Effect of Stochastic Loading on Tensile Damage and Fracture of Fiber-Reinforced Ceramic-Matrix Composites

In this paper, the effect of stochastic loading on tensile damage and fracture of fiber-reinforced ceramic-matrix composites (CMCs) is investigated. A micromechanical constitutive model is developed considering multiple damage mechanisms under tensile loading. The relationship between stochastic stress, tangent modulus, interface debonding and fiber broken is established. The effects of the fiber volume, interface shear stress, interface debonding energy, saturation matrix crack spacing and fiber strength on tensile stress–strain curve, tangent modulus, interface debonding fraction and fiber broken fraction are analyzed. The experimental tensile damage and fracture of unidirectional and 2D SiC/SiC composites subjected to different stochastic loading stress are predicted. When fiber volume increases, the initial composite strain decreases, the initial tangent modulus increases, the transition stress for interface debonding decreases and the initial fiber broken fraction decreases. When fiber strength increases, the initial composite strain and fiber broken fraction decrease.


Introduction
Ceramic-matrix composites (CMCs) have the advantages of high-temperature resistance, corrosion resistance, low density, high specific strength and high specific modulus [1]. The fabrication methods of CMCs include the chemical vapor infiltration (CVI), polymer infiltration and pyrolysis process (PIP) and melt infiltration (MI). The mechanical performance of CMCs depends on the fabrication method. To ensure the reliability and safety of fiber-reinforced CMCs used in hot-section components of an aero engine, it is necessary to develop performance evaluation, damage evolution, strength and life prediction tools for airworthiness certification [2]. Since the applications of fiber-reinforced CMCs involve components with lives that are measured in tens of thousands of hours, the successful design and implementation of CMC components depend on the knowledge of the material behavior over periods of time comparable to the expected service life of the component [3].
(2) Stage II, matrix cracking and interface debonding region. Matrix micro cracking occurs first and the fiber debonding from the matrix, leading to nonlinear behavior of CMCs.   Using stochastic matrix cracking model, the relationship between stochastic stress and matrix crack spacing is given by Equation (2). [12] where ls is the saturation matrix crack spacing, σm is the matrix stress, σR is matrix cracking characteristic stress and m is the matrix Weibull modulus. The fracture mechanics approach is used to determine the fiber/matrix interface debonding length. The interface debonding length is determined by Equation (3).
where Vf and Vm are the fiber and matrix volume fraction, respectively, Em and Ec are the matrix and composite elastic modulus, respectively, and ζd is the interface debonding energy. The Global Load Sharing (GLS) criterion is used to determine the intact fiber stress. The relationship between the fiber intact stress, fiber failure probability and fiber pullout length is given by Equation (4). [21] ( ) ( )  Using stochastic matrix cracking model, the relationship between stochastic stress and matrix crack spacing is given by Equation (2). [12] where ls is the saturation matrix crack spacing, σm is the matrix stress, σR is matrix cracking characteristic stress and m is the matrix Weibull modulus. The fracture mechanics approach is used to determine the fiber/matrix interface debonding length. The interface debonding length is determined by Equation (3).
where Vf and Vm are the fiber and matrix volume fraction, respectively, Em and Ec are the matrix and composite elastic modulus, respectively, and ζd is the interface debonding energy. The Global Load Sharing (GLS) criterion is used to determine the intact fiber stress. The relationship between the fiber intact stress, fiber failure probability and fiber pullout length is given by Equation (4). [21] ( ) ( ) Using stochastic matrix cracking model, the relationship between stochastic stress and matrix crack spacing is given by Equation (2) [12].
where l s is the saturation matrix crack spacing, σ m is the matrix stress, σ R is matrix cracking characteristic stress and m is the matrix Weibull modulus. The fracture mechanics approach is used to determine the fiber/matrix interface debonding length. The interface debonding length is determined by Equation (3).
where V f and V m are the fiber and matrix volume fraction, respectively, E m and E c are the matrix and composite elastic modulus, respectively, and ζ d is the interface debonding energy. The Global Load Sharing (GLS) criterion is used to determine the intact fiber stress. The relationship between the fiber intact stress, fiber failure probability and fiber pullout length is given by Equation (4) [21].
Under stochastic loading, the damages of the matrix cracking, interface debonding and fiber failure occur. The micro stress filed of the damaged CMCs after stochastic loading is given by Equation (1). The fiber axial stress distribution is affected by the stochastic loading stress level, matrix cracking, interface debonding and fiber failure. The stochastic matrix cracking model is used to determine the matrix crack spacing at the applied stress level, as shown in Equation (2), and the fracture mechanics interface debonding criterion is used to determine the interface debonding length, and the GLS criterion is used to determine the load allocation between fracture and intact fibers. The micromechanical constitutive models for the conditions of the matrix cracking, interface debonding and fiber failure are given by Equations (6) and (7). Using the developed micromechanicam constitutive models and damage models, the effects of the fiber volume, interface shear stress, interface debonding energy, matrix crack spacing and fiber strength on tensile stress-strain curve, tangent modulus, interface debonding fraction and fiber broken fraction of SiC/SiC composite subjected to different stochastic loading are analyzed. Li et al. [4] investigated the tensile behavior of 2D woven SiC/SiC composite at room temperature. The composite was fabricated using the CVI method with the PyC interphase. The tensile experiments were conducted under displacement contrl of 0.3 mm/min. The material properties are given by: MPa and m f = 3.

Effect of Fiber Volume on Tensile Damage and Fracture of SiC/SiC Composite with Stochastic Loading
The fiber volume affects the tensile behavior of CMCs. The fiber volume range of SiC/SiC composite is between V f = 27.7 and 40% [9]. In the present analysis, the effect of the fiber volume (i.e., V f = 0.3 and 0.35) on the tensile stress-strain curves, tangent modulus, interface debonding fraction and broken fiber fraction of SiC/SiC composite subjected to the stochastic loading of σ s = 200, 300 and 350 MPa are shown in Figure 3; Figure 4 and Table 1. When the fiber volume increases, the stress carried by the fiber increases, the initial composite strain decreases, the initial tangent modulus increases, the transition stress for the interface debonding decreases and the initial fiber broken fraction decreases.

