Stress-Rupture of Fiber-Reinforced Ceramic-Matrix Composites with Stochastic Loading at Intermediate Temperatures. Part I: Theoretical Analysis

Under stress-rupture loading, stochastic loading affects the internal damage evolution and lifetime of fiber-reinforced ceramic-matrix composites (CMCs) at intermediate temperatures. The damage mechanisms of the matrix cracking, fiber/matrix interface debonding and oxidation, and fiber fracture are considered in the analysis of stochastic loading. The strain, fiber/matrix interface debonding and oxidation length, and the broken fibers fraction versus the time curves of SiC/SiC composite under constant and three different stochastic loading conditions are analyzed. The effects of the stochastic loading stress level, stochastic loading time, and time spacing on the damage evolution and lifetime of SiC/SiC composite are discussed. When the stochastic loading stress level increases, the stress-rupture lifetime decreases, and the time for the interface complete debonding and oxidation decreases. When the stochastic loading time and time spacing increase, the stress-rupture lifetime decreases, and the time for the interface complete debonding and oxidation remains the same.


Introduction
Ceramic-matrix composites (CMCs) are a new type of thermal-structural-functional integrated material with the advantages of metal materials, ceramic materials, and carbon materials [1]. They have the characteristics of material-structural integration. Through the optimization design of each structural unit, synergistic effects can be produced, and high performance and reasonable matching of each performance can be achieved. Therefore, CMCs have high temperature resistance, corrosion resistance, wear resistance, low density, high specific strength, high specific modulus, low thermal expansion coefficient, insensitivity to cracks, no catastrophic damage, and other advantages [2]. Compared to metallic alloys, CMCs can have a density reduction of 30-50% and can exceed the working temperature range [3]. With the increase of thrust-weight ratio and turbine inlet temperature, CMCs have become one of the preferred high-temperature structural materials for aeroengines. When CMCs are used in hot-section components in aeroengines, i.e., turbine, combustion chamber, combustion liner, and nozzles, the amount of cooling air can be significantly reduced or even zero, the combustion efficiency can be improved, and the pollution emission and noise level can be reduced. At present, the application of CMCs in aeroengines follows the development idea from stationary parts to rotating parts, from intermediate temperature parts (i.e., 700-1000 • C) to high temperature parts (i.e., 1000-1300 • C), and gives priority to developing intermediate temperature and intermediate load (i.e., less than 120 MPa) stationary parts (i.e., seals and flaps, etc.), then the high temperature intermediate load (i.e., less than 120 MPa) stationary parts (i.e., flame tube, flame holder, turbine outer ring, guide vane, etc.), and then the high temperature and high load (i.e., higher than 120 MPa) rotating parts (i.e., turbine rotor, turbine

