, and is accelerated by the presence of moisture. However, for the oxide/oxide CMCs, the composite is inherently resistant to oxidation at elevated temperature, but the formation of matrix cracks and degradation of fibers strength also affect the fatigue performance. Whitworth [7] evaluated the residual strength degradation in graphite/epoxy composite laminates subjected to cyclic loading, and the effects of peak stress and applied cycles on the degradation of residual strength have been analyzed. Keiji [8] developed the residual tensile strength model of fiber-reinforced CMCs after fatigue loading, and the effects of fiber strength and the interfacial shear stress on the fatigue life and residual strength of the composite have been investigated. Shah et al. [9] developed a probabilistic modeling approach to quantify the scatter in the first matrix cracking strength and the ultimate tensile strength in CMCs. Murthy et al. [10] investigated the probabilistic analysis and reliability assessment of the turbine vane withstanding a maximum temperature of 1315 °C within the substrate and the hot surface temperature of 1482 °C, considering the random variables of the material properties, strength and pressure loading on the vane. Li [11] developed approaches to model the tensile strength of CMCs subjected to multiple fatigue loading. It was found that the loading sequence, peak stress and cycle number affect the tensile strength of the composite. Li [12] investigated the tensile strength degradation after non-stress oxidation at elevated temperature. It was found that the oxidation temperature and oxidation time affect the damage and fracture in CMCs. However, under cyclic loading at elevated temperature in oxidative environments, the composite strength significantly degrades compared with that under non-stress oxidation at elevated temperature, due to serious oxidation occurred as matrix cracks open under stress, and the wide range of interface wear/interface oxidation/fibers fracture.

The objective of this paper is to investigate the composite strength degradation of non-oxide and oxide/oxide fiber-reinforced CMCs under cyclic loading at elevated temperature in oxidative environments. Considering the damage mechanisms of matrix cracking, interface debonding, interface wear, interface oxidation and fibers fracture, the composite residual strength model has been developed by combining the micro stress field of the damaged composite, damage models and fracture criterion. The effects of peak stress level, initial and steady-state interface shear stress, fibers Weibull modulus and fibers strength, and testing temperature on the degradation of residual strength and fibers failure have been investigated. The evolution of residual strength versus cycle number curves of non-oxide and oxide/oxide CMCs under cyclic loading at elevated temperature have been predicted.

2 Residual Strength Model

Under cyclic fatigue loading, the interface shear stress and fibers strength degrade with increasing applied cycles due to the interface wear and interface oxidation [13,14,15,16,17]. A unit cell is extracted from the ceramic composite system to analyze the stress distributions in the fiber and the matrix, as shown in Fig. 1. The fiber radius is r f and the matrix radius is R(R = r f/V f 1/2). The length of the unit cell is l c/2, which is just the half matrix crack space.

Fig. 1
figure 1

The unit cell of the Budiansky-Hutchinson-Evans shear-lag model

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When fibers break, the stress dropped by broken fibers would be transferred to intact fibers at the cross-section. The two-parameter Weibull model is adopted to describe the fibers strength distribution. The fibers fracture probability P f is determined by Eq. (1).

$$ {P}_{\mathrm{f}}=1- \exp \left(-\underset{L_0}{\int}\frac{1}{l_0}{\left[\frac{\sigma_{\mathrm{f}}(x)}{\sigma_0}\right]}^m dx\right) $$
(1)

where σ 0 denotes the fibers strength at the tested gauge length l 0; m denotes the fiber Weibull modulus; and L 0 denotes the integral length.

