1 Introduction

With the demand for high thrust-weight ratio and more efficient aero engine, the temperature of the turbine sections will be raised to a level exceeding the limit of current metallic materials. New materials will have to be tested and validated at very high temperatures that surpass 1300 °C. Ceramic-matrix composites (CMCs) are lighter than superalloys and maintaining the structural integrity even at higher temperatures, desirable qualities for improving aero engine efficiency, and have already been implemented on some aero engines’ components, i.e., the CMC flaps for exhaust nozzles of SNECMA M53 and M88 aero engines [1, 2]. The CMC turbine vanes have been designed and tested in the aero engine environment under the implementation of Ultra Efficient Engine Technology (UEET) program [3]. A CMC turbine blade has been tested for 4 h by General Electric in a modified GE F414 engine, which represents the first application of the CMC material in a rotating engine component. Incorporating the CMC turbine blades on a GE90 − sized engine, the overall weight can be reduced by 455 kg, which represents approximately 6 % of dry weight of a full sized GE90 − 115 [4]. In these applications, the CMCs will be subjected to varying temperatures, pressures, and degrading environments like water vapor. Such environment can cause rapid oxidation which is a primary mechanism of environmental degradation of CMCs.

Upon first loading to fatigue peak stress, matrix multicracking, fiber/matrix interface debonding, and partially fibers fractured in the interface debonded and bonded region, would occur [5]. With increasing cycle number, interface shear stress decreases due to interface wear when fibers slide relative to matrix during unloading and subsequent reloading, which reduces the load transfer capacity between fibers and the matrix [6]. The fibers strength also degrades with increasing cycle number due to interface wear, which reduces the load carrying capacity of fibers [7]. At elevated temperatures, matrix cracks would serve as avenues for the ingress of environment atmosphere into the composite. When the oxidizing gas ingresses into the composite, a sequence of events is triggered starting first with the oxidation of the interphase, and then fibers. With increasing oxidation time, the oxidation region propagates; and the interface shear stress and fibers strength decrease. Under cyclic fatigue loading, fibers gradually fracture due to the degradation of interface shear stress and fibers strength. When the broken fibers fraction approaches to the critical value, the composites fatigue fail [8, 9].

The objective of this paper is to investigate the effects of temperature, oxidation and fiber preforms on the fatigue life of C/SiC composites. An effective coefficient of the fiber volume fraction along the loading direction (ECFL) was introduced to describe the fiber architecture of the preforms. Under cyclic fatigue loading, the fibers broken fraction was determined by combining the interface wear model and fibers statistical failure model at room temperature, and interface/fibers oxidation model, interface wear model and fibers statistical failure model at elevated temperatures, based on the assumption that the fibers strength is subjected to two-parameter Weibull distribution and the load carried by broken and intact fibers satisfy the Global Load Sharing (GLS) criterion. The fatigue life S–N curves and fatigue limits of unidirectional, cross-ply, 2D, 2.5D and 3D C/SiC composites at room and elevated temperatures have been predicted.

2 Life Prediction Model

Under cyclic fatigue loading, the loading directions were along with the fiber for unidirectional CMCs, 0° fiber ply for cross-ply and plain-weave 2D CMCs, warp yarn for 2.5D CMCs, and axial fibers at a small angle θ for 3D CMCs. An effective coefficient of the fiber volume content along the loading direction (ECFL) is defined as,

$$ \psi =\frac{V_{\mathrm{f}\_\mathrm{axial}}}{V_{\mathrm{f}}} $$
(1)

where V f and V f_axial denote the total fiber volume fraction in the composites and the effective fiber volume fraction in the cyclic loading direction. The values of parameter ψ, which can be used to characterize fiber architectures, corresponding to unidirectional, cross-ply, 2D, 2.5D and 3D C/SiC composites are 1.0, 0.5, 0.5, 0.75 and 0.93. Under cyclic fatigue loading at room and elevated temperatures, the broken fibers fraction in the 0o plies or longitudinal yarns of cross-ply, 2D, 2.5D and 3D CMCs would increase with increasing applied cycles and oxidation time. When the broken fibers fraction in the 0o plies or longitudinal yarns approaches to the critical value, the composites fatigue fracture.