Theoretical model
When stochastic loading occurs during constant stress loading at an elevated temperature, the damage extent inside of fiber-reinforced CMCs becomes much more serious. In the present analysis, the shear-lag model was used to analyze the stress distribution of damaged CMCs under stress-rupture with stochastic loading. The damage mechanisms of the matrix cracking, fiber/matrix interface debonding and oxidation, and broken fibers were considered. The matrix stochastic cracking model, fracture mechanics approach, and Global Load Sharing criterion were used to determine the matrix crack spacing, fiber/matrix interface debonding length, and the broken fibers fraction under stress-rupture with stochastic loading. The constitutive relationship considering the time-dependent damage mechanisms was also developed. Figure 1 shows the stochasic loading sequence under constant stress-rupture laoding of fiber-reinforced CMCs at an elevated temperatrue, which can be divided into four cases, as follows: (1) Case I, constant stress loading; (2) Case II, constant stress loading and stochsatic loading of σ a with ∆t a ; (3) Case III, constant stress loading and stochastic loading of σ a and σ b with ∆t a and ∆t b ; (4) Case IV, constant stress loading and stochastic loading of σ a , σ b and σ c with ∆t a , ∆t b , and ∆t c .    Figure 2 shows a unit cell used for the stress analysis of the fiber and the matrix when the matrix cracking, fiber/matrix interface debonding, and fiber failure appear inside of CMCs. When the fiber fractures under stochastic loading, the fiber axial stress distribution can be determined using the following equation: where r f denotes the fiber radius; τ f denotes the fiber/matrix interface shear stress in the oxidation region; τ i denotes the fiber/matrix interface shear stress in the slip region; T S (t) denotes the intact fiber stress under stochastic loading; l d (t) denotes the time-dependent fiber/matrix interface debonding length under stochastic loading; l c denotes the matrix crack spacing under stochastic loading; ρ denotes the shear-lag model parameter; and ζ(t) denotes the time-dependent fiber/matrix interface oxidation length [21].
where rf denotes the fiber radius; τf denotes the fiber/matrix interface shear stress in the oxidation region; τi denotes the fiber/matrix interface shear stress in the slip region; TS(t) denotes the intact fiber stress under stochastic loading; ld(t) denotes the time-dependent fiber/matrix interface debonding length under stochastic loading; lc denotes the matrix crack spacing under stochastic loading; ρ denotes the shear-lag model parameter; and ζ(t) denotes the time-dependent fiber/matrix interface oxidation length [21].
where b is a delay factor considering the deceleration of reduced oxygen activity, and φ1 and φ2 are parameters dependent on temperature and described using the Arrhenius type laws. The fiber axial stress in the fiber/matrix interface bonded region can be determined using the following equation: where Ef, and Ec denote the fiber and the composite elastic modulus, respectively; αf, and αc denote the fiber and the composite thermal expansion coefficient, respectively; and ΔT denotes the temepratrue difference between the testing temperature and the fabrication temperatrue. The matrix cracking under stochastic loading can be described using the two-parameter Weibull distribution, and the time-dependent matrix crack spacing under stochastic loading can be determined using the following equation [22]: where σS denotes the stochastic loading stress; Em denotes the matrix elastic modulus; σR denotes the matrix cracking characteristic strength; σmc denotes matrix first cracking stress; σth denotes matrix The matrix cracking under stochastic loading can be described using the two-parameter Weibull distribution, and the time-dependent matrix crack spacing under stochastic loading can be determined using the following equation [22]: where σ S denotes the stochastic loading stress; E m denotes the matrix elastic modulus; σ R denotes the matrix cracking characteristic strength; σ mc denotes matrix first cracking stress; σ th denotes matrix thermal residual stress; Λ denotes the final nominal crack space; and m denotes matrix Weibull modulus.
The time-dependent fiber/matrix interface debonding length under stochastic loading can be determined using the fracture mechanics approach [23]: where ξ d denotes the fiber/matrix interface debonding energy; F(= πr f 2 σ/V f ) denotes the fiber stress at the matrix cracking plane; w f (σ S , t) denotes the time-dependent fiber axial displacement under stochastic loading at the matrix cracking plane; and v(σ S , t) denotes the time-dependent relative displacement between the fiber and the matrix under stochastic loading. Substituting the time-dependent fiber axial Materials 2019, 12, 3123 5 of 32 displacement and relative displacement into Equation (5), the time-dependent fiber/matrix interface debonding length under stochastic loading can be determined using the following equation: where η = τ f /τ i . The two-parameter Weibull model was adopted to describe the fiber strength distribution, and the Global Load Sharing criterion was used to determine the stress distributions between the intact and fracture fibers [24].
where L denotes the average fiber pullout length, and P(T S ) denotes the fiber failure probability.
where m f denotes the fiber Weibull modulus, and σ c denotes the fiber characteristic strength of a length δ c of fiber.
where σ 0 denotes the time-dependent fiber strength; K IC denotes the fracture toughness; Y denotes the geometric parameter; and k is the parabolic rate constant. When multiple damage mechanisms form inside of fiber-reinforced CMCs, the average composite strain of ε c (t) can be determined by integration of the axial strain in the fiber.
Substituting the time-dependent fiber axial stress under stochastic loading in Equation (1) into Equation (11), the composite average strain of ε c (σ S , t) can be determined using the following equation:

Results and analysis
The strain, fiber/matrix interface debonding and oxidation length, and broken fibers fraction versus the time curves of SiC/SiC composite were analyzed for the Cases II, III, and IV. The material properties were given by:

Case II
For the stochastic loading of Case II, the strain, interface debonding and oxidation length, and the broken fibers fraction of SiC/SiC composite under stress-rupture loading of constant stress of σ = 120 MPa, σ S = 140, 160, and 180 MPa at t = 36 kseconds and ∆t = 36 kseconds at 800 • C in air atmosphere are shown in Figure 3 and Table 1. When the stochastic loading stress level increases, the stress-rupture lifetime decreases, and the time for the interface complete debonding and oxidation decreases.     Under constant stress loading of σ = 120 MPa, the stress-rupture lifetime is t = 2447.9 kseconds; the time for the interface complete debonding is t = 242.7 kseconds; the time for the interface complete oxidation is t = 295.3 kseconds; the failure strain is ε c = 0.201%; and the broken fibers fraction is P = 0.285. When the stochastic loading stress is σ S = 140 MPa, the stress-rupture lifetime is t = 2446.9 kseconds; the time for the interface complete debonding is t = 205.7 kseconds; the time for the interface complete oxidation is t = 259.4 kseconds; the failure strain is ε c = 0.202%; and the broken fibers fraction is P = 0.285. When the stochastic loading stress is σ S = 160 MPa, the stress-rupture lifetime is t = 2444.3 kseconds; the time for the interface complete debonding is t = 192.6 kseconds; the time for the interface complete oxidation is t = 246.8 kseconds; the failure strain is ε c = 0.201%; and the broken fibers fraction is P = 0.285. When the stochastic loading stress is σ S = 180 MPa, the stress-rupture lifetime is t = 2437 kseconds; the time for the interface complete debonding is t = 189.1 kseconds; the time for the interface complete oxidation is t = 243.1 kseconds; the failure strain is ε c = 0.2%; and the broken fibers fraction is The strain, interface debonding and oxidation length, and the broken fibers fraction of SiC/SiC composite under stress-rupture loading of constant stress of σ = 120 MPa, σ S = 140 MPa at t = 72, 108, 144 kseconds and ∆t = 36 kseconds at 800 • C in air atmosphere are shown in Figure 4 and Table 2.     When the stochastic loading time is t = 72 kseconds, the stress-rupture lifetime is t = 2446.2 kseconds; the time for the interface complete debonding is t = 205.7 kseconds; the time for the interface complete oxidation is t = 259.4 kseconds; the failure strain is ε c = 0.201%; and the broken fibers fraction is P = 0.285. When the stochastic loading time increases from t = 72 to 144 kseconds, the stress-rupture lifetime decreases, and the time for the interface complete debonding and oxidation remains the same.
The strain, fiber/matrix interface debonding and oxidation length, and the broken fibers fraction of SiC/SiC composite under stress-rupture loading of constant stress of σ = 120 MPa, σ S = 140 MPa at t = 36 kseconds and ∆t = 72, 108, 144 kseconds at 800 • C in air atmosphere are shown in Figure 5 and Table 3.
When the stochastic loading time spacing is ∆t = 72 kseconds, the stress-rupture lifetime is t = 2446.2 kseconds; the time for the interface complete debonding is t = 205.7 kseconds; the time for the interface complete oxidation is t = 259.4 kseconds; the failure strain is ε c = 0.201%; and the broken fibers fraction is P = 0.285. When the stochastic loading time spacing increases from ∆t = 72 to 144 kseconds, the stress-rupture lifetime decreases, and the time for the interface complete debonding and oxidation remains the same.