Effect of Fiber Volume on Tensile Damage and Fracture of SiC/SiC Composite with Stochastic Loading
The fiber volume affects the tensile behavior of CMCs. The fiber volume range of SiC/SiC composite is between Vf = 27.7 and 40% [9]. In the present analysis, the effect of the fiber volume (i.e., Vf = 0.3 and 0.35) on the tensile stress-strain curves, tangent modulus, interface debonding fraction and broken fiber fraction of SiC/SiC composite subjected to the stochastic loading of σs = 200, 300 and 350 MPa are shown in Figure 3; Figure 4 and Table 1. When the fiber volume increases, the stress carried by the fiber increases, the initial composite strain decreases, the initial tangent modulus increases, the transition stress for the interface debonding decreases and the initial fiber broken fraction decreases.      When V f = 0.3 under σ s = 200 MPa, the damages of matrix cracking and interface debonding occur at σ s = 200 MPa, leading to the increase of the composite initial strain, decreasing of the tangent modulus and increase of the broken fiber fraction. The initial composite strain is ε 0 = 0.00161% due to the matrix cracking and interface debonding at σ s = 200 MPa; the initial tangent modulus is E p = 268 GPa, the degradation rate of the tangent modulus is 15% compared with the original specimen and the fiber broken fraction is P = 0.005. With increasing stress to σ tr = 79.2 MPa, the interface debonding fraction increases, the tangent modulus decreases to E p = 235 GPa corresponding to η = 0.081. Upon increasing stress from σ tr = 79.2 MPa to σ = 200 MPa, the tangent modulus remains constant of E p = 264 GPa with η = 0.084.
Under σ s = 300 MPa, the initial composite strain is ε 0 = 0.00982% due to the damages of the matrix cracking and interface debonding at σ s = 300 MPa; the initial tangent modulus is E p = 229 GPa, the degradation rate of tangent modulus is 28% compared with the original specimen and the fiber broken fraction is P = 0.028. With increasing stress to σ tr = 178.2 MPa, the interface debonding fraction increases, the tangent modulus decreases to E p = 145 GPa corresponding to η = 0.392. Upon increasing stress from σ tr = 178.2 MPa to σ = 300 MPa, the tangent modulus remains constant of E p = 170.1 GPa with η = 0.4.
Under σ s = 350 MPa, the initial composite strain is ε 0 = 0.023% due to the damages of the matrix cracking and interface debonding at σ s = 350 MPa; the initial tangent modulus is E p = 228 GPa, the degradation rate of tangent modulus is 28% compared with the original specimen and the fiber broken fraction is P = 0.06. With increasing stress to σ tr = 231 MPa, the interface debonding fraction increases, the tangent modulus decreases to E p = 131 GPa corresponding to η = 0.51. Upon increasing stress from σ tr = 131 MPa to σ = 350 MPa, the tangent modulus remains constant of E p = 151 GPa with η = 0.51.
When V f = 0.35 under σ s = 200 MPa, the damages of matrix cracking and interface debonding occur at σ s = 200 MPa, leading to the increase of the composite initial strain, decreasing of the tangent modulus and increase of the broken fiber fraction. The initial composite strain is ε 0 = 0.0009%; the initial tangent modulus is E p = 281 GPa, the degradation rate of tangent modulus is 12% compared with original specimen and the fiber broken fraction is P = 0.0027. With increasing stress to σ tr = 55 MPa, the interface debonding fraction increases, the tangent modulus decreases to E p = 265 GPa corresponding to η = 0.046. Upon increasing stress from σ tr = 55 MPa to σ = 200 MPa, the tangent modulus remains constant of E p = 291 GPa with η = 0.046.
Under σ s = 300 MPa, the initial composite strain is ε 0 = 0.0047% due to the damages of matrix cracking and interface debonding at σ s = 300 MPa; the initial tangent modulus is E p = 249 GPa, the degradation rate of tangent modulus is 22% compared with original specimen and the fiber broken fraction is P = 0.014. With increasing stress to σ tr = 156.2 MPa, the interface debonding fraction increases and the tangent modulus decreases to E p = 201 GPa corresponding to η = 0.27. When the stress increases from σ tr = 156.2 MPa to σ = 250 MPa, the tangent modulus remains constant of E p = 216 GPa with η = 0.27.
Under σ s = 350 MPa, the initial composite strain is ε 0 = 0.0095% due to the damages of the matrix cracking and interface debonding at σ s = 350 MPa; the initial tangent modulus is E p = 248 GPa, the degradation rate of tangent modulus is 22% compared with original specimen and the fiber broken fraction is P = 0.028. With increasing stress to σ tr = 206.8 MPa, the interface debonding fraction increases and the tangent modulus decreases to E p = 191.9 GPa corresponding to the interface debonding fraction η = 0.362. When the stress increases from σ tr = 206.8 MPa to σ = 300 MPa, the tangent modulus remains constant of E p = 196.7 GPa with η = 0.363.

