Synergistic Effects of Stress-Rupture and Cyclic Loading on Strain Response of Fiber-Reinforced Ceramic-Matrix Composites at Elevated Temperature in Oxidizing Atmosphere

In this paper, the synergistic effects of stress rupture and cyclic loading on the strain response of fiber-reinforced ceramic-matrix composites (CMCs) at elevated temperature in air have been investigated. The stress-strain relationships considering interface wear and interface oxidation in the interface debonded region under stress rupture and cyclic loading have been developed to establish the relationship between the peak strain, the interface debonded length, the interface oxidation length and the interface slip lengths. The effects of the stress rupture time, stress levels, matrix crack spacing, fiber volume fraction and oxidation temperature on the peak strain and the interface slip lengths have been investigated. The experimental fatigue hysteresis loops, interface slip lengths, peak strain and interface oxidation length of cross-ply SiC/MAS (magnesium alumino-silicate, MAS) composite under cyclic fatigue and stress rupture at 566 and 1093 °C in air have been predicted.


Introduction
Ceramic materials possess high strength and modulus at elevated temperature. However, their use as structural components is severely limited because of their brittleness. Continuous fiber-reinforced ceramic-matrix composites, by incorporating fibers in ceramic matrices, however, not only exploit their attractive high-temperature strength but also reduce the propensity for catastrophic failure [1]. The critical nature of the application of these advanced materials makes complete characterization necessary. The designers must have information pertaining to not only the strength of the material, but also its fatigue and toughness characteristics. Since ceramic-matrix composites (CMCs) have applications typically in the aerospace industry, these characteristics are especially important due to the severe operating environments encountered and the lower safety factors imposed by the weight considerations. Typically, the tests performed are monotonic loading, cyclic fatigue, and stress rupture at a variety of temperatures and/or atmosphere conditions. Most of the tests are conducted independently. However, these do not accurately represent the loading conditions encountered by an aircraft component. For example, a wing spar will encounter stress rupture-type loading through the flight due to the weight of the airframe it is supporting. It will also encounter cyclic fatigue loading due to the mechanical vibrations from the engines and aerodynamic forces. The temperature will change dramatically with changes in the altitude and flight Mach number. However, the airframe as a whole will not suffer any of these stresses independently, and there will always be some combination of these stresses acting on it at any time. It is possible that the cumulative damage caused to a component

