Micromechanical Modeling Tensile and Fatigue Behavior of Fiber-Reinforced Ceramic-Matrix Composites Considering Matrix Fragmentation and Closure

: In this paper, micromechanical constitutive models are developed to predict the tensile and fatigue behavior of ﬁber-reinforced ceramic-matrix composites (CMCs) considering matrix fragmentation and closure. Damage models of matrix fragmentation, interface debonding, and ﬁber’s failure are considered in the micromechanical analysis of tensile response, and the matrix fragmentation closure, interface debonding and repeated sliding are considered in the hysteresis response. Relationships between the matrix fragmentation and closure, tensile and fatigue response, and interface debonding and ﬁber’s failure are established. Experimental matrix fragmentation density, tensile curves, and fatigue hysteresis loops of mini, unidirectional, cross-ply, and 2D plain-woven SiC/SiC composites are predicted using the developed constitutive models. Matrix fragmentation density changes with increasing or decreasing applied stress, which affects the nonlinear strain of SiC/SiC composite under tensile loading, and the interface debonding and sliding range of SiC/SiC composite under fatigue loading.


Introduction
SiC/SiC ceramic matrix composite (CMC) is a type of composite with the SiC fiber as reinforcement and SiC ceramic as matrix. It not only maintains the advantages of SiC ceramic, such as, high temperature resistance, high strength, low density, and oxidation resistance, but also possesses the toughening effect of the SiC fiber, and effectively overcomes the dangerous shortcoming of monolithic ceramics, such as, brittleness, sensitivity to defects or cracks and poor reliability. Compared with superalloy material, SiC/SiC composites possess lower density (i.e., usually 2.0~3.0 g/cm 3 , which is only 1/3~1/4 of superalloy), and higher temperature resistance (i.e., the operating temperature with cooling is higher than 1200 • C). For the application in aeroengine, SiC/SiC composites can reduce the structural weight, simplify cooling structure, reduce cooling air consumption, improve combustion efficiency, and increase thrust-weight-ratio. It is one of the most potential thermal structural materials for aeroengine hot-section components [1][2][3][4].
Under tensile and cyclic fatigue loading, multiple micro damage mechanisms occur and contribute to the nonlinear behavior of the composite [5][6][7][8][9][10]. With increasing applied tensile stress, matrix fragmentation occurs first, and the matrix fragmentation density increases to saturation. Sevener et al. [11] performed experimental investigation on crack opening behavior in 2D plain-woven melt infiltrated SiC/SiC composite under uniaxial tension using scanning electron microscopy (SEM) combined with digital image correlation (DIC) and manual crack opening displacement (COD) measurements. Crack openings were found to increase linearly with increasing applied stress. Li [12] developed a micromechanical model to predict the COD behavior of mini-SiC/SiC composite. Relationships between COD, crack opening stress, interface debonding stress and interface debonding ratio were established. Chen et al. [13] investigated the crack initiation and propagation in braided SiC/SiC tubes with different braiding angle using in situ tensile tests with synchrotron micro-computed tomography. The results show that braiding angle has no obvious effect on the location of crack onsets, whereas it significantly affects the paths of crack propagation. Li [14] predicted the first matrix fragmentation stress in fiber-reinforced CMCs using the energy balance approach considering fiber's debonding and fracture. Morscher et al. [15], Goulmy et al. [16] performed experimental investigations on matrix multiple cracking using the electrical resistance. Li [17] investigated multiple matrix fragmentation behavior of mini, unidirectional, and 2D plain-woven SiC/SiC composites using critical matrix strain energy (CMSE) criterion. Matrix fragmentation density increased with the increasing of fiber's volume, fiber's elastic modulus, interface shear stress, and interface debonding energy, and the decreasing of the fiber's radius and matrix elastic modulus. Matrix fragmentation affects the tensile and fatigue behavior of fiber-reinforced CMCs [18][19][20][21][22][23][24][25]. Ahn and Curtin [26] investigated matrix stochastic fragmentation on hysteresis loops in unidirectional CMCs. Liu et al. [27] analyzed the hysteresis loops of 3D needle-punched C/SiC composite and predicted the mechanical hysteresis loops for different tensile peak stress. The hysteresis-based damage parameters of inverse tangent modulus and interface debonding ratio were adopted to describe the composite's internal damage evolution. Li [28] predicted the non-closure hysteresis loops in fiber-reinforced CMCs at high tensile stress, due to the change of matrix fragmentation density with loading/unloading tensile stress. However, in the research mentioned above, the synergistic effect of matrix fragmentation and closure with increasing or decreasing tensile stress on tensile and fatigue behavior of fiber-reinforced CMCs have not been analyzed.
The objective of this paper is to develop a micromechanical constitutive model to analyze the effects of matrix fragmentation and closure on tensile and fatigue behavior of fiber-reinforced CMCs. Damage models of matrix fragmentation, interface debonding, and fiber's failure are considered in the micromechanical analysis for tensile response. Matrix fragmentation and closure, interface debonding and repeated sliding are considered in the micromechanical analysis for hysteresis response. Relationships between composite's tensile and fatigue response and matrix fragmentation and closure are established. Experimental tensile curves and fatigue hysteresis loops of mini, unidirectional, cross-ply, and 2D plain-woven SiC/SiC composites are predicted.