Effect of Interface Shear Stress on Tensile Damage and Fracture of SiC/SiC Composite with Stochastic Loading
The interface shear stress transfers the load between the fiber and the matrix when the matrix cracking and interface debonding occur. For the weak interface bonding between the fiber and the Materials 2020, 13, 2469 8 of 28 matrix of SiC/SiC composite, the value of the interface shear stress is between τ i = 10 and 30 MPa [22]. In the present analysis, the effect of the interface shear stress (i.e., τ i = 15 and 20 MPa) on the tensile stress-strain curves, tangent modulus, interface debonding fraction and broken fiber fraction of SiC/SiC composite subjected to stochastic loading of σ s = 200, 230 and 250 MPa are shown in Figures 5 and 6 and Table 2. When the interface shear stress increases, the load transfer capacity between the fiber and the matrix increases, the initial composite strain remains the same, the initial tangent modulus increases, the transition stress for interface debonding remains the same and the initial fiber broken fraction remains the same.  Table 2. When the interface shear stress increases, the load transfer capacity between the fiber and the matrix increases, the initial composite strain remains the same, the initial tangent modulus increases, the transition stress for interface debonding remains the same and the initial fiber broken fraction remains the same.   Table 2. When the interface shear stress increases, the load transfer capacity between the fiber and the matrix increases, the initial composite strain remains the same, the initial tangent modulus increases, the transition stress for interface debonding remains the same and the initial fiber broken fraction remains the same.    Under σs = 230 MPa, the initial composite strain is ε0 = 0.021% due to the damages of matrix cracking and interface debonding at σs = 230 MPa; the initial tangent modulus is Ep = 163 GPa, the degradation rate of tangent modulus is 48% compared with original specimen and the fiber broken fraction is P = 0.056. With increasing stress to σtr = 154 MPa, the interface debonding fraction increases and the tangent modulus decreases to Ep = 71.5 GPa corresponding to η = 0.727. When the stress When τ i = 15 MPa under σ s = 200 MPa, the damages of matrix cracking and interface debonding occur at σ s = 200 MPa, leading to the increase of the composite initial strain, decreasing of the tangent modulus and increase of the broken fiber fraction. The initial composite strain is ε 0 = 0.00943%; the initial tangent modulus is E p = 195.6 GPa, the degradation rate of tangent modulus is 37% compared with original specimen and the fiber broken fraction is P = 0.028. With increasing stress to σ tr = 125.4 MPa, the interface debonding fraction increases, the tangent modulus decreases to E p = 108.1 GPa corresponding to η = 0.374. When the stress increases from σ tr = 125.4 MPa to σ = 200 MPa, the tangent modulus remains constant of E p = 131.3 GPa with η = 0.382.
Under σ s = 230 MPa, the initial composite strain is ε 0 = 0.021% due to the damages of matrix cracking and interface debonding at σ s = 230 MPa; the initial tangent modulus is E p = 163 GPa, the degradation rate of tangent modulus is 48% compared with original specimen and the fiber broken fraction is P = 0.056. With increasing stress to σ tr = 154 MPa, the interface debonding fraction increases and the tangent modulus decreases to E p = 71.5 GPa corresponding to η = 0.727. When the stress increases from σ tr = 154 MPa to σ = 230 MPa, the tangent modulus remains constant of E p = 86.8 GPa with η = 0.72.
Under σ s = 250 MPa, the initial composite strain is ε 0 = 0.038% due to the damages of the matrix cracking and interface debonding at σ s = 250 MPa; the initial tangent modulus is E p = 149.5 GPa, the degradation rate of the tangent modulus is 52% compared with original specimen and the fiber broken fraction is P = 0.092. With increasing stress to σ tr = 173.8 MPa, the interface debonding fraction increases and the tangent modulus decreases to E p = 60.7 GPa corresponding to η = 0.952. When the stress increases from σ tr = 173.8 MPa to σ = 250 MPa, the tangent modulus remains constant of E p = 70.4 GPa with η = 0.975.
When τ i = 20 MPa under σ s = 200 MPa, the damages of matrix cracking and interface debonding occur at σ s = 200 MPa, leading to the increase of the composite initial strain, decreasing of the tangent modulus and increase of the broken fiber fraction. The initial composite strain is ε 0 = 0.00943%; the initial tangent modulus is E p = 215 GPa, the degradation rate of tangent modulus is 31% compared with original specimen and the fiber broken fraction is P = 0.028. With increasing stress to σ tr = 123.2 MPa, the interface debonding fraction increases and the tangent modulus decreases to E p = 130 GPa corresponding to η = 0.275. When the stress increases from σ tr = 123.2 MPa to σ = 200 MPa, the tangent modulus remains constant of E p = 152.9 GPa with η = 0.284.
Under σ s = 230 MPa, the initial composite strain is ε 0 = 0.021% due to the damages of the matrix cracking and interface debonding at σ s = 230 MPa; the initial tangent modulus is E p = 185 GPa, the degradation rate of the tangent modulus is 40% compared with original specimen and the fiber broken fraction is P = 0.056. With increasing stress to σ tr = 154 MPa, the interface debonding fraction increases and the tangent modulus decreases to E p = 88.6 GPa corresponding to η = 0.53. When the stress increases from σ tr = 154 MPa to σ = 230 MPa, the tangent modulus remains constant of E p = 105.5 GPa with η = 0.54.
Under σ s = 250 MPa, the initial composite strain is ε 0 = 0.038% due to the damages of the matrix cracking and interface debonding; the initial tangent modulus is E p = 171.7 GPa, the degradation rate of tangent modulus is 45% compared with original specimen and the fiber broken fraction is P = 0.092. With increasing stress to σ tr = 173.8 MPa, the interface debonding fraction increases and the tangent modulus decreases to E p = 72.8 GPa corresponding to η = 0.714. When the stress increases from σ tr = 173.8 MPa to σ = 250 MPa, the tangent modulus remains constant of E p = 86.3 GPa with η = 0.73.

Effect of Interface Debonding Energy on Tensile Damage and Fracture of SiC/SiC Composite with Stochastic Loading
The interface debonding energy is a key interface property of CMCs. Domergue et al. [23] estimated the interface debonding energy of unidirectional SiC/CAS composite by analyzing the hysteresis loops and obtained the interface debonding energy is in the range of ζ d = 0.1 -0.8 J/m 2 . The effect of the interface debonding energy (i.e., ζ d = 0.1 and 0.3 J/m 2 ) on the tensile stress-strain curves, tangent modulus, interface debonding fraction and broken fiber fraction of SiC/SiC composite subjected to stochastic loading of σ s = 180, 220 and 250 MPa are shown in Figures 7 and 8 and Table 3. When the interface debonding energy increases, the initial composite strain, tangent modulus and broken fiber fraction remain the same, the transition stress for interface debonding decreases.      When ζd = 0.1 J/m 2 under σs = 180 MPa, the damages of matrix cracking and interface debonding occur at σs = 180 MPa, leading to the increase of the composite initial strain, decreasing of the tangent modulus and increase of the broken fiber fraction. The initial composite strain is ε0 = 0.00537%; the initial tangent modulus is Ep = 251.5 GPa, the degradation rate of the tangent modulus is 19% compared with original specimen and the fiber broken fraction is P = 0.018. With increasing stress to σtr = 140.8 MPa, the interface debonding fraction increases and the tangent modulus decreases to Ep = 171.6 GPa corresponding to η = 0.167. When the stress increases from σtr = 140.8 MPa to σ = 180 MPa, the tangent modulus remains constant of Ep = 191 GPa with η = 0.171.
Under σs = 220 MPa, the initial composite strain is ε0 = 0.0161% due to the damages of the matrix cracking and interface debonding; the initial tangent modulus is Ep = 209 GPa, the degradation rate of the tangent modulus is 33% compared with original specimen and the fiber broken fraction is P = 0.045. With increasing stress to σtr = 182.6 MPa, the interface debonding fraction increases and the  Table 3. The effect of interface debonding energy (ζ d = 0.1 and 0.3 J/m 2 ) on tensile stress-strain curve, tangent modulus, interface debonding fraction and broken fiber fraction of SiC/SiC composite subjected to stochastic loading of σ s = 180, 220 and 250 MPa. When ζ d = 0.1 J/m 2 under σ s = 180 MPa, the damages of matrix cracking and interface debonding occur at σ s = 180 MPa, leading to the increase of the composite initial strain, decreasing of the tangent modulus and increase of the broken fiber fraction. The initial composite strain is ε 0 = 0.00537%; the initial tangent modulus is E p = 251.5 GPa, the degradation rate of the tangent modulus is 19% compared with original specimen and the fiber broken fraction is P = 0.018. With increasing stress to σ tr = 140.8 MPa, the interface debonding fraction increases and the tangent modulus decreases to E p = 171.6 GPa corresponding to η = 0.167. When the stress increases from σ tr = 140.8 MPa to σ = 180 MPa, the tangent modulus remains constant of E p = 191 GPa with η = 0.171.
Under σ s = 220 MPa, the initial composite strain is ε 0 = 0.0161% due to the damages of the matrix cracking and interface debonding; the initial tangent modulus is E p = 209 GPa, the degradation rate of the tangent modulus is 33% compared with original specimen and the fiber broken fraction is P = 0.045. With increasing stress to σ tr = 182.6 MPa, the interface debonding fraction increases and the tangent modulus decreases to E p = 103.6 GPa corresponding to η = 0.446. When the stress increases from σ tr = 182.6 MPa to σ = 220 MPa, the tangent modulus remains constant of E p = 118.4 GPa with η = 0.451.
Under σ s = 250 MPa, the initial composite strain is ε 0 = 0.038% due to the damages of the matrix cracking and interface debonding; the initial tangent modulus is E p = 188.5 GPa, the degradation rate of the tangent modulus is 40% compared with the original specimen and the fiber broken fraction is P = 0.092. With increasing stress to σ tr = 211.2 MPa, the interface debonding fraction increases and the tangent modulus decreases to E p = 77 GPa corresponding to η = 0.694. When the stress increases from σ tr = 211.2 MPa to σ = 250 MPa, the tangent modulus remains constant of E p = 88.2 GPa with η = 0.705.
When ζ d = 0.3 J/m 2 under σ s = 180 MPa, the damages of matrix cracking and interface debonding occur at σ s = 180 MPa, leading to the increase of the composite initial strain, decreasing of the tangent modulus and increase of the broken fiber fraction. The initial composite strain is ε 0 = 0.00537%; the initial tangent modulus is E p = 251.5 GPa, the degradation rate of the tangent modulus is 19% compared with the original specimen and the fiber broken fraction is P = 0.018. With increasing stress to σ tr = 118.8 MPa, the interface debonding fraction increases and the tangent modulus decreases to E p = 180.6 GPa corresponding to η = 0.141. When the stress increases from σ tr = 118.8 MPa to σ = 180 MPa, the tangent modulus remains constant of E p = 202 GPa with η = 0.144.
Under σ s = 220 MPa, the initial composite strain is ε 0 = 0.0161%; the initial tangent modulus is E p = 209 GPa, the degradation rate of tangent modulus is 33% compared with original specimen and the fiber broken fraction is P = 0.045. With increasing stress to σ tr = 158.4 MPa, the interface debonding fraction increases and the tangent modulus decreases to E p = 110.4 GPa corresponding to η = 0.387. When the stress increases from σ tr = 158.4 MPa to σ = 220 MPa, the tangent modulus remains constant of E p = 127.6 GPa with η = 0.396.
Under σ s = 250 MPa, the initial composite strain is ε 0 = 0.038% due to the damages of the matrix cracking and interface debonding; the initial tangent modulus is E p = 188.5 GPa, the degradation rate of tangent modulus is 40% compared with the original specimen and the fiber broken fraction is P = 0.092. With increasing stress to σ tr = 189.2 MPa, the interface debonding fraction increases and the tangent modulus decreases to E p = 82 GPa corresponding to η = 0.622. When the stress increases from σ tr = 189.2 MPa to σ = 250 MPa, the tangent modulus remains constant of E p = 95.1 GPa with η = 0.63.