Stress Analysis
When the applied stress σ is higher than the matrix cracking stress, matrix multicracking and interface debonding occur. The unit cell containing a single fiber surrounded by a hollow cylinder of matrix is extracted from the ceramic composite system, as shown in Figure 1. The fiber radius is r f , and the matrix radius is R (R = r f /V f 1/2 ). The length of the unit cell is half the matrix crack spacing l c /2. Budiansky et al. [11] assumed that the matrix axial load is concentrated at R, which is an effective radius such that the region between r f and R only carries the shear stress. At elevated temperatures, matrix cracks will serve as avenues for the ingress of the environment atmosphere into the composite, as shown in Figure 2. The oxygen reacts with the carbon layer along the fiber length at a certain rate of dξ/dt, in which ξ is the length of the carbon lost in each side of the crack [12].
where ϕ 1 and ϕ 2 are parameters dependent on the temperature and described using the Arrhenius-type laws; b is a delay factor considering the deceleration of reduced oxygen activity [12].
Materials 2017, 10, 182 3 of 18 matrix is extracted from the ceramic composite system, as shown in Figure 1. The fiber radius is rf, and the matrix radius is R (R = rf/Vf 1/2 ). The length of the unit cell is half the matrix crack spacing lc/2. Budiansky et al. [11] assumed that the matrix axial load is concentrated at R , which is an effective radius such that the region between rf and R only carries the shear stress. At elevated temperatures, matrix cracks will serve as avenues for the ingress of the environment atmosphere into the composite, as shown in Figure 2. The oxygen reacts with the carbon layer along the fiber length at a certain rate of dξ/dt, in which ξ is the length of the carbon lost in each side of the crack [12].
where φ1 and φ2 are parameters dependent on the temperature and described using the Arrhenius-type laws; b is a delay factor considering the deceleration of reduced oxygen activity [12].  Under cyclic loading, the interface shear stress decreases due to interface wear. The interface debonded region can be divided into two regions, including: 1. The interface oxidation region, i.e., x[0, ξ], where the stress transfer between the fiber and the matrix is controlled by a sliding stress τi(x) = τf. matrix is extracted from the ceramic composite system, as shown in Figure 1. The fiber radius is rf, and the matrix radius is R (R = rf/Vf 1/2 ). The length of the unit cell is half the matrix crack spacing lc/2. Budiansky et al. [11] assumed that the matrix axial load is concentrated at R , which is an effective radius such that the region between rf and R only carries the shear stress. At elevated temperatures, matrix cracks will serve as avenues for the ingress of the environment atmosphere into the composite, as shown in Figure 2. The oxygen reacts with the carbon layer along the fiber length at a certain rate of dξ/dt, in which ξ is the length of the carbon lost in each side of the crack [12].
where φ1 and φ2 are parameters dependent on the temperature and described using the Arrhenius-type laws; b is a delay factor considering the deceleration of reduced oxygen activity [12].  Under cyclic loading, the interface shear stress decreases due to interface wear. The interface debonded region can be divided into two regions, including: 1. The interface oxidation region, i.e., x[0, ξ], where the stress transfer between the fiber and the matrix is controlled by a sliding stress τi(x) = τf. Under cyclic loading, the interface shear stress decreases due to interface wear. The interface debonded region can be divided into two regions, including: 1.
The interface oxidation region, i.e., x∈[0, ξ], where the stress transfer between the fiber and the matrix is controlled by a sliding stress τ i (x) = τ f . 2.
The interface wear region, i.e., x∈[ξ, l d ], where the stress transfer between the fiber and the matrix is controlled by a sliding stress τ i (x) = τ i (N), in which τ i (N) denotes the interface shear stress at the Nth applied cycle [13].
where τ 0 denotes the initial interface shear stress; τ s denotes the steady-state interface shear stress; b 0 is a coefficient; and j is an exponent which determines the rate at which interface shear stress drops with the number of cycles N.
The axial stress distributions of the fiber, the matrix and the interface shear stress in the interface oxidation region (x∈[0, ξ]), the interface wear region (x∈[ξ, l d ]) and the interface bonded region (x∈[l d , l c /2]) are given by Equation (3).
where V m denotes the matrix volume fraction; ρ denotes the shear-lag model parameter; and σ fo and σ mo denote the fiber and matrix axial stress in the interface bonded region, respectively.
where E f , E m and E c denote the fiber, matrix and composite elastic modulus, respectively; α f , α m and α c denote the fiber, matrix and composite thermal expansion coefficients, respectively; and ∆T denotes the temperature difference between the fabricated temperature T 0 and room temperature T 1 (∆T = T 1 − T 0 ).

Matrix Multicracking
Upon loading of CMCs, cracks typically initiate within the matrix since the strain-to-failure of the matrix is usually less than that of the fiber. With increasing applied stress, the matrix cracking density increases and eventually approaches saturation. The brittle nature of the matrix material and the possible formation of the initial crack distribution throughout the microstructure suggest that a statistical approach to matrix multicracking evolution is warranted in CMCs. The tensile strength of the brittle matrix is assumed to be described by the two-parameter Weibull distribution where the probability of the matrix failure P m is determined by Equation (5) [14].
where σ R denotes the matrix cracking characteristic strength; σ mc denotes the first matrix cracking stress; σ th denotes the matrix thermal residual stress; and m denotes the matrix Weibull modulus.
To estimate the instantaneous matrix crack space with the increase of applied stress, it leads to the form of Equation (6). where where Λ denotes the final nominal crack space. The final nominal crack space versus the matrix Weibull modulus is simulated by the Monte Carlo simulation method when σ mc /σ R = 0, 0.5, 0.75 and σ th /σ R = 0, 0.1, 0.2 are plotted in Figure 3; δ R denotes the characteristic interface sliding length.
Upon loading of CMCs, cracks typically initiate within the matrix since the strain-to-failure of the matrix is usually less than that of the fiber. With increasing applied stress, the matrix cracking density increases and eventually approaches saturation. The brittle nature of the matrix material and the possible formation of the initial crack distribution throughout the microstructure suggest that a statistical approach to matrix multicracking evolution is warranted in CMCs. The tensile strength of the brittle matrix is assumed to be described by the two-parameter Weibull distribution where the probability of the matrix failure Pm is determined by Equation (5) [14].
where σR denotes the matrix cracking characteristic strength; σmc denotes the first matrix cracking stress; σth denotes the matrix thermal residual stress; and m denotes the matrix Weibull modulus.
To estimate the instantaneous matrix crack space with the increase of applied stress, it leads to the form of Equation (6).
where Λ denotes the final nominal crack space. The final nominal crack space versus the matrix Weibull modulus is simulated by the Monte Carlo simulation method when σmc/σR = 0, 0.5, 0.75 and σth/σR = 0, 0.1, 0.2 are plotted in Figure 3; δR denotes the characteristic interface sliding length.