Micromechanical Constitutive Models
Under tensile and cyclic loading/unloading tensile testing, matrix fragmentation with interface debonding occurs, and the fragmentation density increases with tensile peak stress. However, with decreasing applied stress, the matrix fragmentation density changes with applied stress, which would affect the mechanical hysteresis loops of fiberreinforced CMCs. In this section, the micromechanical constitutive models for monotonic tensile and cyclic loading/unloading hysteresis loops were developed considering matrix fragmentation and closure.

Micromechanical Tensile Constitutive Model Considering Matrix Fragmentation
For fiber-reinforced CMCs under tensile loading, damage mechanisms of matrix fragmentation, interface debonding and fiber's failure occur, and contribute to the nonlinear damage and fracture of composite's tensile response.
For the initial tensile loading, the composite's strain response is, [5] where ε c is the strain of composite, σ is applied stress, and E c is elastic modulus of composite. ( where E f is the fiber's elastic modulus, Φ represents the intact fiber stress, τ i represents the shear stress at the interface, and r f represents the fiber's radius; l d and l c are the lengths of the debonding and the space between the cracks in the matrix, respectively; η is the ratio between l d and half of l c ; σ fo represents the fiber's axial stress levels in the interface bonding region; ρ represents a shear-lag model parameter; α f and α c represent the coefficients of axial thermal expansion of the fiber and composite, respectively; ∆T is the temperature difference between the test and fabrication temperatures. Curtin [29] developed a stochastic model to analysis matrix stochastic cracking inside CMCs, and the relationship between matrix crack spacing and applied stress can be determined by Equation (3). Gao et al. [30] developed a fracture mechanical approach to determine the interface debonding length when matrix crack propagates to the interface, and the relationship between the interface debonding length and the applied stress can be determined by Equation (4). The Global Load Sharing (GLS) criterion is adopted to determine the stress distribution between the intact and breakage fibers, which is given by Equation (5) [31].
where V f and V m are the volume fraction of the fiber and the matrix, respectively; E m is the elastic modulus of the matrix; l sat is the saturation length of matrix cracking; σ m represents the stress loaded on the matrix; σ R represents the characteristic stress for cracks in the matrix; Γ i represents the debonding energy at the interface; <L> corresponds to mean fiber pullout length, and P represents the probability of fiber fragmentation.