Effect of Saturation Matrix Crack Spacing on Tensile Damage and Fracture of SiC/SiC Composite with Stochastic Loading
Li [24] investigated multiple matrix cracking of CMCs with different fiber preforms and found that the saturation matrix cracking spacing is in the range of l s = 100 and 500 µm. In the present analysis, the effect of the saturation matrix crack spacing (i.e., l s = 200 and 250 µm) on the tensile stress-strain curves, tangent modulus, interface debonding fraction and broken fiber fraction of SiC/SiC composite subjected to stochastic loading of σ s = 180, 220 and 250 MPa are shown in Figures 9 and 10 and Table 4. When saturation matrix crack spacing increases, the initial composite strain decreases, the initial tangent modulus increases, the transition stress for interface debonding and initial fiber broken fraction remain the same. Under σs = 250 MPa, the initial composite strain is ε0 = 0.038% due to the damages of the matrix cracking and interface debonding; the initial tangent modulus is Ep = 188.5 GPa, the degradation rate of tangent modulus is 40% compared with the original specimen and the fiber broken fraction is P = 0.092. With increasing stress to σtr = 189.2 MPa, the interface debonding fraction increases and the tangent modulus decreases to Ep = 82 GPa corresponding to η = 0.622. When the stress increases from σtr = 189.2 MPa to σ = 250 MPa, the tangent modulus remains constant of Ep = 95.1 GPa with η = 0.63.

Effect of Saturation Matrix Crack Spacing on Tensile Damage and Fracture of SiC/SiC Composite with Stochastic Loading
Li [24] investigated multiple matrix cracking of CMCs with different fiber preforms and found that the saturation matrix cracking spacing is in the range of ls = 100 and 500 μm. In the present analysis, the effect of the saturation matrix crack spacing (i.e., ls = 200 and 250 μm) on the tensile stress-strain curves, tangent modulus, interface debonding fraction and broken fiber fraction of SiC/SiC composite subjected to stochastic loading of σs = 180, 220 and 250 MPa are shown in Figures  9 and 10 and Table 4. When saturation matrix crack spacing increases, the initial composite strain decreases, the initial tangent modulus increases, the transition stress for interface debonding and initial fiber broken fraction remain the same.         Under σ s = 220 MPa, the initial composite strain is ε 0 = 0.0168% due to the damages of the matrix cracking and the interface debonding; the initial tangent modulus is E p = 180.3 GPa, the degradation rate of tangent modulus is 42% compared with original specimen and the fiber broken fraction is P = 0.045. With increasing stress to σ tr = 143 MPa, the interface debonding fraction increases and the tangent modulus decreases to E p = 88.1 GPa corresponding to η = 0.524. When the stress increases from σ tr = 143 MPa to σ = 220 MPa, the tangent modulus remains constant of E p = 105.2 GPa with η = 0.536.
Under σ s = 250 MPa, the initial composite strain is ε 0 = 0.038% due to the damages of the matrix cracking and the interface debonding; the initial tangent modulus is E p = 157.7 GPa, the degradation rate of tangent modulus is 50% compared with original specimen and the fiber broken fraction is P = 0.092. With increasing stress to σ tr = 173.8 MPa, the interface debonding fraction increases and the tangent modulus decreases to E p = 64 GPa corresponding to η = 0.857. When the stress increases from σ tr = 173.8 MPa to σ = 250 MPa, the tangent modulus remains constant of E p = 75.3 GPa with η = 0.869.
When l s = 250 µm under σ s = 180 MPa, the damages of matrix cracking and interface debonding occur at σ s = 180 MPa, leading to the increase of the composite initial strain, decreasing of the tangent modulus and increase of the broken fiber fraction. The initial composite strain is ε 0 = 0.00549% due to the damages of the matrix cracking and the interface debonding; the initial tangent modulus is E p = 242.3 GPa, the degradation rate of tangent modulus is 22% compared with original specimen and the fiber broken fraction is P = 0.018. With increasing stress to σ tr = 103.4 MPa, the interface debonding fraction increases and the tangent modulus decreases to E p = 173.7 GPa corresponding to η = 0.147.
When the stress increases from σ tr = 103.4.5 MPa to σ = 180 MPa, the tangent modulus remains constant of E p = 198.7 GPa with η = 0.151.
Under σ s = 220 MPa, the initial composite strain is ε 0 = 0.0164% due to the damages of the matrix cracking and the interface debonding; the initial tangent modulus is E p = 196.8 GPa, the degradation rate of tangent modulus is 37% compared with original specimen and the fiber broken fraction is P = 0.045. With increasing stress to σ tr = 143 MPa, the interface debonding fraction increases and the tangent modulus decreases to E p = 102.8 GPa corresponding to η = 0.42. When the stress increases from σ tr = 143 MPa to σ = 220 MPa, the tangent modulus remains constant of E p = 121.2 GPa with η = 0.43.
Under σ s = 250 MPa, the initial composite strain is ε 0 = 0.038% due to the damages of the matrix cracking and the interface debonding; the initial tangent modulus is E p = 174.8 GPa, the degradation rate of tangent modulus is 44% compared with original specimen and the fiber broken fraction is P = 0.092. With increasing stress to σ tr = 173.8 MPa, the interface debonding fraction increases and the tangent modulus decreases to E p = 75 GPa corresponding to η = 0.685. When the stress increases from σ tr = 173.8 MPa to σ = 250 MPa, the tangent modulus remains constant of E p = 88.6 GPa with η = 0.695.