Interface Debonding
When the matrix crack propagates to the fiber/matrix interface, it deflects along the interface. The fracture mechanics approach is adopted in the present analysis. The interface debonding criterion is given by Equation (10) [15].
where ζ d denotes the interface debonded energy; F( = πr f 2 σ/V f ) denotes the fiber load at the matrix cracking plane; w f (0) denotes the fiber axial displacement on the matrix cracking plane; and v(x) denotes the relative displacement between the fiber and the matrix. The axial displacements of the fiber and the matrix, i.e., w f (x) and w m (x), are given by Equation (11).
The relative displacement between the fiber and the matrix, i.e., v(x), is determined by Equation (12).
Substituting w f (x = 0) and v(x) into Equation (10), it leads to the form of Equation (13).
Solving Equation (13), the interface debonded length l d is given by Equation (14).

Stress-Strain Relationship
Under cyclic fatigue loading at elevated temperatures, the interface wear and interface oxidation will affect the degradation of the interface shear stress, the interface debonding and slipping length, and the strain response of fiber-reinforced CMCs. Based on interface debonding and interface slipping between the fiber and the matrix inside of the composite, the interface debonding and slipping can be divided into four different cases, including: Case 1: the interface oxidation region and the interface wear region are less than the matrix crack spacing, and the interface counter-slip upon unloading and the interface new-slip upon reloading are equal to the interface debonded length.

2.
Case 2: the interface oxidation region and the interface wear region are less than the matrix crack spacing, and the interface counter-slip upon unloading and the interface new-slip upon reloading are less than the interface debonded length.

3.
Case 3: the interface oxidation region and the interface wear region are equal to the matrix crack spacing, and the interface counter-slip upon unloading and the interface new-slip upon reloading are less than the matrix crack spacing.

4.
Case 4: the interface oxidation region and the interface wear region are equal to the matrix crack spacing, and the interface counter-slip upon unloading and the interface new-slip upon reloading are equal to the matrix crack spacing.

Case 1
When the interface oxidation region and the interface wear region are less than matrix crack spacing, upon unloading, the interface debonded region can be divided into three regions, i.e., the interface counter-slip region with the interface shear stress of τ f (x∈[0, ξ]), the interface counter-slip region with the interface shear stress of τ i (N) (x∈[ξ, y]) and the interface slip region with the interface shear stress of τ i (N) (x∈[y, l d ]), in which y denotes the interface counter-slip length. Upon unloading to the unloading transition stress of σ tr_pu , the interface counter-slip length approaches the interface debonded length, i.e., y(σ = σ tr_pu ) = l d . When σ < σ tr_pu , counter-slip occurs over the entire interface debonded region, i.e., y(σ < σ tr_pu ) = l d . The unloading strain is divided into two regions, as shown in Equation (15).
Upon reloading, the interface debonded region can be divided into four regions, i.e., the interface new-slip region with the interface shear stress of τ f (x∈[0, z]), the interface counter-slip region with the interface shear stress of τ f (x∈[z, ξ]), the interface counter-slip region with the interface shear stress of τ i (N) (x∈[ξ, y]) and the interface slip region with the interface shear stress of τ i (N) (x∈[y, l d ]). Upon reloading to the reloading transition stress of σ tr_pr , the interface new-slip length approaches the interface debonded length, i.e., z(σ tr_pr ) = l d . When σ > σ tr_pr , new-slip occurs over the entire interface debonded length, i.e., z(σ > σ tr_pr ) = l d . The reloading strain is divided into two regions, as shown in Equation (17). where

Case 2
When the interface oxidation region and the interface wear region are less than the matrix crack spacing, upon unloading to the fatigue valley stress, the interface counter-slip length is less than the interface debonded length, i.e., y(σ = σ min ) < l d , and the unloading strain is determined by Equation (15a). Upon reloading to the fatigue peak stress, the interface new-slip length is less than the interface debonded length, z(σ = σ max ) < l d , and the reloading strain is determined by Equation (17a).