The fracture probabilities of oxidized fibers in the oxidation region, unoxidized fibers in the oxidation region, fibers in the interface debonded region and interface bonded region of P fa , P fb , P fc and P fd are determined by Eq. (2).

$$ {P}_{fa}=1- \exp \left\{-2\frac{l_t}{l_0}{\left[\frac{T}{\sigma_0(t)}\right]}^m\right\} $$
(2)
$$ {P}_{fb}=1- \exp \left\{-2\frac{l_t}{l_0}{\left(\frac{T}{\sigma_0}\right)}^m\right\} $$
(3)
$$ {P}_{fc}=1- \exp \left\{-\frac{r_{\mathrm{f}}{T}^{m+1}}{l_0{\left({\sigma}_0(N)\right)}^m{\tau}_{\mathrm{i}}(N)\left( m+1\right)}\left[1-{\left(1-\frac{l_{\mathrm{d}}(N)}{l_{\mathrm{f}}(N)}\right)}^{m+1}\right]\right\} $$
(4)
$$ {P}_{fd}=1- \exp \left\{-\frac{2{r}_{\mathrm{f}}{T}^m}{\rho {l}_0{\left({\sigma}_0(N)\right)}^m\left( m+1\right)\left(1-\frac{\sigma_{\mathrm{f}\mathrm{o}}}{T}-\frac{l_{\mathrm{d}}(N)}{l_{\mathrm{s}}(N)}\right)}\times \left[\left(1-\frac{l_{\mathrm{d}}(N)}{l_{\mathrm{f}}(N)}-\right.\right.\right.\left.\left.{\left.\left(1-\frac{\sigma_{\mathrm{f}\mathrm{o}}}{T}-\frac{l_{\mathrm{d}}(N)}{l_{\mathrm{f}}(N)}\right)\frac{\rho {l}_{\mathrm{d}}(N)}{r_{\mathrm{f}}}\right)}^{m+1}-{\left(1-\frac{l_{\mathrm{d}}(N)}{l_{\mathrm{f}}(N)}-\left(1-\frac{\sigma_{\mathrm{f}\mathrm{o}}}{T}-\frac{l_{\mathrm{d}}(N)}{l_{\mathrm{f}}(N)}\right)\frac{\rho {l}_c}{2{r}_{\mathrm{f}}}\right)}^{m+1}\right]\right\} $$
(5)

where l d denotes the interface debonded length; σ fo denotes the fiber stress at the interface bonded region; l f denotes the interface slip length; l t denotes the interface oxidation region length; and σ 0(t) denotes the time dependence of fibers strength.

$$ {\tau}_{\mathrm{i}}(N)={\tau}_{\mathrm{i}\mathrm{o}}+\left[1- \exp \left(-\omega {N}^{\lambda}\right)\right]\left({\tau}_{\mathrm{i} \min }-{\tau}_{\mathrm{i}\mathrm{o}}\right) $$
(6)
$$ {\sigma}_{\mathrm{o}}(N)={\sigma}_{\mathrm{o}}\left[1-{p}_1{\left( \log N\right)}^{p_2}\right] $$
(7)
$$ {\sigma}_0(t)={\sigma}_0, t\le \frac{1}{k}{\left(\frac{K_{IC}}{Y{\sigma}_0}\right)}^4 $$
(8)
$$ {\sigma}_0(t)=\frac{K_{IC}}{Y\sqrt[4]{ k t}}, t>\frac{1}{k}{\left(\frac{K_{IC}}{Y{\sigma}_0}\right)}^4 $$
(9)

where τio denotes the initial interface shear stress; τimin denotes the steady-state interface shear stress; ω and λ are empirical constants; p 1 and p 2 are empirical parameters; K IC denotes the critical stress intensity factor; Y is a geometric parameter; and k is the parabolic rate constant.

The relationship between the applied stress, the stress carried by intact fibers and broken fibers is determined by Eq. (10).

$$ \frac{\sigma}{V_{\mathrm{f}}}=\left[1-{P}_{\mathrm{f}}\left(1+\frac{2{l}_{\mathrm{f}}}{l_c}\right)\right] T+{P}_{\mathrm{r}}\frac{2{l}_{\mathrm{f}}}{l_c}\left\langle {T}_{\mathrm{b}}\right\rangle $$
(10)

where

$$ {P}_f=\varphi \left[\eta {P}_{f a}+\left(1-\eta \right){P}_{f b}\right]+{P}_{f c}+{P}_{f d} $$
(11)
$$ {P}_r={P}_{fc}+{P}_{fd} $$
(5b)

where η denotes the oxidation fibers fraction in the oxidized region; and φ denotes the fraction of oxidation in the multiple matrix cracks.