When fibers begin to break, the load dropped by broken fibers would be transferred to intact fibers in the cross-section. The two-parameter Weibull model is adopted to describe the fibers strength distribution. The fibers fracture probability P f is [10]

$$ {P}_{\mathrm{f}}=1- \exp \left(-{\displaystyle \underset{L_0}{\int}\frac{1}{l_0}{\left[\frac{\sigma_{\mathrm{f}}(x)}{\sigma_0}\right]}^mdx}\right) $$
(2)

in which 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.

2.1 Life Prediction Model at Room Temperature

The global load sharing (GLS) assumption is used to determine the load carried by intact and fracture fibers.

$$ \frac{\sigma }{V_{\mathrm{f}}}=\left(1-{P}_{\mathrm{f}}\right)T+\left\langle {T}_{\mathrm{b}}\right\rangle {P}_{\mathrm{f}} $$
(3)

in which σ denotes the applied stress; V f denotes the fiber volume fraction; T denotes the stress carried by intact fibers; and <T b> denotes the average stress carried by broken fibers.

$$ \begin{array}{l}\left\langle {T}_{\mathrm{b}}\right\rangle =\frac{T}{P_{\mathrm{f}}}{\left(\frac{\sigma_{\mathrm{c}}}{T}\right)}^{m+1}{\left(\frac{\sigma_{\mathrm{o}}(N)}{\sigma_{\mathrm{o}}}\right)}^m\frac{\tau_{\mathrm{i}}(N)}{\tau_{\mathrm{i}}}\left\{1- \exp \left[-{\left(\frac{T}{\sigma_{\mathrm{c}}}\right)}^{m+1}{\left(\frac{\sigma_{\mathrm{o}}}{\sigma_{\mathrm{o}}(N)}\right)}^m\frac{\tau_{\mathrm{i}}}{\tau_{\mathrm{i}}(N)}\right]\right\}\\ {}\kern3.2em -\frac{T}{P_{\mathrm{f}}} \exp \left\{-{\left(\frac{T}{\sigma_{\mathrm{c}}}\right)}^{m+1}{\left(\frac{\sigma_{\mathrm{o}}}{\sigma_{\mathrm{o}}(N)}\right)}^m\frac{\tau_{\mathrm{i}}}{\tau_{\mathrm{i}}(N)}\right\}\end{array} $$
(4)

and

$$ {P}_{\mathrm{f}}=1- \exp \left\{-{\left(\frac{T}{\sigma_{\mathrm{c}}}\right)}^{m+1}{\left(\frac{\sigma_{\mathrm{o}}}{\sigma_{\mathrm{o}}(N)}\right)}^m\frac{\tau_{\mathrm{i}}}{\tau_{\mathrm{i}}(N)}\right\} $$
(5)

The interface shear stress at different cycle number τi(N) is determined by Eq. (6). [11]

$$ {\tau}_{\mathrm{i}}(N)={\tau}_{\mathrm{i}\mathrm{o}}+\left[1- \exp \left(-\omega {N}^{\lambda}\right)\right]\left({\tau}_{\mathrm{i} \min }-{\tau}_{\mathrm{i}\mathrm{o}}\right) $$
(6)

in which τio denotes the initial interface shear stress; τimin denotes the steady-state interface shear stress under cyclic loading; and ω and λ are empirical constants.

The variation in fibers strength σ o(N) is given by Eq. (7). [12]

$$ {\sigma}_{\mathrm{o}}(N)={\sigma}_{\mathrm{o}}\left[1-{p}_1{\left( \log N\right)}^{p_2}\right] $$
(7)

where p 1 and p 2 are empirical parameters.