Micromechanical Cyclic Hysteresis Loops Constitutive Model Considering Matrix Fragmentation and Closure
Upon unloading, the composite's strain is divided into two stages, as following: Stage I, when σ unloading > σ tr_unloading , the unloading interface reverse slip length is less than the interface debonding length. Stage II, when σ min < σ unloading < σ tr_unloading , the unloading interface reverse slip length is equal to the interface debonding length.
Upon unloading, when σ unloading > σ tr_unloading , the composite's strain can be determined by Equation (6), [28] where When σ min < σ unloading < σ tr_unloading , the composite's strain can be determined by Equation (8), [28] Upon reloading, the composite's strain is divided into two stages, as following: Stage I, when σ reloading > σ > σ min , the reloading interface new slip length is less than the interface debonding length. Stage II, when σ > σ reloading , the reloading interface new slip length is equal to the interface debonding length.

Prediction of Tensile and Fatigue Hysteresis Loops of Mini-SiC/SiC Composite
Sauder et al. [33] performed experimental study on the tensile and cyclic loading/unloading fatigue hysteresis behavior of mini-SiC/SiC composite. The mini-SiC/SiC composite was fabricated using the chemical vapor infiltration (CVI) method. Tyranno TM SA3 (UBE Industries, Tokyo, Japan) tows coated by a single layer of pyrocarbon (PyC) were used as reinforcement. Figure 1 shows the experimental and predicted tensile stress-strain curves and matrix fragmentation density versus applied stress curves. Matrix fragmentation starts from approximately σ mc = 600 MPa, and approaches saturation at approximately σ sat =1000 MPa, and the saturation fragmentation density is approximately λ sat = 25/mm. During tensile experiment, acoustic emission was used to monitor the matrix crack evolution, and after the fracture of tensile specimen, the saturation matrix cracking density was obtained [33]. The occurrence of matrix fragmentation causes nonlinear behavior of tensile curve, and the composite tensile strength is approximately σ UTS = 1116 MPa with the fracture strain ε f = 0.58%.

Prediction of Tensile and Fatigue Hysteresis Loops of Mini-SiC/SiC Composite
Sauder et al. [33] performed experimental study on the tensile and cyclic loading/unloading fatigue hysteresis behavior of mini-SiC/SiC composite. The mini-SiC/SiC composite was fabricated using the chemical vapor infiltration (CVI) method. Tyranno TM SA3 (UBE Industries, Tokyo, Japan) tows coated by a single layer of pyrocarbon (PyC) were used as reinforcement. Figure 1 shows the experimental and predicted tensile stress-strain curves and matrix fragmentation density versus applied stress curves. Matrix fragmentation starts from approximately σmc = 600 MPa, and approaches saturation at approximately σsat =1000 MPa, and the saturation fragmentation density is approximately λsat = 25/mm. During tensile experiment, acoustic emission was used to monitor the matrix crack evolution, and after the fracture of tensile specimen, the saturation matrix cracking density was obtained [33]. The occurrence of matrix fragmentation causes nonlinear behavior of tensile curve, and the composite tensile strength is approximately σUTS = 1116 MPa with the fracture strain εf = 0.58%.