The stress carried by broken fibers is determined by Eq. (12).

$$ \left\langle {T}_{\mathrm{b}}\right\rangle =\frac{T}{P_r}{\left(\frac{\sigma_{\mathrm{c}}}{T}\right)}^{m+1}{\left(\frac{\sigma_{\mathrm{o}}(N)}{\sigma_{\mathrm{o}}}\right)}^m\frac{\tau_{\mathrm{i}}(N)}{\tau_{\mathrm{i}}}\left\{1- \exp \left[-{\left(\frac{T}{\sigma_{\mathrm{c}}}\right)}^{m+1}{\left(\frac{\sigma_{\mathrm{o}}}{\sigma_{\mathrm{o}}(N)}\right)}^m\frac{\tau_{\mathrm{i}}}{\tau_{\mathrm{i}}(N)}\right]\right\}-\frac{T}{P_r} \exp \left\{-{\left(\frac{T}{\sigma_{\mathrm{c}}}\right)}^{m+1}{\left(\frac{\sigma_{\mathrm{o}}}{\sigma_{\mathrm{o}}(N)}\right)}^m\frac{\tau_{\mathrm{i}}}{\tau_{\mathrm{i}}(N)}\right\} $$
(12)

Substituting Eqs. (2), (8), (11) and (12) into the Eq. (10), the composite residual strength can be determined subjected to cyclic loading at elevated temperature in oxidative environments considering interface wear, interface oxidation, fibers oxidation and fracture.

3 Discussions

The ceramic composite system of SiC/SiC is used for the case study and its basic material properties are given by [18]: V f = 30%, E f = 230 GPa, E m = 350 GPa, r f = 7.5 μm, m = 3, l 0 = 25 × 10−3 m and σ 0 = 0.5 GPa; the interface shear stress degradation model parameters are given by: τio = 50 MPa, τimin = 1.0 MPa, ω = 0.0001 and λ = 1.2; and the fiber strength degradation model parameters are given by: p 1 = 0.02 and p 2 = 1.4. The effects of fatigue peak stress level, interface shear stress, fiber Weibull modulus, fibers strength and testing temperature on the composite strength degradation versus applied cycles have been analyzed.

3.1 Effect of Peak Stress Level

The effects of peak stress level, i.e., σ max = 180 and 200 MPa, on the evolution of residual strength versus applied cycles, and the fibers failure versus applied stress are illustrated in Fig. 2.

Fig. 2
figure 2

a The residual strength versus cycle number curves under σ max = 180 and 200 MPa; b the broken fibers fraction versus applied stress curves after experiencing different applied cycles under σ max = 180 MPa; and c the broken fibers fraction versus applied stress curves after experiencing different applied cycles under σ max = 200 MPa

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When σ max = 180 MPa, the composite residual strength decreases with increasing applied cycles, i.e., 364, 343, 314 and 194 MPa corresponding to the cycle number of N = 1, 100, 1000 and 3000, as shown in Fig. 2a; and the broken fibers fraction versus applied stress curves after experiencing 1, 100, 1000 and 3000 applied cycles under σ max = 180 MPa are illustrated in Fig. 2b, in which the residual strength decreases with increasing cycle number.

When σ max = 200 MPa, the composite residual strength decreases with increasing applied cycles, i.e., 364, 338, 310 and 166 MPa corresponding to the cycle number of N = 1, 100, 1000 and 3000, as shown in Fig. 2a; and the broken fibers fraction versus applied stress curves after experiencing 1, 100, 1000 and 3000 applied cycles under σ max = 200 MPa are illustrated in Fig. 2c, in which the residual strength decreases with increasing cycle number.

With increasing peak stress level, the composite residual strength degrades due to the increasing interface wear and interface debonded region.

3.2 Effect of Interface Shear Stress

The effects of interface shear stress, i.e., τi = 20 and 40 MPa, on the evolution of composite residual strength versus applied cycles, and the broken fibers fraction versus applied stress after experiencing cyclic loading at σ max = 180 MPa are illustrated in Fig. 3.

