A Micromechanical Fatigue Limit Stress Model of Fiber-Reinforced Ceramic-Matrix Composites under Stochastic Overloading Stress

Abstract

Fatigue limit stress is a key design parameter for the structure fatigue design of composite materials. In this paper, a micromechanical fatigue limit stress model of fiber-reinforced ceramic-matrix composites (CMCs) subjected to stochastic overloading stress is developed. The fatigue limit stress of different carbon fiber-reinforced silicon carbide (C/SiC) composites (i.e., unidirectional (UD), cross-ply (CP), 2D, 2.5D, and 3D C/SiC) is predicted based on the micromechanical fatigue damage models and fatigue failure criterion. Under cyclic fatigue loading, the fatigue damage and fracture under stochastic overloading stress at different applied cycle numbers are characterized using two parameters of fatigue life decreasing rate and broken fiber fraction. The relationships between the fatigue life decreasing rate, stochastic overloading stress level and corresponding occurrence applied cycle number, and broken fiber fraction are analyzed. Under the same stochastic overloading stress level, the fatigue life decreasing rate increases with the occurrence applied cycle of stochastic overloading, and thus, is the highest for the cross-ply C/SiC composite and lowest for the 2.5D C/SiC composite. Among the UD, 2D, and 3D C/SiC composites, at the initial stage of cyclic fatigue loading, under the same stochastic overloading stress, the fatigue life decreasing rate of the 3D C/SiC is the highest; however, with the increasing applied cycle number, the fatigue life decreasing rate of the UD C/SiC composite is the highest. The broken fiber fraction increases when stochastic overloading stress occurs, and the difference of the broken fiber fraction between the fatigue limit stress and stochastic overloading stress level increases with the occurrence applied cycle.

1. Introduction

Ceramic-matrix composites (CMCs) possess high specific strength and specific modulus, high temperature resistance, and have already been applied on hot section components of commercial aero engines [1,2,3]. To ensure the reliability and safety of CMC components, it is necessary to develop performance evaluation, damage evolution, strength, and life prediction tools for airworthiness certification [4].

Under cyclic fatigue loading, matrix cracking, interface debonding, interface wear, and fiber fracture occur with the applied cycle, and these fatigue damage mechanisms degrade the mechanical performance of fiber-reinforced CMCs [5,6,7]. Fatigue limit stress is a key parameter for the design of CMC components. However, fatigue limit stress of CMCs depends on many factors, i.e., fiber characteristic and fiber properties [8,9], loading frequency [10,11], temperature [12,13], and testing conditions [14,15,16]. Under cyclic fatigue loading, stochastic overloading stress may occur due to a special operation condition of the aero engine, which can affect the internal fatigue damage evolution and lifetime of CMCs [17,18]. Reynaud [5], Evans [19], and Li [20,21,22] developed micromechanical fatigue life prediction methods for fiber-reinforced CMCs considering different fatigue damage mechanisms. The degradation rate of the fiber/matrix interface shear stress and fiber strength affects the fatigue life and fatigue limit stress. However, in the developed micromechanical model, the effect of stochastic overloading stress on fatigue limit stress has not been considered.

In this paper, a micromechanical fatigue limit stress model of fiber-reinforced CMCs subjected to stochastic overloading stress is developed. The fatigue limit stress for different carbon fiber-reinforced silicon carbide (C/SiC) composites is predicted based on the micromechanical fatigue damage models and fatigue failure criterion. The relationships between the fatigue life decreasing rate, stochastic overloading stress level and corresponding occurrence applied cycle number, and broken fiber fraction are analyzed.

2. Theoretical Model

When stochastic overloading stress occurs under cyclic fatigue loading, the fatigue damage evolution of matrix cracking, interface debonding, and fiber failure are affected. Figure 1 shows stochastic overloading stress σ_s occurred at different applied cycle numbers. In the present analysis, the overloading stress σ_s remains the same at applied cycle numbers N₁, N₂, and N₃.

Based on the global load sharing (GLS) criterion, under stochastic overloading stress, the stress carried by intact and broken fiber is determined by Equation (1) [23].

$$\frac{{\sigma }_{\mathrm{s}}}{V_{\mathrm{f}}}={\mathsf{\Phi }}_{\mathrm{s}}\left(1-P_{\mathrm{f}}\right)+{\mathsf{\Phi }}_{\mathrm{b}}{P}_{\mathrm{f}}$$

(1)

where V_f is the fiber volume, Φ_s is the intact fiber stress under stochastic overloading stress, Φ_b is the stress carried by broken fiber, and P_f is the fiber failure probability, and can be determined by Equation (2).

$$P_{\mathrm{f}}=1-\exp \left[-\mathsf{\Theta }{\mathsf{\Omega }}^m{\left(\frac{{\mathsf{\Phi }}_{\mathrm{s}}}{{\sigma }_{\mathrm{c}}}\right)}^{m+1}\right]$$

(2)

$${\mathsf{\Phi }}_{\mathrm{b}}=\frac{{\mathsf{\Phi }}_{\mathrm{s}}}{\mathsf{\Theta }{\mathsf{\Omega }}^m{P}_{\mathrm{f}}}{\left(\frac{{\sigma }_{\mathrm{c}}}{{\mathsf{\Phi }}_{\mathrm{s}}}\right)}^{m+1}\left\{1-\exp \left[-\mathsf{\Theta }{\mathsf{\Omega }}^m{\left(\frac{{\mathsf{\Phi }}_{\mathrm{s}}}{{\sigma }_{\mathrm{c}}}\right)}^{m+1}\right]\right\}-\frac{{\mathsf{\Phi }}_{\mathrm{s}}}{P_{\mathrm{f}}}\exp \left[-\mathsf{\Theta }{\mathsf{\Omega }}^m{\left(\frac{{\mathsf{\Phi }}_{\mathrm{s}}}{{\sigma }_{\mathrm{c}}}\right)}^{m+1}\right]$$

