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

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 σs occurred at different applied cycle numbers. In the present analysis, the overloading stress σs remains the same at applied cycle numbers N1, N2, and N3.

Figure 1

Diagram of stochastic overloading stress under cyclic fatigue loading.

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]. where Vf is the fiber volume, Φs is the intact fiber stress under stochastic overloading stress, Φb is the stress carried by broken fiber, and Pf is the fiber failure probability, and can be determined by Equation (2). where m is the fiber Weibull modulus, and σc 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. 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).

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 Nlimit.

The fatigue life decreasing rate is defined by Equation (7). where Nf(σlimit) is the fatigue failure cycle number under fatigue limit stress, and Nf(σs) 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. 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.

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
τio3/(MPa)	8	6.2	25	20	20
τimin4/(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

1σuts is composite tensile strength; 2 rf is the fiber radius;.3 τio is the interface shear stress upon initial loading; 4 τimin 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.

3.1. Unidirectional C/SiC Composite

The unidirectional T−700TM 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 (σmin/σmax) 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 Nlimit = 106, the corresponding predicted fatigue limit stress is approximately σlimit = 241 MPa (approximately 89.2%σuts).

Figure 2

(a) Experimental and predicted fatigue life S−N curves; (b) the fatigue life decreasing rate versus occurrence applied cycle number curve for different stochastic loading stress levels; (c) the broken fiber fraction versus applied cycle number curves under σlimit = 241 MPa and stochastic overloading stress σs = 245 MPa at Ns = 10, 102, 103, 104, and 105; (d) the broken fiber fraction versus applied cycle number curves under σlimit = 241 MPa and stochastic overloading stress σs = 250 MPa at Ns = 10, 102, 103, 104, and 105; and, (e) the broken fiber fraction versus applied cycle number curves under σlimit = 241 MPa and stochastic overloading stress σs = 255 MPa at Ns = 10, 102, and 103 of unidirectional C/SiC composite.

Figure 2b shows the fatigue life decreasing rate versus the occurrence cycle number of stochastic overloading curves for different stochastic overloading stress levels of σs = 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 σs = 245, 250, and 255 MPa, the fatigue life decreasing rate increases with the occurrence applied cycle of stochastic overloading (i.e., Ns = 10, 102, 103, 104, and 105). Under σs = 245 MPa, the fatigue life decreasing rate increases from Λ = 0.14392 at Ns = 10 to Λ = 0.8074 at Ns = 105; under σs = 250 MPa, the fatigue life decreasing rate increases from Λ = 0.3091 at Ns = 10 to Λ = 0.97479 at Ns = 104; and under σs = 255 MPa, the fatigue life decreasing rate increases from Λ = 0.4559 at Ns = 10 to Λ = 0.9966 at Ns = 103. When the applied cycle is between Ns = 10 and 102, the fatigue life decreasing rate increases rapidly with the occurrence applied cycle; however, when the applied cycle is higher than Ns = 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 shows the fatigue limit stress and broken fiber fraction at different occurrence cycle numbers and stochastic overloading stress. When stochastic overloading stress σs = 245 MPa occurs at applied cycles Ns = 10, 102, 103, 104, and 105, the broken fiber fraction increases from Pf = 0.02113, 0.17991, 0.20077, 0.22431, and 0.25085 under σlimit = 241 MPa to Pf = 0.02329, 0.19662, 0.21915, 0.2445, and 0.27298; when stochastic overloading stress σs = 250 MPa occurs at applied cycles Ns = 10, 102, 103, and 104, the broken fiber fraction increases to Pf = 0.02626, 0.21897, 0.24365, and 0.27131; finally, when stochastic overloading stress σs = 255 MPa occurs at applied cycles N = 10, 102, and 103, the broken fiber fraction increases to Pf = 0.02952, 0.24295, and 0.26984.

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

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

3.2. Cross-Ply C/SiC Composite

The cross-ply T−700TM 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 Vf = 0.4, and the average tensile strength is approximately σuts = 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 (σmin/σmax) 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 Nlimit = 106, the corresponding predicted fatigue limit stress is approximately σlimit = 103 MPa (approximately 83%σuts).

Figure 3

(a) Experimental and predicted fatigue life S−N curves; (b) the fatigue life decreasing rate versus occurrence applied cycle curve; and, (c) the broken fiber fraction versus applied cycle curves under σlimit = 103 MPa and stochastic overloading stress σs = 105 MPa at Ns = 10, 102, 103, and 104 of cross-ply C/SiC composite.

Figure 3b shows the fatigue life decreasing rate versus the occurrence applied cycle number of stochastic overloading stress curves under σs = 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 σs = 105 MPa, the fatigue life decreasing rate increases from Λ = 0.97114 at Ns = 10 to Λ = 0.98696 at Ns = 104. When the occurrence applied cycle number is between Ns = 10 and 102, the fatigue life decreasing rate increases rapidly; however, when the occurrence applied cycle is between Ns = 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., Ns = 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 shows the fatigue limit stress and broken fiber fraction under a stochastic overloading stress of σs = 105 MPa at different occurrence applied cycle numbers. When stochastic overloading stress σs = 105 MPa occurs at applied cycle numbers of Ns = 10, 102, 103, and 104, the broken fiber fraction increases from Pf = 0.20047, 0.22748, 0.24133, and 0.25508 under σlimit = 103 MPa to Pf = 0.22205, 0.25148, 0.26653, and 0.28143.