Fig. 3
figure 3

a The residual strength versus cycle number curves corresponding to τi = 20 and 40 MPa under σ max = 180 MPa; b the broken fibers fraction versus applied stress curves after experiencing different applied cycles corresponding to τi = 40 MPa under σ max = 180 MPa; and c the broken fibers fraction versus applied stress curves after experiencing different applied cycles corresponding to τi = 20 MPa under σ max = 180 MPa

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When τi = 40 MPa, the composite residual strength decreases with increasing applied cycles, i.e., 338, 318, 294 and 160 MPa corresponding to the cycle number of N = 1, 100, 1000 and 3000 at σ max = 180 MPa, as shown in Fig. 3a; and the broken fibers fraction versus applied stress curves after experiencing 1, 100, 1000 and 3000 applied cycles under σ max = 180 MPa are illustrated in Fig. 3b, in which the residual strength decreases with increasing cycle number.

When τi = 20 MPa, the composite residual strength decreases with increasing applied cycles, i.e., 270, 261, 236 and 175 MPa corresponding to the cycle number of N = 1, 100, 1000 and 3000 at σ max = 180 MPa, as shown in Fig. 3a; and the broken fibers fraction versus applied stress curves after experiencing 1, 100, 1000 and 3000 applied cycles under σ max = 180 MPa are illustrated in Fig. 3c, in which the residual strength decreases with increasing cycle number.

The effects of interface shear stress, i.e., τf = 5 and 10 MPa, on the evolution of composite residual strength versus applied cycles, and the broken fibers fraction versus applied stress after experiencing cyclic loading at σ max = 180 MPa are illustrated in Fig. 4.

Fig. 4
figure 4

a The residual strength versus cycle number curves corresponding to τf = 5 and 10 MPa under σ max = 180 MPa; b the broken fibers fraction versus applied stress curves after experiencing different applied cycles corresponding to τf = 10 MPa under σ max = 180 MPa; and c the broken fibers fraction versus applied stress curves after experiencing different applied cycles corresponding to τf = 5 MPa under σ max = 180 MPa

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When τf = 10 MPa, the composite residual strength decreases with increasing applied cycles, i.e., 292, 277, 259 and 228 MPa corresponding to the cycle number of N = 1, 100, 1000 and 3000 at σ max = 180 MPa, as shown in Fig. 4a; and the broken fibers fraction versus applied stress curves after experiencing 1, 100, 1000 and 3000 applied cycles under σ max = 180 MPa are illustrated in Fig. 4b, in which the residual strength decreases with increasing cycle number.

When τf = 5 MPa, the composite residual strength decreases with increasing applied cycles, i.e., 292, 276, 256 and 198 MPa corresponding to the cycle number of N = 1, 100, 1000 and 3000 at σ max = 180 MPa, as shown in Fig. 4a; and the broken fibers fraction versus applied stress curves after experiencing 1, 100, 1000 and 3000 applied cycles under σ max = 180 MPa are illustrated in Fig. 4c, in which the residual strength decreases with increasing cycle number.

With increasing of initial and steady-state interface shear stress, the composite residual strength increases due to the decrease of the interface wear and interface debonding range.

3.3 Effect of Fiber Weibull Modulus

The effects of fiber Weibull modulus, i.e., m = 3 and 5, on the evolution of composite residual strength versus applied cycles, and the broken fibers fraction versus applied stress after experiencing cyclic loading at σ max = 180 MPa are illustrated in Fig. 5.

Fig. 5
figure 5

a The residual strength versus cycle number curves corresponding to m = 3 and 5 under σ max = 180 MPa; b the broken fibers fraction versus applied stress curves after experiencing different applied cycles corresponding to m = 3 under σ max = 180 MPa; and c the broken fibers fraction versus applied stress curves after experiencing different applied cycles corresponding to m = 5 under σ max = 180 MPa

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When m = 3, the composite residual strength decreases with increasing applied cycles, i.e., 360, 338, 310 and 166 MPa corresponding to the cycle number of N = 1, 100, 1000 and 3000 at σ max = 180 MPa, as shown in Fig. 5a; and the broken fibers fraction versus applied stress curves after experiencing 1, 100, 1000 and 3000 applied cycles under σ max = 180 MPa are illustrated in Fig. 5b, in which the composite residual strength decreases with increasing cycle number.