Substituting Eqs. (4) and (5) into Eq. (3), it leads to the form of

$$ \frac{\sigma }{V_{\mathrm{f}}}=T{\left(\frac{\sigma_{\mathrm{c}}}{T}\right)}^{m+1}{\left(\frac{\sigma_{\mathrm{o}}(N)}{\sigma_{\mathrm{o}}}\right)}^m\frac{\tau_{\mathrm{i}}(N)}{\tau_{\mathrm{i}}}\left\{1- \exp \left[-{\left(\frac{T}{\sigma_{\mathrm{c}}}\right)}^{m+1}{\left(\frac{\sigma_{\mathrm{o}}}{\sigma_{\mathrm{o}}(N)}\right)}^m\frac{\tau_{\mathrm{i}}}{\tau_{\mathrm{i}}(N)}\right]\right\} $$
(8)

Using Eqs. (6), (7) and (8), the stress T carried by intact fibers at the matrix cracking plane can be determined for different fatigue peak stresses. Substituting Eqs. (6), (7) and the intact fibers stress T into Eq. (5), the fibers failure probability corresponding to different cycle number can be determined. When the broken fibers fraction approaches to the critical value q, the composites fatigue fracture.

2.2 Life Prediction Model at Elevated Temperature

When fiber-reinforced CMCs subjected to oxidation, the notch would form at the fibers surface leading to degradation of fibers strength and the increase of fiber stress concentration and fracture probability. 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

$$ {P}_{fa}=1- \exp \left\{-2\frac{l_t}{l_0}{\left[\frac{T}{\sigma_0(t)}\right]}^m\right\} $$
(9a)
$$ {P}_{fb}=1- \exp \left\{-2\frac{l_t}{l_0}{\left(\frac{T}{\sigma_0}\right)}^m\right\} $$
(9b)
$$ {P}_{fc}=1- \exp \left\{-\frac{r_{\mathrm{f}}{T}^{m+1}}{l_0{\left({\sigma}_0(N)\right)}^m{\tau}_{\mathrm{i}}(N)\left(m+1\right)}\left[1-{\left(1-\frac{l_{\mathrm{d}}(N)}{l_{\mathrm{f}}(N)}\right)}^{m+1}\right]\right\} $$
(9c)
$$ \begin{array}{l}{P}_{fd}=1- \exp \left\{-\frac{2{r}_{\mathrm{f}}{T}^m}{\rho {l}_0{\left({\sigma}_0(N)\right)}^m\left(m+1\right)\left(1-\frac{\sigma_{\mathrm{f}\mathrm{o}}}{T}-\frac{l_{\mathrm{d}}(N)}{l_{\mathrm{s}}(N)}\right)}\times \left[\left(1-\frac{l_{\mathrm{d}}(N)}{l_{\mathrm{f}}(N)}-\right.\right.\right.\\ {}\left.\left.{\left.\left(1-\frac{\sigma_{\mathrm{f}\mathrm{o}}}{T}-\frac{l_{\mathrm{d}}(N)}{l_{\mathrm{f}}(N)}\right)\frac{\rho {l}_{\mathrm{d}}(N)}{r_{\mathrm{f}}}\right)}^{m+1}-{\left(1-\frac{l_{\mathrm{d}}(N)}{l_{\mathrm{f}}(N)}-\left(1-\frac{\sigma_{\mathrm{f}\mathrm{o}}}{T}-\frac{l_{\mathrm{d}}(N)}{l_{\mathrm{f}}(N)}\right)\frac{\rho {l}_c}{2{r}_{\mathrm{f}}}\right)}^{m+1}\right]\right\}\end{array} $$
(9d)

where r f denotes the fiber radius; l d denotes the interface debonded length; l c denotes the matrix crack spacing; ρ denotes the shear-lag model parameter; σ fo denotes the fiber stress at the interface bonded region; and l f denotes the slip length over which the fiber stress would decay to zero if not interrupted by the far-field equilibrium stresses; and l t denotes the oxidation region length. [13]

$$ {l}_t={\varphi}_1\left(1-{e}^{-{\varphi}_2t}\right) $$
(10)

where φ 1 and φ 2 are fitting parameters dependent on temperature. σ 0(t) denotes the time dependence of fibers strength. [14]

$$ {\sigma}_0(t)={\sigma}_0,t\le \frac{1}{k}{\left(\frac{K_{IC}}{Y{\sigma}_0}\right)}^4 $$
(11a)
$$ {\sigma}_0(t)=\frac{K_{IC}}{Y\sqrt[4]{kt}},t>\frac{1}{k}{\left(\frac{K_{IC}}{Y{\sigma}_0}\right)}^4 $$
(11b)

where K IC denotes the critical stress intensity factor; Y is a geometric parameter; and k is the parabolic rate constant.