Effect of Fiber Strength on Tensile Damage and Fracture of SiC/SiC Composite with Stochastic Loading
Guo et al. [25] investigated the SiC fiber strength and found that the SiC fiber strength is in the range between σ c = 2.3 and 3.7 GPa. In the present analysis, the effect of the fiber strength (i.e., σ c = 2.0 and 2.5 GPa) on the tensile stress-strain curves, tangent modulus, interface debonding fraction and broken fiber fraction of SiC/SiC composite subjected to stochastic loading of σ s = 180, 220 and 250 MPa are shown in Figures 11 and 12 and Table 5. When the fiber strength increases, the initial composite strain and fiber broken fraction decrease and the initial tangent composite modulus and transition stress for interface debonding remains the same. Under σs = 250 MPa, the initial composite strain is ε0 = 0.038% due to the damages of the matrix cracking and the interface debonding; the initial tangent modulus is Ep = 174.8 GPa, the degradation rate of tangent modulus is 44% compared with original specimen and the fiber broken fraction is P = 0.092. With increasing stress to σtr = 173.8 MPa, the interface debonding fraction increases and the tangent modulus decreases to Ep = 75 GPa corresponding to η = 0.685. When the stress increases from σtr = 173.8 MPa to σ = 250 MPa, the tangent modulus remains constant of Ep = 88.6 GPa with η = 0.695.

Effect of Fiber Strength on Tensile Damage and Fracture of SiC/SiC Composite with Stochastic Loading
Guo et al. [25] investigated the SiC fiber strength and found that the SiC fiber strength is in the range between σc = 2.3 and 3.7 GPa. In the present analysis, the effect of the fiber strength (i.e., σc = 2.0 and 2.5 GPa) on the tensile stress-strain curves, tangent modulus, interface debonding fraction and broken fiber fraction of SiC/SiC composite subjected to stochastic loading of σs = 180, 220 and 250 MPa are shown in Figures 11 and 12 and Table 5. When the fiber strength increases, the initial composite strain and fiber broken fraction decrease and the initial tangent composite modulus and transition stress for interface debonding remains the same.    When σ c = 2.0 GPa under σ s = 180 MPa, the damages of matrix cracking and interface debonding occur at σ s = 180 MPa, leading to the increase of the composite initial strain, decreasing of the tangent modulus and increase of the broken fiber fraction. The composite initial strain is ε 0 = 0.0065% due to the damages of the matrix cracking and the interface debonding; the initial tangent modulus is E p = 251.5 GPa, the degradation rate of tangent modulus is 19% compared with original specimen and the fiber broken fraction is P = 0.022. With increasing stress to σ tr = 103.4 MPa, the interface debonding fraction increases and the tangent modulus decreases to E p = 187.4 GPa corresponding to η = 0.122. When the stress increases from σ tr = 103.4 MPa to σ = 180 MPa, the tangent modulus remains constant of E p = 211.3 GPa with η = 0.126.
Under σ s = 220 MPa, the initial composite strain is ε 0 = 0.02% due to the damages of the matrix cracking and the interface debonding; the initial tangent modulus is E p = 209.6 GPa, the degradation rate of tangent modulus is 33% compared with original specimen and the fiber broken fraction is P = 0.058. With increasing stress to σ tr = 143 MPa, the interface debonding fraction increases and the tangent modulus decreases to E p = 115.7 GPa corresponding to η = 0.349. When the stress increases from σ tr = 143 MPa to σ = 220 MPa, the tangent modulus remains constant of E p = 134.9 GPa with η = 0.357.
Under σ s = 250 MPa, the initial composite strain is ε 0 = 0.059%; the initial tangent modulus is E p = 188.5 GPa, the degradation rate of tangent modulus is 40% compared with original specimen and the fiber broken fraction is P = 0.138. With increasing stress to σ tr = 173.8 MPa, the interface debonding fraction increases and the tangent modulus decreases to E p = 86 GPa corresponding to η = 0.571. When the stress increases from σ tr = 173.8 MPa to σ = 250 MPa, the tangent modulus remains constant of E p = 100.6 GPa with η = 0.579.
When σ c = 2.5 GPa under σ s = 180 MPa, the damages of matrix cracking and interface debonding occur at σ s = 180 MPa, leading to the increase of the composite initial strain, decreasing of the tangent modulus and increase of the broken fiber fraction. The initial composite strain is ε 0 = 0.0028% due to the damages of the matrix cracking and the interface debonding; the initial tangent modulus is E p = 251.5 GPa, the degradation rate of tangent modulus is 19% compared with original specimen and the fiber broken fraction is P = 0.0087. With increasing stress to σ tr = 103.4 MPa, the interface debonding fraction increases and the tangent modulus decreases to E p = 187.4 GPa corresponding to η = 0.122. When the stress increases from σ tr = 103.4 MPa to σ = 180 MPa, the tangent modulus remains constant of E p = 211.3 GPa with η = 0.126.
Under σ s = 220 MPa, the initial composite strain is ε 0 = 0.007% due to the damages of the matrix cracking and the interface debonding; the initial tangent modulus is E p = 209.6 GPa, the degradation rate of tangent modulus is 33% compared with original specimen and the fiber broken fraction is P = 0.02. With increasing stress to σ tr = 143 MPa, the interface debonding fraction increases and the tangent modulus decreases to E p = 115.7 GPa corresponding to η = 0.344. When the stress increases from σ tr = 140.8 MPa to σ = 220 MPa, the tangent modulus remains constant of E p = 134.9 GPa with η = 0.357.
Under σ s = 250 MPa, the initial composite strain is ε 0 = 0.015%; the initial tangent modulus is E p = 188.5 GPa, the degradation rate of the tangent modulus is 40% compared with original specimen and the fiber broken fraction is P = 0.036. With increasing stress to σ tr = 173.8 MPa, the interface debonding fraction increases and the tangent modulus decreases to E p = 86 GPa corresponding to η = 0.571. When the stress increases from σ tr = 173.8 MPa to σ = 250 MPa, the tangent modulus remains constant of E p = 100.6 GPa with η = 0.579.