When m = 5, the composite residual strength decreases with increasing applied cycles, i.e., 242, 230, 202 and 147 MPa corresponding to the cycle number of N = 1, 100, 1000 and 3000 at σ max = 180 MPa, as shown in Fig. 5a; and the broken fibers fraction versus applied stress curves after experiencing 1, 100, 1000 and 3000 applied cycles under σ max = 180 MPa are illustrated in Fig. 5c, in which the composite residual strength decreases with increasing cycle number.

With increasing of fiber Weibull modulus, the composite residual strength after experiencing cyclic loading decreases.

3.4 Effect of Fiber Strength

The effects of fiber strength, i.e., σ 0 = 0.3 and 0.5 GPa, on the evolution of composite residual strength versus applied cycles, and the broken fibers fraction versus applied stress after experiencing cyclic loading at σ max = 180 MPa are illustrated in Fig. 6.

Fig. 6
figure 6

a The residual strength versus cycle number curves corresponding to σ 0 = 0.3 and 0.5 GPa under σ max = 180 MPa; b the broken fibers fraction versus applied stress curves after experiencing different applied cycles corresponding to σ 0 = 0.3 GPa under σ max = 180 MPa; and c the broken fibers fraction versus applied stress curves after experiencing different applied cycles corresponding to σ 0 = 0.5 GPa under σ max = 180 MPa

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When σ 0 = 0.3 GPa, the composite residual strength decreases with increasing applied cycles, i.e., 303, 289, 258 and 179 MPa corresponding to the cycle number of N = 1, 100, 1000 and 3000 at σ max = 180 MPa, as shown in Fig. 6a; and the broken fibers fraction versus applied stress curves after experiencing 1, 100, 1000 and 3000 applied cycles under σ max = 180 MPa are illustrated in Fig. 6b, in which the residual strength decreases with increasing cycle number.

When σ 0 = 0.5 GPa, the composite residual strength decreases with increasing applied cycles, i.e., 427, 410, 380 and 285 MPa corresponding to the cycle number of N = 1, 100, 1000 and 3000 at σ max = 180 MPa, as shown in Fig. 6a; and the broken fibers fraction versus applied stress curves after experiencing 1, 100, 1000 and 3000 applied cycles under σ max = 180 MPa are illustrated in Fig. 6c, in which the residual strength decreases with increasing cycle number.

With increasing of fibers strength, the composite residual strength after experiencing cyclic loading increases.

3.5 Effect of Testing Temperatures

The effects of testing temperature, i.e., Tem = 600 °C and 800 °C, on the evolution of composite residual strength versus applied cycles, and the broken fibers fraction versus applied stress after experiencing cyclic loading at σ max = 180 MPa are illustrated in Fig. 7.

Fig. 7
figure 7

a The residual strength versus cycle number curves corresponding to Tem = 600 °C and 800 °C under σ max = 180 MPa; b the broken fibers fraction versus applied stress curves after experiencing different applied cycles corresponding to Tem = 600 °C under σ max = 180 MPa; and c the broken fibers fraction versus applied stress curves after experiencing different applied cycles corresponding to Tem = 800 °C under σ max = 180 MPa

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When Tem = 600 °C, the composite residual strength decreases with increasing applied cycles, i.e., 363, 353, 326 and 260 MPa corresponding to the cycle number of N = 1, 100, 1000 and 3000 at σ max = 180 MPa, as shown in Fig. 7a; and the broken fibers fraction versus applied stress curves after experiencing 1, 100, 1000 and 3000 applied cycles under σ max = 180 MPa are illustrated in Fig. 7b, in which the residual strength decreases with increasing cycle number.