The GLS assumption is used to determine the load carried by intact and fracture fibers [10].

$$ \frac{\sigma }{V_{\mathrm{f}}}=\left[1-{P}_{\mathrm{f}}\left(1+\frac{2{l}_{\mathrm{f}}}{l_c}\right)\right]T+{P}_{\mathrm{r}}\frac{2{l}_{\mathrm{f}}}{l_c}\left\langle {T}_{\mathrm{b}}\right\rangle $$
(12)

where

$$ {P}_f=\varphi \left[\eta {P}_{fa}+\left(1-\eta \right){P}_{fb}\right]+{P}_{fc}+{P}_{fd} $$
(13a)
$$ {P}_r={P}_{fc}+{P}_{fd} $$
(13b)

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

$$ \varphi =\frac{l_{\mathrm{sat}}}{l_{\mathrm{f}}-2{l}_{\mathrm{t}}} $$
(14)

where l sat denotes the saturation matrix crack spacing.

The average stress carried by broken fibers is given by Eq. (15).

$$ \begin{array}{l}\left\langle {T}_{\mathrm{b}}\right\rangle ={\displaystyle {\int}_0^{l_{\mathrm{f}}}{T}_{\mathrm{b}}(x)f(x)}dx\\ {}\kern1.8em =\frac{T}{P_r}{\left(\frac{\sigma_{\mathrm{c}}}{T}\right)}^{m+1}{\left(\frac{\sigma_{\mathrm{o}}(N)}{\sigma_{\mathrm{o}}}\right)}^m\frac{\tau_{\mathrm{i}}(N)}{\tau_{\mathrm{i}}}\left\{1- \exp \left[-{\left(\frac{T}{\sigma_{\mathrm{c}}}\right)}^{m+1}{\left(\frac{\sigma_{\mathrm{o}}}{\sigma_{\mathrm{o}}(N)}\right)}^m\frac{\tau_{\mathrm{i}}}{\tau_{\mathrm{i}}(N)}\right]\right\}\\ {}\kern3.1em -\frac{T}{P_r} \exp \left\{-{\left(\frac{T}{\sigma_{\mathrm{c}}}\right)}^{m+1}{\left(\frac{\sigma_{\mathrm{o}}}{\sigma_{\mathrm{o}}(N)}\right)}^m\frac{\tau_{\mathrm{i}}}{\tau_{\mathrm{i}}(N)}\right\}\end{array} $$
(15)

Substituting Eqs. (13) and (15) into Eq. (12), the stress T carried by intact fibers at the matrix cracking plane can be determined for different cycle number and fatigue stress. Substituting Eqs. (6), (7), (10) and (11) and the intact fiber stress T into Eq. (9), the fibers failure probabilities of oxidized fibers in the oxidation region, unoxidized fibers in the oxidation region, fibers in the interface debonded region and interface bonded region can be determined for different number of applied cycles.

3 Results and Discussion

3.1 Unidirectional C/SiC Composite

The monotonic tensile strength is 270 MPa at room temperature, and the fatigue peak stresses are 0.51, 0.66, 0.74, 0.88 and 0.96 tensile strength; and the monotonic tensile strength is 320 MPa at 800 °C in air, and the fatigue peak stresses are 0.37, 0.43, 0.56, 0.65 and 0.78 tensile strength [15].

The experimental and theoretical fatigue life S−N curves at room temperature and 800 °C in air are illustrated in Fig. 1a. The fatigue limit at room temperature is 0.88 tensile strength, and 0.22 tensile strength at 800 °C in air. The broken fibers fraction versus cycle number curves under 0.88 and 0.96 tensile strength at room temperature are given in Fig. 1b. When σ max = 0.96 and 0.88 σ UTS, the composite experienced 93,424 and 1,000,000 cycles, respectively. The broken fibers fraction versus cycle number curves under 0.37, 0.43, 0.56, 0.65 and 0.78 tensile strength at 800 °C in air are given in Fig. 1c. When σ max = 0.78, 0.65, 0.56, 0.43 and 0.37 σ UTS, the composite experienced 3133, 9957, 21,245, 48,578 and 102,952 cycles, respectively. The broken fibers fraction versus cycle number curves under 0.9 and 0.95 σ UTS at room temperature and 800 °C in air are given in Fig. 1d. When σ max = 0.95σ UTS, the composite experienced 277,006 cycles at room temperature, and 30 cycles at 800 °C in air; and when σ max = 0.9σ UTS, the composite experienced 1,000,000 cycles at room temperature, and 374 cycles at 800 °C in air.