(3)

where m is the fiber Weibull modulus, and σ_c is the fiber characteristic strength, and Θ and Ω denote the degradation rate of the interface shear stress and fiber strength and can be determined by Equations (4) and (5), respectively.

$$\mathsf{\Theta }\left(N\right)=\frac{1}{1-\left(1-\phi \right)\left[1-\exp \left(-\omega N^{\lambda }\right)\right]}$$

(4)

$$\mathsf{\Omega }\left(N\right)=\frac{1}{1-p_1{\left(\log N\right)}^{p_2}}$$

(5)

where φ is the ratio between the steady interface shear stress and initial interface shear stress, ω and λ are the interface wear model parameter, and p₁ and p₂ are the fiber strength degradation model parameter.

Substituting Equations (2) and (3) into Equation (1), the relation between the applied stress and fiber intact stress is determined by Equations (6).

$$\frac{{\sigma }_{\mathrm{s}}}{V_{\mathrm{f}}}=\frac{{\mathsf{\Phi }}_{\mathrm{s}}}{\mathsf{\Theta }{\mathsf{\Omega }}^m}{\left(\frac{{\sigma }_{\mathrm{c}}}{{\mathsf{\Phi }}_{\mathrm{s}}}\right)}^{m+1}\left\{1-\exp \left[-\mathsf{\Theta }{\mathsf{\Omega }}^m{\left(\frac{{\mathsf{\Phi }}_{\mathrm{s}}}{{\sigma }_{\mathrm{c}}}\right)}^{m+1}\right]\right\}$$

(6)

Using Equations (4)–(6), the intact fiber stress under stochastic overloading stress can be obtained with the occurrence applied cycle number. Substituting the intact fiber stress under stochastic overloading stress into Equation (2), the fraction of broken fiber under stochastic overloading stress can be obtained. Under cyclic fatigue loading, when the fiber failure probability approaches the critical value, the composite fatigue fractures. The fatigue limit stress of CMCs at room temperature can be obtained using the developed life prediction model and fatigue limit cycle number N_limit.

The fatigue life decreasing rate is defined by Equation (7).

$$\mathsf{\Lambda }=\frac{N_{\mathrm{f}}\left({\sigma }_{\mathrm{limit}}\right)-N_{\mathrm{f}}\left({\sigma }_{\mathrm{s}}\right)}{N_{\mathrm{f}}\left({\sigma }_{\mathrm{limit}}\right)}$$

(7)

where N_f(σ_limit) is the fatigue failure cycle number under fatigue limit stress, and N_f(σ_s) is the fatigue failure cycle number under stochastic overloading stress.

3. Experimental Comparisons

Under cyclic fatigue loading, stochastic overloading stress affects the fatigue damage evolution, i.e., increasing the fiber failure probability, and decreasing the fatigue life. In this section, the fatigue limit stress of different C/SiC composites is predicted. The material properties and fatigue damage model parameters of the C/SiC composite are listed in

Table 1. Material properties of carbon fiber-reinforced silicon carbide (C/SiC) composite.

Items	Unidirectional [7]	Cross-Ply [13]	2D [10]	2.5D [8]	3D [9]
Manufacturing Process	Hot Pressing	Hot Pressing	Chemical Vapor Infiltration (CVI)	CVI	CVI
Stress Ratio	0.1	0.1	0.1	0.1	0.1
Frequency/(Hz)	10	10	10	10	60
Fiber Type	T−700TM	T−700TM	T−300TM	T−300TM	T−300TM
Vf	0.4	0.4	0.45	0.4	0.4
σuts1/(MPa)	270	124	420	225	276
rf2/(μm)	3.5	3.5	3.5	3.5	3.5
τio 3/(MPa)	8	6.2	25	20	20
τimin 4/(MPa)	0.3	1.5	8	8	5
ω5	0.04	0.06	0.002	0.001	0.02
Λ 5	1.5	1.8	1.0	1.0	1.0
p16	0.01	0.01	0.018	0.02	0.012
p26	1.0	0.8	1.0	1.2	1.0
m7	5	5	5	5	5

¹σ_uts is composite tensile strength; ² r_f is the fiber radius;.³ τ_io is the interface shear stress upon initial loading; ⁴ τ_imin is the steady-state interface shear stress; ⁵ ω and λ are the interface degradation model parameters; ⁶ p₁ and p₂ are the fiber strength degradation model parameters; ⁷ m is the fiber Weibull modulus.

. Under fatigue limit stress, stochastic overloading occurring in different applied cycle numbers can decrease the fatigue life. Using the developed fiber failure model in Equation (2) and the fatigue damage models in Equations (4) and (5), the effect of the stochastic overloading stress level and corresponding occurrence cycle number on the fatigue limit stress and corresponding fatigue life is analyzed. The relationships between the stochastic overloading stress level, occurrence cycle number, broken fiber fraction, and fatigue limit stress are established.