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	P f	0.05349	0.20047	0.22748	0.24133	0.25508	0.26892
σs = 105 MPaN = 10	N f		N = 1	N = 10	N = 102	N = 103	N = 104	N = 45,256
	45,256	P f	0.05349	0.22205	0.24906	0.26291	0.27666	0.28571
σs = 105 MPaN = 102	N f		N = 1	N = 10	N = 102	N = 103	N = 104	N = 30,243
	30,243	P f	0.05349	0.20047	0.25148	0.26533	0.27908	0.28571
σs = 105 MPaN = 103	N f		N = 1	N = 10	N = 102	N = 103	N = 104	N = 24,787
	24,787	P f	0.05349	0.20047	0.22748	0.26653	0.28028	0.28571
σs = 105 MPaN = 104	N f		N = 1	N = 10	N = 102	N = 103	N = 104	N = 20,455
	20,455	P f	0.05349	0.20047	0.22748	0.24133	0.28143	0.28571

3.3. The 2D C/SiC Composite

The 2D T−300TM 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 (σmin/σmax) was R = 0.1. The average tensile strength was approximately σuts = 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 Nlimit = 106, the corresponding predicted fatigue limit stress is approximately σlimit = 348 MPa (approximately 82.8%σuts).

Figure 4

(a) Experimental and predicted fatigue life S−N curves; (b) the fatigue life decreasing rate versus occurrence applied cycle curve for different stochastic overloading stress levels; (c) the broken fiber fraction versus applied cycle curves under σlimit = 348 MPa and stochastic overloading stress σs = 350 MPa at Ns = 10, 102, 103, 104, and 105; (d) the broken fiber fraction versus applied cycle curves under σlimit = 348 MPa and stochastic overloading stress σs = 355 MPa at Ns = 10, 102, 103, 104, and 105; and, (e) the broken fiber fraction versus applied cycle curves under σlimit = 348 MPa and stochastic overloading stress σs = 360 MPa at Ns = 10, 102, 103, 104, and 105 of 2D C/SiC composite.

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 σs = 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 σs = 350 MPa, the fatigue life decreasing rate increases from Λ = 0.03699 at Ns = 10 to Λ = 0.21481 at Ns = 105; under σs = 355 MPa, the fatigue life decreasing rate increases from Λ = 0.12826 at Ns = 10 to Λ = 0.59226 at Ns = 105; under σs = 360 MPa, the fatigue life decreasing rate increases from Λ = 0.21723 at Ns = 10 to Λ = 0.80517 at Ns = 105. When the occurrence applied cycle is between Ns = 10 and 102, the fatigue life decreasing rate increases slowly with the occurrence applied cycle; however, when the occurrence applied cycle is between Ns = 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 shows the fatigue limit stress and broken fiber fraction at different occurrence cycle numbers and stochastic overloading stress. When stochastic overloading stress σs = 350 MPa occurs at applied cycles Ns = 10, 102, 103, 104, and 105, the broken fiber fraction increases from Pf = 0.03153, 0.04397, 0.11358, 0.1795, and 0.22527 under σlimit = 348 MPa to Pf = 0.03261, 0.04547, 0.11731, 0.18516, and 0.23215; when stochastic overloading stress σs = 355 MPa occurs at applied cycles Ns = 10, 102, 103, 104, and 105, the broken fiber fraction increases to Pf = 0.03546, 0.04941, 0.12704, 0.19983, and 0.24996; finally, when stochastic overloading stress σs = 360 MPa occurs at applied cycles N = 10, 102, 103, 104, and 105, the broken fiber fraction increases to Pf = 0.0385, 0.05361, 0.13736, 0.2153, and 0.26861.

Table 4

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

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

3.4. The 2.5D C/SiC Composite

The 2.5D T−300TM 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 Vf = 0.4, and the average tensile strength was approximately σuts = 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 (σmin/σmax) 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 Nlimit = 106, the corresponding predicted fatigue limit stress is approximately σlimit = 143 MPa (approximately 63.5%σuts). 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 5

(a) Experimental and predicted fatigue life S−N curves; (b) the fatigue life decreasing rate versus occurrence applied cycle curve for different stochastic overloading stress levels; (c) the broken fiber fraction versus applied cycle curves under σlimit = 143 MPa and stochastic overloading stress of σs = 145 MPa at Ns = 10, 102, 103, 104, and 105; (d) the broken fiber fraction versus applied cycle curves under σlimit = 143 MPa and stochastic overloading stress of σs = 150 MPa at Ns = 10, 102, 103, 104, and 105; and, (e) the broken fiber fraction versus applied cycle curves under σlimit = 143 MPa and stochastic overloading stress of σs = 155 MPa at Ns = 10, 102, 103, 104, and 105 of 2.5D C/SiC composite.