Prediction of Fatigue Hysteresis Loops of Unidirectional SiC/SiC Composite
Gordon [34] performed experimental study on cyclic loading/unloading hysteresis behavior of unidirectional SiC/SiC composite. The composite was fabricated using the pre-impregnated (pre-preg) melt-infiltrated (MI) method. The Hi-Nicalon Type S TM fibers were coated with a boron-nitride (BN) interphase. Experimental and predicted tensile loading/unloading curves, ITM, and ICSR/INSR curves are shown in Figure 3. Theoretical predicted hysteresis loops considering closure of matrix fragmentation agreed with experimental data, as shown in Figure 3a. (i.e., B2-C2 in Figure 3d); and the reloading INSR increases with reloading stress due to the increase of the interface new slip length, i.e., from INSR = zero at the valley stress σmin = zero MPa to INSR = 0.14 at the reloading stress σreloading = 236 MPa (i.e., A-B2 in Figure 3e), and then remains constant till the tensile peak stress σmax = 415 MPa (i.e., B2-C in Figure 3e).  Table 3 shows the hysteresis loops related parameters of unidirectional SiC/SiC composite under σmax = 415 and 449 MPa. When the peak stress increases from σmax = 415 to 449 MPa, the interface debonding ratio increases from η = 0.14 to 0.295; the unloading transition stress decreases from σtr_unloading = 209 to 170 MPa; the unloading ITM increases from ITM = 5.34 to 7.21 TPa −1 ; the unloading peak ICSR increases from 0.11 to 0.1888; the reloading transition stress increases from σtr_reloading = 206 to 279 MPa; the reloading peak  Table 3 shows the hysteresis loops related parameters of unidirectional SiC/SiC composite under σ max = 415 and 449 MPa. When the peak stress increases from σ max = 415 to 449 MPa, the interface debonding ratio increases from η = 0.14 to 0.295; the unloading transition stress decreases from σ tr_unloading = 209 to 170 MPa; the unloading ITM increases from ITM = 5.34 to 7.21 TPa −1 ; the unloading peak ICSR increases from 0.11 to 0.1888; the reloading transition stress increases from σ tr_reloading = 206 to 279 MPa; the reloading peak INSR increases from 0.14 to 0.295; and the reloading ITM increases from ITM = 4.88 to 6.76 TPa −1 .

Prediction of Fatigue Hysteresis Loops of Cross-Ply SiC/SiC Composite
Gordon [34] performed experimental study on the cyclic loading/unloading hysteresis behavior of cross-ply [0/90] 2s SiC/SiC composite. The composite was fabricated using the pre-impregnated (pre-preg) melt-infiltrated (MI) method. The Hi-Nicalon Type S TM fibers were coated with a boron-nitride (BN) interphase. Experimental and predicted tensile loading/unloading hysteresis loops, ITM, and ICSR/INSR are shown in Figure 4.
Theoretical predicted hysteresis loops considering closure of matrix fragmentation agreed with experimental data, as shown in Figure 4a. stress σmax = 189 MPa to ICSR = 0.09 at the unloading stress σunloading = 110 MPa (i.e., A-B2 in Figure 4d), and decreases to ICSR = zero the valley stress σmin = zero MPa (i.e., B2-C2 in Figure 4d); and the reloading INSR increases with reloading stress due to the increase of the interface new slip length, i.e., from INSR = zero at the valley stress σmin = zero MPa to INSR = 0.214 at the reloading stress σreloading = 164 MPa (i.e., A-B2 in Figure 4e), and then remains constant till the peak stress σmax = 189 MPa (i.e., B2-C in Figure 4e).    Figure 4c), and increases to ITM = 6.29 TPa −1 at the tensile peak stress σ max = 189 MPa (i.e., B 1 -C 1 in Figure 4c); the unloading ICSR increases with unloading stress due to the increase of the interface counter slip length, i.e., from ICSR = zero at the tensile peak stress σ max = 189 MPa to ICSR = 0.214 at the unloading transition stress σ tr_unloading = 52 MPa (i.e., A-B 1 in Figure 4d), and remains constant till the valley stress σ min = zero MPa (i.e., B 1 -C 1 in Figure 4d); and the reloading INSR increases with reloading stress due to the increase of the interface new slip length, i.e., from INSR = zero at the valley stress σ min = zero MPa to INSR = 0.214 at the transition stress σ tr_reloading = 137 MPa (i.e., A-B 1 in Figure 4e), and then remains constant till the peak stress σ max = 189 MPa (i.e., B 1 -C in Figure 4e).  Table 4 shows the hysteresis loops related parameters of cross-ply SiC/SiC composite under σ max = 189 and 206 MPa. When the peak stress increases from σ max = 189 to 206 MPa, the interface debonding ratio increases from η = 0.214 to 0.35; the unloading transition stress decreases from σ tr_unlooading = 52 to 34 MPa; the unloading ITM increases from ITM = 8.67 to 10.32 TPa −1 ; the unloading peak ICSR increases from 0.09 to 0.137; the reloading transition stress increases from σ tr_relooading = 137 to 172 MPa; the reloading ITM increases from ITM = 7.69 to 9.68 TPa −1 ; and the reloading peak INSR increases from 0.214 to 0.35.