3.1. Unidirectional C/SiC Composite

The unidirectional T−700^TM carbon fiber-reinforced silicon carbide composite was fabricated using the hot-pressing (HP) method. Low pressure chemical vapor infiltration was employed to deposit approximately 5−20 layers of PyC/SiC with the mean thickness of 0.2 μm. The nano-SiC powder and sintering additives were ball milled for 4 h using SiC balls. After drying, the powders were dispersed in xylene with polycarbonsilane (PCS) to form the slurry. Carbon fiber tows were infiltrated by the slurry and wound to form aligned unidirectional composite sheets. After drying, the sheets were cut to a size of 150 mm × 150 mm and pyrolyzed in argon. Then the sheets were stacked in a graphite die and sintered by hot pressing. The dog-bone shaped specimens were cut from 150 mm × 150 mm panels by water cutting. The tension–tension fatigue tests were conducted on a MTS Model 809 servo hydraulic load-frame (MTS Systems Corp., Minneapolis, MN, USA). The fatigue experiments were in a sinusoidal wave form with a loading frequency f = 10 Hz. The fatigue load ratio (σ_min/σ_max) was R = 0.1. The fatigue tests were conducted under load control at room temperature.

Figure 2 shows the experimental and predicted fatigue life S−N curves of the unidirectional C/SiC composite. When the fatigue limit applied cycle number is defined to be N_limit = 10⁶, the corresponding predicted fatigue limit stress is approximately σ_limit = 241 MPa (approximately 89.2%σ_uts).

Figure 2b shows the fatigue life decreasing rate versus the occurrence cycle number of stochastic overloading curves for different stochastic overloading stress levels of σ_s = 245, 250, and 255 MPa (i.e., approximately 1.016, 1.037, 1.058 of fatigue limit stress). During the application of CMC components, the overloading stress level is not high, and the low overloading stress level is chosen for analysis. Under the same stochastic overloading stress level of σ_s = 245, 250, and 255 MPa, the fatigue life decreasing rate increases with the occurrence applied cycle of stochastic overloading (i.e., N_s = 10, 10², 10³, 10⁴, and 10⁵). Under σ_s = 245 MPa, the fatigue life decreasing rate increases from Λ = 0.14392 at N_s = 10 to Λ = 0.8074 at N_s = 10⁵; under σ_s = 250 MPa, the fatigue life decreasing rate increases from Λ = 0.3091 at N_s = 10 to Λ = 0.97479 at N_s = 10⁴; and under σ_s = 255 MPa, the fatigue life decreasing rate increases from Λ = 0.4559 at N_s = 10 to Λ = 0.9966 at N_s = 10³. When the applied cycle is between N_s = 10 and 10², the fatigue life decreasing rate increases rapidly with the occurrence applied cycle; however, when the applied cycle is higher than N_s = 10², the fatigue life decreasing rate increases slowly with the applied cycle. At the initial stage of cyclic fatigue loading, the fatigue damage mechanisms of matrix cracking, interface debonding and wear depend on the fatigue peak stress level. The occurrence of stochastic overloading stress at the initial stage of cyclic fatigue loading deteriorates fatigue damage evolution, i.e., decreasing matrix crack spacing, increasing interface debonding length and broken fiber fraction; however, when matrix cracking and interface wear approach a steady-state, the effect of stochastic overloading stress on fatigue damage or fatigue life decreasing rate decreases.

Figure 2c–e shows the broken fiber fraction versus the applied cycle number curves for different stochastic overloading stress levels and occurrence applied cycle numbers. The broken fiber fraction increases when stochastic overloading stress occurs, and the difference of the broken fiber fraction between original peak stress and stochastic overloading stress level increases with the applied cycle number.

Table 2. Fatigue limit stress and broken fiber fraction of unidirectional C/SiC composite under stochastic overloading stress.

σmax = 241 MPa	Nf2		N3 = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	1,431,993	Pf	0.00609	0.02113	0.17991	0.20077	0.22431	0.25085
σs1 = 245 MPa N = 10	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	1,225,895	Pf	0.00609	0.02329	0.18207	0.20294	0.22648	0.25302
σs = 245 MPa N = 102	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	418,087	Pf	0.00609	0.02113	0.19662	0.21749	0.24103	0.26757
σs = 245 MPa N = 103	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	368,186	Pf	0.00609	0.02113	0.17991	0.21915	0.24269	0.26923
σs = 245 MPa N = 104	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	320,486	Pf	0.00609	0.02113	0.17991	0.20077	0.2445	0.27104
σs = 245 MPa N = 105	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	275,798	Pf	0.00609	0.02113	0.17991	0.20077	0.22431	0.27298
σs = 250 MPa N = 10	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	989,365	Pf	0.00609	0.02626	0.18504	0.2059	0.22944	0.25598
σs = 250 MPa N = 102	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	70,700	Pf	0.00609	0.02113	0.21897	0.23983	0.26338	0.28571
σs = 250 MPa N = 103	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	51,311	Pf	0.00609	0.02113	0.17991	0.24365	0.26719	0.28571
σs = 250 MPa N = 104	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	36,095	Pf	0.00609	0.02113	0.17991	0.20077	0.27131	0.28571
σs = 255 MPa N = 10	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	779,144	Pf	0.00609	0.02952	0.1883	0.20917	0.23271	0.25925
σs = 255 MPa N = 102	Nf		N = 1	N = 10	N = 102	N = 103	N = 8593
	8593	Pf	0.00609	0.02113	0.24295	0.26381	0.28571
σs = 255 MPa N = 103	Nf		N = 1	N = 10	N = 102	N = 103	N = 4870
	4870	Pf	0.00609	0.02113	0.17991	0.26984	0.28571