When Tem = 800 °C, the composite residual strength decreases with increasing applied cycles, i.e., 363, 341, 316 and 214 MPa corresponding to the cycle number of N = 1, 100, 1000 and 3000 at σ max = 180 MPa, as shown in Fig. 7a; and the broken fibers fraction versus applied stress curves after experiencing 1, 100, 1000 and 3000 applied cycles under σ max = 180 MPa are illustrated in Fig. 7c, in which the residual strength decreases with increasing cycle number.

With increasing of testing temperature, the composite residual strength after experiencing cyclic loading decreases.

4 Experimental Comparisons

4.1 Non-oxide CMCs

Lee et al. [2] investigated the tension − tension fatigue behavior of SiC/Si-N-C composite at room temperature and 1000 °C. At room temperature, the composite tensile strength was about 197 MPa, after experiencing 1,000,000 applied cycles under the fatigue peak stress of 125 MPa, the tensile strength degrades to approximately 167 MPa, and the strength decreases about 15%. The experimental and theoretical predicted composite residual strength versus applied cycles curves are illustrated in Fig. 8a. The residual strength decreases with increasing applied cycles, i.e., 197, 192, 186, 179 and 170 MPa corresponding to the cycle number of N = 1, 1000, 10,000, 100,000 and 1,000,000 at σ max = 125 MPa. The broken fibers fraction versus applied stress curves after experiencing 1, 1000, 10,000, 100,000 and 1,000,000 applied cycles under σ max = 125 MPa are illustrated in Fig. 8b. At 1000 °C, the composite tensile strength was about 214 MPa, after experiencing 100,000 under the fatigue peak stress of 100 MPa, the tensile strength degrades to approximately 157 MPa, and the strength decreases about 26%. The experimental and theoretical predicted composite residual strength versus applied cycles curves are illustrated in Fig. 9a. The residual strength decreases with increasing applied cycles, i.e., 214, 206, 193, 113 and 99 MPa corresponding to the cycle number of N = 1, 1000, 10,000, 50,000 and 100,000 at σ max = 100 MPa. The broken fibers fraction versus applied stress curves after experiencing 1, 1000, 10,000 and 100,000 applied cycles under σ max = 100 MPa are illustrated in Fig. 9b.

Fig. 8
figure 8

a The residual strength versus applied cycles; and b the broken fibers versus applied cycles of SiC/Si-N-C composites under fatigue peak stress of σ max = 125 MPa at room temperature

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Fig. 9
figure 9

a The residual strength versus applied cycles; and b the broken fibers versus applied cycles of SiC/Si-N-C composites under fatigue peak stress of σ max = 100 MPa at 1000 °C

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Bertrand et al. [3] investigated the tension − tension fatigue behavior of SiC/SiC composite at 1250 °C and 1350 °C. The composite tensile strength was about 274 MPa at elevated temperature. At 1250 °C, the composite residual strength decreases to 116 MPa after experiencing 100,000 applied cycles under the fatigue peak stress of 125 MPa, and the composite strength degrades approximately 57%. The experimental and theoretical predicted composite residual strength versus applied cycles curves are illustrated in Fig. 10a. The residual strength decreases with increasing applied cycles, i.e., 274, 264, 251, 140 and 117 MPa corresponding to the cycle number of N = 1, 1000, 10,000, 50,000 and 100,000 at σ max = 125 MPa. The broken fibers fraction versus applied stress curves after experiencing 1, 1000, 10,000 and 100,000 applied cycles under σ max = 125 MPa are illustrated in Fig. 10b. At 1350 °C, the composite residual strength decreases to 247 MPa after experiencing 100,000 applied cycles under the fatigue peak stress of 90 MPa, and the composite strength degrades approximately 10%. The experimental and theoretical predicted composite residual strength versus applied cycles curves are illustrated in Fig. 11a. The residual strength decreases with increasing applied cycles, i.e., 274, 262, 248, and 247 MPa corresponding to the cycle number of N = 1, 10,000, 50,000 and 100,000 at σ max = 90 MPa. The broken fibers fraction versus applied stress curves after experiencing 1, 10,000 and 100,000 applied cycles under σ max = 90 MPa are illustrated in Fig. 11b.