Prediction of Tensile and Fatigue Hysteresis Loops of 2D Plain-Woven SiC/SiC Composite
Smith [35] performed experimental study on the tensile and cyclic loading/unloading hysteresis behavior of 2D plain-woven SiC/SiC composite at room temperature. The composite was fabricated using the CVI+MI method at temperature near 1400 • C. The Sylramic TM fibers were coated with a Boron Nitride (BN) interphase. Figure 5 shows experimental and predicted tensile stress-strain curves. The composite exhibited obvious nonlinear behavior and fractured at approximately σ UTS = 430 MPa with the failure strain of ε f = 0.36%.

Prediction of Tensile and Fatigue Hysteresis Loops of 2D Plain-Woven SiC/SiC Composite
Smith [35] performed experimental study on the tensile and cyclic loading/unloading hysteresis behavior of 2D plain-woven SiC/SiC composite at room temperature. The composite was fabricated using the CVI+MI method at temperature near 1400 o C. The Sylramic TM fibers were coated with a Boron Nitride (BN) interphase. Figure 5 shows experimental and predicted tensile stress-strain curves. The composite exhibited obvious nonlinear behavior and fractured at approximately σUTS = 430 MPa with the failure strain of εf = 0.36%.    Figure 6e), and then remains constant till the tensile peak stress σ max = 360 MPa (i.e., B 1 -C in Figure 6e). (b) When the closure of matrix fragmentation is considered, the unloading ITM increases with unloading stress due to internal damage of interface debonding and counter sliding, i.e., from ITM = 4.63 TPa −1 at the tensile peak stress σ max = 360 MPa to ITM = 7.08 TPa −1 at the unloading stress σ unloading = 54 MPa (i.e., A-B 2 in Figure 6b), and then decreases to ITM = 6.87 TPa −1 at the valley stress σ min = zero MPa (i.e., B 2 -C 2 in Figure 6b); the reloading ITM increases with reloading stress due to internal damage of interface debonding and new sliding, i.e., from ITM = 4.58 TPa −1 at the valley stress σ min = zero MPa to ITM = 6.67 TPa −1 at the reloading stress σ reloading = 269 MPa (i.e., A-B 2 in Figure 6c), and increases slowly to ITM = 6.88 TPa −1 at the tensile peak stress σ max = 360 MPa (i.e., B 2 -C 2 in Figure 6c); the unloading ICSR increases with unloading stress due to the increase of the interface counter slip length, i.e., from ICSR = zero at the peak stress σ max = 360 MPa to ICSR = 0.082 at the unloading stress σ unloading = 169 MPa (i.e., A-B 2 in Figure 6d), and decreases to ICSR = zero the valley stress σ min = zero MPa (i.e., B 2 -C 2 in Figure 6d Table 5 shows the hysteresis loops parameters under σmax = 312, 360, and 410 MPa. When the tensile peak stress increases from σmax = 312 to 410 MPa, the interface debonding ratio increases from η = 0.084 to 0.23; the unloading transition stress decreases from σtr_unlooading = 138 to 41 MPa; the unloading ITM increases from ITM = 6.32 to 8.01 TPa −1 ; the unloading peak ICSR increases from 0.054 to 0.098; the reloading transition stress increases from σtr_relooading = 174 to 369 MPa; the reloading ITM increases from ITM = 6.12 to 7.73 TPa −1 ; and the reloading peak INSR increases from 0.084 to 0.23.  Table 5 shows the hysteresis loops parameters under σ max = 312, 360, and 410 MPa. When the tensile peak stress increases from σ max = 312 to 410 MPa, the interface debonding ratio increases from η = 0.084 to 0.23; the unloading transition stress decreases from σ tr_unlooading = 138 to 41 MPa; the unloading ITM increases from ITM = 6.32 to 8.01 TPa −1 ; the unloading peak ICSR increases from 0.054 to 0.098; the reloading transition stress increases from σ tr_relooading = 174 to 369 MPa; the reloading ITM increases from ITM = 6.12 to 7.73 TPa −1 ; and the reloading peak INSR increases from 0.084 to 0.23.