Case III
For the stochastic loading of Case III, the strain, fiber/matrix interface debonding and oxidation length, and the broken fibers fraction of SiC/SiC composite under stress-rupture loading of constant stress of σ = 120 MPa, σ S = 130/140, 140/150, 150/160 MPa at t = 36/108 kseconds and ∆t = 36 kseconds at 800 • C in air atmosphere are shown in Figure 6 and Table 4. When the stochastic loading stress increases, the stress-rupture lifetime decreases, and the time for the interface complete debonding and oxidation decreases.
When the stochastic loading stress is σ S = 130, 140 MPa, the stress-rupture lifetime is t = 2444.1 kseconds; the time for the interface complete debonding is t = 205.7 kseconds; the time for the interface complete oxidation is t = 259.4 kseconds; the failure strain is ε c = 0.2%; and the broken fibers fraction is P = 0.285. When the stochastic loading stress is σ S = 140, 150 MPa, the stress-rupture lifetime is t = 2438.1 kseconds; the time for the interface complete debonding is t = 197.3 kseconds; the time for the interface complete oxidation is t = 251.3 kseconds; the failure strain is ε c = 0.2%; and the broken fibers fraction is P = 0.285. When the stochastic loading stress is σ S = 150, 160 MPa, the stress-rupture lifetime is t = 2425.9 kseconds; the time for the interface complete debonding is t = 192.6 kseconds; the time for the interface complete oxidation is t = 246.8 kseconds; the failure strain is ε c = 0.199%; and the broken fibers fraction is P = 0.285.     The strain, fiber/matrix interface debonding and oxidation length, and the broken fibers fraction of SiC/SiC composite under stress-rupture loading of constant stress of σ = 120 MPa, σ S = 140/160 MPa at t = 72/144, 108/180, 144/216 kseconds and ∆t = 36 kseconds at 800 • C in air atmosphere are shown in Figure 7 and Table 5. When the stochastic loading time increases, the stress-rupture lifetime decreases, and the time for the interface complete debonding increases.  Figure 7 and Table 5. When the stochastic loading time increases, the stress-rupture lifetime decreases, and the time for the interface complete debonding increases.
When the stochastic loading time is t = 72 and 144 kseconds, the stress-rupture lifetime is t = 2418.4 kseconds; the time for the interface complete debonding is t = 160 kseconds at stochastic loading stress of σS = 160 MPa; the time for the interface complete oxidation is t = 246.8 kseconds at constant loading stress of σ = 120 MPa; the failure strain is εc = 0.198%; and the broken fibers fraction is P = 0.285. When the stochastic loading time is t = 144 and 216 kseconds, the stress-rupture lifetime is t = 2376.8 kseconds; the time for the interface complete debonding is t = 205.7 kseconds at constant stress of σ=120 MPa; the time for the interface complete oxidation is t = 246.8 kseconds; the failure strain is εc = 0.195%; and the broken fibers fraction is P = 0.285.   When the stochastic loading time is t = 72 and 144 kseconds, the stress-rupture lifetime is t = 2418.4 kseconds; the time for the interface complete debonding is t = 160 kseconds at stochastic loading stress of σ S = 160 MPa; the time for the interface complete oxidation is t = 246.8 kseconds at constant loading stress of σ = 120 MPa; the failure strain is ε c = 0.198%; and the broken fibers fraction is P = 0.285. When the stochastic loading time is t = 144 and 216 kseconds, the stress-rupture lifetime is t = 2376.8 kseconds; the time for the interface complete debonding is t = 205.7 kseconds at constant stress of σ = 120 MPa; the time for the interface complete oxidation is t = 246.8 kseconds; the failure strain is ε c = 0.195%; and the broken fibers fraction is P = 0.285.
The strain, fiber/matrix interface debonding and oxidation length, and the broken fibers fraction of SiC/SiC composite under stress-rupture loading of constant stress of σ = 120 MPa, σ S = 140/160 MPa at t = 36/144, 36/180, 36/216 kseconds and ∆t = 72, 108, 144 kseconds at 800 • C in air atmosphere are shown in Figure 8 and Table 6. When the stochastic loading time spacing increases, the stress-rupture lifetime decreases, and the time for the interface complete debonding increases.    When the stochastic loading time spacing is ∆t = 72 kseconds, the stress-rupture lifetime is t = 2407.2 kseconds; the time for the interface complete debonding is t = 160 kseconds at stochastic stress of σ S = 160 MPa; the time for the interface complete oxidation is t = 246.8 kseconds at constant stress of σ = 120 MPa; the failure strain is ε c = 0.197%; and the broken fibers fraction is P = 0.285. When the stochastic loading time spacing is ∆t = 144 kseconds, the stress-rupture lifetime is t = 2294.8 kseconds; the time for the interface complete debonding is t = 205.7 kseconds at constant stress of σ = 120 MPa; the time for the interface complete oxidation is t = 246.8 kseconds at stochastic stress of σ S = 160 MPa; the failure strain is ε c = 0.192%; and the broken fibers fraction is P = 0.285.

Case IV
For the stochastic loading of Case IV, the strain, fiber/matrix interface debonding and oxidation length, and the broken fibers fraction of SiC/SiC composite under stress-rupture loading of constant stress of σ = 120 MPa, σ S = 130/140/150, 140/150/160, 150/160/170 MPa at t = 36/108/180 kseconds and ∆t = 36 kseconds at 800 • C in air atmosphere are shown in Figure 9 and Table 7. When the stochastic loading stress increases, the stress-rupture lifetime decreases, and the time for the interface complete oxidation decreases.
When the stochastic loading stress is σ S = 130, 140, 150 MPa, the stress-rupture lifetime is t = 2416.2 kseconds; the time for the interface complete debonding is t = 180 kseconds at stochastic loading stress of σ S = 150 MPa; the time for the interface complete oxidation is t = 251.3 kseconds at constant stress of σ = 120 MPa; the failure strain is ε c = 0.198%; and the broken fibers fraction is P = 0.285. When the stochastic loading stress is σ S = 140, 150, 160 MPa, the stress-rupture lifetime is t = 2357.2 kseconds; the time for the interface complete debonding is t = 180 kseconds at stochastic loading stress of σ S = 160 MPa; the time for the interface complete oxidation is t = 246.8 kseconds at constant stress of σ = 120 MPa; the failure strain is ε c = 0.194%; and the broken fibers fraction is P = 0.285. When the stochastic loading stress is σ S = 150, 160, 170 MPa, the stress-rupture lifetime is t = 2209.1 kseconds; the time for the interface complete debonding is t = 180 kseconds at stochastic loading stress of σ S = 170 MPa; the time for the interface complete oxidation is t = 244.4 kseconds at constant stress of σ = 120 MPa; the failure strain is ε c = 0.189%; and the broken fibers fraction is P = 0.285. Table 7. The strain, fiber/matrix interface debonding and oxidation length, and broken fibers fraction of SiC/SiC composite under stress-rupture loading of constant stress of σ = 120 MPa, σ S = 130/140/150, 140/150/160, 150/160/170 MPa at t = 36/108/180 kseconds and ∆t = 36 kseconds at 800 • C in air atmosphere.