¹σ_s is stochastic overloading stress; ² N_f is the cycle number corresponding to fatigue fracture; ³ N is applied cycle.

shows the fatigue limit stress and broken fiber fraction at different occurrence cycle numbers and stochastic overloading stress. When stochastic overloading stress σ_s = 245 MPa occurs at applied cycles N_s = 10, 10², 10³, 10⁴, and 10⁵, the broken fiber fraction increases from P_f = 0.02113, 0.17991, 0.20077, 0.22431, and 0.25085 under σ_limit = 241 MPa to P_f = 0.02329, 0.19662, 0.21915, 0.2445, and 0.27298; when stochastic overloading stress σ_s = 250 MPa occurs at applied cycles N_s = 10, 10², 10³, and 10⁴, the broken fiber fraction increases to P_f = 0.02626, 0.21897, 0.24365, and 0.27131; finally, when stochastic overloading stress σ_s = 255 MPa occurs at applied cycles N = 10, 10², and 10³, the broken fiber fraction increases to P_f = 0.02952, 0.24295, and 0.26984.

3.2. Cross-Ply C/SiC Composite

The cross-ply T−700^TM carbon fiber-reinforced silicon carbide composite was fabricated using the hot-pressing (HP) method, which offered the ability to fabricate dense composites via a liquid phase sintering method at a low temperature. The fiber volume is V_f = 0.4, and the average tensile strength is approximately σ_uts = 124 MPa. The dog-bone shaped specimens were cut from 150 mm × 150 mm panels by water cutting. The tension–tension fatigue tests were conducted on an MTS Model 809 servo hydraulic load-frame (MTS Systems Corp., Minneapolis, MN, USA). The fatigue experiments were in a sinusoidal wave form with a loading frequency f = 10 Hz. The fatigue load ratio (σ_min/σ_max) was R = 0.1. The fatigue tests were conducted under load control at room temperature.

Figure 3a shows experimental and predicted fatigue life S−N curves of the cross-ply C/SiC composite. When the fatigue limit applied cycle number is defined to be N_limit = 10⁶, the corresponding predicted fatigue limit stress is approximately σ_limit = 103 MPa (approximately 83%σ_uts).

Figure 3b shows the fatigue life decreasing rate versus the occurrence applied cycle number of stochastic overloading stress curves under σ_s = 105 MPa (approximately 1.019 fatigue limit stress). During the application of CMC components, the overloading stress level is not high, and the low overloading stress level is chosen for analysis. The fatigue life decreasing rate increases with the occurrence applied cycle number of stochastic overloading. Under σ_s = 105 MPa, the fatigue life decreasing rate increases from Λ = 0.97114 at N_s = 10 to Λ = 0.98696 at N_s = 10⁴. When the occurrence applied cycle number is between N_s = 10 and 10², the fatigue life decreasing rate increases rapidly; however, when the occurrence applied cycle is between N_s = 10² and 10⁴, the fatigue life decreasing rate increases slowly. The occurrence of stochastic overloading stress at the initial stage of cyclic fatigue loading deteriorates the fatigue damage evolution, i.e., decreasing matrix crack spacing in transverse and longitudinal plies, increasing interface debonding length, and broken fiber fraction; however, when matrix cracking and interface wear approach a steady-state, the effect of stochastic overloading on the fatigue damage or fatigue life decreasing rate decreases.

Figure 3c shows the broken fiber fraction versus applied cycle number curves for different occurrence cycle numbers (i.e., N_s = 10, 10², 10³, and 10⁴). The broken fiber fraction increases when stochastic overloading stress occurs, and the difference of the broken fiber fraction between the original peak stress and stochastic overloading stress level increases with the applied cycle.

Table 3. Fatigue limit stress and broken fiber fraction of cross-ply C/SiC composite under stochastic overloading stress.

σmax = 103 MPa	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	1,568,296	Pf	0.05349	0.20047	0.22748	0.24133	0.25508	0.26892
σs = 105 MPa N = 10	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 45,256
	45,256	Pf	0.05349	0.22205	0.24906	0.26291	0.27666	0.28571
σs = 105 MPa N = 102	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 30,243
	30,243	Pf	0.05349	0.20047	0.25148	0.26533	0.27908	0.28571
σs = 105 MPa N = 103	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 24,787
	24,787	Pf	0.05349	0.20047	0.22748	0.26653	0.28028	0.28571
σs = 105 MPa N = 104	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 20,455
	20,455	Pf	0.05349	0.20047	0.22748	0.24133	0.28143	0.28571

shows the fatigue limit stress and broken fiber fraction under a stochastic overloading stress of σ_s = 105 MPa at different occurrence applied cycle numbers. When stochastic overloading stress σ_s = 105 MPa occurs at applied cycle numbers of N_s = 10, 10², 10³, and 10⁴, the broken fiber fraction increases from P_f = 0.20047, 0.22748, 0.24133, and 0.25508 under σ_limit = 103 MPa to P_f = 0.22205, 0.25148, 0.26653, and 0.28143.