Case 3
When the interface oxidation region and the interface wear region are equal to the matrix crack spacing, upon unloading to the fatigue valley stress, the interface counter-slip length is less than half the matrix crack spacing, i.e., y(σ = σ min ) < l c /2. The unloading strain is determined by Equation (19).
Upon reloading to the fatigue peak stress, the interface new-slip length is less than half the matrix crack spacing, i.e., z(σ = σ max ) < l c /2. The reloading strain is determined by Equation (21).

Case 4
When the interface oxidation region and the interface wear region are equal to the matrix crack spacing, upon unloading to the transition stress of σ tr_fu , the interface counter-slip length approaches half the matrix crack spacing, i.e., y(σ = σ tr_fu ) = l c /2. When σ > σ tr_fu , the interface counter-slip length is less than half the matrix crack spacing, i.e., y(σ > σ tr_fu ) < l c /2, and the unloading strain is determined by Equation (19). When σ < σ tr_fu , the unloading interface counter-slip occurs over the entire matrix crack spacing, i.e., y(σ < σ tr_fu ) = l c /2, and the unloading strain is determined by Equation (23).

Discussions
Under cyclic loading at elevated temperatures, there are two types of loading sequences considered, as shown in Figure 4, including: 1.
Case 1: cyclic fatigue loading without stress rupture, and the interface debonding and frictional slipping between the fiber and the matrix are mainly affected by interface wear.

2.
Case 2: cyclic fatigue loading with stress rupture, and the interface debonding and frictional slipping between the fiber and the matrix are mainly affected by interface oxidation.
The synergistic effects of stress rupture and cyclic loading on the strain response of fiber-reinforced CMCs have been investigated, considering different fatigue peak stresses, matrix crack spacings, fiber volume fractions, oxidation temperatures and stress rupture times. The ceramic composite system of unidirectional SiC/MAS [16] was used for the case study and its basic material properties are given by:  The interface shear stress versus cycle number curve is illustrated in Figure 5, in which the parameters of the interface shear stress degradation model are given by: τ0 = 20 MPa, τs = 5 MPa, b0 = 4 and j = 0.27. The interface shear stress degrades from 20 MPa at the first applied cycle to 7.8 MPa at the 1000th applied cycle.  The interface shear stress versus cycle number curve is illustrated in Figure 5, in which the parameters of the interface shear stress degradation model are given by: τ0 = 20 MPa, τs = 5 MPa, b0 = 4 and j = 0.27. The interface shear stress degrades from 20 MPa at the first applied cycle to 7.8 MPa at the 1000th applied cycle.

Effect of Stress Rupture Time
The peak strain εmax, the interface debonded length 2ld/lc, and the interface oxidation length ξ/ld versus the cycle number curves under σmax = 200 MPa, corresponding to different stress rupture times of t = 1, 5 and 10 s at the oxidation temperature of Tem = 800 °C, are illustrated in Figure 6.
With the increasing stress rupture time, the interface slip lengths increase with applied cycles due to interface oxidation, leading to the increased peak strain.
With the increasing stress rupture time, the interface slip lengths increase with applied cycles due to interface oxidation, leading to the increased peak strain.

Effect of Stress Levels
The peak strain ε max , the interface debonded length 2l d /l c , and the interface oxidation length ξ/l d versus the cycle number curves under σ max = 150 and 250 MPa with a stress rupture time of t = 10 s at the oxidation temperature of Tem = 800 • C are illustrated in Figure 7.