Fig. 1
figure 1figure 1

a The fatigue life S−N curves of experimental data and theoretical analysis at room temperature and 800 °C in air; b the broken fibers fraction versus cycle number under 0.96 and 0.88 σ UTS at room temperature; c the broken fibers fraction versus cycle number under 0.37, 0.43, 0.56, 0.65 and 0.78 σ UTS at 800 °C in air; and d the broken fibers fraction versus cycle number under 0.9 and 0.95 σ UTS at room temperature and 800 °C in air of unidirectional C/SiC composite

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3.2 Cross-ply C/SiC Composite

The monotonic tensile strength of cross-ply C/SiC composite is 124 MPa at room temperature, and the fatigue peak stresses are 0.70, 0.80, 0.85 and 0.9 tensile strength; and the monotonic tensile strength of cross-ply C/SiC composite is 150 MPa at 800 °C in air, and the fatigue peak stresses are 0.6, 0.65 and 0.7 tensile strength [15].

The experimental and theoretical fatigue life S−N curves at room temperature and 800 °C in air are illustrated in Fig. 2a. The fatigue limit at room temperature is 0.85 tensile strength, and 0.26 tensile strength at 800 °C in air. The broken fibers fraction versus cycle number curves under 0.85 and 0.9 tensile strength at room temperature are given in Fig. 2b. When σ max = 0.9 and 0.85 σ UTS, the composite experienced 389,371 and 1,000,000 cycles, respectively. The broken fibers fraction versus cycle number curves under 0.6, 0.65 and 0.7 tensile strength at 800 °C in air are given in Fig. 2c. When σ max = 0.7, 0.65 and 0.6 σ UTS, the composite experienced 1474, 2486 and 3643 cycles, respectively. The broken fibers fraction versus cycle number curves under 0.8 tensile strength at room temperature and 800 °C in air are given in Fig. 2d. When σ max = 0.8 σ UTS, the composite experienced 1,000,000 cycles at room temperature, and 75 cycles at 800 °C in air.

Fig. 2
figure 2figure 2

a The fatigue life S−N curves of experimental data and theoretical analysis at room temperature and 800 °C in air; b the broken fibers fraction versus cycle number under 0.9 and 0.85 σ UTS at room temperature; c the broken fibers fraction versus cycle number under 0.6, 0.65 and 0.7 σ UTS at 800 °C in air; and d the broken fibers fraction versus cycle number under 0.8 σ UTS at room temperature and 800 °C in air of cross-ply C/SiC composite

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3.3 2D C/SiC Composite

The monotonic tensile strength of 2D C/SiC composite is 420 MPa at room temperature, and the fatigue peak stresses are 0.80, 0.83, 0.86, 0.89, 0.91 and 0.96 tensile strength [16]; and the monotonic tensile strength of 2D C/SiC composite is 487 MPa at 550 °C in air, and the fatigue peak stresses are 0.22, 0.36, 0.56 and 0.72 tensile strength [17].