3.3. The 2D C/SiC Composite

The 2D T−300^TM carbon fiber-reinforced silicon carbide composite was fabricated using the chemical vapor infiltration (CVI) method. It contained 26 plies of plain-weave cloth in a (0°/90°) lay-up. Fiber preform was given a pyrolytic carbon coating. The fiber volume was 45%, and density was 1.93–1.98 g/cm³, and the porosity was approximately 22%. The dog-bone shaped specimens were cut from 200 mm × 200 mm panels using diamond tooling. The tension–tension fatigue tests at room temperature were conducted on a servohydraulic load-frame that was equipped with edge-loaded grips. The fatigue experiments were performed under load control at a sinusoidal wave form and a loading frequency f = 10 Hz. The fatigue load ratio (σ_min/σ_max) was R = 0.1. The average tensile strength was approximately σ_uts = 420 MPa.

Figure 4a shows experimental and predicted fatigue life S−N curves of the 2D C/SiC composite. When the fatigue limit applied cycle number is defined to be N_limit = 10⁶, the corresponding predicted fatigue limit stress is approximately σ_limit = 348 MPa (approximately 82.8%σ_uts).

Figure 4b shows the fatigue life decreasing rate versus the occurrence applied cycle number of stochastic overloading stress curves for different stochastic overloading stress levels of σ_s = 350, 355, and 360 MPa (i.e., approximately 1.005, 1.02, 1.034 fatigue limit stress). During the application of CMC components, the overloading stress level is not high, and the low overloading stress level is chosen for analysis. Under the same stochastic overloading stress level, the fatigue life decreasing rate increases with the occurrence applied cycle of stochastic overloading. Under σ_s = 350 MPa, the fatigue life decreasing rate increases from Λ = 0.03699 at N_s = 10 to Λ = 0.21481 at N_s = 10⁵; under σ_s = 355 MPa, the fatigue life decreasing rate increases from Λ = 0.12826 at N_s = 10 to Λ = 0.59226 at N_s = 10⁵; under σ_s = 360 MPa, the fatigue life decreasing rate increases from Λ = 0.21723 at N_s = 10 to Λ = 0.80517 at N_s = 10⁵. When the occurrence applied cycle is between N_s = 10 and 10², the fatigue life decreasing rate increases slowly with the occurrence applied cycle; however, when the occurrence applied cycle is between N_s = 10² and 10⁵, the fatigue life decreasing rate increases rapidly with the applied cycle. For 2D C/SiC, the fatigue damage is not sensitive to stochastic overloading stress at the initial stage of cyclic fatigue loading; however, with the applied cycles increasing, the fatigue damage extent increases, leading to the rapid increase in the fatigue life decreasing rate with the occurrence cycle number of the stochastic overloading stress.

Figure 4c–e shows the broken fiber fraction versus the applied cycle number curves for different stochastic overloading stress levels and occurrence cycle numbers. The broken fiber fraction increases when stochastic overloading stress occurs, and the difference of the broken fiber fraction between the original peak stress and stochastic overloading stress level increases with the applied cycle.

Table 4. Fatigue limit stress and broken fiber fraction of 2D C/SiC composite under stochastic overloading stress.

σmax = 348 MPa	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	1,067,612	Pf	0.02529	0.03153	0.04397	0.11358	0.1795	0.22527
σs = 350 MPa N = 10	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	1,028,121	Pf	0.02529	0.03261	0.04505	0.11466	0.18059	0.22635
σs = 350 MPa N = 102	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	1,013,262	Pf	0.02529	0.03153	0.04547	0.11508	0.18101	0.22677
σs = 350 MPa N = 103	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	937,226	Pf	0.02529	0.03153	0.04397	0.11731	0.18323	0.229
σs = 350 MPa N = 104	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	875,603	Pf	0.02529	0.03153	0.04397	0.11358	0.18516	0.23093
σs = 350 MPa N = 105	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	838,274	Pf	0.02529	0.03153	0.04397	0.11358	0.1795	0.23215
σs = 355 MPa N = 10	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	930,683	Pf	0.02529	0.03546	0.0479	0.11751	0.18342	0.2292
σs = 355 MPa N = 102	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	882,390	Pf	0.02529	0.03153	0.04941	0.11902	0.18493	0.23071
σs = 355 MPa N = 103	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	661,586	Pf	0.02529	0.03153	0.04397	0.12704	0.19295	0.23873
σs = 355 MPa N = 104	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	513,025	Pf	0.02529	0.03153	0.04397	0.11358	0.19983	0.24561
σs = 355 MPa N = 105	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	435,308	Pf	0.02529	0.03153	0.04397	0.11358	0.17949	0.24996
σs = 360 MPa N = 10	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	835,692	Pf	0.02529	0.0385	0.05094	0.12055	0.18646	0.23224
σs = 360 MPa N = 102	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	759,497	Pf	0.02529	0.03153	0.05361	0.12323	0.18914	0.23491
σs = 360 MPa N = 103	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	450,695	Pf	0.02529	0.03153	0.04397	0.13736	0.20327	0.24905
σs = 360 MPa N = 104	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	282,293	Pf	0.02529	0.03153	0.04397	0.11358	0.2153	0.26108
σs = 360 MPa N = 105	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	208,007	Pf	0.02529	0.03153	0.04397	0.11358	0.17949	0.26861

shows the fatigue limit stress and broken fiber fraction at different occurrence cycle numbers and stochastic overloading stress. When stochastic overloading stress σ_s = 350 MPa occurs at applied cycles N_s = 10, 10², 10³, 10⁴, and 10⁵, the broken fiber fraction increases from P_f = 0.03153, 0.04397, 0.11358, 0.1795, and 0.22527 under σ_limit = 348 MPa to P_f = 0.03261, 0.04547, 0.11731, 0.18516, and 0.23215; when stochastic overloading stress σ_s = 355 MPa occurs at applied cycles N_s = 10, 10², 10³, 10⁴, and 10⁵, the broken fiber fraction increases to P_f = 0.03546, 0.04941, 0.12704, 0.19983, and 0.24996; finally, when stochastic overloading stress σ_s = 360 MPa occurs at applied cycles N = 10, 10², 10³, 10⁴, and 10⁵, the broken fiber fraction increases to P_f = 0.0385, 0.05361, 0.13736, 0.2153, and 0.26861.