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 σs = 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 σs = 145 MPa, the fatigue life decreasing rate increases from Λ = 0.00723 at Ns = 10 to Λ = 0.11768 at Ns = 105; under σs = 150 MPa, the fatigue life decreasing rate increases from Λ = 0.02742 at Ns = 10 to Λ = 0.3939 at Ns = 105; and, under σs = 155 MPa, the fatigue life decreasing rate increases from Λ = 0.05087 at Ns = 10 to Λ = 0.63095 at Ns = 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 shows the fatigue limit stress and broken fiber fraction at different occurrence applied cycle numbers and stochastic overloading stress. When stochastic overloading stress σs = 145 MPa occurs at applied cycles Ns = 10, 102, 103, 104, and 105, the broken fiber fraction increases from Pf = 0.00767, 0.01251, 0.03658, 0.07519, and 0.14038 under σlimit = 143 MPa to Pf = 0.00834, 0.01359, 0.03969, 0.08145, and 0.15161; when stochastic overloading stress σs = 150 MPa occurs at applied cycles Ns = 10, 102, 103, 104, and 105, the broken fiber fraction increases to Pf = 0.01021, 0.01663, 0.04842, 0.09888, and 0.1825; and, when stochastic overloading stress σs = 155 MPa occurs at applied cycles N = 10, 102, 103, 104, and 105, the broken fiber fraction increases to Pf = 0.01241, 0.02021, 0.05864, 0.11905, and 0.21754.

Table 5

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

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

3.5. The 3D C/SiC Composite

The 3D T−300TM 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 Vf = 0.4, and the average tensile strength is approximately σuts = 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 (σmin/σmax) 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 Nlimit = 106, the corresponding predicted fatigue limit stress is approximately σlimit = 236 MPa (approximately 85.5%σuts).

Figure 6

(a) Experimental and predicted fatigue life S−N curves; (b) the fatigue life decreasing rate versus occurrence applied cycle curve for different stochastic overloading stress levels; (c) the broken fiber fraction versus applied cycle curves under σlimit = 236 MPa and stochastic overloading stress of σs = 240 MPa at Ns = 10, 102, 103, 104, and 105; (d) the broken fiber fraction versus applied cycle curves under σlimit = 236 MPa and stochastic overloading stress of σs = 245 MPa at Ns = 10, 102, 103, and 104; and, (e) the broken fiber fraction versus applied cycle curves under σlimit = 236 MPa and stochastic overloading stress of σs = 250 MPa at Ns = 10, 102, and 103 of 3D C/SiC composite.

Figure 6b shows the fatigue life decreasing rate versus the occurrence cycle number of stochastic overloading curves for different stochastic overloading stress levels of σs = 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 σs = 240 MPa, the fatigue life decreasing rate increases from Λ = 0.22884 at Ns = 10 to Λ = 0.73423 at Ns = 105; under σs = 245 MPa, the fatigue life decreasing rate increases from Λ = 0.46292 at Ns = 10 to Λ = 0.94456 at Ns = 104; and under σs = 250 MPa, the fatigue life decreasing rate increases from Λ = 0.64272 at Ns = 10 to Λ = 0.98713 at Ns = 103. When the occurrence applied cycle is between Ns = 10 and 102, the fatigue life decreasing rate increases rapidly; however, when the occurrence applied cycle is between Ns = 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 shows the fatigue limit stress and broken fiber fraction at different occurrence cycle numbers and stochastic overloading stress. When stochastic overloading stress σs = 240 MPa occurs at applied cycles Ns = 10, 102, 103, 104, and 105, the broken fiber fraction increases from Pf = 0.04324, 0.11794, 0.18509, 0.2122, and 0.24367 under σlimit = 236 MPa to Pf = 0.04771, 0.12961, 0.2026, 0.23189, and 0.26575; when stochastic overloading stress σs = 245 MPa occurs at applied cycles Ns = 10, 102, 103, and 104, the broken fiber fraction increases to Pf = 0.05383, 0.14538, 0.22602, and 0.25812; and, when stochastic overloading stress σs = 250 MPa occurs at applied cycles N = 10, 102, and 103, the broken fiber fraction increases to Pf = 0.06055, 0.1625, and 0.25116.

Table 6

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

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

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.

Figure 7

Fatigue life decreasing rate versus stochastic overloading stress for different occurrence applied cycle of (a) Ns = 10; (b) Ns = 102; (c) Ns = 103; (d) Ns = 104; and, (e) Ns = 105.

For Ns = 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 Ns = 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., Ns = 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 Ns = 10 and 102, the fatigue life decreasing rate increases rapidly with the occurrence applied cycle; however, when the applied cycle is higher than Ns = 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 Ns = 10 and 102, the fatigue life decreasing rate increases slowly with the occurrence applied cycle; however, when the applied cycle is between Ns = 102 and 105, the fatigue life decreasing rate increases rapidly with the applied cycle.