The experimental and theoretical fatigue life S−N curves at room temperature and 550 °C in air are illustrated in Fig. 3a. The fatigue limit at room temperature is 0.83 tensile strength, and 0.3 tensile strength at 550 °C in air. The broken fibers fraction versus cycle number curves under 0.8, 0.83, 0.86, 0.89, 0.91 and 0.96 tensile strength at room temperature are given in Fig. 3b. When σ max = 0.96, 0.91, 0.89, 0.86, 0.83 and 0.8 σ UTS, the composite experienced 2009, 28,237, 93,029, 550,740, 1,000,000 and 1,000,000 cycles, respectively. The broken fibers fraction versus cycle number curves under 0.22, 0.36, 0.56 and 0.72 tensile strength at 550 °C in air are given in Fig. 3c. When σ max = 0.72, 0.56, 0.36 and 0.22 σ UTS, the composite experienced 3514, 24,840, 183,651 and 1,000,000 cycles, respectively. The broken fibers fraction versus cycle number curves under 0.85 tensile strength at room temperature and 550 °C in air are given in Fig. 3d. When σ max = 0.85σ UTS, the composite experienced 1,000,000 cycles at room temperature, and 42 cycles at 550 °C in air.

Fig. 3
figure 3figure 3

a The fatigue life S−N curves of experimental data and theoretical analysis at room temperature and 550 °C in air; b the broken fibers fraction versus cycle number under 0.8, 0.83, 0.86, 0.89, 0.91 and 0.96 σ UTS at room temperature; c the broken fibers fraction versus cycle number under 0.22, 0.36, 0.56 and 0.72 σ UTS at 550 °C in air; and d the broken fibers fraction versus cycle number under 0.85 σ UTS at room temperature and 550 °C in air of 2D C/SiC composite

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3.4 2.5D C/SiC Composite

The monotonic tensile strength of 2.5D C/SiC composite is 212 MPa at room temperature, and the fatigue peak stresses are 0.85, 0.89, 0.94, 0.96 and 0.99 tensile strength [18]; and the monotonic tensile strength of 2.5D C/SiC composite is 280 MPa at 800 °C in air, and the fatigue peak stresses are 0.5, 0.6, 0.7 and 0.8 tensile strength [19].

The experimental and theoretical fatigue life S−N curves at room temperature and 800 °C in air are illustrated in Fig. 4a. The fatigue limit at room temperature is 0.85 tensile strength, and 0.28 tensile strength at 800 °C in air. The broken fibers fraction versus cycle number curves under 0.85, 0.89, 0.94, 0.96 and 0.99 tensile strength at room temperature are given in Fig. 4b. When σ max = 0.99, 0.96, 0.94, 0.89 and 0.85 σ UTS, the composite experienced 6182, 9084, 12,173, 77,354 and 1,000,000 cycles, respectively. The broken fibers fraction versus cycle number curves under 0.5, 0.6, 0.7 and 0.8 tensile strength at 800 °C in air are given in Fig. 4c. When σ max = 0.8, 0.7, 0.6 and 0.5 σ UTS, the composite experienced 736, 3525, 8314 and 37,308 cycles, respectively. The broken fibers fraction versus cycle number curves under 0.8 tensile strength at room temperature and 800 °C in air are given in Fig. 4d. When σ max = 0.8 σ UTS, the composite experienced 1,000,000 cycles at room temperature, and 736 cycles at 800 °C in air.

Fig. 4
figure 4figure 4

a The fatigue life S−N curves of experimental data and theoretical analysis at room temperature and 800 °C in air; b the broken fibers fraction versus cycle number under 0.85, 0.89, 0.94, 0.96 and 0.99 σ UTS at room temperature; c the broken fibers fraction versus cycle number under 0.5, 0.6, 0.7 and 0.8 σ UTS at 800 °C in air; and d the broken fibers fraction versus cycle number under 0.8 σ UTS at room temperature and 800 °C in air of 2.5D C/SiC composite

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3.5 3D C/SiC Composite

The monotonic tensile strength is 276 MPa at room temperature, and the fatigue peak stress are 0.80, 0.83, 0.87, 0.89, 0.90 and 0.94 tensile strength [20]; the monotonic tensile strength is 360 MPa at 1100 °C in 10−4 Pa vacuum, and the fatigue peak stresses are 0.97 and 0.88 tensile strength [21]; the monotonic tensile strength is 304 MPa at 1300 °C in 10−4 Pa vacuum, and the fatigue peak stresses are 0.5, 0.83, 0.93, 0.98 and 0.99 tensile strength [20]; and the monotonic tensile strength is 261 MPa at 1500 °C in 10−4 Pa vacuum, and the fatigue peak stresses are 0.98, 0.96, 0.95, 0.92, 0.9, 0.87 and 0.83 tensile strength [22].