3.4. The 2.5D C/SiC Composite

The 2.5D T−300^TM carbon fiber-reinforced silicon carbide composite was fabricated using the chemical vapor infiltration (CVI) method. Low pressure CVI was employed to deposit a pyrolytic carbon layer and a silicon matrix. A thin pyrolytic carbon layer was deposited on the surface of the carbon fiber as the interfacial layer with C₃H₈ at 800 °C. Methyltrichlorosilane (MTS, CH₃ SiCl₃) was used as a gas source for the deposition of the SiC matrix. The conditions for deposition were 1000 °C. Argon was employed as a diluent gas to slow down the chemical reaction rate of deposition. The test specimens were machined from fabricated composites and further coated with SiC by isothermal CVI under the same conditions. The fiber volume was V_f = 0.4, and the average tensile strength was approximately σ_uts = 225 MPa. The dog-bone shaped specimens were cut from composite panels using diamond tooling. The tension–tension fatigue tests were conducted on an MTS Model 809 servo hydraulic load-frame (MTS Systems Corp., Minneapolis, MN, USA). The fatigue experiments were performed under load control at a loading frequency f = 10 Hz. The fatigue load ratio (σ_min/σ_max) was R = 0.1.

Figure 5a shows the experimental and predicted fatigue life S−N curves of the 2.5D C/SiC composite. When the fatigue limit applied cycle number is defined to be N_limit = 10⁶, the corresponding predicted fatigue limit stress is approximately σ_limit = 143 MPa (approximately 63.5%σ_uts). The fatigue limit stress of the 2.5D C/SiC composite is lower than the other CMCs, i.e., unidirectional, cross-ply, 2D, and 3D CMCs. The low fatigue limit stress of the 2.5D C/SiC composite is mainly due to yarns bending inside of composites.

Figure 5b shows the fatigue life decreasing rate versus the occurrence applied cycle number of stochastic overloading curves for different stochastic overloading stress levels of σ_s = 145, 150, and 155 MPa (i.e., 1.014, 1.049, 1.084 fatigue limit stress). During the application of CMC components, the overloading stress level is not high, and the low overloading stress level is chosen for analysis. Under the same stochastic overloading stress level, the fatigue life decreasing rate increases with the occurrence applied cycle of stochastic overloading stress. Under σ_s = 145 MPa, the fatigue life decreasing rate increases from Λ = 0.00723 at N_s = 10 to Λ = 0.11768 at N_s = 10⁵; under σ_s = 150 MPa, the fatigue life decreasing rate increases from Λ = 0.02742 at N_s = 10 to Λ = 0.3939 at N_s = 10⁵; and, under σ_s = 155 MPa, the fatigue life decreasing rate increases from Λ = 0.05087 at N_s = 10 to Λ = 0.63095 at N_s = 10⁵. For the 2.5D C/SiC, with fatigue cycles increasing, the fatigue damage extent increases, leading to the increase in the fatigue life decreasing rate with the occurrence cycle number of stochastic overloading stress.

Figure 5c–e shows the broken fiber fraction versus applied cycle curves for different stochastic overloading stress levels and occurrence applied cycle numbers. The broken fiber fraction increases when stochastic overloading stress occurs, and the difference of the broken fiber fraction between the original peak stress and stochastic overloading stress level increases with the applied cycle.

Table 5. Fatigue limit stress and broken fiber fraction of 2.5D C/SiC composite under stochastic overloading stress.

σmax = 143 MPa	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	1,012,346	Pf	0.00574	0.00767	0.01251	0.03658	0.07519	0.14038
σs = 145 MPa N = 10	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	1,005,026	Pf	0.00574	0.00834	0.01317	0.03724	0.07585	0.14105
σs = 145 MPa N = 102	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	1,000,463	Pf	0.00574	0.00767	0.01359	0.03766	0.07627	0.14146
σs = 145 MPa N = 103	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	978,311	Pf	0.00574	0.00767	0.01251	0.03969	0.0783	0.1435
σs = 145 MPa N = 104	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	944,705	Pf	0.00574	0.00767	0.01251	0.03658	0.08145	0.14664
σs = 145 MPa N = 105	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	893,212	Pf	0.00574	0.00767	0.01251	0.03658	0.07519	0.15161
σs = 150 MPa N = 10	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	984,589	Pf	0.00574	0.01021	0.01504	0.03911	0.07772	0.14292
σs = 150 MPa N = 102	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	967,500	Pf	0.00574	0.00767	0.01663	0.0407	0.07931	0.1445
σs = 150 MPa N = 103	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	886,901	Pf	0.00574	0.00767	0.01251	0.04842	0.08703	0.15223
σs = 150 MPa N = 104	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	772,084	Pf	0.00574	0.00767	0.01251	0.03658	0.09888	0.16408
σs = 150 MPa N = 105	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	613,579	Pf	0.00574	0.00767	0.01251	0.03658	0.07519	0.1825
σs = 155 MPa N = 10	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	960,848	Pf	0.00574	0.01241	0.01725	0.04132	0.07993	0.14512
σs = 155 MPa N = 102	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	929,613	Pf	0.00574	0.00767	0.02021	0.04427	0.08288	0.14808
σs = 155 MPa N = 103	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	787,301	Pf	0.00574	0.00767	0.01251	0.05864	0.09725	0.16245
σs = 155 MPa N = 104	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	599,723	Pf	0.00574	0.00767	0.01251	0.03658	0.11905	0.18425
σs = 155 MPa N = 105	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	373,608	Pf	0.00574	0.00767	0.01251	0.03658	0.07519	0.21754