Effect of Matrix Crack Spacing
The peak strain εmax, the interface debonded length 2ld/lc, and the interface oxidation length ξ/ld versus the cycle number curves corresponding to different matrix crack spacings of lc = 200 and 300 μm under σmax = 200 MPa with a stress rupture time of t = 10 s at the oxidation temperature of Tem = 800 °C are illustrated in Figure 8.
With the increase of the fatigue peak stress, the interface slip lengths increase, leading to the increase of the peak strain.

Effect of Matrix Crack Spacing
The peak strain ε max , the interface debonded length 2l d /l c , and the interface oxidation length With the increasing matrix crack spacing, the interface slip lengths decrease, leading to the decrease of the peak strain.

Effect of Fiber Volume Content
The peak strain εmax, the interface debonded length 2ld/lc, and the interface oxidation length ξ/ld versus the cycle number curves corresponding to different fiber volume fractions of Vf = 35% and 45% under σmax = 200 MPa with a stress rupture time of t = 10 s at the oxidation temperature of Tem = 800 °C are illustrated in Figure 9.
With the increasing matrix crack spacing, the interface slip lengths decrease, leading to the decrease of the peak strain.

Effect of Fiber Volume Content
The peak strain ε max , the interface debonded length 2l d /l c , and the interface oxidation length ξ/l d versus the cycle number curves corresponding to different fiber volume fractions of V f = 35% and 45% under σ max = 200 MPa with a stress rupture time of t = 10 s at the oxidation temperature of Tem = 800 • C are illustrated in Figure 9.

Effect of Oxidation Temperature
The peak strain εmax, the interface debonded length 2ld/lc, and the interface oxidation length ξ/ld versus the cycle number curves corresponding to different oxidation temperatures of Tem = 700 °C and 900 °C under σmax = 200 MPa with a stress rupture time of t = 10 s are illustrated in Figure 10.
With the increase of the fiber volume fraction, the interface slip lengths decrease, leading to the decrease of the peak strain.

Effect of Oxidation Temperature
The peak strain ε max , the interface debonded length 2l d /l c , and the interface oxidation length ξ/l d versus the cycle number curves corresponding to different oxidation temperatures of Tem = 700 • C and 900 • C under σ max = 200 MPa with a stress rupture time of t = 10 s are illustrated in Figure 10.
With increasing the oxidation temperature, the interface slip lengths increase, leading to the increase of the peak strain.

Experimental Comparisons
Grant [16] investigated the stress rupture and cyclic fatigue behavior of cross-ply SiC/MAS composite at elevated temperatures in air. The fatigue hysteresis loops, the interface slip lengths, the peak strain and the interface oxidation lengths of cross-ply SiC/MAS composite under different fatigue peak stresses and stress rupture times at elevated temperatures were predicted using the present analysis.

Experimental Comparisons
Grant [16] investigated the stress rupture and cyclic fatigue behavior of cross-ply SiC/MAS composite at elevated temperatures in air. The fatigue hysteresis loops, the interface slip lengths, the peak strain and the interface oxidation lengths of cross-ply SiC/MAS composite under different fatigue peak stresses and stress rupture times at elevated temperatures were predicted using the present analysis.

Strain Response under Cyclic Fatigue and Stress Rupture at 566 • C in Air
The experimental and theoretical fatigue hysteresis loops under the fatigue peak stress of σ max = 138 MPa with the stress rupture time t = 10 s are given in Figure 11a. The fatigue hysteresis loops at the cycle numbers of N = 1, 133 and 265 corresponded to the interface slip Cases 3, 3 and 4, respectively. The residual strain increased with applied cycles, and the area of fatigue hysteresis loops decreased with the cycle number. The interface slip lengths, i.e., the unloading interface counter-slip length and the reloading interface new-slip length, increased with the cycle number, i.e., from 2y/l c = 2z/l c = 0.33 at the first applied cycle to 2y/l c = 2z/l c = 1.0 at the 265th applied cycle, as shown in Figure 11b. The experimental and theoretical peak strains versus the cycle number curves are illustrated in Figure 11c. The experimental peak strain increased from 0.34% at the first applied cycle to 0.446% at the 265th applied cycle; the theoretical peak strain increased from 0.34% at the first applied cycle to the peak value of 0.463% at the 677th applied cycle, i.e., the A-B part in Figure 11c, corresponding to the interface completely debonding and the interface partially oxidizing, i.e., the A-B part in Figure 11d. It remained constant at the value 0.463% with increasing applied cycles, i.e., the B-C part in Figure 11c, corresponding to the interface completely debonding and the interface completely oxidizing, i.e., the B-C part in Figure 11d. The theoretical predicted results agreed with the experimental data.