Experimental Comparisons
Li et al. [4], Liu [5], Guo and Kagawa [6] and Morscher [7] investigated tensile behavior of unidirectional and 2D SiC/SiC composites at room temperature. In this section, using the developed damage models and micromechanical constitutive models for the conditions of matrix cracking, interface debonding and fiber failure, the experimental tensile stress-strain curves are predicted.
The comparisons between tensile stress-strain curves with and without stochastic loading are analyzed. The relationships between the stochastic loading stress levels, tangent modulus, interface debonding fraction and fiber broken fraction are established. Li et al. [4] investigated the tensile behavior of 2D SiC/SiC composite at room temperature. The composite was fabricated using chemical vapor infiltration (CVI) method. The tensile test was performed under displacement control with the speed of 0.3 mm/min. The experimental tensile stress-strain curves, tangent modulus versus strain curves, interface debonding fraction and broken fiber fraction versus stress curves of 2D SiC/SiC composite without stochastic loading and with stochastic loading at σ s = 140, 180, 200 and 240 MPa at room temperature are shown in Figure 13 and Table 6. When stochastic loading stress increases, the initial composite strain increases, the initial tangent modulus decreases, the transition stress for interface debonding increases and the initial fiber broken fraction increases.
Materials 2020, 13, x 19 of 27 The relationships between the stochastic loading stress levels, tangent modulus, interface debonding fraction and fiber broken fraction are established.

2D SiC/SiC under Stochastic Loading of 140, 180, 200 and 240 MPa
Li et al. [4] investigated the tensile behavior of 2D SiC/SiC composite at room temperature. The composite was fabricated using chemical vapor infiltration (CVI) method. The tensile test was performed under displacement control with the speed of 0.3 mm/min. The experimental tensile stress-strain curves, tangent modulus versus strain curves, interface debonding fraction and broken fiber fraction versus stress curves of 2D SiC/SiC composite without stochastic loading and with stochastic loading at σs = 140, 180, 200 and 240 MPa at room temperature are shown in Figure 13 and Table 6. When stochastic loading stress increases, the initial composite strain increases, the initial tangent modulus decreases, the transition stress for interface debonding increases and the initial fiber broken fraction increases.    Under σ s = 140 MPa, the initial strain is ε 0 = 0.001%; the initial tangent modulus is E p = 288 GPa, the degradation rate of tangent modulus is 7% compared with original specimen and the fiber broken fraction is P = 0.006. With increasing stress to σ tr = 63.8 MPa, the tangent modulus decreases to E p = 267.4 GPa corresponding to η = 0.024. When the stress increases from σ tr = 63.8 MPa to σ = 140 MPa, the tangent modulus remains constant of E p = 282.3 GPa with η = 0.025.
Under σ s = 180 MPa, the initial strain is ε 0 = 0.005%; the initial tangent modulus is E p = 251 GPa, the degradation rate of tangent modulus is 19% compared with original specimen and the fiber broken fraction is P = 0.018. With increasing stress to σ tr = 103.4 MPa, the tangent modulus decreases to E p = 187.4 GPa corresponding to η = 0.122. When the stress increases from σ tr = 103.4 MPa to σ = 180 MPa, the tangent modulus remains constant of E p = 211.3 GPa with η = 0.126.
Under σ s = 200 MPa, the initial strain is ε 0 = 0.009%; the initial tangent modulus is E p = 229.5 GPa, the degradation rate of tangent modulus is 26% compared with original specimen and the fiber broken fraction is P = 0.028. With increasing stress to σ tr = 123.2 MPa, the tangent modulus decreases to E p = 147.1 GPa corresponding to η = 0.22. When the stress increases from σ tr = 123.2 MPa to σ = 200 MPa, the tangent modulus remains constant of E p = 169.7 GPa with η = 0.226.
Under σ s = 240 MPa, the initial strain is ε 0 = 0.028%; the initial tangent modulus is E p = 194.2 GPa, the degradation rate of tangent modulus is 38% compared with original specimen and the fiber broken fraction is P = 0.078. With increasing stress to σ tr = 162.8 MPa, the tangent modulus decreases to E p = 94 GPa corresponding to η = 0.49. When the stress increases from σ tr = 162.8 MPa to σ = 240 MPa, the tangent modulus remains constant of E p = 109.8 GPa with η = 0.5.