shows the fatigue limit stress and broken fiber fraction at different occurrence applied cycle numbers and stochastic overloading stress. When stochastic overloading stress σ_s = 145 MPa occurs at applied cycles N_s = 10, 10², 10³, 10⁴, and 10⁵, the broken fiber fraction increases from P_f = 0.00767, 0.01251, 0.03658, 0.07519, and 0.14038 under σ_limit = 143 MPa to P_f = 0.00834, 0.01359, 0.03969, 0.08145, and 0.15161; when stochastic overloading stress σ_s = 150 MPa occurs at applied cycles N_s = 10, 10², 10³, 10⁴, and 10⁵, the broken fiber fraction increases to P_f = 0.01021, 0.01663, 0.04842, 0.09888, and 0.1825; and, when stochastic overloading stress σ_s = 155 MPa occurs at applied cycles N = 10, 10², 10³, 10⁴, and 10⁵, the broken fiber fraction increases to P_f = 0.01241, 0.02021, 0.05864, 0.11905, and 0.21754.

3.5. The 3D C/SiC Composite

The 3D T−300^TM carbon fiber-reinforced silicon carbide composite was fabricated using the chemical vapor infiltration (CVI) method. Low pressure I-CVI was employed to deposit a pyrolytic carbon layer and the silicon carbide matrix. A thin pyrolytic carbon layer was deposited on the surface of the carbon fiber as the interfacial layer with C₄H₁₀ at 950–1000 °C. The thickness of the pyrolytic carbon layer was approximately 0.2 μm. The fiber volume is V_f = 0.4, and the average tensile strength is approximately σ_uts = 276 MPa. The dog-bone shaped specimens were cut from composite panels using the diamond tooling, and then coated with a SiC coating. The tension–tension fatigue tests at room temperature were conducted on a servohydraulic mechanical testing machine. The fatigue experiments were performed under load control at a loading frequency f = 60 Hz. The fatigue load ratio (σ_min/σ_max) was R = 0.1. The loading frequency affects the fatigue life and fatigue limit stress. At room temperature, When the loading frequency increases, the fatigue limit stress also increases.

Figure 6a shows experimental and predicted fatigue life S−N curves of 3D C/SiC composite. When the fatigue limit applied cycle number is defined to be N_limit = 10⁶, the corresponding predicted fatigue limit stress is approximately σ_limit = 236 MPa (approximately 85.5%σ_uts).

Figure 6b shows the fatigue life decreasing rate versus the occurrence cycle number of stochastic overloading curves for different stochastic overloading stress levels of σ_s = 240, 245, and 250 MPa (i.e., approximately 1.017, 1.038, and 1.059 fatigue limit stress). During application of CMC components, the overloading stress level is not high, and the low overloading stress level is chosen for analysis. Under the same stochastic overloading stress level, the fatigue life decreasing rate increases with the occurrence applied cycle of stochastic overloading. Under σ_s = 240 MPa, the fatigue life decreasing rate increases from Λ = 0.22884 at N_s = 10 to Λ = 0.73423 at N_s = 10⁵; under σ_s = 245 MPa, the fatigue life decreasing rate increases from Λ = 0.46292 at N_s = 10 to Λ = 0.94456 at N_s = 10⁴; and under σ_s = 250 MPa, the fatigue life decreasing rate increases from Λ = 0.64272 at N_s = 10 to Λ = 0.98713 at N_s = 10³. When the occurrence applied cycle is between N_s = 10 and 10², the fatigue life decreasing rate increases rapidly; however, when the occurrence applied cycle is between N_s = 10² and 10⁵, the fatigue life decreasing rate increases slowly. The occurrence of stochastic overloading stress at the initial stage of cyclic fatigue loading deteriorates the fatigue damage evolution, i.e., decreasing matrix crack spacing in transverse and longitudinal yarns, increasing interface debonding length, and broken fiber fraction; however, when matrix cracking and interface wear approach a steady-state, the effect of stochastic overloading on the fatigue damage or the fatigue life decreasing rate decreases.

Figure 6c–e shows the broken fiber fraction versus the applied cycle curves for different stochastic overloading stress levels and occurrence cycle numbers. The broken fiber fraction increases when stochastic overloading stress occurs, and the difference of the broken fiber fraction between the original peak stress and stochastic overloading stress level increases with the applied cycle.

Table 6. Fatigue limit stress and broken fiber fraction of 3D C/SiC composite under stochastic overloading stress.