Case IV
For the stochastic loading of Case IV, the strain, fiber/matrix interface debonding and oxidation length, and the broken fibers fraction of SiC/SiC composite under stress-rupture loading of constant stress of σ = 120 MPa, σS = 130/140/150, 140/150/160, 150/160/170 MPa at t = 36/108/180 kseconds and Δt = 36 kseconds at 800 °C in air atmosphere are shown in Figure 9 and Table 7. When the stochastic loading stress increases, the stress-rupture lifetime decreases, and the time for the interface complete oxidation decreases.
When the stochastic loading stress is σS = 130, 140, 150 MPa, the stress-rupture lifetime is t = 2416.     Figure 10 and Table 8. When the stochastic loading time increases, the stress-rupture lifetime decreases, and the time for the interface complete debonding and oxidation increases.     When the stochastic loading time is t = 72, 144, 216 kseconds, the stress-rupture lifetime is t = 1794 kseconds; the time for the interface complete debonding is t = 160 kseconds at stochastic loading stress of σ S = 160 MPa; the time for the interface complete oxidation is t = 243.3 kseconds at stochastic loading stress of σ S = 180 MPa; the failure strain is ε c = 0.182%; and the broken fibers fraction is P = 0.285. When the stochastic loading time is t = 144, 216, 288 kseconds, the stress-rupture lifetime is t = 324 kseconds; the time for the interface complete debonding is t = 205.7 kseconds at constant stress of σ = 120 MPa; the time for the interface complete oxidation is t = 246.8 kseconds at stochastic loading stress of σ S = 160 MPa; the failure strain is ε c = 0.29%; and the broken fibers fraction is P = 0.25.
The strain, fiber/matrix interface debonding and oxidation length, and the broken fibers fraction of SiC/SiC composite under stress-rupture loading of constant stress of σ = 120 MPa, σ S = 140/160/180 MPa at t = 36/144/216, 36/180/324, 36/216/360 kseconds and ∆t = 72, 108, 144 kseconds at 800 • C in air atmosphere are shown in Figure 11 and Table 9. When the stochastic loading time spacing increases, the stress-rupture lifetime decreases, and the time for the interface complete debonding increases.  When the stochastic loading time spacing is ∆t = 72 kseconds, the stress-rupture lifetime is t = 572.4 kseconds; the time for the interface complete debonding is t = 160 kseconds at stochastic loading stress of σ S = 160 MPa; the time for the interface complete oxidation is t = 246.8 kseconds at constant stress of σ = 120 MPa; the failure strain is ε c = 0.173%; and the broken fibers fraction is P = 0.285. When the stochastic loading time spacing is ∆t = 144 kseconds, the stress-rupture lifetime is t = 396 kseconds; the time for the interface complete debonding is t = 205.7 kseconds at constant stress of σ = 120 MPa; the time for the interface complete oxidation is t = 246.8 kseconds at stochastic loading stress of σ S = 160 MPa.

Conclusions
In this paper, the damage evolution and lifetime of fiber-reinforced CMCs under stress-rupture with stochastic loading at intermediate temperatures were investigated. The relationships between the stochastic loading stress level, time, time spacing, damage mechanisms of matrix cracking, interface debonding and oxidation, and fiber failure were established. The strain, fiber/matrix interface debonding and oxidation length, and the broken fibers fraction versus the time curves of SiC/SiC composite under constant stress and three different stochastic loading conditions were analyzed. The effects of the stochastic loading stress level, stochastic loading time, and time spacing on the damage evolution and lifetime of SiC/SiC composite were discussed. For the stochastic loading of Cases II, III, and IV, the stress-rupture lifetime decreases with increasing stochastic loading stress level, time, and time spacing. The time for the interface complete debonding and oxidation is affected by the loading mode, stress level, loading time, and time spacing.