Strain Response under Cyclic Fatigue and Stress-Rupture at 1093 °C in Air
The experimental and theoretical fatigue hysteresis loops under the fatigue peak stress of σmax = 103 MPa with the stress rupture time t = 10 s are given in Figure 12a. The fatigue hysteresis loops at the cycle numbers of N = 1, 108 and 216 corresponded to the interface slip Cased 3 and 4, respectively. The residual strain increased with applied cycles, and the area of fatigue hysteresis loops decreased with the cycle number. The interface slip lengths, i.e., the unloading interface

Strain Response under Cyclic Fatigue and Stress-Rupture at 1093 • C in Air
The experimental and theoretical fatigue hysteresis loops under the fatigue peak stress of σ max = 103 MPa with the stress rupture time t = 10 s are given in Figure 12a. The fatigue hysteresis loops at the cycle numbers of N = 1, 108 and 216 corresponded to the interface slip Cased 3, 3 and 4, respectively. The residual strain increased with applied cycles, and the area of fatigue hysteresis loops decreased with the cycle number. The interface slip lengths, i.e., the unloading interface counter-slip length and the reloading interface new-slip length, increased with the cycle number, i.e., from 2y/l c = 2z/l c = 0.51 at the first applied cycle to 2y/l c = 2z/l c = 1 at the 216th applied cycle, as shown in Figure 12b. The experimental and theoretical peak strains versus cycle number curves are illustrated in Figure 12c. The experimental peak strain increased from 0.46% at the first applied cycle to 0.52% at the 216th applied cycle; the theoretical peak strain increased from 0.46% at the first applied cycle to the peak value of 0.546% at the 590th applied cycle, i.e., the A-B part in Figure 12c, corresponding to the interface completely debonding and the interface partially oxidizing, i.e., the A-B part in Figure 12d. It remained constant at the value of 0.546% with increasing applied cycles, i.e., the B-C part in Figure 12c, corresponding to the interface completely debonding and the interface completely oxidizing, i.e., the B-C part in Figure 12d. The theoretical predicted results agreed with the experimental data.

Conclusions
The synergistic effects of stress rupture and cyclic loading on the strain response of fiber-reinforced CMCs at elevated temperature were investigated. The effects of the stress rupture time, stress levels, matrix crack spacing, fiber volume fraction and oxidation temperature on the peak strain and the interface slip lengths were investigated. The experimental fatigue hysteresis loops, interface slip lengths, peak strain and interface oxidation length of cross-ply SiC/MAS composite under cyclic fatigue and stress rupture at 566 °C and 1093 °C in air were predicted.
1. With the increase of the stress rupture time, fatigue peak stress and oxidation temperature, the interface slip lengths increase with applied cycles due to interface oxidation, leading to the increase of the peak strain. 2. With the increase of the matrix crack spacing and fiber volume fraction, the interface slip lengths decrease, leading to the decrease of the peak strain.
Acknowledgments: The work reported here is supported by the Natural Science Fund of Jiangsu Province (Grant no. BK20140813), and the Fundamental Research Funds for the Central Universities (Grant no. NS2016070). The author also wishes to thank two anonymous reviewers and editors for their helpful comments on an earlier version of the paper.

Conclusions
The synergistic effects of stress rupture and cyclic loading on the strain response of fiber-reinforced CMCs at elevated temperature were investigated. The effects of the stress rupture time, stress levels, matrix crack spacing, fiber volume fraction and oxidation temperature on the peak strain and the interface slip lengths were investigated. The experimental fatigue hysteresis loops, interface slip lengths, peak strain and interface oxidation length of cross-ply SiC/MAS composite under cyclic fatigue and stress rupture at 566 • C and 1093 • C in air were predicted.
With the increase of the stress rupture time, fatigue peak stress and oxidation temperature, the interface slip lengths increase with applied cycles due to interface oxidation, leading to the increase of the peak strain.
With the increase of the matrix crack spacing and fiber volume fraction, the interface slip lengths decrease, leading to the decrease of the peak strain.