σmax = 236 MPa	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	1,383,192	Pf	0.03314	0.04324	0.11794	0.18509	0.2122	0.24367
σs = 240 MPa N = 10	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	1,066,666	Pf	0.03314	0.04771	0.12242	0.18957	0.21668	0.24815
σs = 240 MPa N = 102	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	696,321	Pf	0.03314	0.04324	0.12961	0.19676	0.22387	0.25534
σs = 240 MPa N = 103	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	488,262	Pf	0.03314	0.04324	0.11794	0.2026	0.22971	0.26118
σs = 240 MPa N = 104	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	426,789	Pf	0.03314	0.04324	0.11794	0.18509	0.23189	0.26336
σs = 240 MPa N = 105	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	367,609	Pf	0.03314	0.04324	0.11794	0.18509	0.2122	0.26575
σs = 245 MPa N = 10	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	742,891	Pf	0.03314	0.05383	0.12853	0.19568	0.22279	0.25426
σs = 245 MPa N = 102	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	262,035	Pf	0.03314	0.04324	0.14538	0.21253	0.23963	0.2711
σs = 245 MPa N = 103	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	107,854	Pf	0.03314	0.04324	0.11794	0.22602	0.25313	0.2846
σs = 245 MPa N = 104	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	76,689	Pf	0.03314	0.04324	0.11794	0.18509	0.25812	0.28571
σs = 250 MPa N = 10	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	494,186	Pf	0.03314	0.06055	0.13525	0.2024	0.22951	0.26098
σs = 250 MPa N = 102	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	84,191	Pf	0.03314	0.04324	0.1625	0.22966	0.25676	0.28571
σs = 250 MPa N = 103	Nf		N = 1	N = 10	N = 102	N = 103	N = 104	N = 105
	17,798	Pf	0.03314	0.04324	0.11794	0.25116	0.27827	0.28571

shows the fatigue limit stress and broken fiber fraction at different occurrence cycle numbers and stochastic overloading stress. When stochastic overloading stress σ_s = 240 MPa occurs at applied cycles N_s = 10, 10², 10³, 10⁴, and 10⁵, the broken fiber fraction increases from P_f = 0.04324, 0.11794, 0.18509, 0.2122, and 0.24367 under σ_limit = 236 MPa to P_f = 0.04771, 0.12961, 0.2026, 0.23189, and 0.26575; when stochastic overloading stress σ_s = 245 MPa occurs at applied cycles N_s = 10, 10², 10³, and 10⁴, the broken fiber fraction increases to P_f = 0.05383, 0.14538, 0.22602, and 0.25812; and, when stochastic overloading stress σ_s = 250 MPa occurs at applied cycles N = 10, 10², and 10³, the broken fiber fraction increases to P_f = 0.06055, 0.1625, and 0.25116.

4. Discussion

Figure 7 shows the fatigue life decreasing rate versus stochastic overloading stress for different occurrence cycle numbers of different C/SiC composites. The fatigue life decreasing rate increases with the stochastic overloading stress level for different fiber preforms (i.e., unidirectional, cross-ply, 2D, 2.5D, and 3D). However, with increasing applied cycles, the evolution of the fatigue life decreasing rate with stochastic overloading stress depends on the fiber preforms, which indicates that the fiber preforms affect the fatigue damage evolution process.

For N_s = 10, 10², 10³, and 10⁴, under the same stochastic overloading stress level, the fatigue life decreasing rate is the highest for the cross-ply C/SiC composite, which indicates that the fiber preform of the cross-ply is very sensitive to the stochastic overloading stress.

For N_s = 10, 10², 10³, 10⁴, and 10⁵, under the same stochastic overloading stress level, the fatigue life decreasing rate is the lowest for the 2.5D C/SiC composite, which indicates that the fiber preform of 2.5D has high resistance to the stochastic overloading stress.

Among UD, 2D, and 3D C/SiC composites, at the initial stage of cyclic fatigue loading, i.e., N_s = 10, under the same stochastic overloading stress, the fatigue life decrease rate of the 3D C/SiC is the highest; however, with the increasing applied cycle number, the fatigue life decreasing rate of the UD C/SiC composite is the highest under the same stochastic overloading stress.

5. Conclusions

In this paper, a micromechanical fatigue limit stress model of fiber-reinforced CMCs subjected to stochastic overloading stress is developed. The fatigue limit stress for different C/SiC composites is predicted. The relationships between fatigue life decreasing rate, stochastic overloading stress and corresponding occurrence cycle number, and broken fiber fraction are analyzed.

Under the same stochastic overloading stress level, the fatigue life decreasing rate increases with the stochastic overloading stress level and occurrence applied cycle of stochastic overloading for different fiber preforms. The broken fiber fraction increases when the stochastic overloading stress occurs, and the difference of the broken fiber fraction between the fatigue limit stress and stochastic overloading stress level increases with the applied cycle.

Under the same stochastic overloading stress level and occurrence applied cycle, the fatigue life decreasing rate is the highest for the cross-ply C/SiC composite, and lowest for the 2.5D C/SiC composite.

For UD, CP, and 3D C/SiC composites, when the applied cycle is between N_s = 10 and 10², the fatigue life decreasing rate increases rapidly with the occurrence applied cycle; however, when the applied cycle is higher than N_s = 10², the fatigue life decreasing rate increases slowly with the applied cycle.

For the 2D and 2.5D C/SiC composites, when the applied cycle is between N_s = 10 and 10², the fatigue life decreasing rate increases slowly with the occurrence applied cycle; however, when the applied cycle is between N_s = 10² and 10⁵, the fatigue life decreasing rate increases rapidly with the applied cycle.