UD and 2D SiC/SiC under Stochastic Loading
Liu [5] investigated the tensile behavior of unidirectional and 2D SiC/SiC composites at room temperature. The tensile test was performed under displacement control with the loading rate of 0.2 mm/min.
For unidirectional SiC/SiC composite, the tensile stress-strain curves, tangent modulus versus strain curves, interface debonding fraction and broken fiber fraction versus stress curves without stochastic loading and with stochastic loading at σ s = 140, 180, 200 and 220 MPa at room temperature are shown in Figure 14 and Table 7. temperature. The tensile test was performed under displacement control with the loading rate of 0.2 mm/min.
For unidirectional SiC/SiC composite, the tensile stress-strain curves, tangent modulus versus strain curves, interface debonding fraction and broken fiber fraction versus stress curves without stochastic loading and with stochastic loading at σs = 140, 180, 200 and 220 MPa at room temperature are shown in Figure 14 and Table 7.  Under σs = 140 MPa, the initial strain is ε0 = 0.002%; the initial tangent modulus is Ep = 235.9 GPa, the degradation rate of tangent modulus is 1% compared with original specimen and the fiber broken fraction is P = 0.01. With increasing stress to σtr = 58.8 MPa, the tangent modulus decreases to Ep = 235.7 GPa corresponding to η = 0.0001. When the stress increases from σtr = 58.8 MPa to σ = 140 MPa, the tangent modulus remains constant of Ep = 235.8 GPa with η = 0.0001.
Under σs = 180 MPa, the initial strain is ε0 = 0.009%; the initial tangent modulus is Ep = 234 GPa, the degradation rate of tangent modulus is 1.1% compared with original specimen and the fiber broken fraction is P = 0.03. With increasing stress to σtr = 98.4 MPa, the tangent modulus decreases to Ep = 230.1 GPa corresponding to η = 0.005. When the stress increases from σtr = 98.4 MPa to σ = 180  Under σ s = 140 MPa, the initial strain is ε 0 = 0.002%; the initial tangent modulus is E p = 235.9 GPa, the degradation rate of tangent modulus is 1% compared with original specimen and the fiber broken fraction is P = 0.01. With increasing stress to σ tr = 58.8 MPa, the tangent modulus decreases to E p = 235.7 GPa corresponding to η = 0.0001. When the stress increases from σ tr = 58.8 MPa to σ = 140 MPa, the tangent modulus remains constant of E p = 235.8 GPa with η = 0.0001.
Under σ s = 180 MPa, the initial strain is ε 0 = 0.009%; the initial tangent modulus is E p = 234 GPa, the degradation rate of tangent modulus is 1.1% compared with original specimen and the fiber broken fraction is P = 0.03. With increasing stress to σ tr = 98.4 MPa, the tangent modulus decreases to E p = 230.1 GPa corresponding to η = 0.005. When the stress increases from σ tr = 98.4 MPa to σ = 180 MPa, the tangent modulus remains constant of E p = 230.9 GPa with η = 0.005.
Under σ s = 200 MPa, the initial strain is ε 0 = 0.0174%; the initial tangent modulus is E p = 229.6 GPa, the degradation rate of tangent modulus is 2.8% compared with original specimen and the fiber broken fraction is P = 0.048. With increasing stress to σ tr = 118.8 MPa, the tangent modulus decreases to E p = 215.4 GPa corresponding to η = 0.023. When the stress increases from σ tr = 118.8 MPa to σ = 200 MPa, the tangent modulus remains constant of E p = 217.8 GPa with η = 0.023.
Under σ s = 220 MPa, the initial strain is ε 0 = 0.033%; the initial tangent modulus is E p = 220.1 GPa, the degradation rate of tangent modulus is 6.8% compared with original specimen and the fiber broken fraction is P = 0.082. With increasing stress to σ tr = 139 MPa, the tangent modulus decreases to E p = 184.5 GPa corresponding to η = 0.07. When the stress increases from σ tr = 139 MPa to σ = 220 MPa, the tangent modulus remains constant of E p = 189.2 GPa with η = 0.071.
For 2D SiC/SiC composite, the tensile stress-strain curves, tangent modulus versus strain curves, interface debonding fraction and broken fiber fraction versus stress curves without stochastic loading and with stochastic loading at σ s = 80, 100 and 120 MPa at room temperature are shown in Figure 15 and Table 8. For 2D SiC/SiC composite, the tensile stress-strain curves, tangent modulus versus strain curves, interface debonding fraction and broken fiber fraction versus stress curves without stochastic loading and with stochastic loading at σs = 80, 100 and 120 MPa at room temperature are shown in Figure 15 and Table 8.    Under σ s = 80 MPa, the initial strain is ε 0 = 0.002%; the initial tangent modulus is E p = 138.3 GPa, the degradation rate of tangent modulus is 1.2% compared with original specimen and the fiber broken fraction is P = 0.01. With increasing stress to σ tr = 13.2 MPa, the tangent modulus decreases to E p = 137.6 GPa corresponding to η = 0.001. When stress increases from σ tr = 13.2 MPa to σ = 80 MPa, the tangent modulus remains constant of E p = 137.7 GPa with η = 0.002.
Under σ s = 100 MPa, the initial strain is ε 0 = 0.008%; the initial tangent modulus is E p = 135.6 GPa, the degradation rate of tangent modulus is 3.1% compared with original specimen and the fiber broken fraction is P = 0.027. With increasing stress to σ tr = 32.4 MPa, the tangent modulus decreases to E p = 131 GPa corresponding to η = 0.012. When stress increases from σ tr = 32.4 MPa to σ = 100 MPa, the tangent modulus remains constant of E p = 131.3 GPa with η = 0.013.
Under σ s = 120 MPa, the initial strain is ε 0 = 0.025%; the initial tangent modulus is E p = 131.9 GPa, the degradation rate of tangent modulus is 5.8% compared with original specimen and the fiber broken fraction is P = 0.068. With increasing stress to σ tr = 52.8 MPa, the tangent modulus decreases to E p = 119.5 GPa corresponding to η = 0.038. When stress increases from σ tr = 52.8 MPa to σ = 120 MPa, the tangent modulus remains constant of E p = 119.5 GPa with η = 0.039.