Comparative Analysis
Under tensile and fatigue loading, composite's internal damage evolution depends on the fiber's preform and volume fraction. For minicomposite, unidirectional, cross-ply and 2D plain-woven SiC/SiC composites with different fiber volume fraction, it can be found that the composite's unloading and reloading peak ITMs decrease with increasing fiber's volume faction.
For mini-SiC/SiC composite with the fiber' volume of V f = 0.43, when the tensile peak stress increases from σ max = 890 to 1078 MPa, the unloading peak ITM increases from ITM = 3.58 to 5.18 TPa −1 ; the reloading ITM increases from ITM = 3.486 to 4.87 TPa −1 . For cross-ply SiC/SiC composite with fiber's volume of V f = 0.25, when the peak stress increases from σ max = 189 to 206 MPa, the unloading ITM increases from ITM = 8.67 to 10.32 TPa −1 ; the reloading ITM increases from ITM = 7.69 to 9.68 TPa −1 .
The composite's interface debonding and slip parameters, i.e., the interface debonding ratio (η), the interface counter slip ratio (ICSR) and interface new slip ratio (INSR) decrease with increasing fiber's volume fraction.
For mini-SiC/SiC composite with the fiber' volume of V f = 0.43, when the tensile peak stress increases from σ max = 890 to 1078 MPa, the interface debonding ratio increases from η = 0.075 to 0.38; the unloading peak ICSR increases from 0.07 to 0.17; and the reloading peak INSR increases from 0.075 to 0.38. For unidirectional SiC/SiC composite with fiber's volume of V f = 0.25, when the peak stress increases from σ max = 415 to 449 MPa, the interface debonding ratio increases from η = 0.14 to 0.295; the unloading peak ICSR increases from 0.11 to 0.1888; and the reloading peak INSR increases from 0.14 to 0.295.

Summary and Conclusions
In this paper, micromechanical constitutive models were developed to analyze the effects of matrix fragmentation and closure on tensile and fatigue behavior of fiber-reinforced CMCs. Relationships between composite's tensile and fatigue response, matrix fragmentation and closure and related interface debonding and fiber's failure were established. Experimental matrix fragmentation density, tensile curves, and fatigue hysteresis loops of mini, unidirectional, cross-ply, and 2D plain-woven SiC/SiC composites are predicted.
(1) The Matrix fragmentation density changes with increasing or decreasing tensile stress and affects the tensile nonlinear strain and the interface debonding ratio.
(2) The closure of matrix fragmentation affects the fatigue hysteresis loops. Upon unloading, the inverse tangent modulus increases to the peak value, and then decreases with unloading stress, and the interface counter slip ratio increases to the peak value, and then decreases to zero with unloading stress; upon reloading, the inverse tangent modulus increases with reloading stress, and the interface new slip ratio increases slowly during initial stage of reloading, then increases to the peak value, and remains constant till peak stress. (3) Theoretical predicted matrix fragmentation density, tensile nonlinear curves, and the fatigue hysteresis loops agreed with experimental data.
There is inherent variation across samples tested in CMCs, however, the paper focused on the micromechanical constitutive models for tensile and fatigue behavior of CMCs without considering the inherent variation for different tested samples. The author will consider the inherent variation for different samples tested in CMCs in the further study.