Fig. 10
figure 10

a The residual strength versus applied cycles; and b the broken fibers versus applied cycles of SiC/Si-N-C composites under fatigue peak stress of σ max = 125 MPa at 1250 °C

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Fig. 11
figure 11

a The residual strength versus applied cycles; and b the broken fibers versus applied cycles of SiC/Si-N-C composites under fatigue peak stress of σ max = 90 MPa at 1350 °C

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4.2 Oxide/Oxide CMCs

Ruggles-Wrenn et al. [4] investigated the tension − tension fatigue behavior of 2D Nextel™ 720/Alumina composite under the loading frequency of 1.0 Hz at 1200 °C in steam atmosphere. The ultimate tensile strength was 190 MPa. Under the fatigue peak stress of 100 MPa, the composite residual strength decreases to 174 MPa after experiencing 100,000 applied cycles, and the composite strength degrades approximately 8.4%; and under the fatigue peak stress of 125 MPa, the composite residual strength decreases to 168 MPa after experiencing 100,000 applied cycles, and the composite strength degrades approximately 11.5%. The experimental and theoretical predicted composite residual strength versus applied cycles curves under σ max = 100 MPa are illustrated in Fig. 12a. The residual strength decreases with increasing applied cycles, i.e., 190, 188, 185 and 173 MPa corresponding to the cycle number of N = 1, 1000, 10,000 and 100,000 at σ max = 100 MPa. The broken fibers fraction versus applied stress curves after experiencing 1, 1000, 10,000 and 100,000 applied cycles under σ max = 100 MPa are illustrated in Fig. 12b. The experimental and theoretical predicted composite residual strength versus applied cycles curves under σ max = 125 MPa are illustrated in Fig. 13a. The residual strength decreases with increasing applied cycles, i.e., 190, 183, 182 and 167 MPa corresponding to the cycle number of N = 1, 1000, 10,000 and 100,000 at σ max = 125 MPa. The broken fibers fraction versus applied stress curves after experiencing 1, 1000, 10,000 and 100,000 applied cycles under σ max = 125 MPa are illustrated in Fig. 13b.

Fig. 12
figure 12

a The residual strength versus applied cycles; and b the broken fibers versus applied cycles of Nextel™ 720/Alumina composites under fatigue peak stress of σ max = 100 MPa at 1200 °C

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Fig. 13
figure 13

a The residual strength versus applied cycles; and b the broken fibers versus applied cycles of Nextel™ 720/Alumina composites under fatigue peak stress of σ max = 125 MPa at 1200 °C

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4.3 Conclusions

In this paper, the strength degradation of non-oxide and oxide/oxide fiber-reinforced ceramic-matrix composites (CMCs) subjected to cyclic loading at elevated temperature in oxidative environments has been investigated. Considering the damage mechanisms of matrix cracking, interface debonding, interface wear, interface oxidation and fibers fracture, the composite residual strength model has been established by combining the micro stress field of the damaged composites, the damage models and the fracture criterion. The effects of peak stress level, initial and steady-state interface shear stress, fiber Weibull modulus and fibers strength, and testing temperature on the degradation of composite strength and fibers failure have been investigated. The evolution of residual strength versus cycle number curves of SiC/Si-N-C, SiC/SiC and Nextel™ 720/Alumina composites under cyclic loading at elevated temperatures in oxidative environments have been predicted.

  1. (1)

    With increasing of peak stress level, the composite residual strength after experiencing cyclic loading decreases, due to the increasing range of interface debonding and interface wear.

  2. (2)

    With increasing of initial and steady-state interface shear stress, the composite residual strength after experiencing cyclic loading increases, due to the decrease of interface debonding and interface wear range.

  3. (3)

    With increasing of fiber Weibull modulus, the composite residual strength after experiencing cyclic loading decreases.

  4. (4)

    With increasing of fiber strength, the composite residual strength after experiencing cyclic loading increases.

  5. (5)

    With increasing of testing temperature, the composite residual strength after experiencing cyclic loading decreases due to the interface oxidation and fibers strength degradation.