2D SiC/SiC under Stochastic Loading of 80, 100 and 120 MPa
Guo and Kagawa [6] investigated the tensile behavior of 2D plain-woven fabric SiC/SiC composite fabricated by the PIP process. The quasi-static tensile test was conducted under displacement control with the rate of 0.5 mm/min. The tensile stress-strain curves, tangent modulus versus strain curves, interface debonding fraction and broken fiber fraction versus stress curves without stochastic loading and with stochastic loading at σ s = 80, 100 and 120 MPa at room temperature are shown in Figure 16 and Table 9.
Guo and Kagawa [6] investigated the tensile behavior of 2D plain-woven fabric SiC/SiC composite fabricated by the PIP process. The quasi-static tensile test was conducted under displacement control with the rate of 0.5 mm/min. The tensile stress-strain curves, tangent modulus versus strain curves, interface debonding fraction and broken fiber fraction versus stress curves without stochastic loading and with stochastic loading at σs = 80, 100 and 120 MPa at room temperature are shown in Figure 16 and Table 9.  Table 9. The tensile stress-strain curve, tangent modulus, interface debonding fraction and broken fiber fraction of 2D SiC/SiC composite subjected to stochastic loading of σs = 80, 100 and 120 MPa.  Under σ s = 80 MPa, the initial strain is ε 0 = 0.005%; the initial tangent modulus is E p = 56 GPa, the degradation rate of tangent modulus is 6.7% compared with original specimen and the fiber broken fraction is P = 0.01. With increasing stress to σ tr = 33.6 MPa, the tangent modulus decreases to E p = 49.4 GPa corresponding to η = 0.07. When stress increases from σ tr = 33.6 MPa to σ = 80 MPa, the tangent modulus remains constant of E p = 50.4 GPa with η = 0.074.
Under σ s = 100 MPa, the initial strain is ε 0 = 0.016%; the initial tangent modulus is E p = 53.5 GPa, the degradation rate of tangent modulus is 11% compared with original specimen and the fiber broken fraction is P = 0.027. With increasing stress to σ tr = 52.8 MPa, the tangent modulus decreases to E p = 39.7 GPa corresponding to η = 0.19. When stress increases from σ tr = 52.8 MPa to σ = 100 MPa, the tangent modulus remains constant of E p = 40.7 GPa with η = 0.2.
Under σ s = 120 MPa, the initial strain is ε 0 = 0.046%; the initial tangent modulus is E p = 51 GPa, the degradation rate of tangent modulus is 15% compared with original specimen and the fiber broken fraction is P = 0.068. With increasing stress to σ tr = 73.2 MPa, the tangent modulus decreases to E p = 30.8 GPa corresponding to η = 0.39. When stress increases from σ tr = 73.2 MPa to σ = 120 MPa, the tangent modulus remains constant of E p = 31.7 GPa with η = 0.395. Morscher [7] investigated the tensile behavior of 2D SiC/SiC composite at room temperature. The tensile test was conducted under load control. The tensile stress-strain curves, tangent modulus versus strain curves, interface debonding fraction and broken fiber fraction versus stress curves without stochastic loading and with stochastic loading at σ s = 180, 220, 260 and 300 MPa are shown in Figure 17 and Table 10.
degradation rate of tangent modulus is 6.7% compared with original specimen and the fiber broken fraction is P = 0.01. With increasing stress to σtr = 33.6 MPa, the tangent modulus decreases to Ep = 49.4 GPa corresponding to η = 0.07. When stress increases from σtr = 33.6 MPa to σ = 80 MPa, the tangent modulus remains constant of Ep = 50.4 GPa with η = 0.074.
Under σs = 100 MPa, the initial strain is ε0 = 0.016%; the initial tangent modulus is Ep = 53.5 GPa, the degradation rate of tangent modulus is 11% compared with original specimen and the fiber broken fraction is P = 0.027. With increasing stress to σtr = 52.8 MPa, the tangent modulus decreases to Ep = 39.7 GPa corresponding to η = 0.19. When stress increases from σtr = 52.8 MPa to σ = 100 MPa, the tangent modulus remains constant of Ep = 40.7 GPa with η = 0.2.
Under σs = 120 MPa, the initial strain is ε0 = 0.046%; the initial tangent modulus is Ep = 51 GPa, the degradation rate of tangent modulus is 15% compared with original specimen and the fiber broken fraction is P = 0.068. With increasing stress to σtr = 73.2 MPa, the tangent modulus decreases to Ep = 30.8 GPa corresponding to η = 0.39. When stress increases from σtr = 73.2 MPa to σ = 120 MPa, the tangent modulus remains constant of Ep = 31.7 GPa with η = 0.395.

2D SiC/SiC under Stochastic Loading of 180, 220, 260 and 300 MPa
Morscher [7] investigated the tensile behavior of 2D SiC/SiC composite at room temperature. The tensile test was conducted under load control. The tensile stress-strain curves, tangent modulus versus strain curves, interface debonding fraction and broken fiber fraction versus stress curves without stochastic loading and with stochastic loading at σs = 180, 220, 260 and 300 MPa are shown in Figure 17 and Table 10.     Under σ s = 180 MPa, the initial strain is ε 0 = 0.003%; the initial tangent modulus is E p = 142.3 GPa, the degradation rate of tangent modulus is 50% compared with original specimen and the fiber broken fraction is P = 0.004. With increasing stress to σ tr = 122 MPa, the tangent modulus decreases to E p = 53.9 GPa corresponding to η = 0.63. When stress increases from σ tr = 122 MPa to σ = 180 MPa, the tangent modulus remains constant of E p = 63.9 GPa with η = 0.638.
Under σ s = 220 MPa, the initial strain is ε 0 = 0.007%; the initial tangent modulus is E p = 131.4 GPa, the degradation rate of tangent modulus is 54% compared with original specimen and the fiber broken fraction is P = 0.01. With increasing stress to σ tr = 162 MPa, the tangent modulus decreases to E p = 42 GPa corresponding to η = 0.63. When stress increases from σ tr = 122 MPa to σ = 180 MPa, the tangent modulus remains constant of E p = 63.9 GPa with η = 0.97.
Under σ s = 260 MPa, the initial strain is ε 0 = 0.014%; the initial tangent modulus is E p = 128.7 GPa, the degradation rate of tangent modulus is 55% compared with original specimen and the fiber broken fraction is P = 0.021. With increasing stress to σ tr = 150 MPa, the tangent modulus decreases to E p = 41.3 GPa corresponding to η = 0.93. When stress increases from σ tr = 150 MPa to σ = 260 MPa, the tangent modulus remains constant of E p = 46 GPa with η = 1.0.
Under σ s = 300 MPa, the initial strain is ε 0 = 0.029%; the initial tangent modulus is E p = 128.3 GPa, the degradation rate of tangent modulus is 55.1% compared with original specimen and the fiber broken fraction is P = 0.04. With increasing stress to σ tr = 148 MPa, the tangent modulus decreases to E p = 41.3 GPa corresponding to η = 0.93. When stress increases from σ tr = 148 MPa to σ = 300 MPa, the tangent modulus remains constant of E p = 46 GPa with η = 1.0.

Conclusions
In this paper, the effect of stochastic loading on tensile damage and fracture of fiber-reinforced CMCs is investigated. A micromechanical constitutive model is developed considering multiple damage mechanisms under tensile loading. The relationship between stochastic stress, tangent modulus, interface debonding and fiber broken is established. The effects of fiber volume, interface shear stress, interface debonding energy, saturation matrix crack spacing and fiber strength on tensile stress-strain curve, tangent modulus, interface debonding fraction and fiber broken fraction are analyzed. The experimental tensile damage and fracture of unidirectional and 2D SiC/SiC composites subjected to different stochastic loading stress are predicted.
(1) When fiber volume increases, the initial composite strain decreases, the initial tangent modulus increases, the transition stress for interface debonding decreases and the initial fiber broken fraction decreases; (2) When the interface shear stress increases, the initial composite strain remains the same, the initial tangent modulus increases, the transition stress for interface debonding remains the same and the initial fiber broken fraction remains the same; (3) When the interface debonding energy increases, the initial composite strain, tangent modulus and broken fiber fraction remain the same and the transition stress for interface debonding decreases; (4) When saturation matrix crack spacing increases, the initial composite strain decreases, the initial tangent modulus increases and the transition stress for interface debonding and initial fiber broken fraction remain the same; (5) When the fiber strength increases, the initial composite strain and fiber broken fraction decrease and the initial tangent composite modulus and transition stress for interface debonding remains the same.