The experimental and theoretical fatigue life S−N curves at room temperature, 1100, 1300 and 1500 °C in vacuum are illustrated in Fig. 5a. The fatigue limit at room temperature is 0.86 tensile strength, 0.95 tensile strength at 1100 °C in 10−4 Pa vacuum, 0.93 tensile strength at 1300 °C in 10−4 Pa vacuum, and 0.9 tensile strength at 1500 °C in 10−4 Pa vacuum. The broken fibers fraction versus cycle number curves under 0.94, 0.9, 0.89, 0.87 and 0.83 σ UTS at room temperature are given in Fig. 5b. When σ max = 0.94, 0.9, 0.89, 0.87 and 0.83 σ UTS, the composite experienced 349, 17,233, 45,914, 323,787 and 1,000,000 cycles, respectively. The broken fibers fraction versus cycle number curves under 0.88 and 0.97 tensile strength at 1100 °C in 10−4 Pa vacuum are given in Fig. 5c. When σ max = 0.97 and 0.88 σ UTS, the composite experienced 171,281 and 1,000,000 cycles, respectively. The broken fibers fraction versus cycle number curves under 0.99, 0.98, 0.93 and 0.83 tensile strength at 1300 °C in 10−4 Pa vacuum are given in Fig. 5d. When σ max = 0.99, 0.98, 0.93 and 0.83 σ UTS, the composite experienced 14,008, 17,740, 870,731 and 1,000,000 cycles, respectively. The broken fibers fraction versus cycle number curves under 0.98, 0.96, 0.95, 0.92 and 0.9 tensile strength at 1500 °C in 10−4 Pa vacuum are given in Fig. 5e. When σ max = 0.98, 0.96, 0.95, 0.92 and 0.9 σ UTS, the composite experienced 2493, 11,799, 25,608, 259,213 and 1,000,000 cycles, respectively. The broken fibers fraction versus cycle number curves under 0.95 tensile strength at room temperature, 1100, 1300 and 1500 °C in 10−4 Pa vacuum are given in Fig. 5f. When σ max = 0.95 σ UTS, the composite experienced 219 cycles a at room temperature, 1,000,000 cycles at 1100 °C, 143,725 cycles at 1300 °C, and 25,608 cycles at 1500 °C in 10−4 Pa vacuum.

Fig. 5
figure 5figure 5figure 5

a The fatigue life S−N curves of experimental data and theoretical analysis at room temperature, 1100, 1300 and 1500 °C in vacuum; b the broken fibers fraction versus cycle number under 0.83, 0.87, 0.89, 0.9 and 0.94 σ UTS at room temperature; c the broken fibers fraction versus cycle number under 0.88 and 0.97 σ UTS at 1100 °C in vacuum; d the broken fibers fraction versus cycle number under 0.83, 0.93, 0.98 and 0.99 σ UTS at 1300 °C in vacuum; e the broken fibers fraction versus cycle number under 0.9, 0.92, 0.95, 0.96 and 0.98 σ UTS at 1500 °C in vacuum; and f the broken fibers fraction versus cycle number under 0.95 σ UTS at room temperature, 1100, 1300 and 1500 °C in vacuum of 3D C/SiC composite

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3.6 Discussion

At room temperature, the fatigue limit corresponding to unidirectional, cross-ply, 2D, 2.5D and 3D C/SiC composites are 0.88, 0.85, 0.83, 0.85 and 0.86 tensile strength. The total fiber volume fraction of unidirectional, cross-ply, 2D, 2.5D and 3D C/SiC composites are 40, 40, 45, 40 and 40 %, respectively. The ECFL for unidirectional, cross-ply, 2D, 2.5D and 3D C/SiC are 40, 20, 22.5, 30 and 37 %. For unidirectional and 3D C/SiC composites with high ECFL, i.e., 40 and 37 %, the fatigue limits are 0.88 and 0.86 tensile strength, respectively. For cross-ply, 2D and 2.5D C/SiC composites, the fatigue limit of cross-ply C/SiC composite is 0.85 tensile strength with ECFL of 20 %; the fatigue limit of 2.5D C/SiC composite is also 0.85 tensile strength, however with higher ECFL of 30 % due to fibers bending inside of the composite; and the fatigue limit of 2D C/SiC composite is 0.83 tensile strength with ECFL of 20 % also due to fibers bending in the preform.

At 550 °C in air, the fatigue limit of 2D C/SiC composite is 0.3 tensile strength; and at 800 °C in air condition, the fatigue limits of unidirectional, cross-ply, and 2.5D C/SiC composites are 0.22, 0.26 and 0.28 tensile strength. With increasing oxidation tempereture, the fatigue limit of C/SiC composite decreases. And at the same oxidation temperature, the fatigue limit of unidirectional C/SiC composite decreases the most rapidly, i.e., from 0.88 tensile strength at room temperature to 0.22 tensile strength at 800 °C in air, with the highest fiber volume fraction along the loading direction; the fatigue limit of 2.5D C/SiC composite is the highest among the three C/SiC composites at 800 °C in air, due to high ECFL and the protection of transverse yarns from the outside environment; and the fatigue limit of cross-ply C/SiC composite lies in the middle, i.e., between unidirectional and 2.5D C/SiC composite, due to low ECFL compared with 2.5D C/SiC composite, and the protection of transverse plies from the outside environment compared with unidirectional C/SiC composite.

For 3D C/SiC composite, the fatigue limits at 1100, 1300 and 1500 °C in 10−4 Pa vacuum are 0.95, 0.93 and 0.9 tensile strength, which are all higher than the fatigue limit at room temperature, i.e., 0.86 tensile strength. As the the radial thermal residual tensile stress existed in the fiber/matrix interface due to the large mismatch of the radial thermal expansion coefficient between carbon fibers and silicon carbide matrix, i.e., 7 × 10−6/°C vs 4.8 × 10−6/°C, when the composite was cooled down from the high fabricated temperature, i.e., 950 to 1000 °C, the interface shear stress is low, and decreases rapidly at room temperature; at 1100, 1300 and 1500 °C, the interface shear stress increases and decreases slowly, due to radial thermal residual compressive stress; however, at 1300 and 1500 °C, the interface shear stress is such high that matrix cracks propagate through the fiber/matrix interface, leading to the decrease of fatigue limit compared with that at 1100 °C.

4 Conclusions

The effects of temperature, oxidation and fiber preforms on the fatigue life of C/SiC composites have been analyzed. An effective coefficient of the fiber volume fraction along the loading direction (ECFL) was introduced to describe the fiber architecture of the preforms. The broken fibers fraction under cyclic fatigue loading considering the degradation of interface shear stress and fibers strength was obtained. The fatigue life S−N curves and fatigue limits of unidirectional, cross-ply, 2D, 2.5D and 3D C/SiC composites have been predicted.

  1. (1)

    At room temperature, the fatigue limits of unidirectional, cross-ply, 2D, 2.5D and 3D C/SiC composites are 0.88, 0.85, 0.83, 0.85 and 0.86 tensile strength. The unidirectional and 3D C/SiC composites with high ECFL have the high fatigue limit compared with that of cross-ply, 2D and 2.5D C/SiC composites.

  2. (2)

    At 550 °C in air, the fatigue limit of 2D C/SiC composite is 0.3 tensile strength; and at 800 °C in air, the fatigue limits of unidirectional, cross-ply, and 2.5D C/SiC composites are 0.22, 0.26 and 0.28 tensile strength. The fatigue limit decreases with increasing oxidation temperature.

  3. (3)

    For 3D C/SiC composite, the fatigue limits at 1100, 1300 and 1500 °C in 10−4 Pa vacuum are 0.95, 0.93 and 0.9 tensile strength, which are all higher than the fatigue limit at room temperature, i.e., 0.86 tensile strength, due to radial thermal compressive stress at elevated temperatures.

  4. (4)

    Compared with S−N curves at room temperature or at elevated temperature in vacuum environments, the fatigue life S−N curves at elevated temperature in oxidized atmosphere exhibit an apparent region, in which the fatigue life decreases rapidly due to oxidation.